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     

On the Least Core and the Mas-Colell Bargaining Set

We show that the least core of a TU coalitional game with a finite set of players is contained in the Mas-Colell bargaining set. This result is extended to games with a measurable space of players in which the worth of the grand coalition is at least that of any other coalition in the game. As a consequence, we obtain an existence theorem for the Mas-Colell bargaining set in TU games with a measurable space of players. Journal of Economic Literature Classification Number: C71.     

von Neumann–Morgenstern stable sets of income tax rates in public good economies

This paper studies the von Neumann–Morgenstern stable sets in a public good economy, where any allocation is achieved through a proportional income tax system. The outcome of the economy is defined to be a set of tax rates. We show that whenever the core as the set of tax rates is nonempty, it is a unique stable set. When the core is empty, the set of tax rates is a stable set if and only if the set is a singleton consisting of a tax rate satisfying a certain condition. We show the existence of such stable sets. The author is grateful to Mikio Nakayama, Jun Wako and an anonymous referee for their helpful comments and suggestions. He is also grateful to Youngsub Chun, Takuya Masuzawa and Shigeo Muto for their helpful comments.     

The bargaining set of a large game

We study the equivalence between the Mas-Colell bargaining set and the core in the general context of TU games with a measurable space of players. In the first part of the paper, we study the problem without imposing any restriction on the class of games we consider. In the second part, we first introduce a new class of exact games, which we call thin games. For these games, we show not only that the Mas-Colell bargaining set is equal to the core, but also that it is the unique stable set in the sense of von Neumann and Morgenstern. We then study the relation between thin games, exact non-atomic market games and non-atomic convex games. Finally, by further developing “thinness” related ideas, we prove new equivalence results for a class of non-exact market games as well as a class of non-exact, non-market games.     

Strategic approximations of discontinuous games

An infinite game is approximated by restricting the players to finite subsets of their pure strategy spaces. A strategic approximationof an infinite game is a countable subset of pure strategies with the property that limits of all equilibria of all sequences of approximating games whose finite strategy sets eventually include each member of the countable set must be equilibria of the infinite game. We provide conditions under which infinite games admit strategic approximations.     

Farsightedly stable networks

A set of networks G is pairwise farsightedly stable (i) if all possible farsighted pairwise deviations from any network g G to a network outside G are deterred by the threat of ending worse off or equally well off, (ii) if there exists a farsighted improving path from any network outside the set leading to some network in the set, and (iii) if there is no proper subset of G satisfying conditions (i) and (ii). A non-empty pairwise farsightedly stable set always exists. We provide a full characterization of unique pairwise farsightedly stable sets of networks. Contrary to other pairwise concepts, pairwise farsighted stability yields a Pareto dominant network, if it exists, as the unique outcome. Finally, we study the relationship between pairwise farsighted stability and other concepts such as the largest pairwise consistent set and the von Neumann–Morgenstern pairwise farsightedly stable set.     

Voluntary Contributions to Multiple Public Projects

The problem of financing a set of discrete public goods (facilities, projects) by private contributions is studied. The corresponding cooperative game, the realization game , is shown to be convex. For the noncooperative setting we study a realization scheme that induces a strategic game. This contribution game is shown to be a generalized ordinal potential game ; a best–response in the contribution game implies a best response in a coordination game in which the payoff to all players is the utilitarian collective welfare function, i.e., the sum of the utility functions of the players. Strategy profiles maximizing utilitarian welfare are strong Nash equilibria of the contribution game. Each strong Nash equilibrium corresponds in a natural way with a core element of the realization game, and vice versa. Moreover, each strong Nash equilibrium is coalitional proof.     

Equilibrium for Overlapping Generations in Continuous Time

We prove the existence of general equilibrium for continuous-time overlapping-generations models. Previous theorems exclude all non-linear C.E.S. and von Neumann–Morgenstern preferences and exclude production. Our primitive assumptions are satisfied by such preferences and by all Markovian production technologies satisfying Bewley's assumptions for Arrow–Debreu models provided that storage is possible, at some finite rate of depreciation and some positive capacity. A non-existence example shows our Markovian and storage assumptions cannot be dropped.Journal of Economic LiteratureClassification Numbers: C62, D51, D90.     

Cooperative games in permutational structure

Summary. By a cooperative game in coalitional structure or shortly coalitional game we mean the standard cooperative non-transferable utility game described by a set of payoffs for each coalition being a nonempty subset of the grand coalition of all players. It is well-known that balancedness is a sufficient condition for the nonemptiness of the core of such a cooperative non-transferable utility game. In this paper we consider non-transferable utility games in which for any coalition the set of payoffs depends on a permutation or ordering upon any partition of the coalition into subcoalitions. We call such a game a cooperative game in permutational structure or shortly permutational game. Doing so we extend the scope of the standard cooperative game theory in dealing with economic or political problems. Next we define the concept of core for such games. By introducing balancedness for ordered partitions of coalitions, we prove the nonemptiness of the core of a balanced non-transferable utility permutational game. Moreover we show that the core of a permutational game coincides with the core of an induced game in coalitional structure, but that balancedness of the permutational game need not imply balancedness of the corresponding coalitional game. This leads to a weakening of the conditions for the existence of a nonempty core of a game in coalitional structure, induced by a game in permutational structure. Furthermore, we refine the concept of core for the class of permutational games. We call this refinement the balanced-core of the game and show that the balanced-core of a balanced permutational game is a nonempty subset of the core. The proof of the nonemptiness of the core of a permutational game is based on a new intersection theorem on the unit simplex, which generalizes the well-known intersection theorem of Shapley. Received: October 31, 1995; revised version: February 5, 1997     

