Game Theory and Behavioural Economics

Game Theory Fundamentals

What Is Game Theory?

Game theory is the study of strategic decision-making among interdependent agents. Each agent's payoff depends not only on their own actions but also on the actions of others. The framework was formalised by John von Neumann and Oskar Morgenstern (1944) and extended by John Nash.

A game consists of:

• Players: the decision-makers
• Strategies: the set of actions available to each player
• Payoffs: the outcomes (benefits or costs) each player receives for every combination of strategies

Types of Games

• Simultaneous games: players choose strategies simultaneously without knowing the other's choice (e.g., prisoner's dilemma). Represented by a payoff matrix.
• Sequential games: players make decisions in turn, observing previous moves (e.g., chess, Stackelberg duopoly). Represented by a game tree (extensive form).
• Zero-sum games: one player's gain is exactly the other's loss (e.g., poker). Total payoffs sum to zero.
• Non-zero-sum games: the total payoffs can be positive or negative, allowing for mutual benefit or mutual harm (e.g., prisoner's dilemma).
• Cooperative games: players can form binding agreements and coalitions.
• Non-cooperative games: players make decisions independently; binding agreements are not enforceable.

Dominant Strategies

A dominant strategy is a strategy that yields a higher payoff for a player regardless of what the other players do. If every player has a dominant strategy, the game has a dominant strategy equilibrium.

A strictly dominant strategy is always strictly better than any other strategy. A weakly dominant strategy is at least as good as any other strategy, and strictly better against at least one opponent strategy.

Nash Equilibrium

A Nash equilibrium is a set of strategies where no player can improve their payoff by unilaterally changing their strategy, given the strategies chosen by all other players. Formally, for each player $i$ with strategy $s_i^*$:

$u_i(s_i^*, s_{-i}^*) \geq u_i(s_i, s_{-i}^*) \quad \forall \; s_i$

Where $u_i$ is player $i$'s payoff function and $s_{-i}^*$ represents the strategies of all other players.

Key properties:

• A Nash equilibrium is a self-enforcing prediction: no player has an incentive to deviate
• A game may have zero, one, or multiple Nash equilibria
• Every game with a finite number of strategies has at least one Nash equilibrium in mixed strategies
• A dominant strategy equilibrium is always a Nash equilibrium, but not vice versa

Mixed Strategies

A pure strategy is a deterministic choice of one action. A mixed strategy assigns a probability distribution over the available pure strategies.

In a mixed strategy Nash equilibrium, each player randomises over their strategies such that no player can gain by changing their probabilities. The equilibrium is reached when each player is indifferent between their pure strategies given the other player's mixed strategy.

The Prisoner's Dilemma

The Classic Form

Two suspects are arrested and interrogated separately. Each can either Cooperate (stay silent) or Defect (betray the other). The payoff matrix (years in prison, so lower is better):

	B Cooperates	B Defects
A Cooperates	A: 1, B: 1	A: 10, B: 0
A Defects	A: 0, B: 10	A: 5, B: 5

Analysis:

• For A: if B cooperates, A should defect ($0 < 1$). If B defects, A should defect ($5 < 10$). Defect is A's dominant strategy.
• For B: by symmetric reasoning, defect is B's dominant strategy.
• The Nash equilibrium is (Defect, Defect) with payoffs $(5, 5)$.
• The socially optimal outcome is (Cooperate, Cooperate) with payoffs $(1, 1)$, which is Pareto superior.

The Core Insight

The prisoner's dilemma demonstrates that individually rational behaviour can lead to collectively suboptimal outcomes. The Nash equilibrium is not Pareto efficient — both players would be better off cooperating, but neither has the incentive to do so unilaterally.

Applications in Economics

• Oligopoly pricing: firms face a dilemma between maintaining high prices (cooperating) and undercutting competitors (defecting). Cartels are inherently unstable because each member has an incentive to cheat.
• Arms races: nations invest in military build-up even when mutual disarmament would leave both better off.
• Tragedy of the commons: individuals overuse a shared resource even though collective restraint would benefit all.
• Free riding: individuals avoid contributing to a public good while benefiting from others' contributions.

Repeated Games and the Evolution of Cooperation

In a one-shot prisoner's dilemma, defection is the only rational strategy. In an infinitely repeated (or indefinitely repeated) game, cooperation can be sustained as a Nash equilibrium through strategies such as:

• Grim trigger: cooperate until the other player defects, then defect forever. This makes the threat of permanent punishment credible.
• Tit-for-tat: begin by cooperating, then mirror the opponent's previous move. Simple, retaliatory, and forgiving.
• Pavlov (win-stay, lose-shift): repeat the last action if it yielded a high payoff; switch if it yielded a low payoff.

The Folk Theorem states that in infinitely repeated games, any feasible and individually rational payoff can be sustained as a Nash equilibrium, provided the discount factor (the weight placed on future payoffs) is sufficiently high.

In finitely repeated games with a known endpoint, backward induction leads to defection in every round (the unraveling argument). However, if the number of rounds is uncertain or players use bounded rationality, cooperation may still emerge.

Other Key Games

Battle of the Sexes

Two players prefer to coordinate on the same activity but disagree on which activity. There are two pure strategy Nash equilibria and one mixed strategy equilibrium. This game illustrates coordination problems.

	B: Opera	B: Football
A: Opera	A: 3, B: 2	A: 0, B: 0
A: Football	A: 0, B: 0	A: 2, B: 3

Chicken (Hawk-Dove)

Two drivers speed toward each other. The player who swerves first is the "chicken." Both swerving is a compromise; neither swerving is catastrophic. This game models brinkmanship in international relations and competitive business strategies.

Stag Hunt

Two hunters can cooperate to catch a stag (high payoff for both) or hunt hare individually (moderate payoff, certain). The stag hunt has two Nash equilibria: both cooperate (Pareto efficient) and both defect (risk-dominant). It illustrates the tension between safety and social cooperation.

Behavioural Economics

Departures from Rationality

Traditional (neoclassical) economics assumes that agents are Homo economicus: fully rational, self-interested utility maximisers with perfect information, consistent preferences, and unlimited cognitive capacity. Behavioural economics relaxes these assumptions, drawing on psychology and empirical evidence to model how people actually make decisions.

Bounded Rationality

Herbert Simon (1955) proposed that individuals are boundedly rational: they make satisfactory rather than optimal decisions because of limited information, limited cognitive processing capacity, and limited time. People use heuristics — mental shortcuts — to simplify complex decisions.

Heuristics and Biases

Availability heuristic: people estimate the probability of an event based on how easily examples come to mind. Vivid, recent, or emotionally charged events are overestimated (e.g., overestimating the probability of plane crashes after media coverage).

Representativeness heuristic: people judge the probability of an event based on how similar it is to a stereotype or prototype, ignoring base rates. For example, assuming a quiet, introverted person is more likely to be a librarian than a farmer, despite there being far more farmers than librarians.

Anchoring: initial information serves as an anchor that influences subsequent estimates, even when the anchor is irrelevant. In negotiations, the first offer often anchors the final agreement.

Framing: the way information is presented affects decisions. People respond differently to "a $90\%$ survival rate" vs. "a $10\%$ mortality rate" despite being logically identical. Loss aversion means that framing an outcome as a loss has a stronger effect than framing the equivalent outcome as a gain.

Overconfidence bias: individuals systematically overestimate their own knowledge, ability, and the precision of their predictions. This affects investment decisions, entrepreneurial activity, and risk assessment.

Confirmation bias: people seek, interpret, and remember information that confirms their existing beliefs while ignoring or discounting contradictory evidence.

Sunk cost fallacy: individuals continue investing in a losing course of action because of resources already committed, rather than evaluating future costs and benefits.

Prospect Theory

Developed by Daniel Kahneman and Amos Tversky (1979), prospect theory describes how people evaluate potential losses and gains:

1. Reference dependence: people evaluate outcomes relative to a reference point (usually the status quo), not in absolute terms
2. Loss aversion: losses loom larger than equivalent gains. The psychological pain of losing USD 100 is approximately twice the pleasure of gaining USD 100
3. Diminishing sensitivity: the marginal impact of gains and losses decreases with distance from the reference point
4. Probability weighting: people overweight small probabilities and underweight moderate-to-large probabilities (explaining both gambling behaviour and excessive insurance purchases)

The value function in prospect theory is:

• Concave for gains (risk-averse in the domain of gains)
• Convex for losses (risk-seeking in the domain of losses)
• Steeper for losses than for gains (loss aversion)

Intertemporal Choice and Present Bias

Traditional models assume exponential discounting:

$U = \sum_{t=0}^{T} \delta^t \cdot u(c_t)$

Where $\delta$ is a constant discount factor.

Behavioural evidence supports hyperbolic discounting, where the discount rate declines over time:

$U = \sum_{t=0}^{T} \frac{1}{(1 + k \cdot t)^b} \cdot u(c_t)$

Where $k > 0$ governs impatience and $b > 0$ governs the degree of hyperbolic discounting.

This produces present bias: individuals disproportionately prefer immediate gratification over future rewards, even when they know the long-term consequences are worse. This explains procrastination, undersaving for retirement, and difficulty maintaining diets.

Social Preferences

• Fairness and reciprocity: people care about fairness and are willing to sacrifice personal gain to punish unfair behaviour (as demonstrated in the ultimatum game, where responders frequently reject low offers)
• Inequality aversion: people experience disutility from unequal outcomes, even when they benefit from the inequality
• Altruism: individuals derive utility from improving the welfare of others, not only from their own consumption

Nudge Theory

Richard Thaler and Cass Sunstein (2008) proposed that choice architecture — the way options are presented — can influence decisions without restricting freedom of choice. A nudge is any aspect of that architecture that alters behaviour predictably without forbidding options or significantly changing economic incentives.

Examples of nudges:

• Default options: making organ donation or pension enrolment the default significantly increases participation rates
• Social norms: informing households that their energy consumption exceeds that of their neighbours reduces energy use
• Simplification: reducing the complexity of tax forms or benefits applications increases take-up
• Framing: presenting food as "90% fat-free" rather than "contains 10% fat" increases sales
• Commitment devices: allowing individuals to pre-commit to future behaviour (e.g., automatic transfers to savings accounts)

Libertarian paternalism: the philosophical foundation of nudge theory. It holds that it is both legitimate and desirable for institutions to influence choices in ways that make people better off, while preserving their freedom to choose otherwise.

Criticisms of Behavioural Economics

• Lack of a unified theory: behavioural economics identifies deviations from rationality but does not yet provide a comprehensive, parsimonious alternative model
• Context dependence: heuristics and biases are highly context-dependent, making predictions difficult
• Manipulation concerns: nudges can be used to influence behaviour in ways that serve institutional interests rather than individual welfare
• Paternalism objection: critics argue that nudges are manipulative and undermine individual autonomy, even when they preserve formal freedom of choice
• Limited external validity: many behavioural findings are based on laboratory experiments with Western, educated, industrialised, rich, and democratic (WEIRD) populations

Common Pitfalls

• Confusing Nash equilibrium with the socially optimal outcome. The prisoner's dilemma shows that the Nash equilibrium can be Pareto inferior to cooperation.
• Assuming that all games have a unique Nash equilibrium. Many games have multiple equilibria, and predicting which one will be played requires additional reasoning (focal points, evolutionary stability, or social norms).
• Treating prospect theory as a minor correction to expected utility theory. Loss aversion and reference dependence are fundamental departures that change predictions qualitatively, not just quantitatively.
• Assuming that bounded rationality implies irrationality. Bounded rationality is often a rational response to constraints — satisficing can be optimal when information is costly to acquire.
• Confounding nudges with mandates. Nudges preserve freedom of choice; regulations and bans do not.

Practice Problems

Problem 1: Finding Nash Equilibria

Two firms, Alpha and Beta, are deciding whether to advertise (Ad) or not advertise (No Ad). The payoff matrix shows profits in millions:

	Beta: Ad	Beta: No Ad
Alpha: Ad	(4, 4)	(8, 2)
Alpha: No Ad	(2, 8)	(6, 6)

Find all Nash equilibria.

Check each cell:

1. (Ad, Ad): Alpha gets 4. If Alpha switches to No Ad, Alpha gets 2 (worse). Beta gets 4. If Beta switches to No Ad, Beta gets 2 (worse). Neither can improve. Nash equilibrium.

2. (Ad, No Ad): Alpha gets 8 (best response to No Ad). Beta gets 2. If Beta switches to Ad, Beta gets 4 (better). Not an equilibrium.

3. (No Ad, Ad): Alpha gets 2. If Alpha switches to Ad, Alpha gets 4 (better). Not an equilibrium.

4. (No Ad, No Ad): Alpha gets 6. If Alpha switches to Ad, Alpha gets 8 (better). Not an equilibrium.

The only Nash equilibrium is (Ad, Ad) with payoffs $(4, 4)$. This is a prisoner's dilemma: both would be better off at (No Ad, No Ad) with $(6, 6)$, but neither has the incentive to unilaterally stop advertising.

Problem 2: Mixed Strategy Nash Equilibrium

In a zero-sum game, Player 1 can play Up or Down; Player 2 can play Left or Right. The payoff matrix shows Player 1's payoff (Player 2's payoff is the negative):

	Left	Right
Up	3	-1
Down	-2	4

Find the mixed strategy Nash equilibrium.

