Finitely repeated games with social preferences

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 $$\delta _A+\delta _B < 1$$ .

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.

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.

Union–firm bargaining: Order of play and efficiency

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.

Uncertainty in Spatial Duopoly with Possibly Asymmetric Distributions: A State Space Approach

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.

Bertrand competition and Cournot outcomes: further results

Bilateral bargaining in networks

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.

Subgame Perfect Equilibria in Stage Games

Author index

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References and Further Reading