Question 1: Expected Utility Theory and Rank-Dependent Decision-Weighting
(a) Implications of Expected Utility Theory (20 points):
Under expected utility theory, an agent’s preferences can be represented by parallel, linear indifference curves in the unit probability triangle. The theory implies that the agent evaluates lotteries based on the expected value of their utility, considering the probabilities of each outcome.
The utility function is linear, representing a constant marginal utility of wealth. The slope of the indifference map reflects the marginal utility of wealth, indicating the rate at which the agent is willing to exchange wealth for a change in probability.
(b) Inconsistency of Pattern 1 with Expected Utility Theory (10 points):
Pattern 1, where B is preferred to C but not to A, violates expected utility theory. This violates the independence axiom, suggesting that preferences are inconsistent with the principle of maintaining preferences regardless of irrelevant alternatives. It’s a violation of regularity, revealing that the agent’s preferences are not solely driven by expected utility considerations.
(c) Rank-Dependent Decision-Weighting (25 points):
To accommodate Pattern 2 under rank-dependent decision-weighting, a probability-weighting function, π(.), is introduced. The utility function is multiplied by a decision-weighting function, incorporating non-linear probability weighting. Necessary conditions for Pattern 2 involve the utility and probability-weighting functions being specified in terms of q and r.
For instance, if the agent’s preferences maximize the rank-dependent decision-weighted sum of utility, conditions on the functions need to be established. The exact conditions depend on the chosen functional forms of the utility and probability-weighting functions. Conditions must ensure that Pattern 2 is attainable.
(d) Comparison of Models (45 points):
Comparing rank-dependent decision-weighting to expected utility theory, the former allows for non-linear probability weighting, capturing phenomena like probability distortions. It provides a more flexible framework that accommodates empirical findings. However, its complexity raises challenges in estimation.
Strengths include better alignment with observed behavior, capturing risk attitudes more accurately. Weaknesses involve increased complexity and the need for specific conditions to achieve certain preference patterns.
Transitivity is a strength as it aligns with rational decision-making. However, it might oversimplify human decision processes, ignoring complexity.
Question 2: Prisoner’s Dilemma and Inequity Aversion
(a) Nash Equilibria in the Prisoner’s Dilemma (5 points):
Nash equilibria occur where no player can unilaterally deviate for higher payoff. In this case, (Defect, Defect) is a Nash equilibrium as neither player can benefit by changing their strategy unilaterally.
(b) Fehr-Schmidt Model of Inequity Aversion (20 points):
Fehr-Schmidt’s model introduces parameters α and β, where α captures aversion to disadvantageous inequality, and β captures aversion to advantageous inequality. Individuals care not only about their absolute payoff but also about inequality. It combines self-interest and social preferences.
(c) Nash Equilibria with Fehr-Schmidt Preferences (10 points):
Identifying pure-strategy Nash equilibria involves players choosing the strategy that maximizes their Fehr-Schmidt utility. The equilibria depend on players’ preferences.
(d) Sequential Game with Fehr-Schmidt Preferences (25 points):
Solving the sequential game involves backward induction. The subgame-perfect Nash equilibria are determined by players’ strategic choices considering their preferences and knowledge of each other’s preferences.
(e) Limitations of Fehr-Schmidt Model (40 points):
Theoretical limitations involve the challenge of precisely measuring and modeling inequity aversion parameters. Experimental evidence suggests significant individual variation in inequity aversion, challenging the universality of the model.
The model may oversimplify social preferences, and some empirical findings diverge from its predictions, highlighting potential limitations in capturing the complexity of human behavior.
Overall, the Fehr-Schmidt model provides a valuable framework, but its limitations must be acknowledged in understanding and predicting individual behavior in strategic interactions.