Subgame perfection in ultimatum bargaining trees

In typical experiments on ultimatum bargaining, the game is described verbally and the majority of subjects deviate from subgame-perfect behavior. Proposers typically offer significantly more than the minimum possible and Responders reject “unfair” offers. In this work, we show that when the ultimatum bargaining game is presented as an abstract game tree, the vast majority of behavior is consistent with individualistic preferences and subgame-perfection. This finding raises doubts about theories that ignore the potential influence of social context and experiments that do not control for social context.     

Evolutionary Stability in Alternating-Offers Bargaining Games

This paper characterizes modified evolutionarily stable strategies (messes) in Rubinstein's alternating-offers, infinite-horizon bargaining game. We show that a mess causes agreement to be achieved immediately, with neither player willing to delay the agreement by one period in order to achieve the other player's share of the surplus. Each player's share of the surplus is then bounded between the shares received by the two players in the unique subgame-perfect equilibrium of Rubinstein's game. As the probability of a break-down in negotiations becomes small (or discount factors become large), these bounds collapse on the subgame-perfect equilibrium.Journal of Economic LiteratureClassification Numbers C70, C78.     

Game Theory via Revealed Preferences

We investigate equilibrium notions in game theory from the revealed preference approach. For extensive game forms with complete information, we derive a set of independent necessary and sufficient conditions for the observed outcomes to be rationalized by subgame-perfect Nash equilibrium. Journal of Economic Literature Classification Numbers: C72, C92.     

Cooperation in an Overlapping Generations Experiment

Recent theoretical work shows that folk theorems can be developed for infinite overlapping generations games. Cooperation in such games can be sustained as a Nash equilibrium. But, of course, there are other equilibria. This paper investigates experimentally whether cooperation actually occurs in a simple overlapping generations game. Subjects both play the game and formulate strategies. Our main finding is that subjects fail to exploit the intertemporal structure of the game. Even when we provided subjects with a recommendation to play the grim trigger strategy, most of the subjects still employed safe history-independent strategies. Journal of Economic Literature Classification Numbers: C72, C92, D90.     

Bargaining over loan contracts with signaling

This paper combines a sequential bargaining game between an enterprise and a fixed number of banks with a signaling game through which the enterprise reveals her project quality as well as her market-speed on the lending market. We characterize subgame-perfect Nash equilibrium loan contracts that are supported by separating perfect Bayesian equilibria in the signaling game. In contrast to existing models of lending markets, low-quality investment projects might be rewarded with more favorable equilibrium loan contracts than high-quality projects. Also in contrast to existing models, an increase in the competitive pressure between banks reduces the aggregate welfare in our model. The reason is that more favorable loan conditions come with a greater incentive for the ‘strong’ entrepreneur to distinguish herself from her ‘weak’ counterpart through socially wasteful signaling costs.     

OPEC's Response to International Climate Agreements

This paper studies a game between a group of countries that have agreed to participate in an international climate agreement (the signatories) and OPEC. The purpose of the signatories is to design carbon taxes that maximize their total net income, given a goal on global carbon emissions. As a response to the climate agreement, OPEC imposes an oil tax on its member states that maximizes OPEC's profits. Within a numerical model we find the subgame-perfect equilibrium of a game in which each player chooses when to fix his decision variables. It is shown that in equilibrium the group of signatories chooses to be the leader and OPEC chooses to be the follower. It is demonstrated, however, that for both agents the order of move is of minor (numerical) importance. Hence, the players have limited incentives for strategic behaviour.     

Existence and efficiency of a stationary subgame-perfect equilibrium in coalitional bargaining models with nonsuperadditive payoffs

We provide the existence theorem of stationary subgame-perfect equilibrium (SSPE) in a noncooperative coalitional bargaining game model with random proposers. Our model contains a bargaining situation where the coalitional game is nonsuperadditive. We also provide a necessary and sufficient condition for the existence of a pure-strategy SSPE satisfying the efficiency property when the discount factor is close to one. Furthermore, we provide examples where the delay in agreement occurs, even in a random-proposers model, when the game is nonsuperadditive. I am grateful to Akira Okada and an anonymous referee for their useful comments and helpful suggestions.     

