Stationary Subgame Perfect Equilibrium Research Articles

Stochastic bargaining in the lab

We study novel multi-stage bargaining games that differ in the timing and efficiency of Stationary Subgame Perfect Equilibrium (SSPE) outcomes, but include the possibility of an equal split in every stage, based on the theory of Merlo and Wilson (1995). Despite this theory’s wide application to bargaining in natural environments, our study is the first to rigorously test it using laboratory experiments in both gain and loss domains. In our design, the total surplus to be split within a stage is not a fixed amount, and it can expand or shrink between stages, making delay and inefficiency possible in equilibrium. We find that players regularly agree on the largest available equal split regardless of SSPE predictions. These patterns are well captured by an “Efficient Equality Equilibrium,” built on the idea that people share lexicographic preferences for equal-split outcomes. Our results highlight the important interplay of equality and efficiency in bargaining and other conflict resolution environments. In addition, we find that men and women are equally likely to propose equal splits over gains, while male proposers are significantly more selfish over losses.

Coalition-then-allocation legislative bargaining

We investigate legislative bargaining where players first bargain over coalitions and after one coalition is formed, players in this coalition bargain over allocations. We show that if discount factors in coalition bargaining are smaller than those in allocation bargaining and sufficiently large, in any stationary subgame perfect equilibrium (SSPE), relatively impatient proposers immediately form a minimal winning coalition of relatively impatient players, but relatively patient proposers fail to form a coalition and bargaining delays occur. We also show that if the discount factors in coalition bargaining are smaller than those in allocation bargaining and sufficiently small, delays do not occur. Furthermore, we show that if the discount factors in allocation bargaining are smaller than those in coalition bargaining and sufficiently similar across players, there exist multiple SSPEs exhibiting delays and having different payoff tuples. We also introduce leader-dependent hedonic games, where each player has a preference relation over pairs of a coalition and its leader. We view a truncated game with replacing subgames of allocation bargaining by their SSPE playoff tuples to be based on a leader-dependent hedonic game.

Group cooperation against a hegemon

In this paper, I study the ability of a group of citizens to cooperate against a hegemon in a repeated contest game and where group members and the hegemon have different valuations of the prize. I first consider that group members use grim trigger strategies (GTSs) to support cooperative behavior and show that full cooperation within the group is more easily sustained as a stationary subgame perfect (Nash) equilibrium (SSPE) as either group size or the heterogeneity in the valuation of the prize increases. In turn, I show that full cooperation within the group can also be sustained as a weakly renegotiation-proof equilibrium (WRPE). However, an increase in group size makes it more difficult to sustain within-group cooperation, but an increase in the relative valuation of the prize by group members still facilitates group cooperation.

A noncooperative bargaining game with endogenous protocol and partial breakdown

In this paper, we present a noncooperative sequential bargaining game with transferable utility. Instead of assuming exogenous protocol or random-proposer recognition, we provide a mechanism in which the protocol is generated endogenously. Besides, a partial breakdown probability is considered rather than a discount factor when the proposal is rejected by some player. We show that for each partial breakdown probability, stationary subgame perfect equilibria exist if the characteristic function game is totally balanced. Moreover, the outcomes of stationary subgame perfect equilibria in our model coincide with the core allocations.

Bilateral bargaining with endogenous status quo

We consider a non-cooperative bargaining game where in each round, if a proposal is rejected, with a probability, the allocation implemented in the previous round is implemented as a status quo and the game proceeds to the next round, and with the complementary probability, no allocations are implemented and the negotiations break down. We show that there exists a unique stationary subgame perfect equilibrium such that in any round, a proposer offers a proposal such that the responder is indifferent between accepting and rejecting it and an agreement is achieved. If the probability of breakdown is close to zero, the sequence of equilibrium allocations evolves from the initial status quo to an allocation near the equilibrium allocation of the Rubinstein bargaining game over time. The effect of the initial status quo on the equilibrium allocations vanishes over time.

