Payoff dominance and risk dominance in the observable delay game: a note

We examine whether the payoff dominant sequential-move (Stackelberg) outcome is realized when timing is endogenized. We adopt the observable delay game formulated by Hamilton and Slutsky [Games Econ Behav 2(1):29–46, 1990]. We find that if one sequential-move outcome is payoff dominant, either (i) the outcome both players prefer is the unique equilibrium; or (ii) two sequential-move outcomes are equilibria and the one both players prefer is risk dominant. In other words, no conflict between payoff dominance and risk dominance in the observable delay game exists, in contrast to other games such as (non pure) coordination games. We also find that even if one of two sequential-move outcomes is the unique equilibrium outcome in the observable delay game, it does not imply that the equilibrium outcome is payoff dominant to the other sequential-move outcome.      

Efficiency in coalition games with externalities

A natural extension of superadditivity is not sufficient to imply that the grand coalition is efficient when externalities are present. We provide a condition, analogous to convexity, that is sufficient for the grand coalition to be efficient and show that this also implies that the (appropriately defined) core is nonempty. Moreover, we propose a mechanism which implements the most efficient partition for all coalition formation games and characterizes the resulting payoff division.     

Efficiency in coalition games with externalities

A natural extension of superadditivity is not sufficient to imply that the grand coalition is efficient when externalities are present. We provide a condition, analogous to convexity, that is sufficient for the grand coalition to be efficient and show that this also implies that the (appropriately defined) core is nonempty. Moreover, we propose a mechanism which implements the most efficient partition for all coalition formation games and characterizes the resulting payoff division.     

On the Coalition-Proofness of the Pareto Frontier of the Set of Nash Equilibria

This article shows that the Pareto efficient frontier of the Nash equilibrium set of games with strategic substitutes is coalition-proof under the following conditions: (1) the game has three players, or, alternatively, a player's payoff depends on her own strategy and on the sum (but not on the composition) of other players' strategies; (2) an increase in a player's strategy either raises all other players' payoffs monotonically or reduces them monotonically; and (3) each player's payoff is strictly concave in her own strategy. Under these conditions, the Pareto dominance refinement is equivalent to the coalition-proof Nash equilibrium refinement.Journal of Economic LiteratureClassification Number: C72.     

Innovative leadership: First-mover advantages in new product adoption

Summary. This paper analyzes innovation adoption when uncertainty about its profitability cannot be resolved immediately. Firms begin with a common estimate of the probability of high demand. If any adopts, all observe realized demand. An increase in the initial estimate can decrease the equilibrium number of initial adopters, because it results in higher updated estimates that can induce future adoption by additional firms that reduces the initial adoption payoff. Moreover, innovative leadership does not imply initial adoption because leadership implies a greater waiting payoff as well as a greater adoption payoff. Leadership does, however, still provide a higher expected payoff. Received: July 16, 2001; revised version: January 13, 2002     

The Partnered Core of a Game without Side Payments

A payoff for a game is partnered if it admits no asymmetric dependencies. We introduce the partnered core of a game without side payments and show that the partnered core of a balanced game is nonempty. The result is a strengthening of Scarf's Theorem on the nonemptiness of the core of a balanced game without side payments. In addition, it is shown that if there are at most a countable number of points in the partnered core of a game then at least one core point isminimallypartnered, meaning that no player requires any other player in particular to obtain his part of the core payoff.Journal of Economic LiteratureClassification Number: C71.     

On the existence of pure-strategy perfect equilibrium in discontinuous games

We provide sufficient conditions for a (possibly) discontinuous normal-form game to possess a pure-strategy trembling-hand perfect equilibrium. We first show that compactness, continuity, and quasiconcavity of a game are too weak to warrant the existence of a pure-strategy perfect equilibrium. We then identify two classes of games for which the existence of a pure-strategy perfect equilibrium can be established: (1) the class of compact, metric, concave games satisfying upper semicontinuity of the sum of payoffs and a strengthening of payoff security; and (2) the class of compact, metric games satisfying upper semicontinuity of the sum of payoffs, strengthenings of payoff security and quasiconcavity, and a notion of local concavity and boundedness of payoff differences on certain subdomains of a player's payoff function. Various economic games illustrate our results.     

Correlated Equilibrium in Stochastic Games

We study the existence of uniform correlated equilibrium payoffs in stochastic games. The correlation devices that we use are either autonomous (they base their choice of signal on previous signals, but not on previous states or actions) or stationary (their choice is independent of any data and is drawn according to the same probability distribution at every stage). We prove that any n-player stochastic game admits an autonomous correlated equilibrium payoff. When the game is positive and recursive, a stationary correlated equilibrium payoff exists. Journal of Economic Literature Classification Numbers: C72, C73.     

