Subgame perfect equilibrium in a bargaining model with deterministic procedures

A well-known result from the theory of finitely repeated games states that if the stage game has a unique equilibrium, then there is a unique subgame perfect equilibrium in the finitely repeated game in which the equilibrium of the stage game is being played in every period. Here I show that this result does in general not hold anymore if players have social preferences of the form frequently assumed in the recent literature, for example in the inequity aversion models of Fehr and Schmidt (Quartely Journal of Economics 114:817–868, 1999) or Bolton and Ockenfels (American Economic Review 100:166–193, 2000). In fact, repeating the unique stage game equilibrium may not be a subgame perfect equilibrium at all. This finding should have relevance for all experiments with repeated interaction, whether with fixed, random or perfect stranger matching.

A non--<:ooperative model of coalition bargaining in the context of characteristic functions with side-payments is introduced. Temporary commitments to payoff demands are a salient feature of the model. For every coalition game, the model generates a finite extensive game with perfect information. For two basic cases - unanimity games and 3-person quota games without the grand coalition - the non--<:ooperative analysis yields results akin to cooperative game theory. The analysis is based on subgame perfectness alone without the use of further selection criteria. The equal payoff division is the only subgame perfect equilibrium outcome in the case of the zero-normalized unanimity game. In the case of the 3-person quota game without the grand coalition, the players receive their quotas in 2-person coalitions, if a subgame perfect equilibrium is played; each of the three 2-person coalitions occurs with probability 1/3.

This paper describes the results of an experiment applying the strategy method to analyze the behavior of subjects in an 8-player common pool resource (CPR) game. The CPR game consists of a constituent game played for 20 periods. The CPR game has a unique optimum and a unique subgame perfect equilibrium; the latter involves overinvestment in the appropriation from the CPR. Sixteen students, all experienced in game theory, were recruited to play the CPR game over the course of 6 weeks. In the first phase of the experiment, they played the CPR game on-line 3 times. In the second phase of the experiment, the tournament phase, they designed strategies which were then played against each other. At the aggregate level, subgame perfect equilibrium organizes the data fairly well. At the individual level, however, fewer than 5% of subjects play in accordance with the game equilibrium prediction.

We study a game of complete information with multiple principals and multiple common agents. Each agent takes an action that can affect the payoffs of all principals. Prat and Rustichini (Econometrica, 2003) who introduce this model assume first price contracts: each principal offers monetary transfers to each agent conditional on the action taken by the agent. We define a notion of VCG contracts which are a restricted natural class of contractible contracts and study its effect on the existence of efficient pure subgame perfect equilibrium outcomes. We identify a "unitary balancedness'' condition that is necessary and sufficient for the existence of a pure subgame perfect equilibrium (SPE) with VCG contracts. As a consequence, we show that the class of instances of this game that admit an efficient SPE with VCG contracts strictly contains the class of instances of this game that admit an efficient SPE with first price contracts. Although VCG contracts broaden the existence of pure subgame perfect equilibria, we show that the worst case welfare loss in any SPE outcome with VCG contracts is not worse than the respective worst case loss with first price contracts.

We study Hotelling's two-stage model of spatial competition, in which two firms first simultaneously choose locations in the unit interval, then simultaneously choose prices. Under Hotelling's assumptions (uniform distribution of consumers, travel cost proportional to distance, inelastic demand of one unit by each consumer) the price-setting subgames possess equilibria in pure strategies for only a limited set of location pairs. Because of this problem (pointed out independently by Vickrey (1964) and d'Aspremont et al. (1979)), Hotelling's claim that there is an equilibrium of the two-stage game in which the firms locate close to each other is incorrect. A result of Dasgupta and Maskin (1986) guarantees that each price-setting subgame has an equilibrium in mixed strategies. We first study these mixed strategy equilibria. We are unable to provide a complete characterization of them, although we show that for a subset of location pairs all equilibria are of a certain type. We reduce the problem of finding an equilibrium of this type to that of solving three or fewer highly nonlinear equations. At each of a large number of location pairs we have computed approximate solutions to the system of equations. Next, we use our analytical results and computations to study the equilibrium location choices of the firms. There is a unique (up to symmetry) subgame perfect equilibrium in which the location choices of the firms are pure; in it, the firms locate 0.27 from the ends of the market. At this equilibrium, the support of the subgame equilibrium price strategy is the union of two short intervals. Most of the probability weight is in the upper interval, so that this strategy is reminiscent of occasional sales by the firms. We also find a subgame perfect equilibrium in which each firm uses a mixed strategy in locations. In fact, in the class of strategy pairs in which the firms use the same mixed strategy over locations, and this strategy is symmetric about 0.5, there is a single equilibrium. In this equilibrium most of the probability weight of the common strategy is between 0.2 and 0.4, and between 0.6 and 0.8. There is a wide range of pure Nash (as opposed to subgame perfect) equilibrium location pairs: the subgame strategies in which each firm threatens to charge a price of zero in response to a deviation support all but those location pairs in which the firms are very close.

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.