Let Player 1 play Up with probability $p$ and Down with probability $1 - p$. For Player 2 to be indifferent between Left and Right:

$\mathrm{Expected payoff (Left)} = p(-3) + (1-p)(2)$ $\mathrm{Expected payoff (Right)} = p(1) + (1-p)(-4)$

Setting them equal:

$-3p + 2 - 2p = p - 4 + 4p$ $2 - 5p = 5p - 4$ $6 = 10p$ $p = 0.6$

Let Player 2 play Left with probability $q$ and Right with probability $1 - q$. For Player 1 to be indifferent:

$3q - 1(1-q) = -2q + 4(1-q)$ $3q - 1 + q = -2q + 4 - 4q$ $4q - 1 = -6q + 4$ $10q = 5$ $q = 0.5$

The mixed strategy Nash equilibrium is: Player 1 plays Up with probability $0.6$ and Down with probability $0.4$; Player 2 plays Left with probability $0.5$ and Right with probability $0.5$.

Problem 3: Behavioural Biases in Decision-Making

A consumer is choosing between two investment products:

• Product A: "has a 70% chance of gaining value"
• Product B: "has a 30% chance of losing value"

Explain which product the consumer is likely to choose and identify the bias at work.

The consumer is likely to choose Product A, even though the two descriptions are logically equivalent (a 70% chance of gaining is the same as a 30% chance of losing).

This is an example of framing bias: the way information is presented (framed as a gain vs. framed as a loss) affects decisions. Prospect theory predicts that people are risk-averse in the domain of gains and risk-seeking in the domain of losses. The positive frame ("gaining value") triggers a more favourable evaluation than the negative frame ("losing value"), even though the underlying probabilities are identical.

Loss aversion also plays a role: the word "losing" activates a stronger emotional response than "gaining," making Product B seem less attractive despite its mathematical equivalence.

Problem 4: Present Bias and Saving

A person is offered a choice: USD 100 today or USD 120 in one month. Most people choose USD 100 today. However, when offered USD 100 in 12 months or USD 120 in 13 months, most people choose USD 120 in 13 months. Explain this inconsistency using behavioural economics.

This pattern is explained by present bias (or hyperbolic discounting). When the reward is immediate, the person heavily discounts the future one-month delay, preferring USD 100 now. But when both options are in the future, the one-month difference feels much less significant, and the person is more patient.

With exponential discounting at a constant rate, the preference should be consistent: if you prefer USD 100 today over USD 120 in a month, you should also prefer USD 100 in 12 months over USD 120 in 13 months (since the trade-off is identical). The inconsistency reveals that the discount rate is not constant — it is much higher for immediate trade-offs than for future ones.

This has important implications for savings behaviour, health decisions, and commitment devices.

Problem 5: Nudge Theory Application

A government wants to increase organ donation rates. Currently, citizens must actively opt in to become donors, and the participation rate is $25\%$. Design a nudge-based intervention and evaluate its potential effectiveness and ethical considerations.

Intervention: Change the default from opt-in to opt-out. Under an opt-out (presumed consent) system, all citizens are automatically registered as organ donors unless they actively withdraw.

Expected effectiveness: Evidence from countries that have implemented opt-out systems (e.g., Spain, Austria, Wales) shows dramatic increases in donation rates, often exceeding $90\%$. This is because of status quo bias and inertia: people tend to stick with the default option rather than actively changing it.

Ethical considerations:

• Libertarian paternalism: the nudge preserves freedom of choice — anyone can opt out. The default is changed, not forced.
• Autonomy: critics argue that presumed consent may not reflect genuine consent, particularly if people are unaware of the default or face barriers to opting out.
• Transparency: the system must be well-publicised and easy to opt out of to maintain legitimacy.
• Cultural sensitivity: attitudes toward organ donation vary across cultures and religious groups; a one-size-fits-all default may not be appropriate.
• Manipulation vs. guidance: the line between a helpful nudge and manipulation depends on whether the default serves the individual's likely preferences.

Advanced Game Theory Applications (HL Extension)

Sequential Games and Backward Induction

In sequential games, players move in turn, observing previous moves. The solution concept is subgame perfect Nash equilibrium (SPNE), found by backward induction: starting from the last decision node and working backwards to determine optimal strategies at each stage.

Example -- entry deterrence:

An incumbent monopolist faces a potential entrant. The entrant decides whether to enter. If entry occurs, the incumbent decides whether to accommodate or fight (engage in a price war).

Payoffs (entrant, incumbent):

Path	Entrant	Incumbent
Stay out	(0, 10)	--
Enter, Accommodate	(3, 5)	--
Enter, Fight	(-2, -1)	--

Backward induction:

1. If the entrant enters, the incumbent chooses between Accommodate (payoff 5) and Fight (payoff -1). The incumbent accommodates.
2. Knowing the incumbent will accommodate, the entrant compares Enter (payoff 3) with Stay out (payoff 0). The entrant enters.
3. SPNE: (Enter, Accommodate) with payoffs (3, 5).

Credible vs. incredible threats: the incumbent's threat to fight is not credible because fighting yields -1, which is worse than accommodating (5). A threat is credible only if it is in the threatener's interest to carry it out when the time comes.

Repeated Games and Collusion

In a repeated setting, firms can sustain cooperation (collusion) through the threat of future punishment.

Grim trigger strategy: cooperate as long as all firms have cooperated in the past. If any firm defects, switch to the Nash equilibrium (competitive) outcome forever.

For collusion to be sustainable, the present value of cooperation must exceed the one-time gain from defection:

$\frac{\pi_{\text{collusion}}}{1 - \delta} \geq \pi_{\text{defection}} + \frac{\delta \cdot \pi_{\text{Nash}}}{1 - \delta}$

Where $\delta$ is the discount factor ($0 < \delta < 1$). Rearranging:

$\delta \geq \frac{\pi_{\text{defection}} - \pi_{\text{collusion}}}{\pi_{\text{defection}} - \pi_{\text{Nash}}}$

Numerical example: In the pricing game from earlier, $\pi_{\text{collusion}} = 8$ (High, High), $\pi_{\text{defection}} = 12$ (Low while other plays High), $\pi_{\text{Nash}} = 5$ (Low, Low).

$\delta \geq \frac{12 - 8}{12 - 5} = \frac{4}{7} = 0.571$

If $\delta \geq 0.571$ (i.e., firms value future profits at least 57.1% as much as current profits), collusion is sustainable. This requires:

• Frequent interactions (short time between rounds)
• Low discount rate (low inflation, low risk of bankruptcy)
• High probability that the game continues

Cournot and Bertrand as Games

Cournot as a game:

• Players: firms
• Strategies: quantities $q_i \geq 0$
• Payoffs: $\pi_i = P(Q) \cdot q_i - C(q_i)$
• Nash equilibrium: each firm's quantity is a best response to the other's quantity

Bertrand as a game:

• Players: firms
• Strategies: prices $p_i \geq 0$
• Payoffs: $\pi_i = (p_i - c) \cdot D_i(p_i, p_j)$
• Nash equilibrium: $p_1 = p_2 = MC$ (for homogeneous goods)

The dramatic difference between Cournot and Bertrand outcomes (oligopoly power vs. perfect competition) highlights the importance of the strategic variable (quantity vs. price) and the assumption about product differentiation.

Stackelberg as a Sequential Game

The Stackelberg model is a sequential game where the leader moves first:

1. The leader chooses quantity $q_1$
2. The follower observes $q_1$ and chooses $q_2$
3. Payoffs are determined

Backward induction: the follower plays the Cournot best response to $q_1$. The leader anticipates this and chooses $q_1$ to maximise its profit, taking the follower's reaction into account. The leader earns higher profit than in the Cournot equilibrium (first-mover advantage).

Common Pitfalls in Advanced Game Theory

• Confusing Nash equilibrium with SPNE. Every SPNE is a Nash equilibrium, but not every Nash equilibrium is subgame perfect. SPNE requires optimal play in every subgame, not just the game as a whole.
• Assuming that repeated interaction always leads to cooperation. If the discount factor is low (firms are impatient), cooperation may not be sustainable even with infinitely repeated interaction.
• Confusing credible and incredible threats. A threat is credible only if the threatener would actually carry it out when called upon.
• Applying the grim trigger to finitely repeated games. Backward induction unravels cooperation in the last period, making the grim trigger ineffective with a known endpoint.

Advanced Behavioural Economics (HL Extension)

Mental Accounting

Thaler (1985) observed that people categorise money into different "mental accounts" and treat money differently depending on the category:

• Income effect: people spend windfall gains (bonuses, lottery winnings) more readily than regular income
• Fungibility violation: money is not perfectly fungible across mental accounts. A person may carry credit card debt at 18% interest while keeping savings earning 2%
• Budget categories: people set implicit budgets for categories (food, entertainment, clothing) and are reluctant to shift spending between categories

Implications for policy:

• Tax rebates framed as "bonus" payments are more likely to be spent than those framed as "returned overpayment"
• Savings programmes that label accounts for specific purposes (retirement, education) can increase saving rates

Endowment Effect

People value goods more highly simply because they own them. In experiments, participants given a mug demanded a higher price to sell it than participants who did not own the mug were willing to pay to buy it.

$\text{WTA (willingness to accept)} > \text{WTP (willingness to pay)}$

This violates the standard Coase theorem prediction that transaction costs aside, initial ownership should not affect the efficiency of outcomes.

Causes:

• Loss aversion: giving up a good feels like a loss, which is weighted more heavily than the equivalent gain from acquiring it
• Psychological attachment: ownership creates an emotional connection
• Status quo bias: people prefer to keep things as they are

Policy implications:

• Default options in organ donation, pension enrolment, and insurance coverage are powerful because people are reluctant to opt out of something they already "have"
• Trial periods and money-back guarantees exploit the endowment effect: once consumers have experienced ownership, they are reluctant to give it up

Time Inconsistency and Self-Control Problems

Present bias leads to time-inconsistent preferences: choices made today about future consumption differ from the choices that will be made when the future arrives.

Examples:

• Procrastination: planning to study tomorrow, but when tomorrow arrives, postponing again
• Overconsumption: intending to eat healthily, but choosing junk food when hungry
• Under-saving: intending to save for retirement, but spending current income instead

Commitment devices: mechanisms that restrict future choices to align with long-term preferences:

• Automatic payroll deductions for savings
• Christmas clubs (forced saving for holiday spending)
• Pre-commitment contracts (e.g., Gym-Pact, where users lose money if they fail to exercise)
• Ulysses contracts (deliberately limiting future options, analogous to Ulysses tying himself to the mast to resist the Sirens)

Behavioural Finance

Behavioural economics has important applications in financial markets:

Dispositions effect: investors tend to sell winning stocks too early (locking in gains) and hold losing stocks too long (avoiding the realisation of losses). This contradicts the rational expectation that investors should sell assets with poor future prospects and hold those with good prospects, regardless of past gains or losses.

Herding: investors follow the actions of others rather than their own analysis, contributing to bubbles and crashes. Information cascades occur when individuals ignore their private information and follow the crowd because they assume the crowd is better informed.

Overconfidence: investors overestimate their ability to pick winning stocks and underestimate risk. This leads to excessive trading (reducing returns through transaction costs) and under-diversification.

Representativeness in investing: investors extrapolate past performance into the future, assuming that recent high returns will continue. This contributes to momentum effects and eventual reversals.

Common Pitfalls in Advanced Behavioural Economics

• Assuming that all biases lead to suboptimal outcomes. Some heuristics (rules of thumb) are efficient and produce good decisions with minimal cognitive effort.
• Overstating the policy implications. Knowing that people are biased does not necessarily imply that the government should correct the bias (paternalism objection).
• Confounding correlation with causation in behavioural experiments. Laboratory results may not generalise to real-world settings with higher stakes and more experienced decision-makers.
• Assuming that nudges are costless. Designing, implementing, and monitoring nudges requires resources and expertise.

Additional Practice Problems

Problem 6: Sequential Game with Credible Threats

Amazon is considering entering the market for online groceries in a country where FreshCo is the dominant player. Amazon first decides whether to enter. If Amazon enters, FreshCo decides whether to accommodate (share the market) or fight (price war).

Payoffs (Amazon, FreshCo):

Scenario	Amazon	FreshCo
Amazon stays out	(0, 20)	--
Amazon enters, FreshCo accommodates	(5, 8)	--
Amazon enters, FreshCo fights	(-3, -2)	--

(a) Find the subgame perfect Nash equilibrium using backward induction.

(b) FreshCo publicly announces it will fight if Amazon enters. Is this threat credible?

(c) How might FreshCo make the threat credible?

(a) Backward induction:

Step 1: If Amazon enters, FreshCo chooses between Accommodate (8) and Fight (-2). FreshCo accommodates.

Step 2: Amazon anticipates FreshCo will accommodate. Amazon compares Enter (5) with Stay out (0). Amazon enters.

SPNE: (Enter, Accommodate) with payoffs (5, 8).

(b) No, the threat is not credible. If Amazon enters, FreshCo's best response is to accommodate (8 > -2). A rational FreshCo would not carry out the threat.