Simultaneous-offers bargaining with a mediator

An arbitrator is often introduced into bargaining to avoid disagreement. The arbitrator's role is to impose some agreement when negotiators cannot reach an agreement. However, in models with an arbitrator, a fair agreement for negotiators is eliminated from equilibrium outcomes if the arbitrator is biased. To avoid disagreement without eliminating the achievability of a fair agreement in equilibrium, we consider introducing a mediator. While an arbitrator imposes an agreement, a mediator can only give advice. We analyze a simultaneous-offers bargaining model with a mediator and obtain the following desirable results. First, disagreement is not supported as an outcome of a stationary subgame perfect equilibrium (SSPE). Second, even if a mediator is biased, the fair agreement in the sense of the Nash bargaining solution (NBS) is always one of the SSPE agreements. Finally, if a mediator is fair, negotiators always reach an agreement with the NBS in SSPE.

Bargaining delay under partial breakdowns and externalities

We consider a noncooperative coalitional bargaining game where partial breakdowns, such as the rejecter of a proposal exiting from the game, take place with a positive probability, and externalities accompany the formation of coalitions. We present an example in which there exists a stationary subgame perfect equilibrium (SSPE) such that a bargaining delay occurs. We also show that under no partial breakdowns or no externalities, delay does not occur in any SSPE.

Bargaining foundation for ratio equilibrium in public‐good economies

AbstractWe provide a bargaining foundation for the concept of ratio equilibrium in public‐good economies. We define a bargaining game of alternating offers, in which players bargain to determine their cost shares of public‐good production and a level of public good. We study the stationary subgame perfect equilibrium (SSPE) without delay of the bargaining game. We demonstrate that when the players are perfectly patient, they are indifferent between the equilibrium offers of all players. We also show that every SSPE without delay in which the ratios offered by all players are the same leads to a ratio equilibrium. In addition, we demonstrate that all equilibrium ratios are offered by the players at some SSPE without delay. We use these results to discuss the case when the assumption of perfectly patient players is relaxed and the cost of delay vanishes.

Bargaining Foundation for Ratio Equilibrium in Public Good Economies

We provide a bargaining foundation for the concept of ratio equilibrium in public good economies. We define a bargaining game of alternating offers in which players bargain to determine their cost shares of public good production and a level of public good. We study the stationary subgame perfect equilibrium without delay of the bargaining game. We demonstrate that when the players are perfectly patient, they are indifferent between the equilibrium offers of all players. We also show that every stationary subgame perfect equilibrium without delay in which the ratios offered by all players are the same leads to a ratio equilibrium. In addition, we demonstrate that all equilibrium ratios are offered by the players at some stationary subgame perfect equilibrium without delay. We use these results to discuss the case when the assumption of perfectly patient players is relaxed and the cost of delay vanishes.

Multilateral Bargaining in Networks: On the Prevalence of Inefficiencies

We introduce a noncooperative multilateral bargaining model for a network-restricted environment, in which players can communicate only with their neighbors. Each player strategically chooses the bargaining partners among the neighbors to buy out their communication links with upfront transfers. The main theorem characterizes a condition on network structures for efficient equilibria and shows the prevalence of strategic delays. If the underlying network is either complete or circular, then an efficient stationary subgame perfect equilibrium exists for all discount factors: all the players always try to reach an agreement as soon as practicable and hence no strategic delay occurs. In any other network, however, an efficient equilibrium is impossible for sufficiently high discount factors because some players strategically delay an agreement. We also provide an example of a Braess-like paradox, in which the more links are available, the less links are actually used. Thus, network improvements may decrease social welfare.

Multilateral Bargaining in Networks: On the Prevalence of Inefficiencies

We introduce a new noncooperative multilateral bargaining model for network-restricted environments in which players can bargain only with their neighbors. The main theorem characterizes a condition on network structures for efficient equilibria. If the underlying network is either complete or circular, an efficient stationary subgame perfect equilibrium exists for all discount factors—all the players always try to reach an agreement as soon as practicable, and hence no strategic delay occurs. In any other network, however, an efficient equilibrium is impossible if a discount factor is greater than a certain threshold, as some players strategically delay an agreement. We also provide an example of a Braess-like paradox, in which network improvements decrease social welfare. The e-companion is available at https://doi.org/10.1287/opre.2018.1725 .

Equivalent conditions for the existence of an efficient equilibrium in coalitional bargaining with externalities and renegotiations

We consider a noncooperative coalitional bargaining game with externalities and renegotiations. We provide the necessary and sufficient condition for an efficient stationary subgame perfect equilibrium to exist. This condition states that a Nash bargaining solution is immune to any blocking.

