Reputation with Multiple Opponents, and Commitment Bias in Voting

The first chapter of my thesis is the first attempt in the game theory literature to study the impact of reputation in a game with an arbitrary number of nonmyopic players, not just two. I consider a reputation game with perfect monitoring and multiple long-lived opponents indexed 1, 2, · · · , n. Player 0, who attempts to build reputation, could be either the normal type (maximizing the discounted sum of payoffs) or one of many commitment types. For a single opponent (n = 1) the previous literature finds that, when all players are arbitrarily patient and 0 is patient relative to others, in any equilibrium player 0 gets arbitrarily close to g∗∗ 0 , the maximum feasible payoff for 0 subject to giving player 1 at least her minmax. In other words, 0 can appropriate ‘everything’. For n > 1, in stark contrast, I find that player 0’s minimum equilibrium payoff is typically strictly below g∗∗ 0 and could be as low as his minmax value. My first chapter makes two important simplifying assumptions: (i) the type space that 0 can mimic contains only pure strategy types, and (ii) an announcement stage is available before the dynamic game is played. In the second chapter I show that the announcement stage is without loss of generality; although different strategies are called for, the same bound works when player 0 does not have an opportunity to signal his type. This chapter also shows that the qualitative result proved in the first chapter is robust to a very large set of types I extend the result to include types that, in addition to playing arbitrarily complex history-dependant actions, may be committed to mixing. I state and prove a new upper bound on vmin 0 that works even in the presence of types that mix. My third chapter, co-authored with Vinayak Tripathi, draws attention to a paradoxical commitment bias that may be present in voting games. Consider an electorate whose individual rankings of alternative policies may change between the time they vote for a candidate and the date a policy is implemented. Rankings may change following common or idiosyncratic shocks. Voters choose, via a simple majority election,