Subgame Perfect Equilibrium Payoffs Research Articles

A complete folk theorem for finitely repeated games

This paper analyzes the set of pure strategy subgame perfect Nash equilibria of any finitely repeated game with complete information and perfect monitoring. The main result is a complete characterization of the limit set, as the time horizon increases, of the set of pure strategy subgame perfect Nash equilibrium payoff vectors of the finitely repeated game. This model includes the special case of observable mixed strategies.

Equilibrium payoffs and proposal ratios in bargaining models

We analyze a bargaining model which is a generalization of the model of Rubinstein (Econometrica 50(1):97–109, 1982) from the viewpoint of the process of how a proposer is decided in each period. In our model, a player’s probability to be a proposer depends on the history of proposers and players divide a pie of size 1. We derive a subgame perfect equilibrium (SPE) and analyze how its SPE payoffs are related to the process. In the bilateral model, there is a unique SPE. In the n-player model, although SPE may not be unique, a Markov perfect equilibrium (MPE) similar to the SPE in the bilateral model exists. In the case where the discount factor is sufficiently large, if the ratio of opportunities to be a proposer converges to some value, players divide the pie according to the ratio of this convergent value under these equilibria. This result implies that although our process has less regularity than a Markov process, the same result as in the model that uses a Markov process holds. In addition to these results, we show that the limit of the SPE (or the MPE) payoffs coincides with the asymmetric Nash bargaining solution weighted by the convergent values of the ratio of the opportunities to be a proposer.

Implementation of Nash bargaining solutions with non-convexity

Nash solutions for two-player bargaining problems with non-convexity are shown to be dictatorial selections of Nash product maximizers in recent literature. In this paper we show that these solutions are implementable as unique subgame perfect equilibrium payoff allocations of a sequential game.

Subgame perfect equilibria in majoritarian bargaining

We study the division of a surplus under majoritarian bargaining in the three-person case. In a stationary equilibrium as derived by Baron and Ferejohn (1989), the proposer offers one third times the discount factor of the surplus to a second player and allocates no payoff to the third player, a proposal which is accepted without delay. Laboratory experiments show various deviations from this equilibrium, where different offers are typically made and delay may occur before acceptance. We address the issue to what extent these findings are compatible with subgame perfect equilibrium and characterize the set of subgame perfect equilibrium payoffs for any value of the discount factor. We show that for any proposal in the interior of the space of possible agreements there exists a discount factor such that the proposal is made and accepted. We characterize the values of the discount factor for which equilibria with one-period delay exist. We show that any amount of equilibrium delay is possible and we construct subgame perfect equilibria such that arbitrary long delay occurs with probability one.

Subgame perfect equilibrium in a bargaining model with deterministic procedures

Two players, A and B, bargain to divide a perfectly divisible pie. In a bargaining model with constant discount factors, $$\delta _A$$ and $$\delta _B$$ , we extend Rubinstein (Econometrica 50:97–110, 1982)’s alternating offers procedure to more general deterministic procedures, so that any player in any period can be the proposer. We show that each bargaining game with a deterministic procedure has a unique subgame perfect equilibrium (SPE) payoff outcome, which is efficient. Conversely, each efficient division of the pie can be supported as an SPE outcome by some procedure if $$\delta _A+\delta _B\ge 1$$ , while almost no division can ever be supported in SPE if $$\delta _A+\delta _B < 1$$ .

A strategic implementation of the sequential equal surplus division rule for digraph cooperative games

We provide a strategic implementation of the sequential equal surplus division rule (Beal et al. in Theory Decis 79:251–283, 2015). Precisely, we design a non-cooperative mechanism of which the unique subgame perfect equilibrium payoffs correspond to the sequential equal surplus division outcome of a superadditive rooted tree TU-game. This mechanism borrowed from the bidding mechanism designed by Perez-Castrillo and Wettstein (J Econ Theory 100:274–294, 2001), but takes into account the direction of the edges connecting any two players in the rooted tree, which reflects some dominance relation between them. Our proofs rely on interesting properties that we provide for a general class of bidding mechanisms.

Uniqueness of stationary equilibrium payoffs in the Baron–Ferejohn model with risk‐averse players

AbstractI study a multilateral sequential bargaining model among risk‐averse players in which the players may differ in their probability of being selected as the proposer and the rate at which they discount future payoffs. For games in which agreement requires less than unanimous consent, I characterize the set of stationary subgame perfect equilibrium payoffs. With this characterization, I establish the uniqueness of the equilibrium payoffs. For the case where the players have the same discount factor, I show that the payoff to a player is non‐decreasing in his probability of being selected as the proposer. For the case where the players have the same probability of being selected as the proposer, I show that the payoff to a player is non‐decreasing in his discount factor. This generalizes earlier work by allowing the players to be risk averse.