Static and dynamic underinvestment: An experimental investigation

In this paper, we analyze a stylized version of an environment with public goods, dynamic linkages, and legislative bargaining. Our theoretical framework studies the provision of a durable public good as a modified two-period version of Battaglini et al. (2012). The experimental design allows us to disentangle inefficiencies that would result in a one-shot world (static inefficiencies) from additional inefficiencies that emerge in an environment in which decisions in the present affect future periods (dynamic inefficiencies). We solve the first-best solution and compare it to the symmetric stationary subgame-perfect equilibrium of a legislative bargaining game. The experimental results indicate that subjects do react to dynamic linkages, and, as such, there is evidence of both static and dynamic inefficiencies. The quantitative predictions of the bargaining model with respect to the share of dynamic inefficiencies are closest to the data when dynamic linkages are high. To the extent that behavior is different from the model's predictions, a systematic pattern emerges — namely, the use of strategic cooperation, whereby groups increase the efficiency of current proposals by selectively punishing, in future proposals, individuals who propose highly inefficient allocations.

Nash bargaining solution under externalities

We define a Nash bargaining solution (NBS) of partition function games. Based on a partition function game, we define an extensive game, which is a propose–respond sequential bargaining game where the rejecter of a proposal exits from the game with some positive probability. We show that the NBS is supported as the expected payoff profile of any stationary subgame perfect equilibrium (SSPE) of the extensive game such that in any subgame, a coalition of all active players forms immediately. We provide a necessary and sufficient condition for such an SSPE to exist. Moreover, we consider extensions to the cases of nontransferable utilities, time discounting and multiple-coalition formation.

Costless Delay in Negotiations

We study strategic negotiation models featuring costless delay, general recognition procedures, endogenous voting orders, and finite sets of alternatives. Two examples show: 1. non-existence of stationary subgame-perfect equilibrium (SSPE). 2. the recursive equations and optimality conditions are necessary for SSPE but insufficient because these equations can be singular. Strategy profiles excluding perpetual disagreement guarantee non-singularity. The necessary and sufficient conditions for existence of stationary best responses additionally require either an equalizing condition or a minimality condition. Quasi SSPE only satisfy the recursive equations and optimality conditions. These always exist and are SSPE if either all equalizing conditions or all minimality conditions hold.

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.

Hart–Mas-Colell implementation of the discounted Shapley value

We consider an implementation of the discounted Shapley value. We modify the Hart–Mas-Colell model such that each player discounts future payoffs and proposes not only an allocation, but also a coalition. We show that the discounted Shapley value is supported by each stationary subgame perfect equilibrium of the modified game such that in each subgame, the coalition that consists of all active players immediately forms. We also provide conditions for such a stationary subgame perfect equilibrium to exist.

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.

Destroying Surplus and Buying Time in Unanimity Bargaining

In non-cooperative bargaining games in the tradition of Rubinstein, the proposer derives bargaining power from the prospect of a costly delay which would follow the rejection of a proposal. We consider a unanimity bargaining game in which the proposer can strategically choose to prolong this delay. Prolonging the delay increases the proposer's bargaining power, but is assumed to come at a cost and thus cause an inefficiency. We use an appropriate refinement of stationary subgame-perfect equilibrium as the solution concept. We characterize equilibrium strategies and payoffs. We establish conditions on model Parameters under which equilibrium is or is not efficient. For inefficient equilibria, we quantify the extent of the inefficiency. Moreover, we study the relation between the number of players and the degree of inefficiency. We find that inefficient equilibria become more inefficient the more players there are. Moreover, the parameter region in which an efficient equilibrium is possible shrinks when the number of players increases.

Veto players and equilibrium uniqueness in the Baron–Ferejohn model

In political economy, the seminal contribution of the Baron–Ferejohn bargaining model constitutes an important milestone for the study of legislative policy making. In this paper, we analyze a particular equilibrium characteristic of this model, equilibrium uniqueness. The Baron–Ferejohn model yields a class of payoff-unique stationary subgame perfect equilibria (SSPE) in which players’ equilibrium strategies are not uniquely determined. We first provide a formal proof of the multiplicity of equilibrium strategies. This also enables us to establish some important properties of SSPE. We then introduce veto players into the original Baron–Ferejohn model. We state the conditions under which the new model has a unique SSPE not only in terms of payoffs but also in terms of players’ equilibrium strategies.