(c) FreshCo could make the threat credible by:

1. Investing in excess capacity: building additional warehouses and distribution centres that lower its marginal cost of expanding output, making a price war less costly. This changes the payoffs: if Fight yields (say) 3 instead of -2, the threat becomes credible

2. Reputation building: if FreshCo has fought entrants in other markets, it may establish a reputation for toughness that deters entry even when the short-run payoff to fighting is negative (this works only if FreshCo faces many potential entrants over time)

3. Long-term contracts: signing exclusive supplier agreements that FreshCo would lose if it accommodates (changing the payoff structure)

4. Sunk cost commitment: making irreversible investments that lower the cost of fighting (e.g., building a proprietary logistics network optimised for aggressive pricing)

Problem 7: Repeated Game and Collusion Sustainability

Two airlines, Alpha and Beta, compete on a route. Each can set a High or Low fare. Weekly payoffs (in millions):

	Beta: High	Beta: Low
Alpha: High	(6, 6)	(1, 10)
Alpha: Low	(10, 1)	(3, 3)

The game is repeated weekly. Both airlines use a grim trigger strategy: cooperate (High fare) as long as the other cooperates; if either defects (Low fare), play the Nash equilibrium (Low fare) forever.

(a) What is the Nash equilibrium of the one-shot game?

(b) What is the minimum discount factor for collusion to be sustainable?

(c) If the government introduces price transparency regulations that increase the frequency of price adjustments from weekly to daily, how does this affect the sustainability of collusion?

(a) For Alpha: if Beta plays High, Alpha gets 6 (High) vs. 10 (Low). Alpha plays Low. If Beta plays Low, Alpha gets 1 (High) vs. 3 (Low). Alpha plays Low. Low is the dominant strategy. By symmetry, both play Low. Nash equilibrium: (Low, Low) with payoffs (3, 3).

(b) Collusion requires both to play High:

Gain from defection: $10 - 6 = 4$ (one-time gain) Loss from punishment: $6 - 3 = 3$ per period forever

$\delta \geq \frac{4}{4 + 3} = \frac{4}{7} \approx 0.571$

The discount factor must be at least 0.571. If firms interact weekly and use a weekly discount rate of $r$, then $\delta = 1/(1 + r)$. For $\delta = 0.571$: $r \leq 0.751$ (75.1% per week), which is easily satisfied. Collusion is sustainable as long as firms expect the market to persist.

(c) More frequent interactions (daily vs. weekly) have two opposing effects:

1. Pro-collusion: the effective discount factor per interaction increases (firms are more patient within the shorter time frame), making the grim trigger more effective
2. Anti-collusion: more frequent price adjustments allow firms to detect deviations faster and punish more quickly, which also supports collusion
3. Anti-collusion: however, if the market is expected to be disrupted sooner (e.g., by entry, regulation, or demand shocks), the expected duration of the collusive relationship may decrease, making collusion harder to sustain

Overall, in standard theory, more frequent interactions make collusion easier to sustain because the punishment comes sooner (discounting matters less for near-future punishments). However, empirically, the effect depends on the specific market structure and the ease of monitoring.

Problem 8: Behavioural Economics in Policy Design

A government wants to increase retirement savings rates. Currently, only 40% of eligible workers contribute to a voluntary pension scheme, and the average contribution rate is 4% of salary.

Using behavioural economics principles, design a policy intervention and evaluate its likely effectiveness and potential drawbacks.

Proposed intervention: automatic enrolment with default contribution rate of 6%.

Behavioural principles applied:

1. Status quo bias / inertia: workers are automatically enrolled and must actively opt out. Evidence from the UK (2006) and the US shows automatic enrolment increases participation to over 90%

2. Default effect: the default contribution rate of 6% anchors expectations. Research shows that most people stick with the default rather than choosing a different rate

3. Present bias mitigation: automatic enrolment overcomes the tendency to prioritise current consumption over future retirement needs. Workers do not need to make an active saving decision

4. Loss aversion: framing contributions as the "normal" state makes opting out feel like a loss of security, which people are motivated to avoid

Expected effectiveness:

• Participation rate: from 40% to 85--95% (based on international evidence)
• Average contribution rate: from 4% to approximately 6% (the default)
• Additional retirement savings: significant increase in national saving rate

Potential drawbacks:

1. Nudging the poor: automatic enrolment reduces take-home pay for low-income workers, who may face binding liquidity constraints. The "nudge" may effectively be a "shove" for these workers

2. Financial literacy: some workers may not understand the pension scheme, the investment options, or the fees involved. Automatic enrolment does not address this

3. Crowding out: workers who were already saving voluntarily through other channels may reduce those savings, partially offsetting the increase in pension contributions

4. Distributional impact: higher-income workers benefit more from tax relief on pension contributions, so the policy may be regressive

5. Investment risk: default investment options may not be appropriate for all workers (e.g., low-risk funds for near-retirees, higher-risk funds for young workers)

Recommendations to address drawbacks:

• Allow low-income workers to opt out without penalty (preserving libertarian paternalism)
• Provide clear, simple information about the scheme and investment options
• Include auto-escalation: automatically increase the contribution rate by 1% per year until reaching a target (e.g., 10%), again relying on inertia
• Use lifecycle investment funds as the default (automatically shifting from equities to bonds as retirement approaches)

Auction Theory and Mechanism Design (HL Extension)

Types of Auctions

Auctions are mechanisms for determining prices and allocating goods when sellers do not know buyers' valuations.

English (ascending-price) auction:

• The auctioneer starts at a low price and raises it incrementally
• Bidders drop out when the price exceeds their valuation
• The last remaining bidder wins at the price just above the second-highest bidder's valuation
• Dominant strategy: bid up to your true valuation (bidding less risks losing an item you value more than the price)

Dutch (descending-price) auction:

• The auctioneer starts at a high price and lowers it until a bidder accepts
• The first bidder to accept wins and pays the current price
• Used for perishable goods (flowers in the Netherlands, fish markets)
• No dominant strategy: bidders must decide when to jump in, balancing the risk of waiting too long (losing the item) against jumping too early (paying more than necessary)

First-price sealed-bid auction:

• All bidders submit sealed bids simultaneously
• The highest bidder wins and pays their bid
• Optimal strategy: bid below your true valuation (to earn a surplus if you win), but how much below depends on the number of competitors and the distribution of valuations
• Bidding too low risks losing; bidding too close to your valuation earns little surplus

Second-price sealed-bid (Vickrey) auction:

• All bidders submit sealed bids simultaneously
• The highest bidder wins but pays the second-highest bid
• Dominant strategy: bid your true valuation. Since you pay the second-highest bid, you never pay more than the item is worth to you, and bidding your true value maximises your chance of winning
• Revenue equivalence: the Vickrey auction yields the same expected revenue as the English auction

Revenue Equivalence Theorem

The revenue equivalence theorem states that under certain conditions, all standard auction formats (English, Dutch, first-price sealed-bid, second-price sealed-bid) yield the same expected revenue for the seller and the same expected surplus for the buyer with the highest valuation.

Conditions:

1. Bidders are risk-neutral
2. Bidders have independent private values (each bidder's valuation is drawn independently from the same distribution)
3. Bidders are symmetric (same distribution of valuations)
4. Payment depends only on bids (no entry fees, etc.)

When these conditions are violated:

• Risk aversion: first-price auctions yield higher revenue than second-price auctions (risk-averse bidders bid higher in first-price auctions to reduce the risk of losing)
• Common values: when the item has the same underlying value but bidders have different information (e.g., oil leases), the winner's curse can arise
• Correlated values: when bidders' valuations are positively correlated, English auctions yield more revenue than sealed-bid auctions (because bidders can update their valuations based on when others drop out)

The Winner's Curse

In common-value auctions, each bidder has an estimate of the item's true value, but all estimates are imperfect. The bidder with the highest estimate (the winner) is likely to have overestimated the value, because the highest of several noisy estimates tends to exceed the true value.

$E[\text{True value} | \text{Winning} ] < \text{Winning bid}$

Example: bidding for an oil lease. Each firm commissions a geological survey estimating the amount of oil. The firm with the highest estimate wins the auction but discovers that the actual amount of oil is less than estimated (because it had the most optimistic survey).

Rational response to the winner's curse: experienced bidders shade their bids downward to compensate for the expected overestimation. The more bidders competing, the stronger the winner's curse and the more aggressively bidders must shade their bids.

Mechanism Design: The Revelation Principle

Mechanism design is the reverse of game theory: instead of analysing the outcome of a given game, the designer asks "what game (rules) should I create to achieve a desired outcome?"

The revelation principle states that for any mechanism that achieves a particular outcome, there exists an equivalent direct mechanism in which participants truthfully reveal their private information.

Implications:

• The designer can focus on direct revelation mechanisms without loss of generality
• A mechanism is incentive-compatible if truth-telling is a dominant strategy for each participant
• The challenge is to design mechanisms that are simultaneously incentive-compatible, budget-balanced, and efficient (maximising total surplus)

Common Pitfalls in Auction Theory

• Confusing first-price and second-price auction strategies. In a first-price auction, you should bid below your valuation; in a second-price auction, you should bid your true valuation
• Ignoring the winner's curse in common-value auctions. Bidders who do not account for the winner's curse systematically overpay
• Assuming revenue equivalence always holds. It depends on specific assumptions about risk neutrality, independence, and symmetry
• Confusing private values with common values. Private value auctions (art, collectibles) differ fundamentally from common value auctions (mineral rights, spectrum licences)

Advanced Behavioural Topics (HL Extension)

Altruism and Public Goods Games

The standard economic model predicts that rational, self-interested individuals will not contribute to public goods (free riding). Experimental evidence consistently contradicts this:

• In one-shot public goods games, individuals typically contribute 40--60% of their endowment
• Contributions decline with repetition but do not reach zero
• Some individuals consistently contribute at high levels ("conditional cooperators": they contribute if others do)
• A small minority always free-rides

Fischbacher and Gachter (2010): individuals can be classified as:

1. Conditional cooperators (~50%): contribute more when others contribute more
2. Free riders (~30%): contribute nothing regardless of others' contributions
3. Triangle contributors (~20%): other patterns

Implications for policy:

• Tax compliance cannot be explained solely by deterrence (audits, penalties). Social norms, trust in government, and perceptions of fairness also matter
• Public awareness campaigns that highlight the contributions of others can increase cooperation (social proof)
• Punishment of free-riders (even at a cost to the punisher) can sustain cooperation, suggesting a role for social sanctions

Fairness and the Ultimatum Game

In the ultimatum game, Player 1 proposes a split of a fixed sum (e.g., USD 10). Player 2 can accept (both receive the proposed split) or reject (both receive nothing).

Standard prediction: Player 1 offers the minimum possible (e.g., USD 0.01); Player 2 accepts because something is better than nothing.

Experimental findings:

• Average offers are 40--50% of the total
• Offers below 20% are frequently rejected
• Results are remarkably consistent across cultures, though the exact proportions vary

Interpretation: people care about fairness, not just material payoffs. Rejecting a low offer is a costly signal that unfairness is unacceptable. This behaviour contradicts the standard model of self-interested utility maximisation.

Mental Accounting: Extended Analysis

Thaler's four principles of mental accounting:

1. Segregation: gains are segregated (framed separately), while losses are integrated (combined)
2. Cancellation: when a gain and a loss are linked, they are evaluated together (hedonic framing)
3. Integration: losses are combined with larger gains to reduce the pain of loss
4. Separation: small gains are separated from larger losses to provide a silver lining

Application to pricing:

• Bundling: firms bundle a high-margin product with a low-margin product. Consumers evaluate the bundle as a single gain, reducing their sensitivity to the price of the high-margin component
• Partitioned pricing: quoting a product price plus a separate shipping fee feels cheaper than quoting the all-inclusive price (consumers may not fully add the components)
• Drip pricing: revealing additional fees incrementally reduces the perceived total cost relative to upfront disclosure

Common Pitfalls in Advanced Behavioural Economics

• Overgeneralising laboratory findings to real-world settings. Behavioural effects observed with small stakes may not apply to high-stakes decisions
• Assuming all behavioural biases are "irrational." Many heuristics are ecologically rational -- they produce good decisions in the environments where they evolved
• Neglecting individual heterogeneity. Different people exhibit different biases to different degrees
• Confounding correlation with causation in field experiments. Even randomised controlled trials may have spillover effects, Hawthorne effects, or selection bias

Additional Practice Problems

Problem 9: Auction Strategy

You are bidding in a second-price sealed-bid auction for a painting. Your private valuation of the painting is USD 800. You believe there are 3 other bidders, each with valuations uniformly distributed between USD 0 and USD 1,000.

(a) What is your optimal bid?

(b) What is the probability that you win the auction?

(c) What is your expected surplus if you win?

(a) In a second-price sealed-bid auction, the dominant strategy is to bid your true valuation. Your optimal bid is USD 800.

If you bid less (e.g., USD 700), you risk losing to a bidder with a valuation between 700 and 800 who bids their true valuation (between 700 and 800). You would lose an item you value at 800 and pay less than 800, earning a positive surplus.

If you bid more (e.g., USD 900), you risk winning when the second-highest bid is between 800 and 900. You would pay more than your valuation (800), earning a negative surplus.

(b) You win if your bid (800) is the highest of the four bids. The other three bids are uniformly distributed on [0, 1000]. The probability that all three are below 800 is:

$P(\text{win}) = (800/1000)^3 = 0.8^3 = 0.512$ (51.2%)

(c) Given that you win, the second-highest bid is the maximum of three uniform [0, 800] draws. The expected value of the maximum of $n$ uniform [0, b] draws is:

$E[\text{second bid} | \text{win}] = \frac{3}{4} \times 800 = 600$

Your expected surplus = 800 - 600 = \200$.

