Psychology 288 Game Theory Professor Tasos Kalandrakis Sprin

Psceco 288game Theoryprof Tasos Kalandrakisspring 2015assignment

PSC/ECO 288 Game Theory Prof. Tasos Kalandrakis Spring 2015 Assignment #8: Due in class Wednesday, April 22 Please write clearly and make sure to justify all your answers. 1. (25 points total) Find a WSE in the following dynamic game of incomplete information. Nature S W /2 1/4 U D 2 L R (2, 0) h, 0) (5, 2) U D 2 L R (2, 0) h, 3) (4, 0) U D 2 L R (2, 0) h, 5) (0, M 1/4 h1 h2 h3 I1 (a) (7 points) Report a behavioral strategy for players moving at h1,h2,h3, and I1. (b) (8 points) Derive and report equilibrium beliefs µ(h4),µ(h5),µ(h6) for player 2 at I1. (c) (10 points) Verify that sequential rationality is satisfied at all information sets h1,h2,h3 and I1.

Paper For Above instruction

The assignment involves analyzing a dynamic game of incomplete information, specifically identifying a weak sequential equilibrium (WSE) within the context of the game structure provided. The problem requires three core tasks: first, to formulate a behavioral strategy profile for the players at specified information sets; second, to derive the beliefs that player 2 holds at the information set I1, particularly at histories h4, h5, and h6; and third, to verify that the strategies and beliefs satisfy the principle of sequential rationality throughout all relevant information sets.

In the context of game theory, a weak sequential equilibrium is a refinement of Bayesian Nash equilibrium applicable to dynamic games with imperfect information. It requires strategies that are sequentially rational given beliefs, and beliefs that are consistent with the strategies. The first step, identifying strategies at the nodes h1, h2, h3, and I1, involves specifying how players choose their actions at each information set, considering their information and possible beliefs about the hidden states introduced by Nature’s move.

For the behavioral strategies at h1, h2, h3, and I1, one must consider the possible moves and payoffs, and determine the probability distributions over actions that make each player's choice optimal given their beliefs. For example, at h1, if Player 1 chooses between U and D, the probability assigned to each action will depend on the expected payoffs given the opponent's beliefs. Similar reasoning applies to h2 and h3. Player 2’s strategies at I1 must also be conditioned on their beliefs about the prior moves and Nature's realization.

The second task involves belief updating at I1 concerning the history of plays. Specifically, deriving the beliefs µ(h4), µ(h5), and µ(h6) entails applying Bayes' rule where possible, given the observed histories and the strategies employed by the players. These beliefs are essential for ensuring that strategies are sequentially rational, as they influence Player 2’s decision-making at I1.

Finally, verifying sequential rationality involves checking that, at each information set, the strategies maximize expected payoffs given the beliefs, and that the beliefs are consistent with the strategies and observed histories. This step ensures that no player has an incentive to deviate, given the assumed strategies and beliefs, thus satisfying the criteria for a weak sequential equilibrium.

Overall, the solution requires careful consideration of the payoff structure, the information sets, methodical application of belief updating rules, and systematic verification of rationality conditions. This analysis not only illuminates the strategic interactions within the game but also illustrates the refinement concepts vital to understanding equilibrium behavior in games of incomplete information.

References

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  • Kreps, D., & Wilson, R. (1982). Reputation and imperfect information. Journal of Economic Theory, 27(2), 253-279.

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