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For any game, trembling hand perfect equilibria are a subset of sequential equilibria. What is a simple example where a sequential equilibrium is not a trembling hand perfect equilibrium? Is it possible to create a normal form example?

Links:

Kreps-Wilson: Sequential Equilibria

Selten: Reexamination of the Perfectness Concept for Equilibrium Points in Extensive Games

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  • $\begingroup$ I thought sequential equilibrium are only for extensive-form games? $\endgroup$ – usul Mar 12 '15 at 1:10
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    $\begingroup$ Every normal form game may be rewritten to extensive form if you use non-singleton information sets. In case you meant sequential games: The concept is clearly meant to refine equilibria in sequential games, but I think the technical definition does not require the game to be sequential. Kreps and Wilson specify that you should use Bayes' rule to update your beliefs "whenever possible". If the game is not sequential, there is no updating. $\endgroup$ – Giskard Mar 12 '15 at 23:26
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Yes. In a normal form game, every Nash equilibrium is also a sequential equilibrium. But not every Nash equilibrium is trembling hand perfect. Consider the game in which each of two players has two strategies, A and B. Both players get payoff 0 except in one case: they achieve positive payoffs if they both choose A. Then (A,A) and (B,B) are two Nash equilibria of this game. Both are therefore also sequential equilibria. However, (B,B) is not trembling hand perfect. If there is even the smallest tremble in player 2's choice, player 1 has a strict preference for A. Only (A,A) is trembling hand perfect. The generalization of this is that Nash equilibria in which some players play weakly dominated strategies are not trembling hand perfect.

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    $\begingroup$ Hi TMB, Thank you for your answer. I came up with a similar example, but here is my problem: The definition of sequential equilibrium uses assessments. These require that my beliefs about the actions of the other player are based on strictly positive strategies (i.e. strategies with full support). In the game you describe I would have to believe that there is a positive probability that the other plays A, and I hence I should also play A, just like in trembling hand perfect equilibrium. Am I missing something in the Kreps-Wilson paper? I know the concept is intended for sequential games. $\endgroup$ – Giskard Mar 12 '15 at 23:20
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    $\begingroup$ The beliefs in a sequential equilibrium do not have to have full support. They must be derived as the limit of some full support sequence of beliefs. But in a normal form game every belief can be derived as the limit of some full support sequence of beliefs. The important point is that, although the beliefs must be derived from a full support sequence of beliefs, the equilibrium strategies must be best responses only to the limit beliefs, not to the full support beliefs. ... Let me know if this brief answer is not clear enough, and I'll try to go into more detail. $\endgroup$ – TMB Mar 13 '15 at 15:15

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