Essays on Dynamic Games and Forward Induction

Open Access
Isogai, Shigeki
Graduate Program:
Doctor of Philosophy
Document Type:
Date of Defense:
May 25, 2017
Committee Members:
  • Edward Green, Dissertation Advisor
  • Edward Green, Committee Chair
  • Robert Clifford Marshall, Committee Member
  • Vijay Krishna, Committee Member
  • Lisa Lipowski Posey, Outside Member
  • game theory
  • collusion
In this essay, I study how forward-induction reasoning affect plausibility/stability of agreements in which players in a dynamic interaction enforces cooperation with the threat of mutually destructive punishment. While the traditional theory using equilibrium concept shows that such strategy profile is self-enforcing, under a modification of the model, such strategy profile fails to be consistent with players' rationality. In the first chapter I provide the simplest setting under which this non-rationalizability result of deterrence can be shown. The game is a two-player three-stage game: in the first stage, the players choose whether to enter the strategic interaction by paying some cost; in the second stage, the players play a prisoners' dilemma game; and in the third stage, the players play a coordination game. Each move is simultaneous and the players' past actions are perfectly monitored. While there exists a subgame-perfect equilibrium in which players can cooperate with the threat of punishment provided the punishment is strong enough, I show that the strategy profile does not consists of rationalizable strategies under a certain parameter values. This occurs because choosing to enter, unilaterally defect, and then punish the opponent is strictly dominated by a mixture of the two strategies ``do not enter'' and ``enter, defect, but do not punish.'' This result shows that a simple modification of the game and forward-induction consideration encoded in rationalizability might cast doubt on the idea of deterring defection by the threat of mutual punishment. The other two chapters study to what extent the result in the fist chapter does or does not apply in different settings. The second chapter considers the infinite-horizon extension of the model in the first chapter. In the first period (denoted as period 0), the players choose whether to enter the game. After the players choose to enter, the continuation game is the infinite repetition of the stage game which consists of two phases: in the first phase players play prisoners' dilemma game, after which players simultaneously choose to continue the game, exit from the game without punishing the opponent, or punish the opponent and exit from the game. I show that with a similar condition as in the result in the first chapter, strategy which entails defection and punishment in the first stage is not rationalizable. Moreover, since the exit-without-punishment option works as an outside option in later stages of the game, we also obtain a result which provides conditions under which punishment after defection is excluded by rationalizability. The third chapter extends the model in the first chapter toward an incomplete-information model in that it considers a model of random number of players, who are sequentially matched and play the game as in the first chapter. I assume that while the past actions in the stage games are not observable, occurrences of punishment is publicly observable to all the players (the typical example is the formation of cartels and the occurrence of leniency applications). I explore how this observable punishment works as a signaling device and how this model gives rise to a rationalizable use of punishment. I first show that a simple repetition of games does not give rise to a rationalizable punishment because of the assumption that the players cannot distinguish the non-occurrence of deviation and failure to punishment. I then discuss possible modifications to recover the punishment being an equilibrium action; i.e., that a small perturbation in payoffs can recover the possibility of punishment.