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Complex Systems in Finance and Econometrics pp 549–567Cite as

Market Games and Clubs

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Article Outline

Glossary

Definition of the Subject

Introduction

Transferable Utility Games; Some Standard Definitions

A Market

Market-Game Equivalence

Equivalence of Markets and Games with Many Players

Cores and Approximate Cores

Nonemptiness and Convergence of Approximate Cores of Large Games

Shapley Values of Games with Many Players

Economies with Clubs

With a Continuum of Players

Other Related Concepts and Results

Some Remarks on Markets and More General Classes of Economies

Conclusions and Future Directions

Bibliography

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Notes

  1. 1.

    It is well known and easily demonstrated that the uniform ε-core of a TU game is nonempty if and only if it contains an equal treatment payoff vector. This follows from the fact that the uniform ε-core is a convex set.

  2. 2.

    Exactly the same definition applies to situations with a compact metric space of player types, c.f. Wooders [84,88].

  3. 3.

    Other approaches to economies with clubs/local public goods include Casella and Feinstein [15], Demange [23], Haimanko, O., M. Le Breton and S. Weber [28], and Konishi, Le Breton and Weber [37]. Recent research has treated clubs as networks.

  4. 4.

    See, for example Moulin [47,48] for excellent discussions of these two problems.

Abbreviations

Game:

A (cooperative) game (in characteristic form) is defined simply as a finite set of players and a function or correspondence ascribing a worth (a non-negative real number, interpreted as an idealized money) to each nonempty subset of players, called a group or coalition.

Payoff vector:

A payoff vector is a vector listing a payoff (an amount of utility or money) for each player in the game.

Core:

The core of a game is the set (possibly empty) of feasible outcomes – divisions of the worths arising from coalition formation among the players of the game – that cannot be improved upon by any coalition of players. core

Totally balanced game:

A game is totally balanced if the game and every subgame of the game (a game with player set taken as some subset of players of the initially given game) has a nonempty core.

Market:

A market is defined as a private goods economy in which all participants have utility functions that are linear in (at least) one commodity (money).

Shapley value:

The Shapley value of a game is feasible outcome of a game in which all players are assigned their expected marginal contribution to a coalition when all orders of coalition formation are equally likely.

Pregame:

A pair, consisting of a set of player types (attributes or characteristics) and a function mapping finite lists of characteristics (repetitions allowed) into the real numbers. In interpretation, the pregame function ascribes a worth to every possible finite group of players, where the worth of a group depends on the numbers of players with each characteristic in the group. A pregame is used to generate games with arbitrary numbers of players.

Small group effectiveness:

A pregame satisfies small group effectiveness if almost all gains to collective activities can be realized by cooperation only within arbitrarily small groups (coalitions) of players.

Per capita boundedness:

A pregame satisfies per capita boundedness if the supremum of the average worth of any possible group of players (the per capita payoff) is finite.

Asymptotic negligibility:

A pregame satisfies asymptotic negligibility if vanishingly small groups can have only negligible effects on per capita payoffs.

Market games:

A market game is a game derived from a market. Given a market and a group of agents we can determine the total utility (measured in money) that the group can achieve using only the endowments belonging to the group members, thus determining a game.

Club:

A club is a group of agents or players that forms for the purpose of carrying out come activity, such as providing a local public good.

An economy:

We use the term ‘economy’ to describe any economic setting, including economies with clubs, where the worth of club members may depend on the characteristics of members of the club, economies with pure public goods, local public goods (public goods subject to crowding and/or congestion), economies with production where what can be produced and the costs of production may depend on the characteristics of the individuals involved in production, and so on. A large economy has many participants.

Price taking equilibrium:

A price taking equilibrium for a market is a set of prices, one for each commodity, and an allocation of commodities to agents so that each agent can afford his part of the allocation, given the value of his endowment.

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  1. Department of Economics, Vanderbilt University, Nashville, USA

    Myrna Wooders

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Wooders, M. (2009). Market Games and Clubs. In: Meyers, R. (eds) Complex Systems in Finance and Econometrics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-7701-4_31

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