Subgame perfect equilibria in discounted stochastic games

This paper considers policies and payoffs corresponding to subgame perfect equilibrium strategies in discounted stochastic games with finitely many states. It is shown that a policy is induced by an equilibrium strategy if and only if it can be supported with the threat of reverting to the induced policy that gives the least equilibrium payoff for the deviator. It follows that the correspondence of subgame perfect equilibrium payoffs is the largest fixed-point of a correspondence-valued operator defined by the players' incentive compatibility conditions. Moreover, the fixed-point iteration converges to the equilibrium payoff correspondence.

Characterizing the limit set of perfect and public equilibrium payoffs with unequal discounting

We study repeated games with imperfect public monitoring and unequal dis- counting. We characterize the limit set of perfect and public equilibrium payoffs as discount factors converge to 1 with the relative patience between players fixed. We show that the pairwise and individual full rank conditions are sufficient for the folk theorem. In this paper, we characterize the equilibrium payoffs in repeated games with imperfect public monitoring and unequal discounting as discount factors converge to 1 with rel- ative patience fixed. In particular, we show that the pairwise and individual full rank conditions are sufficient for the folk theorem. Lehrer and Pauzner (1999) (henceforth LP) analyze two-player repeated games with perfect monitoring and unequal discounting. They define the set of feasible and sequen- tially individually rational (henceforth SIR) payoffs and show that in two-player games with perfect monitoring, the limit set of subgame perfect equilibrium payoffs coincides with that of SIR payoffs as discount factors converges to 1 with the relative patience fixed (the folk theorem). Recently, Chen and Takahashi (2012) extend the result to n-player games with perfect monitoring. This paper extends their results to imperfect public monitoring. While the proofs of both Lehrer and Pauzner (1999 )a ndChen and Takahashi (2012) are constructive, we em- ploy a nonconstructive approach using the recursive structure of the perfect and public equilibrium (henceforth PPE). Specifically, we attain a characterization of the set of PPE payoffs as discount factors converge to1. In addition, we characterize SIR payoffs. Given these characterizations, we show that if the pairwise and individual full rank conditions are satisfied, these two sets coincide, that is, the folk theorem holds.

Repeated Sequential Prisoner's Dilemma: The Stackleberg Variant

We study the Stackleberg variant of the repeated Sequential Prisoner's Dilemma (SPD). The game goes in two stages, and the two players, the leader and the follower, are asymmetric in both stages. In the first stage of the game, the leader chooses a strategy (for the repeated SPD of the second stage), which is immediately known to the follower. In the second stage, they play repeated SPD: In each round the follower moves after observing the leader's action. Assuming complete rationality, we find some extraordinary properties of this model. (i) The (subgame perfect) equilibrium payoff profile is unique, which lies on the corner of the region predicted by classical folk theorems: It is best for the leader and at the same time worst for the follower, (ii) the leader has simple optimal strategies that are one-step memory and stationary. These features are in great contrast with classical results, where either uniqueness cannot be guaranteed and equilibrium strategies are often quite complicated, or bounded rationality is required. Although full cooperation, i.e., the outcome is always (cooperate, cooperate), is not attainable in our model, at least a half of the optimal social welfare can be guaranteed. We also do a non-equilibrium analysis which makes the usual equilibrium analysis more convincing.

Public Information in Markov Games

In a Markov game, players engage in a sequence of games determined by a Markov process. In this setting, this paper investigates the impact of varying the informativeness of public information, as defined by Blackwell 1951 and 1953, pertaining to the games that will be played in future periods. In brief, when a curvature condition on payoffs is satisfied, the finding is that, for any fixed discount factor, the set of strongly symmetric subgame perfect equilibrium payoffs of a Markov game with more informative signals is contained in this set of equilibrium payoffs if the Markov game is played with any less informative signals. The second result shows that larger equilibrium payoffs are possible with less informative signals when the curvature condition fails, but only for some discount factors. The third result strengthens the curvature condition, but generalizes the first result to all subgame perfect equilibrium payoffs. Finally, a collusion application is presented to illustrate the curvature condition.

Public information in Markov games

In a Markov game, players engage in a sequence of games determined by a Markov process. In this setting, this paper investigates the impact of varying the informativeness of public information, as defined by Blackwell [8,9], pertaining to the games that will be played in future periods. In brief, when a curvature condition on payoffs is satisfied, the finding is that, for any fixed discount factor, the set of strongly symmetric subgame perfect equilibrium payoffs of a Markov game with more informative signals is contained in this set of equilibrium payoffs if the Markov game is played with any less informative signals. The second result shows that larger equilibrium payoffs are possible with less informative signals when the curvature condition fails, but only for some discount factors. The third result strengthens the curvature condition, but generalizes the first result to all subgame perfect equilibrium payoffs. Finally, a collusion application is presented to illustrate the curvature condition.

FRACTAL GEOMETRY OF EQUILIBRIUM PAYOFFS IN DISCOUNTED SUPERGAMES

This paper examines the pure-strategy subgame-perfect equilibrium payoffs in discounted supergames with perfect monitoring. It is shown that the equilibrium payoffs can be identified as sub-self-affine sets or graph-directed iterated function systems. We propose a method to estimate the Hausdorff dimension of the equilibrium payoffs and relate it to the equilibrium paths and their graph presentation.

