Unique Subgame Perfect Equilibrium Outcome Research Articles

Subgame perfect equilibria under the deferred acceptance algorithm

We analyze a subgame perfect equilibrium (SPE) of an extensive game with perfect information induced by the firm-oriented deferred acceptance (DA) algorithm in a one-to-one matching market between firms and workers. Our game repeats the following procedure until every firm in the market has a partner: (i) an unmatched firm strategically decides to which worker to make an offer or to exit the market, and (ii) the worker receiving the offer strategically decides whether to tentatively accept or reject it. When no agents are strategic, the resulting outcome is the firm-optimal stable matching. We show that the worker-optimal stable matching is the unique SPE outcome when only workers are strategic. By contrast, multiple SPE outcomes may exist, possibly including unstable matchings when only firms are strategic. We show that every firm weakly prefers any SPE outcome to the worker-optimal stable matching and that the matching induced by Kesten’s efficiency-adjusted DA algorithm can be achieved as an SPE. When both workers and firms are strategic, we also show that the worker-optimal stable matching is still the unique SPE outcome.

A strategic justification of the constrained equal awards rule through a procedurally fair multilateral bargaining game

We propose a new game to strategically justify the constrained equal awards rule in claims problems. Our game is “procedurally fair” (claimants are treated equally) and “multilateral” (all claimants negotiate simultaneously). In addition, even if claimants cannot reach an agreement in any period, they can renegotiate in the next period. We show that, for each claims problem, the awards vector chosen by the constrained equal awards rule achieved at period 1 is the unique subgame perfect equilibrium outcome of the game.

Optimal recommendation in two-player bargaining games

This paper extends the traditional two-player noncooperative bargaining game by adding a recommendation stage before the regular bargaining stage. At the recommendation stage, a coordinator recommends a feasible payoff pair to the players. If both players accept the recommendation, the game ends and the recommendation is enforced. Otherwise, the game enters the bargaining stage, where each player is picked as the proposer of any bargaining round according to an exogenous probability characterizing their relative bargaining power, and both players receive their respective reservation payoffs with a certain risk of breakdown if they fail to reach an agreement in this round. We characterize the unique subgame perfect equilibrium outcome of this extended game. The equilibrium recommendation, which will be accepted by both players, is optimal for the coordinator among all acceptable recommendations. A player’s payoff in optimal recommendation increases with her bargaining power. As the risk of breakdown vanishes, the optimal recommendation converges to an asymmetric Nash bargaining solution parameterized by the probability at the bargaining stage.

Strategic Justification in Claims Problems: Procedurally Fair and Multilateral Bargaining Game

In a claims problem, we propose a new game for strategic justification of the constrained equal awards rule. In contrast to Li and Ju (Mimeo, 2016) and Tsay and Yeh (Games and Economic Behavior, 113, 515-532, 2019), our game is procedurally fair (claimants are treated equally) and multilateral (all claimants negotiate simultaneously). In addition, even if claimants cannot reach an agreement in any period, they can renegotiate in the next period. We show that for each claims problem, the awards vector chosen by the constrained equal awards rule is the unique subgame perfect equilibrium outcome of the game.

Subgame Perfect Equilibria Under the Deferred Acceptance Algorithm

We analyze a subgame perfect equilibrium (SPE) of an extensive game with perfect information induced by the firm-oriented deferred acceptance (DA) algorithm in a one-to-one matching market between firms and workers. When no agents are strategic, the resulting outcome is the firm-optimal stable matching. We show that the worker-optimal stable matching is the unique SPE outcome when only workers are strategic. By contrast, multiple SPE outcomes may exist, possibly including unstable matchings when only firms are strategic. We show that every firm weakly prefers any SPE outcome to the worker-optimal stable matching and that the matching induced by Kesten's efficiency-adjusted DA algorithm can be achieved as a SPE. When both workers and firms are strategic, we also show that the worker-optimal stable matching is still the unique SPE outcome. Hence, the strategic behavior of workers yields the worker-optimal stable matching in a dynamic implementation of the firm-oriented DA algorithm. Our results reveal that the workers strategic behavior yields a reversal property in a dynamic implementation of the firm-oriented DA algorithm.

