Cores,aggressiveness and the breakdown of cooperation in economic games

The paper develops the use of the core as a solution concept in game theory in two interrelated directions. In the first place, an indicator of aggressiveness of claims is introduced in a modified definition of the core. The modified core may be smaller than the usual core, and may fail to exist if aggressiveness increases beyond some critical level. In the second place the article gives a formulation of a mixed cooperative/non-cooperative game, in which the game will be played cooperatively within coalitions, but non-cooperatively as between coalitions. A mixed cooperative/non-cooperative solution obtains if the grand coalition of all players fail to materialize because the various claims are incompatible. The two directions referred to are interrelated because the level of aggressiveness may be decisive for whether or not the grand coalition, and possibly other coalitions, will break down. The final section of the paper draws some general conclusions and relates the approach to other ideas in the literature.     

Non-cooperative solutions for estate division problems

In an estate division problem an estate has to be divided among several players whose total entitlement to the estate exceeds its size. This paper extends the non-cooperative approach through a claim game, as initiated by O?Neill (1982), by allowing players to put multiple claims on the same part of the estate, and by considering the case where individual entitlements may exceed the estate. A full characterization of the set of Nash equilibria of the claim game is obtained both for restricted estate division problems, where individual entitlements do not exceed the estate, and for the general case. Variations on the claim game are considered, which result in proportional division in equilibrium.     

Bilateral commitment

We consider non-cooperative environments in which two players have the power to gradually and unilaterally rule out some of their actions. Formally, we embed a strategic-form game into a multi-stage game, in which players can restrict their action spaces in all but the final stage, and select among the remaining actions in the last stage. We say that an action profile is implementable by commitment if this action profile is played in the last stage of a subgame-perfect equilibrium path. We provide a complete characterization of all implementable action profiles and a simple method to find them. It turns out that the set of implementable profiles does not depend on the length of the commitment process. We show, furthermore, that commitments can have social value in the sense that in some games there are implementable action profiles that dominate all Nash equilibria of the original game.     

The Effects of Issue Salience on Political Decision-making

Game-theoretic models of political decision-making regard policy outcomes as a combined result of actor preferences and institutions. This paper explores the effects of adding relative issue salience to the analysis. I focus on non-cooperative models of complete and perfect information that are based on the concept of structure-induced equilibrium (Shepsle in Am J Polit Sci 23(1):27–59, 1979). Assuming a progressive agenda-setter and two conservative voters, I show how a change of the relative issue salience for players in a game results in a different equilibrium outcome in a political decision-making body.     

Cooperative vs non-cooperative truels: little agreement, but does that matter?

It is well-known that non-cooperative and cooperative game theory may yield different solutions to games. These differences are particularly dramatic in the case of truels, or three-person duels, in which the players may fire sequentially or simultaneously, over one round or n rounds. Representative solution concepts (Nash and subgame-perfect equilibrium; two notions of core) are compared, and little agreement is found among them. Although it might be desirable to subsume these different solutions within a common framework, such unification seems unlikely since they are grounded in fundamentally different notions of stability.     

Coalitional bargaining games with random proposers: Theory and application

We consider a non-cooperative coalitional bargaining game with random proposers in a general situation for which players differ in recognition probability and time preference. We characterize an efficient equilibrium as the generalized Nash bargaining solution that belongs to the core. The model is applied to wage bargaining between an employer and multiple workers. Although involuntary unemployment may occur in equilibrium, full employment emerges as players become sufficiently patient.     

Coordination and the Provision of Discrete Public Goods by Correlated Equilibria

The strategic analysis of the private provision of a discrete public good has shown the existence of multiple Nash equilibria with the efficient number of players voluntarily contributing. However the coordination issue is left unexplained by this literature. The experimental evidence shows that communication among players is helpful in achieving cooperation. We claim that, from the theoretical point of view, this is equivalent to playing correlated equilibria in an extended public good game with communication, modeled as Chicken. We characterize such equilibria as feasible coordination mechanisms to achieve public goods provision in the general contribution game. We further introduce a second kind of game characterized by payoff externalities that may persist after the minimal threshold of contributors is achieved. While it is easy to show the existence of Pareto efficient correlated equilibria in the first game, in the second one players face incentive problems such that a first best cannot always be an equilibrium. Nevertheless there exist correlated equilibria that can qualify as incentive efficient mechanisms, once free riding is seen as a moral hazard issue. Finally, with an example, we discuss the impact of coalition formation in our framework.     