Expected surplus unconditional: 0.512 \times 200 = \102.40$.

Problem 10: Public Goods Game and Policy

A village of 100 households is considering contributing to a new well that costs USD 10,000. Each household values the well at USD 200. The village leader proposes that each household contributes USD 100, with the government covering the remaining USD 0 (since $100 \times 100 = 10,000$).

(a) Is this project socially efficient? Explain.

(b) Why might the voluntary contribution scheme fail?

(c) How might the village leader use behavioural insights to ensure the project is funded?

(a) Total value to the village = 100 \times 200 = \20,000$. Total cost$= $10,000$. Net benefit$= $10,000 > 0$. The project is socially efficient.

Each household's individual benefit ($200) exceeds the cost per household ($100), so contributing is individually rational as well. However, the well is a public good (non-excludable, non-rival), so free-riding is possible.

(b) The voluntary scheme may fail because:

• Each household reasons: "If others contribute, the well will be built regardless of my contribution, so I can free-ride." This is the dominant strategy in a one-shot game
• If many households think this way, too few contribute and the well is not built (even though everyone would benefit from it)
• Coordination problems: households may be uncertain whether others will contribute, leading to a "wait and see" approach

(c) Behavioural strategies:

1. Default contribution: make the USD 100 contribution the default (opt-out rather than opt-in). Status quo bias will lead most households to accept the default

2. Social proof: publicise the names of households that have already contributed. People are more likely to contribute when they know others are contributing (conditional cooperation)

3. Public commitment: ask households to publicly pledge their contribution at a village meeting. Social pressure and commitment consistency increase follow-through

4. Framing: frame the contribution as "your share of the well" rather than "a payment." People are more willing to pay when they feel they are getting something specific in return

5. Matching: announce that a donor will match each contribution. Matching increases the perceived value of contributing and creates a sense of partnership

6. Sunk cost leverage: start construction using a small initial fund. Once construction has begun, households will feel committed to completing the project (sunk cost fallacy applied positively)

Bayesian Games and Incomplete Information (HL Extension)

Games with Incomplete Information

In the games discussed so far, all players know the payoffs of all other players. In reality, players often have private information that others do not know. A Bayesian game (or game of incomplete information) models this situation using probability distributions.

Harsanyi's transformation (1967): any game of incomplete information can be represented as a Bayesian game by introducing types and beliefs:

1. Types: each player has a set of possible types (e.g., a firm might have "low cost" or "high cost" technology). A player's type determines their payoffs
2. Beliefs: each player has a prior probability distribution over the types of other players
3. Strategies: a strategy specifies an action for each possible type

Bayesian Nash Equilibrium

A Bayesian Nash equilibrium is a strategy profile where each type of each player plays a best response to the strategies of all other types, given their beliefs.

Formally, for each player $i$ with type $\theta_i$:

$s_i^*(\theta_i) \in \arg\max_{s_i} \sum_{\theta_{-i}} p_i(\theta_{-i} | \theta_i) \cdot u_i(s_i, s_{-i}^*, \theta_i, \theta_{-i})$

Worked Example: Entry Game with Incomplete Information

An incumbent firm (player 2) can be either "strong" (low cost, can fight profitably) or "weak" (high cost, fighting is costly). A potential entrant (player 1) does not know the incumbent's type but believes each is equally likely ($p = 0.5$).

Payoffs (entrant, incumbent):

Scenario	Strong Incumbent	Weak Incumbent
Entrant stays out	(0, 10)	(0, 10)
Entrant enters, Incumbent accommodates	(3, 5)	(3, 5)
Entrant enters, Incumbent fights	(-2, 3)	(-2, -3)

Analysis:

For the strong incumbent: if the entrant enters, compare Accommodate (5) vs. Fight (3). The strong incumbent fights (payoff $3 > 5$ is false; actually $3 < 5$, so the strong incumbent accommodates). Wait -- let me re-examine. With payoffs as stated:

For the strong incumbent: Accommodate $= 5$, Fight $= 3$. Since $5 > 3$, the strong incumbent accommodates.

For the weak incumbent: Accommodate $= 5$, Fight $= -3$. Since $5 > -3$, the weak incumbent also accommodates.

In this case, the entrant knows the incumbent will always accommodate, so the entrant always enters. This is not very interesting. Let me adjust the payoffs to create a meaningful incomplete information game:

Revised payoffs:

Scenario	Strong Incumbent	Weak Incumbent
Entrant stays out	(0, 10)	(0, 10)
Entrant enters, Incumbent accommodates	(3, 5)	(3, 5)
Entrant enters, Incumbent fights	(-2, 6)	(-2, -3)

For the strong incumbent: Accommodate $= 5$, Fight $= 6$. Since $6 > 5$, the strong incumbent fights.

For the weak incumbent: Accommodate $= 5$, Fight $= -3$. Since $5 > -3$, the weak incumbent accommodates.

The entrant's expected payoff from entering:

$E[\text{payoff}] = 0.5 \times (-2) + 0.5 \times 3 = -1 + 1.5 = 0.5$

Since $0.5 > 0$ (the payoff from staying out), the entrant enters.

Bayesian Nash equilibrium:

• Entrant: Enter
• Strong incumbent: Fight
• Weak incumbent: Accommodate

Signalling

When one player has private information, they may try to signal their type to the other player. A signal is an observable action that is costly and may be correlated with the player's type.

Example: education as a signal of productivity. In Spence's (1973) job market signalling model, a worker knows their own productivity (high or low) but the employer does not. Education is costly: for high-productivity workers, education costs $c_H$ per year; for low-productivity workers, it costs $c_L > c_H$.

A separating equilibrium exists if the cost difference is large enough that high-productivity workers obtain education while low-productivity workers do not:

$w_H - w_L > c_H \cdot e \quad \text{but} \quad w_H - w_L < c_L \cdot e$

Where $e$ is the required education level, $w_H$ is the wage paid to educated workers, and $w_L$ is the wage paid to uneducated workers.

The separating equilibrium condition is:

$\frac{w_H - w_L}{c_H} > e > \frac{w_H - w_L}{c_L}$

This interval must be non-empty for a separating equilibrium to exist.

Screening

Screening occurs when the uninformed party designs a mechanism to induce the informed party to reveal their private information through their choices.

Example: insurance markets. An insurance company cannot observe a driver's risk type (safe or reckless). It offers two policies:

• Full coverage at premium $p_H$ (designed for high-risk drivers)
• Partial coverage at premium $p_L < p_H$ with a deductible $d$

Safe drivers prefer partial coverage with the lower premium; reckless drivers prefer full coverage despite the higher premium. The menu of contracts screens the two types.

The screening condition for the safe type to choose partial coverage:

$u_s(\text{partial}) > u_s(\text{full})$

And for the reckless type to choose full coverage:

$u_r(\text{full}) > u_r(\text{partial})$

Where $u$ denotes expected utility under each contract.

Mechanism Design Applications

Mechanism design is the reverse of game theory: the designer specifies the rules of the game to achieve a desired outcome, anticipating how rational agents will respond.

Key concepts:

1. Incentive compatibility: a mechanism is incentive-compatible if truth-telling is a best response for every participant type

2. Individual rationality: each participant must prefer participating to not participating

3. The revelation principle: any outcome achievable by any mechanism can also be achieved by a direct mechanism where participants simply report their types, provided truth-telling is incentive-compatible

Practical applications:

• Spectrum auctions: governments design auction rules to allocate radio frequencies efficiently, raising revenue while preventing monopolisation
• Procurement: governments design bidding processes to select contractors efficiently while preventing collusion
• Matching markets: school choice systems (e.g., Boston's school assignment mechanism) use strategy-proof mechanisms to allocate students to schools
• Regulation: pollution permit allocation, spectrum sharing, airport slot allocation

Worked Examples: Advanced Game Theory (HL Extension)

Problem 11: Bayesian Nash Equilibrium

Two firms are bidding for a government contract. Firm 1's cost of completing the project is either $c_L = 10$ or $c_H = 20$, each with probability 0.5. Firm 2's cost is always $c_2 = 15$. Firm 1 knows its own cost but Firm 2 only knows the probability distribution.

The government uses a first-price sealed-bid auction. The firm with the lowest bid wins and is paid its bid.

(a) Find the Bayesian Nash equilibrium bidding strategy for Firm 1.

(b) What is Firm 2's equilibrium bid?

(a) Firm 1's strategy: if its cost is $c_L = 10$, it bids $b_L$; if its cost is $c_H = 20$, it bids $b_H$.

In a first-price auction with independent private values, the equilibrium bid for a player with cost $c_i$ drawn from distribution $F$ on $[\underline{c}, \overline{c}]$ is:

$b_i(c_i) = \frac{1}{F(c_i)} \int_{\underline{c}}^{c_i} y \, f(y) \, dy$

For Firm 1 with cost $c \sim \text{Uniform}[10, 20]$:

$b_1(c) = \frac{1}{(c - 10)/10} \int_{10}^{c} y \cdot \frac{1}{10} \, dy = \frac{10}{c - 10} \cdot \frac{y^2}{20} \bigg|_{10}^{c} = \frac{10}{c - 10} \cdot \frac{c^2 - 100}{20}$

$b_1(c) = \frac{c^2 - 100}{2(c - 10)} = \frac{(c - 10)(c + 10)}{2(c - 10)} = \frac{c + 10}{2}$

When $c = 10$: $b_1 = (10 + 10)/2 = 10$ (bids at cost, earning zero surplus)

When $c = 20$: $b_1 = (20 + 10)/2 = 15$ (shades bid downward by $5$)

For the low-cost type: $b_L = 10$

For the high-cost type: $b_H = 15$

(b) Firm 2 has a known cost of 15. Firm 2 must bid against Firm 1, whose bid is uncertain from Firm 2's perspective.

Firm 2 believes Firm 1's cost is uniformly distributed on $[10, 20]$. Firm 1's bidding function is $b_1(c) = (c + 10)/2$.

Since $b_1$ is monotonically increasing in $c$, and $c$ is uniform on $[10, 20]$:

$b_1$ is uniform on $[10, 15]$.

Firm 2 wins when its bid is below Firm 1's bid. Firm 2's expected profit when bidding $b_2$:

$E[\pi_2] = (b_2 - 15) \cdot P(b_2 < b_1) = (b_2 - 15) \cdot \frac{b_2 - 10}{5}$

Maximising: $\frac{d}{db_2}\left[(b_2 - 15)(b_2 - 10)\right] = 2b_2 - 25 = 0$

$b_2 = 12.5$

Firm 2's equilibrium bid is 12.5, shading its cost (15) downward by 2.5.

Expected profit for Firm 2: $(12.5 - 15)(12.5 - 10)/5 = (-2.5)(2.5)/5 = -1.25$

Firm 2 earns negative expected profit because it faces the risk of winning when Firm 1 has a very low cost. This illustrates the winner's curse: in common-value auctions, winning is bad news about the winner's bid relative to competitors' costs.

Problem 12: Signalling Equilibrium

A job applicant's productivity is either $\theta_H = 50$ or $\theta_L = 20$. The employer cannot observe productivity but offers wage $w_H = 40$ to college graduates and $w_L = 25$ to non-graduates. Education costs $e$ years; the cost per year is $c_H = 4$ for high-productivity workers and $c_L = 10$ for low-productivity workers.

(a) Find the range of education levels $e$ that can sustain a separating equilibrium.

(b) Explain why a pooling equilibrium may fail.

(a) For a separating equilibrium:

• High-productivity workers must obtain education: $w_H - w_L \geq c_H \cdot e$
• Low-productivity workers must not obtain education: $w_H - w_L < c_L \cdot e$

Substituting:

$40 - 25 = 15 \geq 4e \implies e \leq 3.75$

$40 - 25 = 15 < 10e \implies e > 1.5$

So any education level $e \in (1.5, 3.75]$ can serve as a separating signal. The employer interprets education $e \geq e^*$ as a signal of high productivity and offers $w_H$.

In equilibrium, the minimum signal is $e^* = 1.5 + \epsilon$ (slightly above 1.5). High-productivity workers obtain $e^*$ years of education; low-productivity workers do not.

(b) A pooling equilibrium (where both types make the same education choice) fails because the employer cannot distinguish between types and would offer an average wage: $w_p = 0.5 \times 40 + 0.5 \times 25 = 32.5$

High-productivity workers would deviate: their education cost ($4e$) is lower than their wage gain ($40 - 32.5 = 7.5$), so they would obtain more education to separate themselves.

Therefore, the only stable equilibrium is the separating one.

Problem 13: Prospect Theory Calculations

A student must choose between two exam preparation strategies:

• Strategy A: 70% chance of scoring 70, 30% chance of scoring 40
• Strategy B: 50% chance of scoring 80, 50% chance of scoring 30

The student's reference point is a score of 60.

(a) Calculate the expected value of each strategy.

(b) Using prospect theory (loss aversion coefficient $\lambda = 2$, probability weighting function $\pi(p) = p^{0.65}$), calculate the prospect theory value of each strategy.

(c) Which strategy does prospect theory predict the student will choose?

(a) $EV_A = 0.7 \times 70 + 0.3 \times 40 = 49 + 12 = 61$

$EV_B = 0.50 \times 80 + 0.50 \times 30 = 40 + 15 = 55$

Under expected utility, the student chooses Strategy A ($61 > 55$).

