## Z

In this game, none of the pure strategies of player 1 is dominated by an alternative pure strategy. However, it is clear that the mixed strategy a1 (0, 1 2, 1 2) dominates the pure strategy X it guarantees a larger expected payoff than X, independently of the strategy adopted by player 2. We conclude, therefore, that requiring payoff domination in terms of a pure strategy may, in general, reduce significantly the set of strategies (pure or mixed) that qualify as dominated. Reciprocally, of...

## D

Figure 1.10 Extensive-form structure. That is, the probability that the behavioral strategy yi attributes to playing any action a in some information set h is identified with the conditional probability induced by ai, as long as such strategy renders it possible that the information set h be reached (i.e., as long as Yl s S,(h) (si) > 0)- Otherwise, if the strategy ai itself rules out that h could be visited, then the contemplated conditional probability is not well defined. In those cases,...

## H

Figure 10.4 Simplified repeated prisoner's dilemma, RD. Figure 10.4 Simplified repeated prisoner's dilemma, RD. The primary aim of this chapter has been to explore the extent to which the received notions of rationality and equilibrium that underlie classical game theory can be provided with a robust evolutionary basis. Traditionally, economists have relied on heuristic arguments to defend the idea that, under sufficiently stringent selection (e.g., market) forces, rational behavior should be...

## S

Where B and S are mnemonic for basketball and shopping, respectively, and the pairs of numbers specified quantify the utilities obtained by each individual (first the girl's, second the boy's) for each choice combination. In principle, the couple could agree on implementing any pair of choices on the day in question. However, only (B, B) and (S, S) represent robust (or stable) agreements in the sense that if they settle on any of them and each believes that the other side is going to abide by...

## B S

Supported by the belief pattern * given by X*(B )(tn) 1 x*(S )(ti2) 1. (6.27) If we contrast this SE with the BNE obtained in the counterpart Bayesian game where decisions are simultaneous (cf. Subsection 6.3.2.1), we observe that the girl's strategies are formally identical in both cases - compare (6.8) and (6.26). One might be tempted to argue (wrongly, however) that this coincidence follows from the fact that, in both situations, the girl has a dominant action for each type. Indeed, when...

## My Si si a si

In contrast with the discussion undertaken for the matching-pennies game, Theorem 6.3 underscores the fact that the purification of a mixed-strategy Nash equilibrium does not (generically) presume a specific form for the payoff perturbations, the only requirement being that they should be stochastically independent across players and satisfy some natural regularity conditions. However, again in contrast with our simple example, the desired purification is attained only as a limit result, i.e.,...

## Y

i Formulate a prediction of play. ii Suppose now that player 1 has taken away two payoff units utiles if she adopts strategy X. Does your prediction change iii Consider now the following second possibility. Player 1 may decide, in an initial stage of the game, whether to have the two utiles mentioned in ii removed. Once she has made this decision, both individuals play the resulting game. Represent the full game in both extensive and strategic forms. After finding all its pure-strategy Nash...

## K

J Pk x 1 Pk x gt 0 k K . 3.40 This condition simply reflects the requirement that the price of the consumption good equal to one, since it is the numeraire must equal its unit marginal cost of production. Thus, to summarize, for a price-formation rule pkO to qualify as valid in the postulated competitive environment, it has only to meet the following two conditions a It must prescribe a null price for those intermediate products in excess supply. b The nonnegative prices of the remaining...

## Exercises

Exercise 4.1 Within the game represented in Figure 4.11, find all pure-strategy profiles that define a a Nash equilibrium, b a subgame-perfect equilibrium, c a weak perfect Bayesian equilibrium. Exercise 4.2 Consider the extensive-form game represented in Figure 4.12. a Specify the strategy spaces of each player. b Find every pure-strategy profile that defines i a Nash equilibrium, ii a subgame-perfect equilibrium, iii a weak perfect Bayesian equilibrium. c Construct an extensive-form...

## Weak perfect Bayesian equilibrium

As illustrated in Section 4.2, when the game is not of perfect information i.e., some information set is not a singleton the SPE notion may fail to weed out every unreasonable Nash equilibrium. To address this problem, we outlined the concept called weak perfect Bayesian equilibrium WPBE . As will be recalled, its most notable variation over SPE is that it involves an explicit description of players' beliefs and demands that agents respond optimally to them. 52 If the game exhibits imperfect...