Endogenous timing in the presence of non-monotonicities

The aim of this paper is to generalize the endogenous timing game proposed by Hamilton and Slutsky (Games and Economic Behavior, 1990, 2, pp. 29–46) by allowing the payoff or the marginal payoff of a player to become non-monotonic with respect to the strategy of the opponent. We propose a taxonomy of the subgame-perfect Nash equilibria based on the characteristics of the payoff functions proposed by Eaton (Canadian Journal of Economics, 2004, 37, pp. 805–29). We determine under which conditions of the initial payoff functions commitment has a social value and when the simultaneous-move Nash equilibrium is commitment robust and discuss its Pareto efficiency.     

There are many evolutionary pathways to cooperation

Current sociobiological discussions attribute the evolution of cooperation to only two main influences: kinship and reciprocity. As a baseline, the paper analyzes the extent of incidental cooperation achieved in three important 2 × 2 payoff environments (Prisoners' Dilemma, Chicken, and Tender Trap) and the two simplest 'rules of the game' or protocols of play (single-round simultaneous-move and single-round sequential move). Kinship promotes cooperation beyond these base levels by modifying payoffs of selfish versus unselfish behaviors. Reciprocity may also promote cooperation, but its expression requires protocols that widen available strategy sets (in comparison with the basic strategies in the underlying 2 × 2 payoff matrices). Once payoff modifications and/or more elaborate protocols are allowed, many other pathways to cooperation are opened up. Among them are punishment options, complementary strategy mixes, recognition effects, coordination using external clues, and group selection.     

The least core, kernel and bargaining sets of large games

Summary. We study the least core, the kernel and bargaining sets of coalitional games with a countable set of players. We show that the least core of a continuous superadditive game with a countable set of players is a non-empty (norm-compact) subset of the space of all countably additive measures. Then we show that in such games the intersection of the prekernel and the least core is non-empty. Finally, we show that the Aumann-Maschler and the Mas-Colell bargaining sets contain the set of all countably additive payoff measures in the prekernel. Received: June 6, 1996; revised version: March 1, 1997     

Payoffs in Nondifferentiable Perfectly Competitive TU Economies

We show that a single-valued solution of nonatomic finite-type market games (or perfectly competitive TU economies underling them) is uniquely determined as the Mertens value by four plausible value-related axioms. Since the Mertens value is always in the core of an economy, this result provides an axiomatization of a core-selection (or, alternatively, a competitive payoff selection). Journal of Economic Literature Classification Numbers: C71, D51, D61.     

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.     

The law of supply in games, markets and matching models

Summary In large games with transferable utility, core payoffs satisfy a comparative statics property: If the proportion of one type of player increases, then the core payoff to that type of player decreases (does not increase). Markets with transferable utility satisfy a similar property: if the aggregate supply of a commodity increases, its value relative to the value of all commodities decreases. In market games, if one type of agent becomes more plentiful, his competitive payoff falls, and its decrease is engineered by a decrease in the relative value of his endowment.We thank Bob Anderson, Joe Farrell, Steve Goldman, Chris Shannon, seminar participants of the Mathematical Economics Seminar at Berkeley (August 1994), the University of Pittsburg (November 1994), Tel Aviv University and the Institute on Rationality at the Hebrew University (January 1995), and especially Vince Crawford for useful discussion.     

An existence result for discontinuous games

We introduce a notion of upper semicontinuity, weak upper semicontinuity, and show that it, together with a weak form of payoff security, is enough to guarantee the existence of Nash equilibria in compact, quasiconcave normal form games. We show that our result generalizes the pure strategy existence theorem of Dasgupta and Maskin [P. Dasgupta, E. Maskin, The existence of equilibrium in discontinuous economic games, I: Theory, Rev. Econ. Stud. 53 (1986) 1-26] and that it is neither implied nor does it imply the existence theorems of Baye, Tian, and Zhou [M. Baye, G. Tian, J. Zhou, Characterizations of the existence of equilibria in games with discontinuous and non-quasiconcave payoffs, Rev. Econ. Stud. 60 (1993) 935-948] and Reny [P. Reny, On the existence of pure and mixed strategy equilibria in discontinuous games, Econometrica 67 (1999) 1029-1056]. Furthermore, we show that an equilibrium may fail to exist when, while maintaining weak payoff security, weak upper semicontinuity is weakened to reciprocal upper semicontinuity.     