(b) Prospect theory calculation:

Strategy A:

Outcomes relative to reference point (60):

• Gain of 10 with probability 0.70
• Loss of 20 with probability 0.30

Probability weights: $\pi(0.70) = 0.70^{0.65} = 0.792$, $\pi(0.30) = 0.30^{0.65} = 0.415$

Prospect theory value:

$V_A = \pi(0.70) \cdot v(10) + \pi(0.30) \cdot v(-20)$

Using the value function $v(x) = x^{0.88}$ for gains and $v(x) = -\lambda|x|^{0.88}$ for losses:

$v(10) = 10^{0.88} = 7.59$

$v(-20) = -2 \times 20^{0.88} = -2 \times 15.60 = -31.20$

$V_A = 0.792 \times 7.59 + 0.415 \times (-31.20) = 6.01 - 12.95 = -6.94$

Strategy B:

Outcomes relative to reference point (60):

• Gain of 20 with probability 0.50
• Loss of 30 with probability 0.50

$\pi(0.50) = 0.50^{0.65} = 0.637$, $\pi(0.50) = 0.637$ (same)

$v(20) = 20^{0.88} = 15.60$

$v(-30) = -2 \times 30^{0.88} = -2 \times 22.65 = -45.30$

$V_B = 0.637 \times 15.60 + 0.637 \times (-45.30) = 9.94 - 28.86 = -18.92$

(c) Prospect theory predicts Strategy A ($V_A = -6.94 > V_B = -18.92$), which is the same as expected utility in this case. However, the intuition differs:

• Strategy A has a smaller potential loss (-20 vs. -30) and a higher probability of a gain
• The loss aversion coefficient makes the larger potential loss in Strategy B particularly aversive
• The probability weighting overweights the 50% probability of a large loss in Strategy B

If the reference point were lower (e.g., 40), the analysis would shift significantly toward Strategy A, as both strategies would involve only gains.

Common Pitfalls: Advanced Game Theory (Comprehensive)

• Assuming that Bayesian Nash equilibrium requires players to know each other's types. Players only need to know the probability distribution of others' types
• Confusing signalling with screening. Signalling is initiated by the informed party; screening is initiated by the uninformed party
• Assuming that all separating equilibria are efficient. The equilibrium with the highest education requirement ($e = 3.75$) wastes resources on signalling -- the same separation could be achieved with less education
• Applying the revelation principle without understanding its scope. The revelation principle says an equivalent direct mechanism exists, but it does not guarantee that finding or implementing it is practical
• Ignoring mixed strategies in Bayesian games. Just as complete information games can have mixed strategy equilibria, Bayesian games can have mixed strategy Bayesian Nash equilibria
• Overstating the practical relevance of mechanism design. Many real-world allocation problems involve computational complexity that makes optimal mechanism design intractable

Repeated Games and Cooperation (HL Extension)

The Finitely Repeated Prisoner's Dilemma

When the prisoner's dilemma is repeated a known, finite number of times, backward induction predicts defection in every round:

$\text{Round } T: \text{defect (no future punishment possible)}$ $\text{Round } T-1: \text{defect (round } T \text{ will be defection regardless)}$ $\vdots$ $\text{Round 1: defect}$

The unique subgame perfect equilibrium is (Defect, Defect) in every round.

The Infinitely Repeated Prisoner's Dilemma

When the game is repeated infinitely (or with an unknown endpoint), cooperation can be sustained as a Nash equilibrium.

Grim trigger strategy: cooperate as long as the other player cooperates; if the other player defects once, defect forever.

Cooperation condition: cooperation is sustainable if the present discounted value of cooperating exceeds the one-time gain from defection:

$\frac{g}{1 - \delta} \geq t$

Where:

• $g$ = gain per round from mutual cooperation (vs. mutual defection)
• $t$ = temptation payoff from unilateral defection
• $\delta$ = discount factor ($0 < \delta < 1$)

Rearranging:

$\delta \geq \frac{t - g}{t - p}$

Where $p$ is the punishment payoff (mutual defection).

Numerical example:

Payoff matrix:

	Cooperate	Defect
Cooperate	(3, 3)	(0, 5)
Defect	(5, 0)	(1, 1)

$g = 3 - 1 = 2$, $t = 5$, $p = 1$.

$\delta \geq (5 - 3)/(5 - 1) = 2/4 = 0.5$

If $\delta \geq 0.5$ (players value future payoffs at least half as much as current payoffs), cooperation is sustainable under grim trigger.

Axelrod's Tournament

Robert Axelrod (1984) ran a computer tournament where strategies for the repeated prisoner's dilemma competed against each other. The winner was Tit-for-Tat (Anatol Rapoport):

1. Cooperate in the first round
2. In subsequent rounds, do whatever the opponent did in the previous round

Why Tit-for-Tat works:

1. Nice: never defects first, so it can cooperate with other cooperative strategies
2. Retaliatory: punishes defection immediately, preventing exploitation
3. Forgiving: returns to cooperation as soon as the opponent does, avoiding permanent punishment spirals
4. Clear: simple and predictable, so opponents can understand the consequences of their actions

Applications of Repeated Games

1. Oligopoly pricing: firms in an oligopoly can sustain collusive (monopoly) pricing if the game is repeated and the discount factor is high enough. The condition is:

$\delta \geq \frac{1}{n}$

Where $n$ is the number of firms. More firms make cooperation harder (the discount factor must be higher). With 2 firms, $\delta \geq 0.5$; with 10 firms, $\delta \geq 0.9$.

2. International trade agreements: WTO rules can be understood as a repeated game. Countries comply with trade rules because the long-run benefit of open trade exceeds the short-run gain from protectionism
3. Environmental agreements: the tragedy of the commons can be overcome through repeated interaction. Communities that interact repeatedly can sustain cooperative resource management

Auction Theory (HL Extension)

Types of Auctions

1. English (ascending-price) auction: price rises until only one bidder remains
2. Dutch (descending-price) auction: price falls until a bidder accepts
3. First-price sealed-bid auction: highest bidder wins and pays their bid
4. Second-price sealed-bid (Vickrey) auction: highest bidder wins and pays the second-highest bid

Revenue Equivalence Theorem

Under certain conditions (risk-neutral bidders, independent private values, symmetric bidders), all four auction formats yield the same expected revenue for the seller.

Vickrey auction dominance: the second-price auction has a dominant strategy equilibrium where each bidder bids their true valuation. This is because:

• If you bid above your valuation, you might win but pay more than the item is worth
• If you bid below your valuation, you might lose an item you would have won at a price below your valuation
• Bidding your true valuation is weakly dominant

Winner's Curse

In common-value auctions (where the item has the same value for all bidders, but no one knows the exact value), the winner tends to be the bidder who most overestimated the value.

Formal model:

Suppose the true value of an oil field is $V$, and each bidder receives a signal $s_i = V + \epsilon_i$ where $\epsilon_i$ is noise.

The expected value of $V$ given that you won the auction is:

$E[V | s_i \text{ is the highest signal}] < s_i$

Because winning the auction provides information that your signal was likely the most optimistic.

Mitigation: rational bidders should shade their bids below their signal:

$\text{Optimal bid} = s_i - \frac{\sigma^2}{s_i}$

Where $\sigma^2$ is the variance of the signal noise. The more uncertain the value, the more bidders should shade their bids.

Numerical example:

Three oil companies bid on an offshore oil field. Each estimates the field's value with uncertainty:

• Company A: $s_A = 100$ million
• Company B: $s_B = 80$ million
• Company C: $s_C = 90$ million

In a first-price auction, Company A wins with a bid of 100 million. But $E[V | A \text{ wins}]$ is less than 100 million because A had the highest signal, suggesting A was the most optimistic.

If the signals have standard deviation of 20 million:

Approximate correction for A: $\text{bid} = 100 - 20^2/100 = 100 - 4 = 96$ million.

Company A should bid approximately 96 million rather than 100 million to avoid the winner's curse.

Auction Applications

1. Spectrum auctions: governments use auctions to allocate radio spectrum to telecom companies. The 3G spectrum auction in the UK (2000) raised GBP 22.5 billion
2. Treasury auctions: central banks use auctions to sell government bonds
3. Online advertising: Google's AdWords auction is a generalised second-price auction
4. Art and antiques: Christie's and Sotheby's use English auctions

Exam-Style Questions: Game Theory (HL Extension)

Question 1: Repeated Prisoner's Dilemma and Oligopoly (10 marks)

Two firms, A and B, compete in a market. They can either set a high price (cooperate) or a low price (defect). The payoff matrix (annual profits in USD million) is:

	High Price	Low Price
High Price	(10, 10)	(0, 15)
Low Price	(15, 0)	(3, 3)

The discount factor is $\delta = 0.8$.

(a) Identify the Nash equilibrium of the one-shot game. [2 marks]

(b) Show that cooperation can be sustained using grim trigger. [4 marks]

(c) A third firm enters the market. Can cooperation still be sustained? Explain. [4 marks]

(a) Each firm has a dominant strategy to set a low price (15 > 10 and 3 > 0). The unique Nash equilibrium is (Low Price, Low Price) with payoffs (3, 3).

(b) Gain from cooperation: $g = 10 - 3 = 7$ per year. Temptation: $t = 15$.

Grim trigger condition: $\delta \geq (t - g)/(t - p) = (15 - 10)/(15 - 3) = 5/12 = 0.417$

Since $\delta = 0.8 > 0.417$, cooperation is sustainable. The present value of cooperation: $10 + 10(0.8) + 10(0.8)^2 + \cdots = 10/(1 - 0.8) = 50$.

The present value of defection: $15 + 3(0.8) + 3(0.8)^2 + \cdots = 15 + 3 \times 0.8/(1 - 0.8) = 15 + 12 = 27$.

$50 > 27$: cooperation yields higher lifetime profits.

(c) With three firms using grim trigger, the condition becomes:

$\delta \geq 1/n = 1/3 = 0.333$

Since $\delta = 0.8 > 0.333$, cooperation is still sustainable in theory. However, with more firms, the incentive to defect increases because each firm's share of the cooperative payoff decreases, and monitoring becomes harder. The probability of accidental defection or deviation rises with the number of firms.

Question 2: Auction Design (10 marks)

A government is selling a mining licence. Two mining companies bid. Each company's private valuation is independently drawn from a uniform distribution on [USD 0, USD 100 million].

(a) In a first-price sealed-bid auction, what is the optimal bid for a company with valuation v? [3 marks]

(b) Calculate the optimal bid for valuations of USD 60 million and USD 80 million. [2 marks]

(c) The government considers using a second-price (Vickrey) auction. Explain why the expected revenue is the same as the first-price auction. [5 marks]

(a) In a first-price sealed-bid auction with two bidders and valuations uniformly distributed on $[0, V]$, the symmetric equilibrium bidding strategy is:

$b(v) = \frac{n-1}{n} \cdot v = \frac{1}{2} v$

Each bidder shades their bid to half their valuation.

(b) For $v = 60$: $b = 30$ million. For $v = 80$: $b = 40$ million.

(c) By the Revenue Equivalence Theorem, the expected revenue is the same in both auction formats under the conditions given (risk-neutral bidders, independent private values, symmetric bidders).

In the first-price auction: expected revenue $= E[b(v_{(1)})]$ where $v_{(1)}$ is the highest valuation. $E[v_{(1)}] = 2V/3 = 66.7$ million, so expected revenue $= 33.3$ million.

In the second-price auction: expected revenue $= E[v_{(2)}]$ where $v_{(2)}$ is the second-highest valuation. $E[v_{(2)}] = V/3 = 33.3$ million.

Both yield expected revenue of 33.3 million. The second-price auction is simpler for bidders (bid your true valuation) and avoids the winner's curse, but yields the same expected revenue for the seller.

Practical advantage of Vickrey: bidders do not need to strategise about how much to shade their bids, reducing the complexity of participation and encouraging more bidders to enter. More bidders increase expected revenue in both formats.

Question 3: Behavioural Economics and Policy Design (10 marks)

A government wants to increase pension contributions. Currently, 40% of eligible workers contribute to a pension scheme. Research shows:

• With an opt-in system: 40% enrolment
• With an opt-out system: 92% enrolment
• With an active choice (mandatory decision): 75% enrolment

(a) Which behavioural bias does the opt-in/opt-out difference illustrate? [2 marks]

(b) Calculate the increase in annual pension contributions if 10 million workers each contribute an average of USD 2,000 more per year under opt-out. [2 marks]

(c) Evaluate the use of defaults as a policy tool. [6 marks]

(a) Status quo bias and default effects. Individuals tend to stick with the default option because changing requires active effort, and losses (the effort of enrolment) are weighted more heavily than gains (future pension benefits).

(b) Additional contributors $= (92\% - 40\%) \times 10 = 5.2$ million.

Additional annual contributions $= 5.2 \times 2\,000 = 10.4$ billion.

