Best response dynamics with level-n expectations in two-stage games

Abstract

This paper analyzes behavior in repeatedly played two-stage games, where players choose actions in both stages according to best replies using 'level-n expectations' about the opponent's actions in both stages. Level-n expectations are recursively defined in a way that a player holding level n expectations correctly predicts the action of an opponent holding level n − 1 expectations.A general conceptual framework to study such dynamics for two-stage games is developed and it is shown that, contrary to results for single-stage games, the fixed points of the dynamics depend on the level of the expectations. In particular, for level-zero expectation, fixed points correspond to a Nash equilibrium of a simultaneous move version of the game, whereas (under certain conditions) fixed points converge towards the subgame perfect equilibrium of the two-stage game if the level of expectations goes to infinity. The approach is illustrated using a two-stage duopoly game, where firms in the first stage invest in activities reducing their marginal costs and in the second stage engage in Cournot competition. An increase in the level of expectations leads in the long run to higher cost reducing activities and higher output of the firms, however to lower profits. Level-two expectations are sufficient to move the fixed-point of the dynamics to a close neighbourhood of the subgame-perfect equilibrium.

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“…In [93] it is shown that a Cognitive Hierarchy (CH) equilibrium can be reached by massive amount of devices with various levels of rationality, which follow Poisson distribution. Whereas in [90] expectation of rationality level decreases recursively. As in IC games, the IC strategies mainly depend on strategies of IC target, the rationality setting in [90] might suggest a simple and elegant model for limited rationality two-stage IC games.…”

“…Whereas in [90] expectation of rationality level decreases recursively. As in IC games, the IC strategies mainly depend on strategies of IC target, the rationality setting in [90] might suggest a simple and elegant model for limited rationality two-stage IC games.…”

“…However, such assumption might be impractical for communication application, due to limited computational capability and information accessibility. To develop practical models, limited rationality games with level-k thinking are studied [89][90][91][92][93]. In [93] it is shown that a Cognitive Hierarchy (CH) equilibrium can be reached by massive amount of devices with various levels of rationality, which follow Poisson distribution.…”

“…Accordingly, in this paper we consider limited rationality strategies, where the player does not fully analyze the 2nd stage. We apply different levels of thinking [89,90], related to how deeply the game is analyzed, to estimate whether IC is successful. This level of analysis is used both when choosing the first stage strategy based on assumptions on how the 2nd stage is played, and when playing the 2nd stage game itself.…”

“…Note that here, we treat level-m thinking as a limit on the rationality of the players, and correspondingly on the applicable strategies, when analyzing equilibrium strategies. In [90], level-m thinking was used in the framework of best-response dynamics of two-stage games.…”