Justifiable punishments in repeated games

In repeated games, subgame perfection requires all continuation strategy profiles must be effective to enforce the equilibrium; they serve as punishments should deviations occur. It does not require whether a punishment can be justified for the deviation, which creates a great deal of freedom in constructing equilibrium strategies, resulting the well-known folk theorem. We introduce justifiable punishments in repeated games. After one player deviates, the corresponding continuation or punishment is justifiable if either the deviation is bad to the other player or the continuation itself is good to the other player. We characterize the set of payoff vectors that can be supported by subgame perfect equilibria with justifiable punishments, as the discount factor goes to one. This limiting set of equilibrium payoffs can be quite different from the set of subgame perfect equilibrium payoffs. Any efficient, feasible, and strictly individually rational payoff can be supported by equilibrium with justifiable punishments.

Backward induction and unacceptable offers

How to establish the existence of subgame perfect equilibrium (SPE) in bargaining models if no stationary SPEs (SSPEs) exist? The backward-induction technique of Shaked and Sutton (1984, Econometrica) applies to the cyclical structure of SPE payoffs and provides recursive dynamics on the bounds of SPE payoffs. Acceptable and unacceptable offers have to be incorporated for these dynamics to be necessary and sufficient for extreme SPEs. In this paper, we demonstrate how these recursive dynamics are directly applicable to establish the existence of SPE in a model with no SSPE. Also from these dynamics, the extreme SPE strategy profiles can easily be recovered.

Repeated Sequential Prisoner's Dilemma: The Stackleberg Variant

We study the Stackleberg variant of the repeated Sequential Prisoner’s Dilemma (SPD for short). The game goes in two stages, and the two players, the leader and the follower, are asymmetric in both stages. In the first stage of the game, the leader chooses a strategy (for the repeated SPD of the second stage), which is immediately known to the follower. In the second stage, they play repeated SPD: in each round the follower moves after observing the leader’s action. Assuming complete rationality, we find some extraordinary properties of this model. (i) The (sub-game perfect) equilibrium payoff profile is unique, which lies on the corner of the region predicted by classical folk theorems: it is best for the leader and at the same time worst for the follower. (ii) The leader has simple optimal strategies that are one-step memory and stationary. These features are in great contrast with classical results, where either uniqueness cannot be guaranteed and equilibrium strategies are often quite complicated, or bounded rationality is required. Although full cooperation, i.e. the outcome is always (cooperate, cooperate), is not attainable in our model, at least a half of the optimal social welfare can be guaranteed. We also do a non-equilibrium analysis which makes the usual equilibrium analysis more convincing.

A strategic implementation of the Average Tree solution for cycle-free graph games

Abstract In this note we provide a strategic implementation of the Average Tree solution for zero-monotonic cycle-free graph games. That is, we propose a non-cooperative mechanism of which the unique subgame perfect equilibrium payoffs correspond to the average hierarchical outcome of the game. This mechanism takes into account that a player is only able to communicate with other players (i.e., to make proposals about a division of the surplus of cooperation) when they are connected in the graph.

Subgame Perfect Equilibria in Majoritarian Bargaining

We study the division of a surplus under majoritarian bargaining in the three-person case. In a stationary equilibrium as derived by Baron and Ferejohn (1989), the proposer offers one third times the discount factor of the surplus to a second player and allocates no payoff to the third player, a proposal which is accepted without delay. Laboratory experiments show various deviations from this equilibrium, where different offers are typically made and delay may occur before acceptance. We address the issue to what extent these findings are compatible with subgame perfect equilibrium and characterize the set of subgame perfect equilibrium payoffs for any value of the discount factor. We show that for any proposal in the interior of the space of possible agreements there exists a discount factor such that the proposal is made and accepted. We characterize the values of the discount factor for which equilibria with one-period delay exist. We show that any amount of equilibrium delay is possible and we construct subgame perfect equilibria such that arbitrary long delay occurs with probability one.

A Strategic Implementation of the Average Tree Solution for Cycle-Free Graph Games

In this note we provide a strategic implementation of the average tree solution for zero-monotonic cycle-free graph games. That is, we propose a non-cooperative mechanism of which the unique subgame perfect equilibrium payoffs correspond to the average hierarchical outcome of the game. This mechanism takes into account that a player is only able to communicate with other players (i.e., to make proposals about a division of the surplus of cooperation) when they are connected in the graph.

Non-cooperative support for the asymmetric Nash bargaining solution

We study a model of non-cooperative multilateral unanimity bargaining on a full-dimensional payoff set. The probability distribution with which the proposing player is selected in each bargaining round follows an irreducible Markov process. If a proposal is rejected, negotiations break down with an exogenous probability and the next round starts with the complementary probability. As the risk of exogenous breakdown vanishes, stationary subgame perfect equilibrium payoffs converge to the weighted Nash bargaining solution with the stationary distribution of the Markov process as the weight vector.