(c) Evaluation of defaults as a policy tool:

Advantages:

1. Effectiveness: opt-out systems dramatically increase participation (40% to 92%)
2. Low cost: defaults require minimal administrative change compared to financial incentives or mandates
3. Preserves choice: workers can still opt out if they have strong preferences against saving
4. Evidence-based: extensive empirical evidence from the UK, US, and other OECD countries confirms the effectiveness of automatic enrolment

Disadvantages:

1. Paternalism: defaults manipulate choice architecture without explicit consent. Some argue this undermines autonomy
2. Heterogeneity: the optimal default varies across individuals. A single default may not be appropriate for low-income workers who need current consumption
3. Limited effect on contribution rates: automatic enrolment increases the number of contributors but not necessarily the contribution rate (many stick with the minimum default rate)
4. May crowd out financial literacy: reliance on defaults may reduce individuals' engagement with their financial decisions

Conclusion: defaults are a powerful and cost-effective policy tool, but should be complemented with financial education and flexible options for those with different needs.

Case Studies: Game Theory and Behavioural Economics (HL Extension)

OPEC and Repeated Game Theory

OPEC (Organisation of Petroleum Exporting Countries) is a real-world example of a repeated game with incomplete monitoring:

1. Cooperation (cartel): OPEC sets production quotas to maintain high oil prices
2. Defection: individual members exceed their quotas to increase revenue
3. Punishment: OPEC lacks a formal enforcement mechanism, but Saudi Arabia has historically acted as the "swing producer," increasing output to punish defectors by driving prices down
4. Credibility: Saudi Arabia's willingness to absorb short-term losses to punish defectors enhances the credibility of the grim trigger

Empirical pattern: OPEC cooperation tends to break down when:

• Oil prices are high (temptation to defect increases)
• New members join (more players make cooperation harder)
• Demand shocks create uncertainty about future payoffs

Behavioural Economics in Practice: UK Behavioural Insights Team

The UK's Behavioural Insights Team ("Nudge Unit"), established in 2010, was the world's first government unit dedicated to applying behavioural economics to policy:

1. Tax compliance: sending letters informing taxpayers that "9 out of 10 people in your town pay their tax on time" increased on-time payments by 15 percentage points
2. Organ donation: the shift to an opt-out system (2015) was informed by behavioural evidence on default effects
3. Job seeker compliance: rewriting job centre letters in simpler language and adding social norms increased appointment attendance
4. Energy efficiency: providing households with comparisons of their energy use to neighbours reduced energy consumption by 2--6%

Evaluation: the BIT has demonstrated that low-cost behavioural interventions can achieve significant policy outcomes. However, effects are often modest, may decay over time, and raise questions about the ethics of state-sponsored "nudging."

Evolutionary Game Theory (HL Extension)

Replicator Dynamics

In evolutionary game theory, strategies are not chosen rationally but spread through a population based on their relative fitness (payoff). The replicator equation describes how the frequency of a strategy changes over time:

$\frac{dx_i}{dt} = x_i \left[ f_i(\mathbf{'\{'}x{'\}'}) - \bar{f}(\mathbf{'\{'}x{'\}'}) \right]$

Where:

• $x_i$ = frequency of strategy $i$ in the population
• $f_i(\mathbf{'\{'}x{'\}'})$ = fitness (expected payoff) of strategy $i$
• $\bar{f}(\mathbf{'\{'}x{'\}'})$ = average fitness of the population

Strategies with above-average fitness grow; those with below-average fitness shrink.

Evolutionary Stable Strategies (ESS)

A strategy $s^*$ is an evolutionary stable strategy if, when adopted by the entire population, no mutant strategy can invade:

$u(s^*, s^*) \geq u(s, s^*) \text{ for all } s$

And for any neutral mutant ($u(s, s^*) = u(s^*, s^*)$):

$u(s^*, s) > u(s, s)$

Application to the Prisoner's Dilemma:

In a population of Cooperators and Defectors, the payoff matrix is:

	Cooperate	Defect
Cooperate	(3, 3)	(0, 5)
Defect	(5, 0)	(1, 1)

If the population fraction of Cooperators is $x$:

Fitness of Cooperator: $f_C = 3x + 0(1-x) = 3x$

Fitness of Defector: $f_D = 5x + 1(1-x) = 1 + 4x$

Average fitness: $\bar{f} = x(3x) + (1-x)(1+4x) = 3x^2 + 1 + 4x - x - 4x^2 = 1 + 3x - x^2$

Replicator dynamics:

$\frac{dx}{dt} = x(3x - 1 - 3x + x^2) = x(x^2 - 1) = x(x-1)(x+1)$

Since $x \in [0, 1]$, $\frac{dx}{dt} < 0$ for all $x \in (0, 1)$. Cooperators always decline.

Defect is the unique ESS. This shows that in a well-mixed population, cooperation cannot survive in a one-shot prisoner's dilemma.

Spatial Games and Cooperation

When players interact with neighbours rather than randomly, cooperation can persist because cooperators cluster together and benefit from mutual cooperation, while defectors at the boundary of cooperator clusters are outcompeted.

Spatial prisoner's dilemma: if each cell in a grid plays the prisoner's dilemma with its four neighbours and adopts the strategy of its most successful neighbour, cooperators can survive in clusters because:

• Interior cooperators: surrounded by cooperators, high payoff
• Boundary defectors: some neighbours are cooperators, moderate payoff
• Interior defectors: surrounded by defectors, low payoff

This explains why cooperation is observed in nature despite the prisoner's dilemma logic.

Hawk-Dove Game

The Hawk-Dove game models conflict over a resource of value $V$:

	Hawk	Dove
Hawk	$\left(\frac{V-C}{2}, \frac{V-C}{2}\right)$	$(V, 0)$
Dove	$(0, V)$	$\left(\frac{V}{2}, \frac{V}{2}\right)$

Where $C$ is the cost of fighting.

If $V > C$: Hawk is dominant (the resource is worth fighting for)

If $V < C$: the ESS is a mixed strategy. Let $p$ = probability of playing Hawk:

$p \cdot \frac{V-C}{2} + (1-p) \cdot V = p \cdot 0 + (1-p) \cdot \frac{V}{2}$

$p(V-C)/2 + V - pV = V/2 - pV/2$

$pV/2 - pC/2 + V - pV = V/2 - pV/2$

$V - pV - pC/2 = V/2 - pV/2$

$V - pV + pV/2 - pC/2 = V/2$

$V - pV/2 - pC/2 = V/2$

$p(V + C)/2 = V - V/2 = V/2$

$p = V/(V + C)$

Numerical example: $V = 4$, $C = 6$.

$p = 4/(4+6) = 0.4$.

The ESS is: play Hawk with probability 0.4, Dove with probability 0.6.

In a population at equilibrium, 40% play Hawk and 60% play Dove. The average payoff for both strategies is equal:

Hawk payoff $= 0.4 \times (-1) + 0.6 \times 4 = -0.4 + 2.4 = 2.0$

Dove payoff $= 0.4 \times 0 + 0.6 \times 2 = 0 + 1.2 = 1.2$

Wait -- these should be equal at the mixed ESS. Let me recalculate:

Hawk expected payoff $= p \times \frac{V-C}{2} + (1-p) \times V = 0.4 \times (-1) + 0.6 \times 4 = 2.0$

Dove expected payoff $= p \times 0 + (1-p) \times V/2 = 0.4 \times 0 + 0.6 \times 2 = 1.2$

These are not equal, which indicates an error. The correct ESS condition requires equal payoffs:

$p \times \frac{V-C}{2} + (1-p)V = (1-p)\frac{V}{2}$

$p(V-C)/2 + V - pV = V/2 - pV/2$

Multiply through by 2: $p(V-C) + 2V - 2pV = V - pV$

$pV - pC + 2V - 2pV = V - pV$

$2V - pV - pC = V$

$p(V + C) = V$

$p = V/(V+C) = 4/10 = 0.4$

Now verify: Hawk payoff $= 0.4(-1) + 0.6(4) = 2.0$. Dove payoff $= 0.4(0) + 0.6(2) = 1.2$.

The discrepancy arises because the average population payoff is:

$\bar{f} = 0.4(2.0) + 0.6(1.2) = 0.8 + 0.72 = 1.52$

The replicator dynamics check: $\Delta x = x(f_H - \bar{f}) = 0.4(2.0 - 1.52) = 0.192 > 0$.

Hawks would grow, which contradicts the ESS. This suggests $p = 0.4$ is not the correct ESS. The correct derivation: at the mixed ESS, the expected payoff of Hawk must equal the expected payoff of Dove.

Let me use the correct formula: the expected payoff of Hawk against a population playing Hawk with probability $p$:

$f_H = p \cdot \frac{V-C}{2} + (1-p) \cdot V$

$f_D = p \cdot 0 + (1-p) \cdot \frac{V}{2}$

Setting $f_H = f_D$:

$p(V-C)/2 + (1-p)V = (1-p)V/2$

$pV/2 - pC/2 + V - pV = V/2 - pV/2$

$V - pV/2 - pC/2 = V/2$

$V/2 = p(V + C)/2$

$p = V/(V+C)$

This IS correct. Let me recheck the arithmetic with $V=4, C=6, p=0.4$:

$f_H = 0.4 \times (-6/2) + 0.6 \times 4 = 0.4(-1) + 2.4 = -0.4 + 2.4 = 2.0$

$f_D = 0.4 \times 0 + 0.6 \times 2 = 1.2$

$2.0 \neq 1.2$. Something is wrong. Let me recheck:

$f_H = p(V-C)/2 + (1-p)V$

$= 0.4(4-6)/2 + 0.6(4) = 0.4(-2/2) + 2.4 = 0.4(-1) + 2.4 = -0.4 + 2.4 = 2.0$

$f_D = (1-p)V/2 = 0.6(4)/2 = 0.6(2) = 1.2$

The payoffs are indeed not equal at $p = 0.4$. This means the standard formula needs re-examination. The correct ESS for Hawk-Dove is actually:

$p^* = V/C$ when $V < C$.

$p = 4/6 = 2/3 \approx 0.667$.

Check: $f_H = (2/3)(-1) + (1/3)(4) = -2/3 + 4/3 = 2/3$

$f_D = (2/3)(0) + (1/3)(2) = 2/3$

Now the payoffs are equal at $2/3$. The correct ESS is $p^* = V/C$, not $V/(V+C)$.

(The formula $V/(V+C)$ applies to a different normalisation of the game. With the standard payoffs above, the correct ESS is $p^* = V/C$.)

Policy implication: when the cost of conflict ($C$) is high relative to the value of the resource ($V$), aggression is rare ($p^* = V/C$ is small). This explains why most international disputes are resolved peacefully: the cost of war typically exceeds the value of the disputed resource.

Cognitive Biases and Decision-Making: Extended Analysis (HL Extension)

Confirmation Bias

Definition: the tendency to seek, interpret, and remember information that confirms existing beliefs while ignoring or discounting contradictory evidence.

Economic implications:

1. Investment: investors may hold losing stocks too long (seeking confirming information about the stock's future prospects) and sell winning stocks too early
2. Policy: policymakers may interpret data in ways that confirm their preferred policy approach, leading to suboptimal decisions
3. Hiring: interviewers may ask questions designed to confirm their initial impression of a candidate

Overconfidence Bias

Definition: individuals systematically overestimate their knowledge, ability, and the precision of their predictions.

Evidence:

• 93% of US drivers rate themselves as above average in safety (Svenson, 1981)
• Startup founders consistently overestimate the probability of success
• Stock traders overestimate the accuracy of their predictions

Economic consequences:

1. Excessive trading: overconfident investors trade too frequently, reducing returns through transaction costs (Barber and Odean, 2000: overconfident investors underperform by 3% per year)
2. Under-diversification: overconfident investors hold concentrated portfolios
3. Entrepreneurial entry: overconfidence leads to excessive market entry, reducing average profits for all firms (Camerer and Lovallo, 1999)

Present Bias and Time Inconsistency

Definition: individuals disproportionately discount the near future relative to the distant future, leading to time-inconsistent preferences.

Quasi-hyperbolic discounting (Laibson, 1997):

$U = u(c_t) + \beta \sum_{s=1}^{T} \delta^s u(c_{t+s})$

Where $\beta < 1$ captures present bias (typically $\beta \approx 0.7$) and $\delta$ is the standard exponential discount factor.

Implications:

1. Under-saving: individuals plan to save more in the future but fail to do so when the future arrives. The savings rate is lower than planned
2. Procrastination: tasks with immediate costs and future benefits are postponed
3. Addiction: the present bias explains why addictive behaviours persist despite long-term costs

Numerical example:

An individual with $\beta = 0.7$, $\delta = 0.95$ chooses between:

Option A: receive USD 100 today Option B: receive USD 110 in one week

From today's perspective: $U(A) = 100$, $U(B) = 0.7 \times 0.95 \times 110 = 73.15$.

The individual chooses A (immediate gratification).

But from the perspective of next week: $U(A) = 0.7 \times 0.95 \times 100 = 66.50$, $U(B) = 110$.

The individual would choose B if both options were in the future. This is time inconsistency: preferences reverse as the future becomes the present.

Mental Accounting

Definition: individuals categorise money into different mental "accounts" based on its source or intended use, treating money differently depending on the account.

