Alpha- and beta-characteristic functions in cooperative non-sidepayment games: An experimental test

This article investigates theories of payoff allocation in cooperative non-sidepayment games, that is, games where players can form coalitions and coordinate strategies but cannot redistribute payoffs via sidepayments. A labotatory experiment was conducted with 4-person, 3-strategy non-sidepayment games. The goal of this experiment was to ascertain which characteristic function, V_α(S) or V_β(S), provides the more accurate basis for payoff predictions from solution concepts. Predictions from each of three solution concepts (core, stable set, imputation set) were computed on the basis of both V_α(S) and V_β(S), making a total of six theories under test. The results showed two basic findings. First, for each of the three solutions, the prediction computed on the basis of V_α(S) was more accurate than the prediction computed from V_α(S). Second, the β-core was the most accurate of the six theories tested. Overall, these results support the view that V_β(S) is superior to V_α(S) as a basis for payoff predictions in cooperative non-sidepayment games.     

The cournot duopoly model reconsidered in the presence of costly communication and enforcement agreements

Recently I proposed a general notion of equilibrium for n-person games which is based on mutual threats and promises (responses), and which leads, in general, to a determinate solution with respect to the order of choice moves. The nature of the solution set in our model is a function of the action sets, the payoff functions, and the cost of ‘communication and enforcement of agreements’ (C&E). The main concern of this paper is to show that the Cournot duopoly solution, for the case of a linear demand function facing two identical firms, can be regenerated by our solution concept under a certain assumption about a high cost C&E. On the other hand, it is shown that under a relatively low cost of C&E the Cournot/Nash solution does not belong to our solution set, because it is Pareto inferior for both duopolists relative to some solutions in our set, among them, the standard monopoly solution.     

The core as the set of eventually stable outcomes: A note

As a justification of the core as a set of stable social states, Sengupta and Sengupta [1996. A property of the core. Games Econ. Behav. 12, 266–273] show that for any transferable utility (TU) cooperative game with non-empty core, for every imputation outside the core there is an element in the core that indirectly dominates the imputation in a desirable way. In this note we show that this appealing property of the core no longer holds even for the class of hyperplane games, an immediate generalization of TU games into the environments without side payments.     

The Nash bargaining solution in general n-person cooperative games

We present a noncooperative foundation for the Nash bargaining solution for an n-person cooperative game in strategic form. The Nash bargaining solution should be immune to any coalitional deviations. Our noncooperative approach yields a new core concept, called the Nash core, for a cooperative game based on a consistency principle. We prove that the Nash bargaining solution can be supported (in every subgame) by a stationary subgame perfect equilibrium of the bargaining game if and only if the Nash bargaining solution belongs to the Nash core.     

An admissible set occurring in various bargaining situations

Given a set of alternatives S and a binary relation M on S the admissible set of the pair (S, M) is defined to be the set of maximal elements with respect to the transitive closure of M. It is shown that existing solutions in game theory and mathematical economics are special cases of this concept (they are admissible sets of a natural S and M). These include the core of an n-person cooperative game, Nash equilibria of a noncooperative game, and the max-min solution of a two-person zero sum game. The competitive equilibrium prices of a finite exchange economy are always contained in its admissible set. Special general properties of the admissible set are discussed. These include existence, stability, and a stochastic dynamic process which leads to outcomes in the admissible set with high probability.     

Competitive outcomes and the inner core of NTU market games

We consider the inner core as a solution concept for cooperative games with non-transferable utility (NTU) and its relationship to payoffs of competitive equilibria of markets that are induced by NTU games. An NTU game is an NTU market game if there exists a market such that the set of utility allocations a coalition can achieve in the market coincides with the set of utility allocations the coalition can achieve in the game. In this paper, we introduce a new construction of a market based on a closed subset of the inner core which satisfies a strict positive separability. We show that the constructed market represents the NTU game and, further, has the given closed set as the set of payoff vectors of competitive equilibria. It turns out that this market is not uniquely determined, and thus, we obtain a class of markets. Our results generalize those relating to competitive outcomes of NTU market games in the literature.     