A Non-cooperative Mechanism Yielding the Nucleolus of Airport Problems

A non-cooperative approach is employed to solve the problem of allocating the total cost of building and maintaining an airport runway among its users. More precisely, it is proposed a non-cooperative bargaining game with a unique subgame perfect equilibrium (SPE) outcome, and it is shown that the SPE outcome payoffs coincide with those prescribed by the nucleolus of the corresponding airport cooperative game.

Bargaining orders in a multi-person bargaining game

This paper studies a complete-information bargaining game with one buyer and multiple sellers of different “sizes” or bargaining strengths. The bargaining order is determined endogenously. With a finite horizon, there is a unique subgame perfect equilibrium outcome, in which the buyer purchases in order of increasing size–from the smallest to the largest. With an infinite horizon, if the sellers have sufficiently different sizes, there is a unique equilibrium outcome, which has the same bargaining order. If the sellers have similar sizes with an infinite horizon, there may be multiple equilibrium outcomes with different bargaining orders.

Sequential commitment games

We consider an extensive-form game in which players have the option to commit to actions before the game is played. We focus on commitment procedures where players make voluntary irreversible commitments in a prescribed order over the decision nodes. We study whether such commitment procedures may lead to Pareto-efficient outcomes. Our main result is surprisingly positive: for two-player games, we introduce an order over the decision nodes for which the unique subgame perfect equilibrium outcome is Pareto-efficient for every strict preference of the players over the outcomes. We show that the above result does not hold for games with four players.

Commitments in Extensive Form Games

We consider extensive-form games in which players have the option to commit to actions before the game is played. We focus on commitment procedures where players make voluntary irreversible commitments in a prescribed order over the decision nodes. We study whether such commitment procedures may lead to Pareto-efficient outcomes. Our main positive result is surprisingly positive:For two-player games, we introduce an order over the decision nodes for which the unique sub-game perfect equilibrium outcome is Pareto efficient regardless of the actual payoffs. We show an impossibility result for games with four players.

Commitment in alternating offers bargaining

We extend the Ståhl–Rubinstein alternating-offer bargaining procedure to allow players to simultaneously and visibly commit to some share of the pie prior to, and for the duration of, each bargaining round. If commitment costs are small but increasing in the committed share, then the unique subgame perfect equilibrium outcome exhibits a second mover advantage. In particular, as the horizon approaches infinity, and commitment costs approach zero, the unique bargaining outcome corresponds to the reversed Rubinstein outcome (δ/(1+δ),1/(1+δ)), where δ is the common discount factor.

A bargaining procedure leading to the serial rule in games with veto players

This paper studies an allocation procedure for coalitional games with veto players. The procedure is similar to the one presented by Arin and Feltkamp (J Math Econ 43:855–870, 2007), which is based on Dagan et al. (Games Econ Behav 18:55–72, 1997). A distinguished player makes a proposal that the remaining players must accept or reject, and conflict is solved bilaterally between the rejector and the proposer. We allow the proposer to make sequential proposals over several periods. If responders are myopic maximizers (i.e. consider each period in isolation), the only equilibrium outcome is the serial rule of Arin and Feltkamp (Eur J Oper Res 216:208–213, 2012) regardless of the order of moves. If all players are fully rational, the serial rule still arises as the unique subgame perfect equilibrium outcome if the order of moves is such that stronger players respond to the proposal after weaker ones.

Bargaining Order in a Multi-Person Bargaining Game

This paper studies a bargaining game with one buyer and multiple sellers of different sizes or bargaining strengths. The bargaining order is determined by the buyer. If the buyer can commit to a bargaining order, there is a unique subgame perfect equilibrium outcome where the buyer bargains in order of increasing size -- from the smallest to the largest. If the buyer cannot commit to a bargaining order and the sellers are sufficiently different, there is also a unique subgame perfect equilibrium outcome again with the order of increasing size.

One-Seller Assignment Markets with Multi-Unit Demands

We consider one-seller assignment markets with multi-unit demands and prove that the associated game is buyers-submodular. Therefore the core is non-empty and it has a lattice structure which cont ains the allocation where every buyer receives his marginal contribution. We prove that in this kind of market, every pairwise-stable outcome is associated to a co mpetitive equilibrium and vice versa. We study conditions under which the buyers-optimal and the seller-optimal core allocations are competitive equilibrium payoff vectors. Moreover, we characterize the markets for which the core coincid ences with the set of competitive equilibria payoff vectors. When agents behave strategically, we introduce a procedure that implements the buyers-op timal core allocation as the unique subgame perfect Nash equilibrium outcome.