Evolutionary dynamics and backward induction

The backward induction (or subgame-perfect) equilibrium of a perfect information game is shown to be the unique evolutionarily stable outcome for dynamic models consisting of selection and mutation, when the mutation rate is low and the populations are large.     

On price taking behavior in a nonrenewable resource cartel–fringe game

We consider a nonrenewable resource game with one cartel and a set of fringe members. We show that (i) the outcomes of the closed-loop and the open-loop nonrenewable resource game with the fringe members as price takers (the cartel–fringe game à la Salant, 1976) coincide and (ii) when the number of fringe firms becomes arbitrarily large, the equilibrium outcome of the closed-loop Nash game does not coincide with the equilibrium outcome of the closed-loop cartel–fringe game. Thus, the outcome of the cartel–fringe open-loop equilibrium can be supported as an outcome of a subgame-perfect equilibrium. However the interpretation of the cartel–fringe model, where from the outset the fringe is assumed to be price taker, as a limit case of an asymmetric oligopoly with the agents playing Nash–Cournot, does not extend to the case where firms can use closed-loop strategies.     

Dynamics of International Debt Overhang with Two Lender Banks

This paper presents a dynamic formalization of the behaviour of creditor banks on the secondary market for debts. I formulate the problem as an infinite-horizon game with two banks as players where each bank decides in every period either to sell its loan exposure to the debtor country at the present secondary market price, or to wait and keep its exposure to the next period. There exist three types of subgame-perfect equilibria with the property called the time continuation. I consider the relation ship between our equilibria and those of the Kaneko–Prokop (1993) one-period approach to the same problem and show that their approach does not lose much of the dynamic nature of the problem. In any equilibrium of the game, each bank waits in every period with high probability. I discuss the implications of these results for the long-run behaviour of banks on the secondary market and the resolution of debt overhang
JEL Classification Number: F34     

Unobserved heterogeneity and equilibrium: an experimental study of Bayesian and adaptive learning in normal form games

We describe an experiment based on a simple two-person game designed so that different learning models make different predictions. Econometric analysis of the experimental data reveals clear heterogeneity in the subjects’ learning behavior. But the subjects follow only a few decision rules for basing their play on their information, and these rules have simple cognitive interpretations. There is a unique equilibrium in pure strategies, and many equilibria in mixed strategies. We find that the only equilibrium consistent with the data is one of the mixed strategy equilibria. This equilibrium is shown, surprisingly, to be consistent with Jordan's Bayesian model.     

On Environmental Regulation of Oligopoly Markets: Emission versus Performance Standards

By specializing Montero’s (J Environ Econ Manag 44:23–44, 2002) model of environmental regulation under Cournot competition to an oligopoly with linear demand and quadratic abatement costs, we extend his comparison of firms incentives to invest in R&D under emission and performance standards by solving for a closed form solution of the underlying two-stage game. This allows for a full comparison of the two instruments in terms of their resulting propensity for R&D and equilibrium industry output. In addition, we incorporate an equilibrium welfare analysis. Finally, we investigate a three-stage game wherein a welfare-maximizing regulator sets a socially optimal emission cap under each policy instrument. For the latter game, while closed-form solutions for the subgame-perfect equilibrium are not possible, we establish numerically that the resulting welfare is always larger under a performance standard.     

Endogenous transfers in the Prisoner's Dilemma game: An experimental test of cooperation and coordination

We test a two-stage compensation mechanism for promoting cooperation in Prisoner's Dilemma games. Players first simultaneously choose binding non-negative amounts to pay their counterparts for cooperating, and then play the induced game knowing these amounts. In our games, all payment pairs consistent with mutual cooperation in subgame-perfect equilibrium transform these games into coordination games, with both mutual cooperation and mutual defection as Nash equilibria in the second stage. When endogenous transfer payments are not permitted, cooperation is much less likely. Mutual cooperation is most likely when the (sufficient) payments are identical, and it is also substantially more likely with payment pairs that bring the mutual-cooperation payoffs closer together. Both the Fehr–Schmidt and Charness–Rabin models predict that transfers that make final payoffs closer are preferred; however, they do not explain why equal transfers are particularly effective. Transfers are also effective in sustaining cooperation even when they are imposed and not chosen.     