Renegotiation in repeated games

In repeated games, subgame-perfect equilibria involving threats of punishment may be implausible if punishing one player hurts the other(s). If players can renegotiate after a defection, such a punishment may not be carried out. We explore a solution concept that recognizes this fact, and show that in many games the prospect of renegotiation strictly limits the cooperative outcomes that can be sustained. We characterize those outcomes in general, and in the prisoner's dilemma, Cournot and Bertrand duopolies, and an advertising game in particular.     

The supercore for normal-form games

This paper analyzes the supercore of a system derived from a normal-form game. For the case of a finite game with pure strategies, we define a sequence of games and show that the supercore coincides with the set of Nash equilibria of the last game in that sequence. This result is illustrated with the characterization of the supercore for the n-person prisoner's dilemma. With regard to the mixed extension of a normal-form game, we show that the set of Nash equilibrium profiles coincides with the supercore for games with a finite number of Nash equilibria.     

Subjective evaluations of n-person games

Using an axiomatization of subjective expected utility due to Fishburn, we characterize a class of utility functions over a set of n-person games in characteristic-function form. A probabilistic value is defined as the expectation of some player's marginal contribution with respect to some probability measure on the set of coalitions of other players. We decribe conditions under which a utility function on the set of n-person games is a probabilistic value; we prove as well an analogous result for simple games. We present additional axioms that characterize the semivalues and, in turn, the Shapley and Banzhaf values.     

Stackelberg-solvable games and pre-play communication

The concept of Stackelberg-solvable games is introduced and analyzed as a generalization of 2-person zero-sum games. Then, the problem of sincere pre-play communication is examined and the incentive compatibility of the Nash-equilibrium selection is shown to correspond to the Stackelberg-equilibrium property.     

The number of commodities required to represent a market game

An n-person game with side payments that can arise from an economic market is called a market game. It is proved here that any n-person market game can be obtained from a market with at most n ? 1 commodities. Moreover, no smaller number will suffice in some cases (e.g., the unanimity game). This result is then applied to settle a question regarding market games with a continuum of traders.     

Competition and collusion in bilateral markets

In this paper, the core of a market game which constitutes the set of equilibria in the process of competitive contracting and recontracting is criticized as a solution concept for not being immune against theory absorption in the sense that knowledge of the core on part of the traders may result in a collusive stabilization of some dominated imputation. It is pointed out that a stable set (or, von Neumann-Morgenstern) solution does not suffer from this deficiency. Moreover, it is argued that stable set solutions provide an adequate analytical framework for the study of collusion, and are in this respect superior to the approach (relying on the core concept) chosen by Aumann in his work on disadvantageous monopolies. For symmetric bilateral market games — generated by markets involving the exchange of only two commodities, one of which also serves as a means of side payment and utility transfer, among two types of traders — with one seller and one up to three buyers all symmetric solutions are determined. Furthermore, a symmetric solution for markets with equal, but otherwise arbitrary, numbers of sellers and buyers is given. The symmetric stable sets of imputations are interpreted as rational standards of behavior providing the consistent and defensible rules of division necessary to make a cartel agreement viable.This research was partially supported by a grant given to New York University, Department of Economics, by the Office of Naval Research. (# N00014-76-C-0033). A first version of the paper was presented at the Third World Congress of the Eeconomtric Society, Toronto, Aug. 20–26, 1975, under the title Symmetric Solutions of Bilateral Market Games.     

Stable games and their dynamics

We study a class of population games called stable games. These games are characterized by self-defeating externalities: when agents revise their strategies, the improvements in the payoffs of strategies to which revising agents are switching are always exceeded by the improvements in the payoffs of strategies which revising agents are abandoning. We prove that the set of Nash equilibria of a stable game is globally asymptotically stable under a wide range of evolutionary dynamics. Convergence results for stable games are not as general as those for potential games: in addition to monotonicity of the dynamics, integrability of the agents' revision protocols plays a key role.     

On core stability,vital coalitions,and extendability

If a TU game is extendable, then its core is a stable set. However, there are many TU games with a stable core that are not extendable. A coalition is vital if there exists some core element x such that none of the proper subcoalitions is effective for x. It is exact if it is effective for some core element. If all coalitions that are vital and exact are extendable, then the game has a stable core. It is shown that the contrary is also valid for matching games, for simple flow games, and for minimum coloring games.     