Social norms and trust among strangers

We study the development of a social norm of trust and reciprocity among a group of strangers via the “contagious strategy” as defined in Kandori (1992). Over an infinite horizon, the players anonymously and randomly meet each other and play a binary trust game. In order to provide the investors with proper incentives to follow the contagious strategy, there is a sufficient condition that requires that there exists an outside option for the investors. Moreover, the investorsʼ payoff from the outside option must converge to the payoff from trust and reciprocity as the group size goes to infinity. We show that this sufficient condition is also a necessary condition to sustain any sequential equilibrium in which the trustees adopt the contagious strategy. Our results imply that a contagious equilibrium only supports trust if trust contributes almost nothing to the investorsʼ payoffs.     

The social cost problem,rights, and the (non)empty core

We revisit the “Coase theorem” through the lens of a cooperative game model which takes into account the assignment of rights among agents involved in a problem of social cost. We consider the case where one polluter interacts with many potential victims. Given an assignment or a mapping of rights, we represent a social cost problem by a cooperative game. A solution consists in a payoff vector. We introduce three properties for a mapping of rights. First, core compatibility indicates that the core of the associated cooperative games is nonempty. Second, Kaldor‐Hicks core compatibility indicates that there is a payoff vector in the core where victims are fully compensated for the damage once the negotiations are completed. Third, no veto power for a victim says that no victim has the power to veto an agreement signed by the rest of the society. We then demonstrate two main results. First, core compatibility is satisfied if and only if the rights are assigned either to the polluter or to the entire set of victims. Second, there is no mapping of rights satisfying Kaldor‐Hicks core compatibility and no veto power for a victim.     

The equilibrium of the bargaining game and core convergence theorem on an exchange economy with limited traders

Debreu and Scarf (1963), Hildenbrand (1974), Aumann (1964), Dierker (1975), Bewley (1973), and others have shown that the core of an exchange economy with infinitely many or finitely many traders converges. However, an exchange economy does not always consist of infinitely many or finitely many traders. This note provides proof of the core convergence theorem on an exchange economy with limited traders by a bargaining game methodology. The main contribution of this note is to innovate the equilibrium solution to the bargaining game in the exchange economy. In this note, the concept of common payoff is introduced; in the bargaining game of a coalition on its common payoff, all coalition members will get the same distribution, thus the distribution scheme of the cooperation surplus of the exchange economy is determined. This note shows that the bargaining game among the traders on the distribution of the cooperation surplus will make the pure exchange economy with limited traders converge to the Walrasian equilibrium, all the allocations other than the Walrasian equilibrium will be eliminated from the core of this economy.     

Connecting the cooperative and competitive structures of the multiple-partners assignment game

Multiple-partners assignment game is the name used by Sotomayor [The multiple partners game, in: M. Majumdar (Ed.), Equilibrium and Dynamics: Essays in Honor of David Gale, The Macmillan Press Ltd., New York, 1992; The lattice structure of the set of stable outcomes of the multiple partners assignment game, Int. J. Game Theory 28 (1999) 567-583] to describe the cooperative structure of the many-to-many matching market with additively separable utilities. Stability concept is proved to be different from the core concept. An economic structure is proposed where the concept of competitive equilibrium payoff is introduced in connection to the equilibrium concept from standard microeconomic theory. The paper examines how this equilibrium concept compares with the cooperative equilibrium concept. Properties of interest to the cooperative and competitive markets are derived.     

Residual Claims in Co‐operatives: Design Issues

This paper examines issues in the design of a co‐operative member's contractual relationship with the other agents (including the remaining members) using organizational economics. The paper assumes that the central defining characteristic of a co‐op is the residual claim specification. Agency theory identifies certain inherent problems of the co‐op form, the horizon problem, common property problem, and non‐transferability. Non‐transferability both reduces the incentive to monitor and imposes limits on portfolio diversification. This paper argues that features such as claim incompleteness and non‐transferability are not inherent to the co‐op but may be transaction‐cost economizing. The paper also argues that the pre‐emptive payoff feature by which the residual claimants (the co‐op members) also become fixed payoff agents can affect the risk of other agents, and is an important determinant of co‐op risk. A co‐op may have more than one potential residual claim base. Five generic design choices are available for handling possible multiple claim bases: battleground, pre‐specified allocation, limited return, alignment, and fixed payoff. The paper uses the design of residual claims in sugar co‐ops to show how a co‐op can partly overcome some of the problems identified by agency theory. This illustration ties together the issues of claim incompleteness and non‐transferability, pre‐emptive payoff, and multiple claim bases.