The takeover game

The takeover game is an experimental asset market characterized by three important features: (1) manager-determined dividend payments to shareholders, (2) a takeover offer to shareholders, and (3) a double auction market in which shareholders trade shares. This market mechanism essentially allows shareholders to price their trust in management. Despite the unique subgame perfect equilibrium outcome in which no takeover offer is ever rejected and no dividends are ever paid out, we find that the market often survives takeover offers. Managers pay positive dividends and appear to do so strategically to signal trustworthiness to shareholders, especially in periods in which takeover is to be made. While prices are not directly responsive to dividends, we find that market price is a good predictor of shareholder’s intent to accept takeover offers.

Bargaining Order in a Multi-Person Bargaining Game

This paper studies a complete-information bargaining game with one buyer and multiple sellers of di¤erent ?sizes? or bargaining strengths. The bargaining order is determined by the buyer. If the buyer can commit to a bargaining order, there is a unique subgame perfect equilibrium outcome where the buyer bargains in order of increasing size ? from the smallest to the largest. If the buyer cannot commit to a bargaining order and the sellers are su¢ ciently di¤erent, there is also a unique subgame perfect equilibrium outcome again with the order of increasing size.

Reject and renegotiate: The Shapley value in multilateral bargaining

This paper investigates three distinctive and intuitive renegotiation bargaining protocols that all yield the Shapley value as the unique subgame perfect equilibrium outcome. These protocols, built on the multi-bidding procedure of Pérez-Castrillo and Wettstein (2001), allow more freedom in multilateral bargaining where rejected players can further negotiate and form coalitions. The self-duality of the Shapley value plays a key role in the second and third results. Moreover, these renegotiation protocols allow an actual play along the equilibrium path to restore the Shapley value in case of a ‘mistake’ made before.

Efficiency and compromise: a bid-offer–counteroffer mechanism with two players

A bid-offer–counteroffer mechanism is proposed to solve a fundamental two-person decision choice problem with two alternatives. It yields a unique subgame perfect equilibrium outcome, and leads to an intuitive overall solution that offers a reconciliation between egalitarianism and utilitarianism. We then investigate the axiomatic foundation of the solution. Furthermore, we compare it with several conventional strategic approaches to this setting.

An Axiomatic Approach to Arbitration and its Application in Bargaining Games

We define an arbitration problem as the triplet of a bargaining set and the offers submitted by two players. We characterize the solution to a class of arbitration problems using the axiomatic approach. The axioms we impose on the arbitration solution are Symmetry in Offers,'' Invariance'' and Pareto Optimality.'' The key axiom, Symmetry in Offers,'' requires that whenever players' offers are symmetric, the arbitrated outcome should also be symmetric. We find that there exists a unique arbitration solution, called the symmetric arbitration solution, that satisfies all three axioms. We then analyze a simultaneous-offer game and an alternating-offer game. In both games, the symmetric arbitration solution is used to decide the outcome whenever players cannot reach agreement by themselves. We find that in both games, if the discount factor of players is close to 1, then the unique subgame perfect equilibrium outcome coincides with the Kalai-Smorodinsky solution outcome.

Axiomatic and strategic justifications for the constrained equal benefits rule in the airport problem

We consider the “airport problem”, which is concerned with sharing the cost of an airstrip among agents who need airstrips of different lengths. We investigate the implications of two properties, Left-endpoint Subtraction (LS) bilateral consistency and LS converse consistency, in the airport problem. First, on the basis of the two properties, we characterize the constrained equal benefits rule, which equalizes agentsʼ benefits subject to no one receiving a subsidy. Second, we introduce a 2-stage extensive form game that exploits LS bilateral consistency and LS converse consistency. We show that there is a unique subgame perfect equilibrium outcome of the game and moreover, it is the allocation chosen by the constrained equal benefits rule.

Demand bargaining and proportional payoffs in majority games

We study a majoritarian bargaining model in which players make payoff demands in decreasing order of voting weight. The unique subgame perfect equilibrium outcome is such that the minimal winning coalition of the players that move first forms with payoffs proportional to the voting weights. This result advances previous analysis in terms of one or more of the following: a) the simplicity of the extensive form (finite horizon with a predetermined order of moves); b) the range of the majority games covered; c) the equilibrium concept (subgame perfect equilibrium is sufficient for a unique prediction).