Every Normal-Form Game Has a Pareto-Optimal Nonmyopic Equilibrium

It is well-known that Nash equilibria may not be Pareto-optimal; worse, a unique Nash equilibrium may be Pareto-dominated, as in Prisoners’ Dilemma. By contrast, we prove a previously conjectured result: Every finite normal-form game of complete information and common knowledge has at least one Pareto-optimal nonmyopic equilibrium (NME) in pure strategies, which we define and illustrate. The outcome it gives, which depends on where play starts, may or may not coincide with that given by a Nash equilibrium. We use some simple examples to illustrate properties of NMEs—for instance, that NME outcomes are usually, though not always, maximin—and seem likely to foster cooperation in many games. Other approaches for analyzing farsighted strategic behavior in games are compared with the NME analysis

Subgame perfection in Positive Recursive Games with perfect information

We consider a class of n-player stochastic games with the following properties: (1) in every state, the transitions are controlled by one player; (2) the payoffs are equal to zero in every nonabsorbing state; (3) the payoffs are nonnegative in every absorbing state. We propose a new iterative method to analyze these games. With respect to the expected average reward, we prove the existence of a subgame-perfect epsilon-equilibrium in pure strategies for every epsilon > 0. Moreover, if all transitions are deterministic, we obtain a subgame-perfect 0-equilibrium in pure strategies

The Condorcet Paradox Revisited

We analyze the Condorcet paradox within a strategic bargaining model with majority voting, exogenous recognition probabilities, and no discounting for the case with three players and three alternatives. Stationary subgame perfect equilibria (SSPE) exist whenever the geometric mean of the players' risk coefficients, ratios of utility differences between alternatives, is at most one. SSPEs ensure agreement within finite expected time. For generic parameter values, SSPEs are unique and exclude Condorcet cycles. In an SSPE, at least two players propose their best alternative and at most one player proposes his middle alternative with positive probability. Players never reject best alternatives, may reject middle alternatives with positive probability, and reject worst alternatives. Recognition probabilities represent bargaining power and drive expected delay. Irrespective of utilities, no delay occurs for suitable distributions of bargaining power, whereas expected delay goes to infinity in the limit where one player holds all bargaining power. An increase in the recognition probability of a player may weaken his bargaining position. A player weakly improves his bargaining position when his risk coefficient decreases

The Condorcet paradox revisited

We analyze the Condorcet paradox within a strategic bargaining model with majority voting, exogenous recognition probabilities, and no discounting. Stationary subgame perfect equilibria (SSPE) exist whenever the geometric mean of the players' risk coefficients, ratios of utility differences between alternatives, is at most one. SSPEs ensure agreement within finite expected time. For generic parameter values, SSPEs are unique and exclude Condorcet cycles. In an SSPE, at least two players propose their best alternative and at most one player proposes his middle alternative with positive probability. Players never reject best alternatives, may reject middle alternatives with positive probability, and reject worst alternatives. Recognition probabilities represent bargaining power and drive expected delay. Irrespective of utilities, no delay occurs for suitable distributions of bargaining power, whereas expected delay goes to infinity in the limit where one player holds all bargaining power. Contrary to the case with unanimous approval, a player benefits from an increase in his risk aversion