Contract renegotiation in a continuous state space

Summary. When players are unable to write complete state contingent contracts it is shown, within the context of a non-cooperative contracting-renegotiation game, that the only subgame perfect equilibrium allocations are those that correspond to the set of first-best allocations. Players are able to implement this set of allocations by signing an initial contract that is subsequently renegotiated in all states of the world. The contracting-renegotiation problem is complicated in an interesting way by assuming that the state space is continuous. The issue of the existence of an initial contract, that is subsequently renegotiated to the set of first-best allocations, must be resolved. Unlike Aghion, Dewatripont and Rey [1994], the results here do not require nor depend upon the comonotonicity of the objective functions. Received: January 27, 1995; revised version July 1, 1996     

Games of Status

A status game is a cooperative game in which the outcomes are rank orderings of the players. They are a good model for certain situations in which players care about how their "status" compares with that of other players.
We present several formal models within this class. Included are authoritarian status games (where coalitions may assign positions in the rank ordering to nonmembers) and oligarchic status games (where they are unableto do so). We consider the issues of a value concept for authoritarian games and that of core existence for oligarchic games. We then add a transferable resource to the models, obtaining "games of wealth and status."
Finally, we consider an interesting variant, called a "secession game," where coalitions have the right to secede from the grand coalition and form their own smaller "subsocieties," each with its own hierarchy.     

Climate Change Mitigation with Technology Spillovers

We explore the implications of an increase in clean technology spillovers between developed and developing countries. We build a game of abatements in which players are linked with technology spillovers determined by an initial choice of absorptive capacities by developing countries. We show that, within a non-cooperative framework, the response of clean technology investments in developed countries to an increase in cross-country technology spillovers is ambiguous. If the marginal benefits of these additional abatements are not sufficiently high, developed countries have a strategic incentive to decrease investments. Such a strategic response jeopardizes the initial effects of an increase in technology spillovers on climate change mitigation and decreases the incentives for developing countries to enhance their absorptive capacities.     

Raising juveniles

This paper investigates how families decide how juveniles use their time. The problem is analyzed in three variations: (i) a ‘decentralized’ scheme, in which parents control the main budget, but their children dispose of their time as they see fit, together with any earnings from work on their own account; (ii) ‘hierarchy’, in which parents can enforce, at some cost, particular levels of schooling and supervised work contributing to the main budget; and (iii) the cooperative solution, in which resources are pooled and the threat point is one of the non-cooperative outcomes. Adults choose which game is played. While the subgame perfect equilibrium of the overall game is pareto-efficient, it may yield less education than ‘hierarchy’. Restrictions on child labor and compulsory schooling generally affect both the threat point and the feasible set of bargaining outcomes. Families may choose more schooling than the legal minimum.     

The Shapley Value of a Patent Licensing Game: the Asymptotic Equivalence to Non-cooperative Results

We consider a game in characteristic form played by firms and an outside patent holder of a cost-reducing innovation. The worth of a coalition of players is the total Cournot profit the coalition can guarantee to obtain when it operates an optimal number of its firms while the complement operates any number of its firms as to minimize the profit of the coalition. Only firms in a coalition with the patent holder are allowed to use the efficient technology. We prove that when the number of firms is large, the Shapley value of the patent holder approximates the payoff he obtains in the non-cooperative auction game traditionally studied in the literature.We thank an anonymous referee for very helpful comments that significantly improved the paper. The second author is being partially supported by the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Grant-in-Aid for 21 Century COE Program. He wishes to thank his advisor Yair Tauman and co-advisor Pradeep Dubey for their intellectual guidance, Akira Okada and Haruo Imai for their encouragement, and Shigeo Muto for his helpful comments to the first draft of this paper at the autumn meeting of the Japanese Economic Association in 2003.     