On Forward Induction and Evolutionary and Strategic Stability

We analyze which normal form solution concepts capture the notion of forward induction, as defined by E. van Damme (1989, J. Econ. Theory48, 476-496) in the class of generic two player normal form games preceded by an outside option. We find that none of the known strategic stability concepts (including Mertens stable sets and hyperstable sets) capture this form of forward induction. On the other hand, we show that the evolutionary concept of EES set (J. Swinkels, 1992, J. Econ. Theory57, 306-332) is always consistent with forward induction. Journal of Economic Literature Classification Number: C72.     

Stability of the cooperative equilibrium in N-person prisoners' dilemma with sequential moves

Summary. Nishihara [3] showed that N-person prisoners' dilemma has a cooperative Nash equilibrium, if the players decide their actions sequentially in the order determined by Nature under a certain information structure, and if each player's payoffs satisfy a certain inequality. This paper examines the stability of this cooperative equilibrium against two matters: players' slight mistakes and deviations by coalitions. The main results are as follows: (i) if the inequality on each player's payoffs strictly holds, then the cooperative equilibrium is a strictly proper equilibrium; (ii) if N≤3, and if full cooperation is Pareto efficient in N-person prisoners' dilemma, then the cooperative equilibrium is a strong Nash equilibrium; (iii) the cooperative equilibrium is in general a coalition-proof Nash equilibrium. Received: June 23, 1997; revised version: December 2, 1997     

Coalition-Proof Nash Equilibria II. Applications

In Bernheim, Peleg, and Whinston (“Coalition-Proof Equilibria. I. Concepts,” J. Econ. Theory 42 (1987), 1–12), we proposed the notion of Coalition-Proof Nash equilibrium and Perfectly Coalition-Proof Nash equilibrium as solution concepts for strategic environments in which players can freely discuss their strategies, but cannot make binding commitments. This paper undertakes applications to several economic problems, including the behavior of Cournot oligopolists, oligopolistic entry deterrence, cooperation in finite horizon games, and social choice rule implementation.     

Better-reply dynamics and global convergence to Nash equilibrium in aggregative games

We consider n-person games with quasi-concave payoffs that depend on a player's own action and the sum of all players' actions. We show that a discrete-time, stochastic process in which players move towards better replies—the better-reply dynamics—converges globally to a Nash equilibrium if actions are either strategic substitutes or strategic complements for all players around each Nash equilibrium that is asymptotically stable under a deterministic, adjusted best-reply dynamics. We present an example of a 2-person game with a unique equilibrium where the derivatives of the best-reply functions have different signs and the better-reply dynamics does not converge.     

A weak bargaining set for contract choice problems

In this paper, we consider the problem of choosing a set of multi-party contracts, where each coalition of agents has a non-empty finite set of feasible contracts to choose from. We call such problems, contract choice problems. The main result in this paper states that every contract choice problem has a non-empty weak bargaining set. The need for such a solution concept which is considerably weaker than the core arises, since it is well-known that even for very simple contract choice problems, the core may be empty. We also show by means of an example that an analog of the bargaining set due to Mas-Colell [Mas-Colell, A., 1989. An equivalence theorem for a bargaining set. Journal of Mathematical Economics 18, 129–139], as well as the natural analog of the bargaining set due to Aumann and Mashler [Aumann, R., Maschler, M., 1964. The bargaining set for cooperative games. In: Dresher, M., Shapley, L., Tucker, A. (Eds.), Advances in Game Theory. Princeton University Press, Princeton, NJ] may be empty for contract choice problems.     

Equilibrium Selection inn-Person Coordination Games

This paper investigates several approaches to equilibrium selection and the relationships between them. The class of games we study aren-person generalized coordination games with multiple Pareto rankable strict Nash equilibria. The main result is that all selection criteria select the same outcome (namely the risk dominant equilibrium) in two-person games, and that most equivalences break for games with more than two players. All criteria select the Pareto efficient equilibrium in voting games, of which pure coordination games are special cases.Journal of Economic LiteratureClassification Numbers: C70, C72, D82.