Strategic step-by-step negotiation

We analyze a specific type of negotiation process where parties proceed in stages taking into consideration that negotiation may end prematurely with an inefficient agreement. Parties negotiate only one increasing pie, thereby avoiding inefficiencies which are typical for issue-by-issue negotiations. For ann-stage game, we prove the existence of a unique subgame-perfect equilibrium. We then show that step-by-step negotiation can only improve players' expected payoffs if negotiation in stages reduces the difference between their equilibrium offers. For this to occur, however, the risk of negotiation must be affected by the agenda of the bargaining problem.     

Alternating-offer games with final-offer arbitration

I analyze an alternating-offer model that integrates the common practice of having an arbitrator determine the outcomes if both playersʼ offers are rejected. I assume that the arbitrator uses final-offer arbitration (as in professional baseball). I find that if the arbitrator does not excessively favor one player, then the unique subgame-perfect equilibrium always coincides with the subgame-perfect equilibrium outcome in Rubinsteinʼs infinite-horizon alternating-offer game. However, if the arbitrator sufficiently favors the player making the initial offer, then delay occurs in equilibrium.     

Coalitional Bargaining with Competition to Make Offers

The paper analyzes a simple discrete-time noncooperative coalitional bargaining game in which, at each stage, there is a contest for the right to make a proposal. The pure stationary subgame-perfect equilibrium payoff set is equal to the core.Journal of Economic LiteratureClassification Numbers: 026.     

Extensive-form games and strategic complementarities

I prove the subgame-perfect equivalent of the basic result for Nash equilibria in normal-form games of strategic complements: the set of subgame-perfect equilibria is a nonempty, complete lattice—in particular, subgame-perfect Nash equilibria exist. For this purpose I introduce a device that allows the study of the set of subgame-perfect equilibria as the set of fixed points of a correspondence. My results are limited because extensive-form games of strategic complementarities turn out—surprisingly—to be a very restrictive class of games.     

A Model of Behavior in Coordination Game Experiments

This paper constructs a structural model for behavior in expeiments where subjects play a simple coordination game repeatedly under a rotating partner scheme. The model assumes subjects' actions are stochastic best responses to beliefs about opponents' choices, and these beliefs update as subjects observe actual choices during the experiment. The model accounts for heterogeneity across subjects by regarding prior beliefs as random effects and estimating their distribution. Maximum likelihood estimates from experimental data suggest that distributions of initial beliefs vary across games, but in all games studied imply a convergence dynamic toward risk-dominant equilibrium.     

Strategic market games with a finite horizon and incomplete markets

Summary. We study a strategic market game associated to an intertemporal economy with a finite horizon and incomplete markets. We demonstrate that generically, for any finite number of players, every sequentially strictly individually rational and default-free stream of allocations can be approximated by a full subgame-perfect equilibrium. As a consequence, imperfect competition may Pareto-dominate perfect competition when markets are incomplete. Moreover - and this contrasts with the main message conveyed by the market games literature - there exists a large open set of initial endowments for which full subgame-perfect equilibria do not converge to -efficient allocations when the number of players tends to infinity. Finally, strategic speculative bubbles may survive at full subgame-perfect equilibria.Received: 24 January 2002, Revised: 21 February 2003, JEL Classification Numbers: C72, D43, D52. Correspondence to: Gaël GiraudWe thank Tim Van Zandt for his comments.     

Multistage R&D competition and patent policy

In this paper a model of multistage R&D patent policy is investigated. We study the impact of the duration of patent protection for intermediate products on R&D races when the discovery of the final product requires the accomplishment of an intermediate step. Using a multistage model where firms choose their levels of research investment at each stage, we find all subgame-perfect equilibria of the game. We also determine how competition affects a firm's level of research investment at different stages of the R&D competition.