Game Theory in the Social Sciences: Concepts and Solutions v. 1


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Week 2: Mixed-Strategy Nash Equilibrium

There is also a weaker notion:. Before we make use of these choice rules, we need to address two potentially confusing issues about these definitions. The definitions of strict and weak dominance are given in terms of mixed strategies even though we are assuming that players only select pure strategies.

Game Theory in the Social Sciences: Concepts and Solutions by Martin Shubik

That is, we are not considering situations in which players explicitly randomize. In particular, recall that only pure strategies are associated with states in a game model. That is, we do not replace the above definition with.

This is because both definitions are equivalent. Each state in an epistemic -plausibility model is associated with a such a set of strategy profiles. The precise definition depends on the type of game model:.


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This can be made more precise using the following well-known Lemma. Lemma 3. The proof of this Lemma is given in the supplement, Section 1. The general conclusion is that no dominated strategy can maximize expected utility at a given state; and, conversely, if there is a strategy that is not a best in a specific context, then it is not strictly dominated.

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Similar facts hold about weak dominance , though the situation is more subtle. The crucial observation is that there is a characterization of weak dominance in terms of best response to certain types of probability measures. The following analogue of Lemma 3. The proof of this Lemma is more involved.

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See Bernheim Appendix A for a proof. Comparing these two Lemmas, we see that strict dominance implies weak dominance, but not necessarily vice versa. A strategy might not be a best response to any full-support probability measure while being a best response to some particular beliefs, those assigning probability one to a state where the player is indifferent between the outcome of its present action and the potentially inadmissible one. There is another, crucial, difference between weak and strict dominance. The following observation is immediate from the definition of strict dominance:.

Observation 3. If a strategy is strictly dominated, it remains so if the player gets more information about what her opponents might do. The same observation does not hold for weak dominance. The existential part of the definition of weak dominance means that the analogue of Observation 3. The epistemic approach to game theory focuses on the choices of individual decision makers in specific informational contexts, assessed on the basis of decision-theoretic choice rules.

This is a bottom-up, as opposed to the classical top-down, approach. An important line of research in epistemic game theory asks under what epistemic conditions will players follow the recommendations of particular solution concept? Providing such conditions is known as an epistemic characterization of a solution concept. In this section, we present two fundamental epistemic characterization results.


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The first is a characterization of iterated removal of strictly dominated strategies henceforth ISDS , and the second is a characterization of backward induction. These epistemic characterization results are historically important.

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They mark the beginning of epistemic game theory as we know it today. Furthermore, they are also conceptually important. The developments in later sections build on the ideas presented in this section. For that reason, instead of focusing on the formal details, the emphasis here will be on its significance for the epistemic foundations of game theory.

One important message is that the result highlights the importance of higher-order information. Iterated elimination of strictly dominated strategies ISDS is a solution concept that runs as follows. After having removed the strictly dominated strategies in the original game, look at the resulting sub-game, remove the strategies which have become strictly dominated there, and repeat this process until the elimination does not remove any strategies. The profiles that survive this process are said to be iteratively non-dominated.

That is, iteratively removing strictly dominated strategies generates the following sequence of games:.

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For arbitrary large finite strategic games, if all players are rational and there is common belief that all players are rational , then they will choose a strategy that is iteratively non-dominated. The result is credited to Bernheim and Pearce Before stating the formal result, we illustrate the result with an example.

The next step is to identify the types that believe that the other players are rational. In this context, belief means probability 1. We first need some notation. Putting everything together, we have. Note that, the above process need not generate all strategies that survive iteratively removing strictly dominated strategies. However, for any type space, if a strategy profile is consistent with rationality and common belief of rationality, then it must be a strategy that is in the set of strategies that survive iteratively removing strictly dominated strategies.

Epistemic Foundations of Game Theory

Theorem 4. This result establishes sufficient conditions for ISDS. It has also a converse direction: given any strategy profile that survives iterated elimination of strictly dominated strategies, there is a model in which this profile is played where all players are rational and this is common knowledge. In other words, one can always view or interpret the choice of a strategy profile that would survive the iterative elimination procedure as one that results from common knowledge of rationality. Of course, this form of the converse is not particularly interesting as we can always define a type space where all the players assign probability 1 to the given strategy profile and everyone playing their requisite strategy.

Much more interesting is the question whether the entire set of strategy profiles that survive iteratively removal of strictly dominated strategies is consistent with rationality and common belief in rationality. Analogues of the above results have been proven using different game models e. Many authors have pointed out the strength of the common belief assumption in the results of the previous section see, e.

It requires that the players not only believe that the others are not choosing an irrational strategy, but also to believe that everybody believes that nobody is choosing an irrational strategy, and everyone believes that everyone believes that everyone believes that nobody is choosing an irrational strategy, and so on. It should be noted, however, that this unbounded character is there only to ensure that the result holds for arbitrary finite games. A possible reply to the criticism of the infinitary nature of the common belief assumption is that the result should be seen as the analysis of a benchmark case, rather than a description of genuine game playing situations or a prescription for what rational players should do Aumann The results above show that, once formalized, this assumption does indeed lead to a classical solution concept, although, interestingly, not the well-known Nash equilibrium, as is often informally claimed in early game-theoretic literature.

Epistemic conditions for Nash equilibrium are presented in Section 5. The main message to take away from the results in the previous section is: Strategic reasoning in games involves higher-order information. This means that, in particular,. In general, first-order belief of rationality will not do either.


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There are two further issues we need to address. First of all, how can agents arrive at a context where rationality is commonly believed? The above results do not answer that question. This has been the subject of recent work in Dynamic Epistemic Logic van Benthem Then, we have:. Here, the most well-known solution concept is the so-called subgame perfect equilibrium , also known as backward induction in games of perfect information.

The main point that we highlight in this section, which is by now widely acknowledged in the literature, is:.

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Belief revision policies play a key role in the epistemic analysis of extensive games. The most well-known illustration of this is through the comparison of two apparently contradictory results regarding the consequences of assuming rationality and common knowledge of rationality in extensive games. Aumann showed that this epistemic condition implies that the players will play according to the backward induction solution while Stalnaker argued that this is not necessarily true. Extensive games make explicit the sequential structure of choices in a game situation.

In this section, we focus on games of perfect information in which there is no uncertainty about earlier choices in the game.

Game Theory in the Social Sciences: Concepts and Solutions v. 1 Game Theory in the Social Sciences: Concepts and Solutions v. 1
Game Theory in the Social Sciences: Concepts and Solutions v. 1 Game Theory in the Social Sciences: Concepts and Solutions v. 1
Game Theory in the Social Sciences: Concepts and Solutions v. 1 Game Theory in the Social Sciences: Concepts and Solutions v. 1
Game Theory in the Social Sciences: Concepts and Solutions v. 1 Game Theory in the Social Sciences: Concepts and Solutions v. 1
Game Theory in the Social Sciences: Concepts and Solutions v. 1 Game Theory in the Social Sciences: Concepts and Solutions v. 1
Game Theory in the Social Sciences: Concepts and Solutions v. 1 Game Theory in the Social Sciences: Concepts and Solutions v. 1
Game Theory in the Social Sciences: Concepts and Solutions v. 1 Game Theory in the Social Sciences: Concepts and Solutions v. 1
Game Theory in the Social Sciences: Concepts and Solutions v. 1 Game Theory in the Social Sciences: Concepts and Solutions v. 1
Game Theory in the Social Sciences: Concepts and Solutions v. 1

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