Weak links, good shots and other public good games: Building on BBV

We suggest an alternative way of analyzing the canonical Bergstrom-Blume-Varian model of non-cooperative voluntary contributions to a public good that avoids the proliferation of dimensions as the number of players is increased. We exploit this approach to analyze models in which the aggregate level of public good is determined as a more general social composition function of individual gifts — specifically, as a CES form — rather than as an unweighted sum. We also analyze Hirshleifer's weakest-link and best-shot models. In each case, we characterize the set of equilibria, in some cases establishing existence of a unique equilibrium as well as briefly pointing out some interesting comparative static properties. We also study the weakest-link and best-shot limits of the CES composition function and show how the former can be used for equilibrium selection and the latter to establish that equilibria of some better-shot games are identical to those of the much simpler best-shot game.     

Pareto optimality of non-cooperative equilibrium in a time-dependent multi-period game

This paper deals with the conditions under which cooperation could be advantageous to the various players in a linear-quadratic dynamic game. The conclusions, which are arrived at by comparing the non-cooperative Nash equilibrium with the set of Pareto optimal strategies, point to the presence of strong incentives for cooperation. The analysis is applicable to problems of interdependent decision-making in international or interregional economics, thereby stressing the importance of suitable coordination schemes.     

A learning-based model of repeated games with incomplete information

This paper tests a learning-based model of strategic teaching in repeated games with incomplete information. The repeated game has a long-run player whose type is unknown to a group of short-run players. The proposed model assumes a fraction of ‘short-run’ players follow a one-parameter learning model (self-tuning EWA). In addition, some ‘long-run’ players are myopic while others are sophisticated and rationally anticipate how short-run players adjust their actions over time and “teach” the short-run players to maximize their long-run payoffs. All players optimize noisily. The proposed model nests an agent-based quantal-response equilibrium (AQRE) and the standard equilibrium models as special cases. Using data from 28 experimental sessions of trust and entry repeated games, including 8 previously unpublished sessions, the model fits substantially better than chance and much better than standard equilibrium models. Estimates show that most of the long-run players are sophisticated, and short-run players become more sophisticated with experience.     

A Logic for Suspicious Players: Epistemic Actions and Belief–Updates in Games

This paper introduces a notion of 'epistemic action' to describe changes in the information states of the players in a game. For this, ideas are developed from earlier contributions. The ideas are enriched to cover not just purely epistemic actions, but also fact–changing actions ('real moves', e.g. choosing a card, exchanging cards etc.) and nondeterministic actions and strategies (conditional actions having knowledge tests as conditions). The author considers natural operations with epistemic actions and uses them to describe significant aspects of the interaction between beliefs and actions in a game. A logic is used that combines in a specific way a multiagent epistemic logic with a dynamic logic of 'epistemic actions'. The author presents a complete and decidable proof system for this logic. As an application, the author analyses a specific example of a dialogue game (a version of the Muddy Children Puzzle, in which some of the children can 'cheat' by engaging in secret communication moves, while others may be punished for their credulity). Also presented is a sketch of a 'rule–based' approach to games with imperfect information (allowing 'sneaky' possibilities, such as cheating, being deceived and suspecting the others to be cheating).     

Fiscal and monetary interactions when wage-setters are large: is there a role for corporatist policies?

This paper studies the macroeconomic consequences of alternative policy regimes in a closed economy where a central bank, a fiscal authority and a monopoly union interact via their effects on output and inflation. The analysis compares macroeconomic outcomes in a non-cooperative setting, where players may move sequentially or simultaneously, and in a regime of cooperation between the government and wage-setters. The cooperative regime captures a climate of accord among social parties that is finalised at common macroeconomic targets in the tradition of corporatism, as in the recent experience of “social pacts” in many European countries. The paper makes two main contributions. First, it shows that macroeconomic outcomes are suboptimal in the non-cooperative regime and may deliver extreme (undesirable) results even when all players share common ideal targets for output and inflation. All players would be better off with a less extreme value for output or inflation, yet they fail to reach a more advantageous allocation as long as there is an inherent conflict among their further objectives. Moreover, the result is robust to a change in the degree of central bank’s conservatism. Second, I find that cooperation between the government and the monopoly union towards common ideal targets for inflation, output and taxes enhances social welfare even in the absence of explicit coordination with the central bank.     

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 (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.