When Can Limited Randomness Be Used in Repeated Games?

The central result of classical game theory states that every finite normal form game has a Nash equilibrium, provided that players are allowed to use randomized (mixed) strategies. However, in practice, humans are known to be bad at generating random-like sequences, and true random bits may be unavailable. Even if the players have access to enough random bits for a single instance of the game their randomness might be insufficient if the game is played many times. In this work, we ask whether randomness is necessary for equilibria to exist in finitely repeated games. We show that for a large class of games containing arbitrary two-player zero-sum games, approximate Nash equilibria of the $n$-stage repeated version of the game exist if and only if both players have $\Omega(n)$ random bits. In contrast, we show that there exists a class of games for which no equilibrium exists in pure strategies, yet the $n$-stage repeated version of the game has an exact Nash equilibrium in which each player uses only a constant number of random bits. When the players are assumed to be computationally bounded, if cryptographic pseudorandom generators (or, equivalently, one-way functions) exist, then the players can base their strategies on "random-like" sequences derived from only a small number of truly random bits. We show that, in contrast, in repeated two-player zero-sum games, if pseudorandom generators \emph{do not} exist, then $\Omega(n)$ random bits remain necessary for equilibria to exist

A partial folk theorem for games with private learning

The payoff matrix of a finite stage game is realized randomly, and then the stage game is repeated infinitely. The distribution over states of the world (a state corresponds to a payoff matrix) is commonly known, but players do not observe nature’s choice. Over time, they can learn the state in two ways. After each round, each player observes his own realized payoff (which may be stochastic, conditional on the state), and he observes a noisy public signal of the state (whose informativeness may vary with the actions chosen). Actions are perfectly observable. The result is that for any function that maps each state to a payoff vector that is feasible and individually rational in that state, there is a sequential equilibrium in which patient players learn the realized state with arbitrary precision and achieve a payoff close to the one specified for that state. That result extends to the case where there is no public signal, but instead players receive very closely correlated private signals of the vector of realized payoffs.Repeated games, learning, folk theorem

The New Battle of the Sexes: Understanding the Reversal of the Happiness Gender Gap

In the Paradox of Declining Female Happiness, Stevenson and Wolfers (2007) document a new “gender gap” between the sexes, in which women today generally report lower subjective well-being relative to men. Motivated by recent work on gender-specific preferences, this paper considers whether changes in contraceptive technology, and the Pill especially, may have played some role in the declining relative (self-reported) happiness of women. We examine a simple model in which men and women have different preferences over sex and children. We find that plausible differences in male-female preference structures can yield the observed reversal in relative happiness following the introduction of a single technology which may prevent conception but yields no disutility to men. We attempt to characterize the fundamental tradeoffs in a static game of complete information, and make some extensions to repeated games. We find that preference structures substantially change the way in which the Pill may affect bargaining power and outcomes. The model suggests that men may have benefited more than women from the Pill in particular, and raises the question of whether other forms of family planning might better equalize the relative positions of men and women in partnerships. These results have particular relevance for feminist critiques of the sexual revolution.simultaneous game, contraception, fertility, gender, female happiness

Folk Theorems with Bounded Recall under (Almost) Perfect Monitoring, Third Version

We prove the perfect-monitoring folk theorem continues to hold when attention is restricted to strategies with bounded recall and the equilibrium is essentially required to be strict. As a consequence, the perfect monitoring folk theorem is shown to be behaviorally robust under almost-perfect almost-public monitoring. That is, the same specification of behavior continues to be an equilibrium when the monitoring is perturbed from perfect to highly-correlated private.Repeated games, bounded recall strategies, folk theorem,imperfect monitoring

Inequality and a Repeated Joint Project

Agents voluntarily contribute to an infinitely repeated joint project. We investigate the conditions for cooperation to be a renegotiation-proof and coalition-proof equilibrium before examining the influence of output share inequality on the sustainability of cooperation. When shares are not equally distributed, cooperation requires agents to be more patient than under perfect equality. Beyond a certain degree of share inequality, full efficiency cannot be reached without redistribution. This model also explains the coexistence of one cooperating and one free-riding coalition. In this case, increasing inequality can have a positive or negative impact on the aggregate level of effort.

Efficiency in Games With Markovian Private Information

We study repeated Bayesian games with communication and observable actions in which the players' privately known payoffs evolve according to an irreducible Markov chain whose transitions are independent across players. Our main result implies that, generically, any Pareto-efficient payoff vector above a stationary minmax value can be approximated arbitrarily closely in a perfect Bayesian equilibrium as the discount factor goes to 1. As an intermediate step, we construct an approximately efficient dynamic mechanism for long finite horizons without assuming transferable utility