**Symmetric n-player games **(Revise & resubmit at *Journal of Economic Theory*)

This paper considers symmetry in games with more than two players. It is often noted that a two-player game is symmetric if it looks the same to both players. However, when there are more than two players, the most common definition of a symmetric game requires more than that the game looks the same to all of its players. Previous authors have established that games which are symmetric in the common sense have a number of useful properties. With few exceptions, those properties continue to hold in the richer class of games that look the same to all players.

**Comparative statics in symmetric, concave games**

A concave game is one in which Kakutani’s theorem implies the existence of a pure- strategy equilibrium, by a classic argument. This paper establishes an equilibrium comparative statics result for symmetric, concave games with one-dimensional action spaces. This result mirrors the well-known comparative statics results for supermodular games.

**The finite-horizon paradox with impatient players
**The equilibrium payoff set of a finitely repeated game is often much smaller than the payoff set of the corresponding infinitely repeated game. That is one manifestation of the often studied finite-horizon paradox. A number of previous papers have shown that the paradox is not robust in the limiting case where the players are sufficiently patient. This paper investigates the general case where the players’ patience may be limited. The paradox is not robust in this general case if either publicly randomized strategies are observable, or if the stage game belongs to a natural class of games with continuum action spaces.

**Stability of the Equilibrium Payoff Set in Repeated Games**

The set of equilibrium payoffs of an infinitely repeated game is the greatest fixed point of the generation map, B, as shown by Abreu, Pearce, and Stacchetti (1990). Generally regarding a fixed point, there are several familiar questions of stability: Is the fixed point attractive? Is it essential, in the sense that nearby maps have nearby fixed points? This paper takes up these questions regarding stability of the equilibrium payoff set in repeated games. The payoff set is attractive from above and varies upper semicontinuously in the parameters of the game. Under some conditions it is attractive from below and lower semicontinuous, and under other conditions not. Under some conditions, appending a small continuation payoff set to the finitely repeated game yields approximately the same initial payoff set as that of the infinitely repeated game, because the latter’s payoff set is widely attractive.

**An Explicit Solution for Sannikov’s “Optimality Equation” in the Case of One Signal
**This note presents an explicit solution for Sannikov’s (2007) “optimality equation” in a special case. Sannikov introduces and analyzes a class of continuous-time games with imperfect public monitoring. He presents an ordinary differential equation, the optimality equation, that characterizes the boundary of the set of equilibrium payoffs. A numerical solution is readily obtained by computer, but no explicit solution is apparent. In the general case, the optimality equation is non-linear and seemingly intractable. We find and solve a linear equation in the special case where only one player’s incentive constraint is binding. That case holds when only a single player deviates from the static best response. It also holds under an alternative signal structure where there is just one signal. We derive closed-form comparative statics in a simple game.

**Continuously Repeated Games with Private Opportunities to Adjust**

This paper considers a new class of dynamic, two-player games, where a stage game is continuously repeated but each player can only move at random times that she privately observes. A player’s move is an adjustment of her action in the stage game, for example, a duopolist’s change of price. Each move is perfectly observed by both players, but a foregone opportunity to move, like a choice to leave one’s price unchanged, would not be directly observed by the other player. Moral hazard arises when optimal moves require some sacrifice relative to not moving. For example, to be achieved in equilibrium, a duopolist’s upward jump to the monopoly price would require costly incentives. These incentives may be provided by strategies that condition on the random waiting times between moves; punishing a player for moving slowly, lest she silently choose not to move. In contrast, if maintaining the status quo, perhaps the monopoly price, may be achieved in equilibrium, it does not require costly incentives. Deviation from the status quo would be perfectly observed, so punishment must occur only off the equilibrium path. Similarly, moves like jointly optimal price reductions do not require costly incentives. Again, defection, to a larger price reduction, would be perfectly observed. This paper provides a framework for analyzing these games extending continuous time methods of Sannikov (2007). For a class of stage games with monotone public spillovers, like differentiated-product duopoly, I prove that optimal equilibria have three features corresponding to the discussion above: beginning at a lower position, optimal, upward moves are impeded by moral hazard; beginning at a higher position, optimal, downward moves are unimpeded by moral hazard; beginning at an intermediate position, optimally maintaining the status quo is similarly unimpeded. Corresponding cooperative dynamics are suggested in the older, non-game-theoretic literature on tacit collusion, and are consistent with some more recent empirical findings on oligopoly pricing.

**Consistent Planning in Infinite-Horizon Problems with Symmetries
**A dynamic choice problem faced by a non-exponential discounter is typically modeled as a game played by a sequence of her temporal selves, solved by SPNE. It is recognized that this approach yields troublesomely multiple solutions for infinite-horizon problems, which is often attributed to the existence of implausible equilibria based on self-reward and punishment. I present a new solution concept that yields uniquely valued predictions in “strategically constant” (SC) problems. An infinite-horizon problem is SC if all of its continuation problems are isomorphic. The solution seeks to disallow self-reward and punishment in this class of problems by means of an invariance assumption: isomorphic continuation problems yield isomorphic consistent plans. I show that surprisingly many problems in the literature are SC, including many with state variables. I compare this solution to several others including the standard truncation approach, which seeks to reduce multiplicity in infinite-horizon problems by restriction to the limits of equilibrium strategies in finitely truncated problems as the horizon recedes. I find that the proposed solution coincides with the the results of that truncation approach in many, but not all, SC problems.