Union–firm bargaining: Order of play and efficiency

Extreme Equilibria in a General Negotiation Model

The paper is a kind of generalization of Rubinstein bargaining model. Rubinstein assumed that preferences of the players were constant in time, and he analyzed models in which preferences of each player were defined either by constant discount rate or by constant bargaining cost. In this paper, a bargaining model is presented, in which preferences of each player are expressed simultaneously by sequence of discount rates and sequence of bargaining costs varying in time. The results presented in the paper concern subgame perfect equilibria. There is a theorem concerning sufficient and necessary conditions for the existence of subgame perfect equilibrium of the game. Moreover, some theorems presenting forms of subgame perfect equilibria for various cases of the model analyzed have been proved here. A possibility of delay in reaching an agreement is also considered in the paper. If we analyze a class of strategies, that depend on the former history, a delay can appear for some models. The adequate examples are presented. In the paper, some applications of the bargaining model are also described.

We study the one-seller/two-buyer bargaining problem with negative identity-dependent externalities with an alternating offer bargaining model in which new owners of the object have the opportunity of resale. We identify the generically unique subgame perfect equilibrium outcome. The resale opportunity increases the competition among the buyers and therefore benefits the seller. When competition between buyers is very fierce, the seller may prefer to respond to bids rather than to propose an offer herself: a first-mover disadvantage.

Solution uniqueness is an important property for a bargaining model. Rubinstein's (1982) seminal 2-person alternating-offer bargaining game has a unique Subgame Perfect Equilibrium outcome. Is it possible to obtain uniqueness results in the much enlarged setting of multilateral bargaining with a characteristic function? This paper investigates a random-proposer model first studied in Okada (1993) in which each period players have equal probabilities of being selected to make a proposal and bargaining ends after one coalition forms. Focusing on transferable utility environments and Stationary Subgame Perfect Equilibria (SSPE), we find ex ante SSPE payoff uniqueness for symmetric and convex characteristic functions, considerably expanding the conditions under which this model is known to exhibit SSPE payoff uniqueness. Our model includes as a special case a variant of the legislative bargaining model in Baron and Ferejohn (1989), and our results imply (unrestricted) SSPE payoff uniqueness in this case.

This article considers the regulation of nonpoint source water pollution through the use of a voluntary‐threat approach, which allows a polluter group to meet a standard voluntarily but imposes an ambient tax upon noncompliance. In particular, we develop an “endogenous” voluntary‐threat approach that can be parameterized to induce compliance as a unique subgame perfect Nash equilibrium and eliminate zero‐abatement equilibria. Experimental evidence on the proposed approach as well as the “exogenous” approach of Segerson and Wu (2006) suggests that the severity of the threatened tax and the opportunity for communication are critical in determining policy outcomes.

In this paper, we study an overlay RF-powered cognitive radio network with ambient backscatter communications. In the network, when the channel is occupied, the secondary transmitter (ST) can perform either energy harvesting or data transmission using ambient backscattering technique to a gateway. We consider the case that the gateway charges the ST a certain price if the ST transmits information. This leads to questions of how to determine the best price for the gateway and how to find the optimal backscatter time. To address this problem, we propose a Stackelberg game in which the gateway is the leader adapting the price to maximize its profit in the first stage. Meanwhile, the ST chooses its backscatter time to maximize its utility in the second stage. To analyze the game, we apply the backward induction technique. We show that the game always has a unique subgame perfect Nash equilibrium. Additionally, our results provide insights on the impact of the competition on the players' profit and utility.

Ogawa (Int Tax Public Financ 20(3):474–484, 2013) discusses the analysis of Kempf and Rota-Graziosi (J Public Econ 94(9–10):768–776, 2010a) by taking into account capital ownership in the government’s objective functions. He establishes that the unique subgame perfect Nash equilibrium (SPNE) of the endogenous timing game corresponds to the simultaneous Nash equilibrium. This result contrasts with Kempf and Rota-Graziosi (J Public Econ 94(9–10):768–776, 2010a) who conclude to the existence of two Stackelberg outcomes at the SPNEs. Highlighting the role of plain complementarity or substitutability, we obtain conditions under which leadership still emerges at the equilibrium of the endogenous timing game when capital ownership is considered. Numerical simulations confirm this finding and display a case where plain properties are not monotone and a well-identified Stackelberg outcome is the SPNE.

The alternating offers game due to Rubinstein (1982) had been used by Binmore (1980) and by Binmore et.al. (1986) to provide via its unique subgame perfect equilibrium an approximate non-cooperative support for the Nash bargaining solution of associated cooperative two-person bargaining games. These results had strengthened the prominent role of the Nash bargaining solution in cooperative axiomatic bargaining theory and its application, for instance in labor markets, and have often even be interpreted as a mechanism theoretical implementation of the Nash solution. Our results in the present paper provide exact non-cooperative foundations first, in our Proposition, via weakly subgame perfect equilibria of a game that is a modification of Rubinstein´s game, then in our Theorem, via sub-game perfect equilibria of a game that is a further modification of our first game. Moreover, they provide a general rule how to transform approximate support results into exact ones. Finally, we discuss the relation of the above mentioned support results, including our present ones, with mechanism theoretic implementation in (weakly) subgame perfect equilibrium of the Nash solution. There we come to the conclusion that a sound interpretation as an implementation can hardly be found except in very rare cases of extremely restricted domains of players´ preferences.