Examples:

1. House money effect: gamblers are more willing to take risks with money they have won than with their own money
2. Sunk cost fallacy: continuing a project because of resources already invested, even when future costs exceed future benefits. Rationally, sunk costs should be ignored
3. Windfall spending: people spend tax refunds or bonuses more freely than regular income

Policy implications:

1. Framing tax rebates: presenting tax relief as a "bonus" rather than a "refund of overpayment" increases spending
2. Savings products: earmarking savings accounts for specific goals (education, housing) increases savings rates
3. Budget rules: mental accounting can be harnessed to improve financial decision-making (e.g., "pay yourself first" rules)

Exam-Style Questions: Behavioural Economics (Additional)

Question 4: Present Bias and Savings Behaviour (10 marks)

A worker earns USD 50,000 per year and can save either USD 5,000 or USD 2,000 per year for retirement. At age 25, the worker plans to save USD 5,000 per year for 40 years.

With $\beta = 0.7$ and $\delta = 0.95$, the present bias makes the worker save only USD 2,000 per year.

(a) Calculate the retirement savings at age 65 under both plans, assuming a 5% annual return. [4 marks]

(b) Calculate the annual retirement income under both plans, assuming a 20-year drawdown and a 4% return. [3 marks]

(c) Evaluate two policies to overcome present bias in savings decisions. [3 marks]

(a) Plan A (USD 5,000/year for 40 years at 5%):

$\text{FV} = 5000 \times \frac{1.05^{40} - 1}{0.05} = 5000 \times 120.80 = 604\,000$

Plan B (USD 2,000/year for 40 years at 5%):

$\text{FV} = 2000 \times 120.80 = 241\,600$

Present bias costs the worker $604\,000 - 241\,600 = 362\,400$ in retirement savings.

(b) Plan A: annuity factor (20 years, 4%) $= \frac{1 - 1.04^{-20}}{0.04} = 13.59$.

Annual income $= 604\,000 / 13.59 = 44\,445$

Plan B: annual income $= 241\,600 / 13.59 = 17\,778$

Present bias reduces annual retirement income by USD 26,667.

(c) Policy 1: Automatic enrolment with escalation.

Workers are automatically enrolled in a pension plan with a default contribution rate that escalates by 1 percentage point per year (e.g., from 3% to 10%). This exploits status quo bias and inertia to overcome present bias. Evidence: the US Save More Tomorrow programme increased savings rates from 3.5% to 13.6% over four years.

Policy 2: Commitment savings products.

Banks offer savings accounts with penalties for early withdrawal (illiquidity). By committing future income to an illiquid account, workers pre-commit to saving, overcoming the temptation to spend. Evidence: SEED (Save, Earn, Enjoy Deposit) accounts in the Philippines increased savings by 82 percentage points among participants.

Both policies work by changing the choice architecture or adding commitment mechanisms that counteract present bias.

Question 5: Evolutionary Game Theory and International Cooperation (10 marks)

Two countries are negotiating a climate agreement. Each can either Abide (reduce emissions) or Defect (continue business as usual). The payoff matrix (welfare in USD billion) is:

	Abide	Defect
Abide	(50, 50)	(10, 70)
Defect	(70, 10)	(20, 20)

(a) Find the Nash equilibrium of the one-shot game. [2 marks]

(b) If the game is repeated indefinitely with discount factor $\delta$, find the minimum $\delta$ for cooperation under grim trigger. [4 marks]

(c) Explain how evolutionary game theory predicts the outcome in a population of countries where some always Abide and some always Defect. [4 marks]

(a) Each country has a dominant strategy to Defect (70 > 50 and 20 > 10). The Nash equilibrium is (Defect, Defect) with payoffs (20, 20).

(b) Gain from cooperation: $g = 50 - 20 = 30$. Temptation: $t = 70$.

$\delta \geq (t - g)/(t - p) = (70 - 50)/(70 - 20) = 20/50 = 0.4$

If $\delta \geq 0.4$ (countries value future payoffs at least 40% as much as current payoffs), cooperation is sustainable.

(c) In an evolutionary framework with fraction $x$ of Abiders:

$f_{\text{Abide}} = 50x + 10(1-x) = 10 + 40x$

$f_{\text{Defect}} = 70x + 20(1-x) = 20 + 50x$

$\Delta x = x(f_{\text{Abide}} - \bar{f})$

$\bar{f} = x(10 + 40x) + (1-x)(20 + 50x) = 10x + 40x^2 + 20 + 50x - 20x - 50x^2 = 20 + 40x - 10x^2$

$f_{\text{Abide}} - \bar{f} = 10 + 40x - 20 - 40x + 10x^2 = -10 + 10x^2 = 10(x^2 - 1) < 0$ for all $x \in (0,1)$.

The fraction of Abiders always declines: Defect is the unique ESS.

This explains why international climate cooperation is difficult: even if some countries initially cooperate, the evolutionary dynamics drive the population toward universal defection. The only way to sustain cooperation is through repeated interaction (as in part b) or through institutional mechanisms (enforcement, penalties, reputation).

Bargaining and Negotiation Theory (HL Extension)

Nash Bargaining Solution

The Nash bargaining solution maximises the product of the players' gains over their disagreement (threat) points:

$\max_{(u_1, u_2)} (u_1 - d_1)(u_2 - d_2)$

Subject to $u_1 \geq d_1$, $u_2 \geq d_2$, and $(u_1, u_2)$ being feasible.

Where $d_1$ and $d_2$ are the payoffs each player receives if negotiations break down (the threat point or disagreement point).

Numerical Example: Wage Negotiation

A firm and a union are negotiating wages. The firm's profit is $\pi = 100 - w$ (where $w$ is the wage). The union's utility is $u = w$. If negotiations fail, the firm hires replacement workers at $w = 30$ (profit $= 70$), and the union receives $d_u = 0$.

The Nash bargaining solution:

$\max_w (100 - w - 70)(w - 0) = (30 - w)w$

FOC: $30 - 2w = 0 \implies w = 15$.

The Nash solution gives $w = 15$, with the firm earning profit $= 85$ and the union receiving $u = 15$.

Alternative threat point: if the union can strike and reduce the firm's profit to 20 during the strike (while union members receive strike pay of 10):

$\max_w (100 - w - 20)(w - 10) = (80 - w)(w - 10)$

FOC: $80 - w - (w - 10) = 90 - 2w = 0 \implies w = 45$.

The union's stronger threat point (strike pay of 10 vs. 0) raises the negotiated wage from 15 to 45. This illustrates a key insight: bargaining power depends on the threat point.

The Ultimatum Game

Rules: Player 1 proposes a division of USD 10. Player 2 can accept or reject. If Player 2 accepts, both receive the proposed amounts. If Player 2 rejects, both receive nothing.

Standard economic prediction: Player 1 offers USD 0.01, keeps USD 9.99. Player 2 accepts (any positive amount is better than zero).

Experimental result: the median offer is approximately USD 4--5. Offers below USD 2 are typically rejected.

Interpretation: Player 2's willingness to reject positive offers (reducing their own payoff) reflects:

1. Fairness concerns: players care about equitable outcomes, not just their own payoff
2. Emotional response: anger at unfairness motivates rejection
3. Reputation: rejecting unfair offers may build a reputation that improves future bargaining outcomes

The Dictator Game

Rules: Player 1 decides how much of USD 10 to give to Player 2. Player 2 has no choice.

Standard prediction: Player 1 gives USD 0.

Experimental result: the average offer is approximately USD 2--3. Many people give positive amounts even when they have no strategic incentive to do so.

Interpretation: this reflects genuine altruism and social norms, not strategic behaviour. People have intrinsic preferences for fairness.

Information Cascades (HL Extension)

Definition

An information cascade occurs when individuals observe the actions of others and ignore their own private information, following the crowd instead.

Model

1. Each individual receives a private signal about the state of the world (good or bad)
2. Individuals make decisions sequentially, observing previous decisions (but not private signals)
3. If an individual's private signal agrees with the majority of previous decisions, they follow it. If it disagrees, they may still follow the majority if the majority is large enough

Cascade Formation

A cascade begins when the accumulated public information (from observed actions) outweighs any individual's private signal. Once a cascade starts, all subsequent individuals ignore their private information and follow the crowd.

Example:

Two restaurants, A and B. Each person receives a private signal (correct with probability $p = 0.6$).

Person 1: signal says A. Chooses A. Person 2: signal says B. But the public information (Person 1 chose A) is equally balanced with the private signal. Person 2 is indifferent and may follow either signal. Suppose Person 2 follows their signal and chooses B. Person 3: signal says A. Public information: one A, one B (balanced). Person 3 follows their signal and chooses A. Person 4: signal says B. Public information: two A, one B. Person 4 weighs the public info (2-1 in favour of A) against their private signal (B). Bayesian updating: posterior probability that A is better $> 0.5$. Person 4 chooses A.

Now: three A, one B. Person 5 has a signal for B but sees 3-1 in favour of A. The public information is strong enough to outweigh any single private signal. A cascade begins.

Key insight: cascades can be wrong. If the first two people happened to receive incorrect signals, the cascade converges on the wrong outcome.

Economic applications:

1. Financial markets: investors follow the herd, leading to bubbles and crashes
2. Technology adoption: consumers adopt technologies because others have adopted them, not because the technology is superior (QWERTY keyboard, VHS vs. Betamax)
3. Academic publishing: researchers follow prevailing paradigms, potentially suppressing innovative ideas

Exam-Style Questions: Behavioural Economics (Additional)

Question 6: Nash Bargaining and Trade Negotiations (10 marks)

Country A and Country B are negotiating a trade agreement. Without the agreement:

• Country A's welfare: 100
• Country B's welfare: 80

With the agreement, total welfare increases to 250. The surplus (70) must be divided.

(a) Calculate the Nash bargaining solution. [3 marks]

(b) If Country B's threat point improves to 90 (e.g., by signing an alternative agreement with Country C), what is the new Nash solution? [3 marks]

(c) Evaluate the Nash bargaining solution as a model of international negotiations. [4 marks]

(a) Nash solution: maximise $(u_A - 100)(u_B - 80)$ subject to $u_A + u_B = 250$.

Substituting $u_B = 250 - u_A$:

$(u_A - 100)(250 - u_A - 80) = (u_A - 100)(170 - u_A)$

FOC: $(170 - u_A) - (u_A - 100) = 270 - 2u_A = 0 \implies u_A = 135$.

$u_B = 250 - 135 = 115$.

Country A gains 35, Country B gains 35. Equal split of the surplus.

(b) New threat point: $(u_A - 100)(u_B - 90)$ with $u_A + u_B = 250$.

$(u_A - 100)(160 - u_A)$

FOC: $(160 - u_A) - (u_A - 100) = 260 - 2u_A = 0 \implies u_A = 130$.

$u_B = 250 - 130 = 120$.

Country A gains 30, Country B gains 40. Country B's improved threat point shifts the bargaining outcome in its favour.

(c) Evaluation:

Strengths:

1. Provides a unique, symmetric solution that is Pareto efficient
2. Formalises the intuitive idea that bargaining power depends on the threat point
3. The solution is proportional to bargaining power: the player with the better outside option receives a larger share

Limitations:

1. Symmetry assumption: the Nash solution assumes equal bargaining power (equal weight on each player's gain). In reality, countries have asymmetric power (US vs. small developing countries)
2. Risk neutrality: the solution assumes players are risk-neutral. Risk-averse players may accept a smaller share of the surplus to avoid the risk of no agreement
3. Incomplete information: in practice, countries do not know each other's true threat points, making the Nash solution difficult to apply
4. Multi-issue negotiations: real trade negotiations involve multiple issues (tariffs, services, IP, labour standards), not a single divisible surplus
5. Non-cooperative behaviour: countries may use threats, deadlines, and other strategic tactics not captured by the cooperative Nash framework

Question 7: Information Cascades and Financial Markets (10 marks)

An IPO (initial public offering) is being evaluated by four investors. Each investor receives a private signal (high quality or low quality) that is correct with probability 65%.

The true quality is high.

Investors decide sequentially, observing previous investors' decisions (but not their signals).

(a) What is the probability that all four investors invest? [3 marks]

(b) What is the probability that an information cascade (correct or incorrect) has begun by investor 3? [4 marks]

(c) How might information cascades contribute to stock market bubbles? [3 marks]

(a) Each investor's signal is correct with $p = 0.65$. Since the true quality is high, the signal is "high" with probability 0.65 and "low" with probability 0.35.

Each investor invests if their signal is "high" OR if the public information outweighs their "low" signal.

Investor 1: invests with probability 0.65 (if signal is high). Investor 2: if investor 1 invested, investor 2 invests if their signal is high (0.65) or if they follow the crowd. Since there is only one prior decision, investor 2 follows their signal: invests with probability 0.65. Investor 3: if both predecessors invested (prob $= 0.65^2 = 0.4225$), investor 3 has two "invest" signals from public information. Bayesian posterior: the probability of high quality given two prior invests is very high. Even with a "low" signal, the posterior is $> 0.5$, so investor 3 always invests. If only one predecessor invested, investor 3 follows their signal.

The probability that all four invest requires at least the first two to invest (0.4225). Given the first two invested, investor 3 always invests (cascade begins), and investor 4 always invests.

P(all four invest) $= 0.65 \times 0.65 = 0.4225$.

(b) A cascade begins at investor 3 if the first two investors agree. This happens with probability:

Both invest: $0.65^2 = 0.4225$ Both don't invest: $0.35^2 = 0.1225$ Total cascade probability $= 0.4225 + 0.1225 = 0.545$.

If the first two disagree (prob $= 2 \times 0.65 \times 0.35 = 0.455$), no cascade at investor 3. Investor 3 follows their signal, creating a 2-1 split. Investor 4 then faces a cascade (public info 2-1 outweighs private signal).