Combinatorial Games under Auction Play

ARichman gameis a combinatorial game in which, rather than alternating moves, the two players bid for the privilege of making the next move. The theory of such games is a hybrid between the classical theory of games (von Neumann, Morgenstern, …) and the combinatorial theory of games (Berlekamp, Conway, Guy, …). We expand upon our previous work by considering games with infinitely many positions as well as several variants including thePoorman variantin which the high bidder pays a third party (rather than the other player).Journal of Economic LiteratureClassification Number: C7.     

Pillage and property

This paper introduces a class of coalitional games, called pillage games, as a model of Hobbesian anarchy. Any coalition can pillage, costlessly and with certainty, any less powerful coalition. Power is endogenous, so a pillage game does not have a characteristic function, but pillage provides a domination concept that defines a stable set, which represents an endogenous balance of power. Every stable set contains only finitely many allocations, and can be represented as a farsighted core. Additional results are obtained for particular games, including the game in which the power of each coalition is determined by its total wealth.     

A Property of the Core

For any transferable utility game in coalitional form with nonempty core we show that, given any allocation outside the core, there is an allocation in the core that indirectly dominates it.Journal of Economic LiteratureClassification Number: C71.     

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.     

Aggregation under complete markets

This paper proves a complete market aggregation result for a multiple good economy in which the consumers have time-additive, von Neumann–Morgenstern utility functions. This result applies to all concave intraperiod functions, and provides a contrast to many of the results in the microeconomics literature which depend on such stringent assumptions as linear Engel curves.     

Farsightedness in a Coalitional Great Fish War

We explore the implications of the farsightedness assumption on the conjectures of players in a coalitional Great Fish War model with symmetric players, derived from the seminal model of Levhari and Mirman (Bell J Econ 11:649–661, 1980). The farsightedness assumption for players in a coalitional game acknowledges the fact that a deviation from a single player will lead to the formation of a new coalition structure as the result of possibly successive moves of her rivals in order to improve their payoffs. It departs from mainstream game theory in that it relies on the so-called rational conjectures, as opposed to the traditional Nash conjectures formed by players on the behavior of their rivals. For values of the biological parameter and the discount factor more plausible than the ones used in the current literature, the farsightedness assumption predicts a wide scope for cooperation in non-trivial coalitions, sustained by credible threats of successive deviations that defeat the shortsighted payoff of any prospective deviator. Compliance or deterrence of deviations may also be addressed by acknowledging that information on the fish stock or on the catch policies actually implemented may be available only with a delay (dynamic farsightedness). In that case, the requirements are stronger and the sizes and number of possible farsighted stable coalitions are different. In the sequential move version, which could mimic some characteristics of fishery models, the results are not less appealing, even if the dominant player or dominant coalition with first move advantage assumption provides a case for cooperation with the traditional Nash conjectures.     

Coalitions, agreements and efficiency

If agents negotiate openly and form coalitions, can they reach efficient agreements? We address this issue within a class of coalition formation games with externalities where agents’ preferences depend solely on the coalition structure they are associated with. We derive Ray and Vohra's [Equilibrium binding agreements, J. Econ. Theory 73 (1997) 30-78] notion of equilibrium binding agreements using von Neumann and Morgenstern [Theory of Games and Economic Behavior, Princeton University Press, Princeton, 1944] abstract stable set and then extend it to allow for arbitrary coalitional deviations (as opposed to nested deviations assumed originally). We show that, while the extended notion facilitates the attainment of efficient agreements, inefficient agreements can nevertheless arise, even if utility transfers are possible.     

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.     

Stability and Largeness of the Core

In general, there are examples of TU games where the core is stable but is not large. In this paper, we show that the extendability condition introduced by Kikuta and Shapley (1986, “Core Stability in n-Person Games,” Mimeo) is sufficient for the core to be stable as well as large, for TU games with five or fewer players. We provide a counter example when the number of players is six. We then introduce a stronger extendability condition and show that it is necessary and sufficient for the core to be large. Our proof makes use of a well-known result from the theory of convex sets. Journal of Economic Literature Classification Number: C71.     

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.     

Cooperative Outcomes through Noncooperative Games

This paper shows the equivalence between the stable solution set of any cooperative game in characteristic form (G1) and the subgame perfect Nash equilibria in pure strategies of a certain noncooperative game (G2). Players of G1 are named "agents." G2 is played by different players ("principals") who compete in wages to attract agents. The equivalence result holds when there are enough principals (if the game is superadditive, two principals suffice). Finally, another related cooperative game (G3) is constructed with both principals and agents as players. For G2 and G3 the same result is then proven, for any number of principals. Journal of Economic Literature Classification Numbers: C71 and C72.