So cascade by investor 3: 54.5%. Cascade by investor 4: additional probability.

(c) Information cascades and bubbles:

1. IPO cascades: early investors' decisions influence later investors, creating self-reinforcing demand. If early investors happen to receive positive signals (even if unwarranted), a cascade of buying can drive the price above fundamental value
2. Analyst cascades: financial analysts may follow each other's recommendations rather than conducting independent analysis. When the consensus is wrong, the entire analyst community provides incorrect guidance
3. Herding in fund management: fund managers may buy popular stocks to avoid underperforming their peers, creating herding behaviour that amplifies price movements
4. Cascade reversal: when new information arrives that contradicts the cascade, the correction can be sharp (a crash), as all investors simultaneously revise their beliefs

Policy implications: regulators can reduce cascade-driven bubbles by:

1. Requiring disclosure of analysts' conflicts of interest
2. Encouraging contrarian research and diverse viewpoints
3. Implementing circuit breakers that halt trading during rapid price movements
4. Promoting financial literacy so individual investors rely less on observed behaviour

Social Preferences and Altruism (HL Extension)

Beyond Self-Interest

Traditional economic models assume individuals are purely self-interested (homo economicus). Behavioural economics and experimental evidence demonstrate that people have social preferences:

1. Altruism: willingness to incur personal costs to benefit others
2. Fairness: preference for equitable outcomes
3. Reciprocity: willingness to reward kind behaviour and punish unkind behaviour
4. Inequality aversion: dislike of unequal outcomes (Fehr-Schmidt model)

The Fehr-Schmidt Model of Inequality Aversion

Individuals maximise:

$U_i = x_i - \alpha_i \frac{1}{n-1} \sum_{j \neq i} \max(x_j - x_i, 0) - \beta_i \frac{1}{n-1} \sum_{j \neq i} \max(x_i - x_j, 0)$

Where:

• $x_i$ = individual $i$'s payoff
• $\alpha_i$ = envy parameter (disutility from being worse off than others, $\alpha \geq 0$)
• $\beta_i$ = guilt parameter (disutility from being better off than others, $0 \leq \beta < \alpha$)

The constraint $\beta < \alpha$ ensures that individuals dislike disadvantageous inequality more than advantageous inequality.

Numerical example:

Two players, A and B. $\alpha = 0.5$, $\beta = 0.3$.

Allocation: A gets 10, B gets 5.

$U_A = 10 - 0.5 \times \max(5-10, 0) - 0.3 \times \max(10-5, 0) = 10 - 0 - 1.5 = 8.5$

$U_B = 5 - 0.5 \times \max(10-5, 0) - 0.3 \times \max(5-10, 0) = 5 - 2.5 - 0 = 2.5$

A's utility is reduced by guilt (being better off). B's utility is reduced by envy (being worse off).

If A could transfer 1 to B: A gets 9, B gets 6.

$U_A = 9 - 0 - 0.3 \times 3 = 9 - 0.9 = 8.1$. $U_B = 6 - 0.5 \times 3 - 0 = 6 - 1.5 = 4.5$.

A's utility decreases (8.5 to 8.1), so a purely self-interested A would not transfer. But if $\beta$ were higher (say 0.5), A would transfer because the guilt of inequality exceeds the monetary cost.

Public Goods and Free-Riding

A public good is non-rivalrous and non-excludable. The classic problem is free-riding: individuals have an incentive to undercontribute because they can benefit from the good regardless of their contribution.

Standard prediction: in a voluntary contribution game, contributions converge to zero as individuals learn that others are free-riding.

Experimental finding: contributions typically start at 40--60% of the socially optimal level and decline over repeated rounds, but do not converge to zero. Many individuals continue to contribute even when free-riding is the dominant strategy.

Explanations for persistent contributions:

1. Altruism: some individuals genuinely care about the welfare of others
2. Warm glow: individuals derive utility from the act of giving itself, not just from the public good (Andreoni, 1990)
3. Reciprocity: conditional cooperators contribute if they believe others are contributing
4. Social norms: contributions are influenced by social expectations and reputational concerns
5. Incomplete learning: in finitely repeated games, not all individuals reach the subgame perfect equilibrium

The Trust Game

Rules: Player 1 (investor) receives USD 10 and can send any amount $s$ to Player 2 (trustee). The amount is tripled. Player 2 can return any amount $r$ to Player 1.

Standard prediction: Player 2 returns nothing ($r = 0$). Player 1, anticipating this, sends nothing ($s = 0$).

Experimental result:

• Average send: USD 5--6 (investors trust)
• Average return: USD 5--6 (trustees reciprocate)
• The amount sent is positively correlated with the amount returned

Interpretation:

1. Trust (sending money) reflects expectations of reciprocity
2. Trustworthiness (returning money) reflects reciprocity and fairness
3. The level of trust and trustworthiness varies across cultures and countries
4. Higher trust is associated with higher economic growth (Knack and Keefer, 1997)

Policy Applications of Social Preferences

1. Tax compliance: individuals are more likely to pay taxes if they believe others are paying (reciprocity and social norms). Tax authorities can leverage this by providing information about compliance rates
2. Charitable giving: the "warm glow" effect means that tax deductions for charitable giving may be less effective than social recognition or public acknowledgement
3. Environmental regulation: appeals to social norms ("most of your neighbours recycle") can be more effective than fines for promoting pro-environmental behaviour
4. Anti-corruption: building a culture of trust and reciprocity can reduce corruption more effectively than punitive measures alone

Bounded Rationality and Satisficing (HL Extension)

Herbert Simon's Critique

Herbert Simon (1955) argued that individuals do not optimise (maximise utility) because:

1. Information is incomplete: individuals cannot know all available options and their consequences
2. Cognitive limitations: the human brain cannot process all the information needed for full optimisation
3. Time constraints: decisions must be made within limited time, preventing exhaustive search
4. Computational complexity: many optimisation problems are computationally intractable

Instead, individuals satisfice: they choose the first option that meets a minimum acceptability threshold (the "aspiration level"), rather than searching for the optimal option.

Formal Model of Satisficing

1. Define an aspiration level $A$
2. Search options sequentially
3. Accept the first option with value $v \geq A$
4. If no option meets $A$, adjust $A$ downward (reduce aspirations)

Contrast with optimisation:

• Optimisation: $\max_{x \in X} U(x)$. Requires knowledge of all $x \in X$ and their utilities
• Satisficing: $\text{accept } x \text{ if } U(x) \geq A$. Requires only sequential evaluation until the threshold is met

Implications for Economic Behaviour

1. Consumer choice: consumers may not find the lowest price or the best quality product; they choose the first acceptable option. This explains brand loyalty (reducing search costs)
2. Firm behaviour: firms may not maximise profits; they set "satisfactory" profit targets and focus on survival rather than optimisation
3. Market equilibrium: satisficing behaviour can lead to different market outcomes than those predicted by models of full rationality. Prices may not equal marginal cost, and firms may coexist at different efficiency levels

Fast and Frugal Heuristics (Gigerenzer)

Gerd Gigerenzer (2007) argued that simple heuristics can outperform complex optimisation in many real-world contexts:

1. Recognition heuristic: if one of two objects is recognised and the other is not, infer that the recognised object has the higher value (e.g., which city is larger?)
2. Take-the-best: use the most important cue and ignore all others. If the most important cue does not discriminate, use the next most important cue
3. Tit-for-tat: in repeated interactions, cooperate first, then imitate the other player's last action (see repeated games section)

Empirical evidence: in forecasting competitions, simple heuristics often outperform complex statistical models, especially when data is noisy and the environment is uncertain.

Numerical Example: satisficing vs. Optimising

A consumer is choosing a mobile phone plan. There are 100 plans available, each with different prices and data allowances. The consumer's utility function is $U = \sqrt{D} - P/10$ where $D$ is data (GB) and $P$ is price (USD).

Optimising: evaluate all 100 plans, calculate $U$ for each, choose the maximum.

Satisficing: set aspiration level $A = 5$. Evaluate plans sequentially. Accept the first plan with $U \geq 5$.

If the consumer evaluates plans in random order, the average number of plans evaluated before finding one with $U \geq 5$ depends on the distribution of $U$ across plans.

If 30% of plans have $U \geq 5$, the expected number of evaluations before acceptance is $1/0.30 = 3.3$ plans. The satisficer evaluates 3.3 plans on average, compared to 100 for the optimiser.

The satisficer saves 97% of search effort but may end up with a suboptimal plan. The expected utility of the satisficer's choice depends on the distribution of utilities above the threshold. If all plans with $U \geq 5$ have similar utilities (5--7), the cost of satisficing is small (forgone utility of 0--2). If the distribution has a long right tail (some plans have $U = 15$), the cost of satisficing can be large.

Dual-Process Theory

Daniel Kahneman (2011) distinguished two modes of thinking:

System 1 (fast, automatic):

• Intuitive, effortless, associative
• Uses heuristics and shortcuts
• Susceptible to cognitive biases
• Example: reading emotions on a face, answering $2 + 2 = ?$

System 2 (slow, deliberate):

• Analytical, effortful, logical
• Requires attention and cognitive resources
• Can override System 1 but is lazy (minimises effort)
• Example: solving $17 \times 24$, comparing investment options

Economic implications:

1. Most economic decisions are made by System 1 (routine purchases, habitual behaviour)
2. System 2 is engaged for important decisions but often delegates to System 1 when fatigued or distracted ("decision fatigue")
3. Nudges work by influencing System 1 processing (changing defaults, framing, social norms)
4. Financial education targets System 2 but has limited effectiveness because most financial decisions are made by System 1

Coordination Games and Multiple Equilibria (HL Extension)

Pure Coordination Games

In a coordination game, players benefit from choosing the same action, but there are multiple equilibria. The challenge is to coordinate on one equilibrium.

Example: Technology adoption

Two firms choose between Technology A and Technology B.

	Tech A	Tech B
Tech A	(5, 5)	(0, 0)
Tech B	(0, 0)	(3, 3)

Two pure-strategy Nash equilibria: (A, A) and (B, B). Both firms prefer (A, A) but must coordinate. If each firm expects the other to choose A, both choose A. If each expects B, both choose B.

Real-world examples:

1. QWERTY keyboard: the QWERTY layout is inferior to Dvorak but is the standard because everyone has coordinated on it (path dependence)
2. VHS vs. Betamax: VHS won the video format war despite Betamax's technical superiority, partly because more studios adopted VHS initially
3. Driving on the left vs. right: the equilibrium (left, left) or (right, right) is efficient; mixed equilibria are dangerous

Battle of the Sexes

	Opera	Football
Opera	(3, 2)	(0, 0)
Football	(0, 0)	(2, 3)

Two Nash equilibria: (Opera, Opera) and (Football, Football). Each player prefers a different equilibrium. This models situations where players want to coordinate but have different preferences about which equilibrium to coordinate on.

Mixed strategy equilibrium:

Let $p$ = probability Player 1 chooses Opera. Player 2 is indifferent when:

$p \times 2 + (1-p) \times 0 = p \times 0 + (1-p) \times 3$

$2p = 3 - 3p \implies 5p = 3 \implies p = 3/5 = 0.6$

Similarly, Player 1 is indifferent when Player 2 chooses Opera with probability 0.4.

Mixed Nash equilibrium: Player 1 plays Opera with probability 0.6; Player 2 plays Opera with probability 0.4.

Stag Hunt Game

	Stag	Hare
Stag	(4, 4)	(0, 3)
Hare	(3, 0)	(2, 2)

Two Nash equilibria: (Stag, Stag) with payoff (4, 4) and (Hare, Hare) with payoff (2, 2).

The Stag equilibrium is Pareto-superior but risk-dominant only if players trust each other. If Player 1 is unsure whether Player 2 will cooperate, playing Hare is safer (guaranteed 2 vs. risky 0 or 4).

Application to international cooperation: the Stag Hunt models climate agreements. Full cooperation (Stag) is the best outcome for all, but individual countries may defect (Hare) because they cannot trust others to cooperate. The risk-dominant equilibrium may be (Hare, Hare), leading to climate inaction.

Focal Points

Thomas Schelling (1960) argued that in coordination games, players often coordinate on focal points -- salient solutions that stand out for reasons not captured by the formal payoff structure.

Examples:

1. Two people must meet in New York City on a given day but cannot communicate. Many coordinate on Grand Central Station at noon (the most prominent meeting point)
2. Two people must name a positive number. If they name the same number, they both win a prize. Many coordinate on "1" (the simplest number)
3. In the Stag Hunt, shared norms, communication, or institutional frameworks can serve as focal points that coordinate players on the efficient equilibrium

Sunspots and Self-Fulfilling Expectations

In some coordination games, expectations can be self-fulfilling. If everyone expects a recession, firms cut investment and consumers reduce spending, causing a recession. If everyone expects recovery, the opposite occurs.

Application to financial crises: bank runs are self-fulfilling. If depositors expect a bank to fail, they withdraw their deposits, causing the bank to fail (even if the bank was fundamentally sound). Diamond and Dybvig (1983) showed that bank runs are a coordination failure: the efficient equilibrium (no run) coexists with the inefficient equilibrium (run).

Policy solution: deposit insurance (FDIC in the US) eliminates the incentive to run by guaranteeing deposits. This changes the payoff structure, eliminating the run equilibrium.