Substitute goods, auctions, and equilibrium

This paper identifies two notions of substitutes for auction and equilibrium analysis. Weak substitutes, the usual price-theory notion, guarantees monotonicity of tâtonnement processes and convergence of clock auctions to a pseudo-equilibrium, but only strong substitutes, which treats each unit traded as a distinct good with its own price, guarantees that every pseudo-equilibrium is a Walrasian equilibrium, that the Vickrey outcome is in the core, and that the “law of aggregate demand” is satisfied. When goods are divisible, weak substitutes along with concavity guarantees all of the above properties, except for the law of aggregate demand.     

Approximate tâtonnement processes

Summary We introduce a probabilistic model for price adjustment in an exchange economy which approximates the classical Walras tâtonnement process while avoiding many of its unrealistic features. The model is decentralized in that the trades permitted to an agent and the resulting price changes depend only on the commodity vector currently held by that agent, and not on the commodity vectors held by the other agents in the economy. Our results will show that the Walras tâtonnement process can be decentralized without changing its behavior on the macroeconomic scale. Our model has a finite set of commodities, a market maker who adjusts prices, and a large finite set of agents who trade only with the market maker. Each agent has a demand function depending on his commodity vector and the price vector. At each discrete time, one agent is chosen at random and exchanges his current commodity vector for his demand vector. Then the market maker adjusts the price vector by an amount which depends on the selected agent's commodity vector and the current price. Prices are adjusted rapidly enough to avoid prolonged trading at the wrong price, but slowly enough so that a substantial price change will depend on a significant simple of agents. The main result shows that with probability arbitrarily close to one the price will rapidly approach and then remain close to an equilibrium value, following a path which is close to the price path of the corresponding tâtonnement process.     

A three way equivalence

In view of the well known core equivalence results in atomless economies, coincidence of market game equilibrium allocations with competitive allocations is tantamount to a three way equivalence between market game mechanisms, competitive equilibria and the core. Based on this idea we study equilibrium refinements of market games, which allow us to use the core equivalence machinery in order to provide an exact market game characterization of competitive equilibria.     

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.     

Cores of combined games

This paper studies the core of combined games, obtained by summing different coalitional games when bargaining over multiple independent issues. It is shown that the set of balanced transferable utility games can be partitioned into equivalence classes of component games to determine whether the core of the combined game coincides with the sum of the cores of its components.     

ANTI‐DUAL OF ECONOMIC COALITIONAL TU GAMES*

Some well‐known coalitional TU (transferable utility) games applied to specific economic problems are shown to be connected through the relation defined as the anti‐dual. Solutions such as the core, the Shapley value and the nucleolus of anti‐dual games are obtained straightforwardly from original games.     

Competitive behavior in market games: Evidence and theory

We explore whether competitive outcomes arise in an experimental implementation of a market game, introduced by Shubik (1973) [21]. Market games obtain Pareto inferior (strict) Nash equilibria, in which some or possibly all markets are closed. We find that subjects do not coordinate on autarkic Nash equilibria, but favor more efficient Nash equilibria in which all markets are open. As the number of subjects participating in the market game increases, the Nash equilibrium they achieve approximates the associated competitive equilibrium of the underlying economy. Motivated by these findings, we provide a theoretical argument for why evolutionary forces can lead to competitive outcomes in market games.     

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.     

The type-agent core for exchange economies with asymmetric information

The type-agent core is a new solution concept for exchange economies with asymmetric information. It coincides with the set of subgame-perfect equilibrium outcomes of a simple competitive screening game. Uninformed intermediaries help the agents to cooperate in an attempt to make some profit. The paper extends the work of Perez-Castrillo [Cooperative outcomes through non-cooperative games, Games Econ. Behav. 7 (1994) 428-440] to exchange economies with non-transferable utility and asymmetric information. The type-agent core is a subset of Wilson's coarse core [Wilson, Information, efficiency, and the core of an economy, Econometrica 46 (1978) 807-816]. It is never empty, even though it may be a strict subset of Wilson's fine core. In addition, it converges towards the set of constrained market equilibria as the economy is replicated.     

A necessary and sufficient condition for non-emptiness of the core of a non-transferable utility game

It is well-known that a transferable utility game has a non-empty core if and only if it is balanced. In the class of non-transferable utility games balancedness or the more general π-balancedness due to Billera (SIAM J. Appl. Math. 18 (1970) 567) is a sufficient, but not a necessary condition for the core to be non-empty. This paper gives a natural extension of the π-balancedness condition that is both necessary and sufficient for non-emptiness of the core.     

Generalized Tâtonnement and the Solution of Economic Models*

The theoretical literature on tâtonnement processes leads to the conclusion that convergence is guaranteed only urrder fairly restrictive assumptions. Recently, however, large competitive equilibrium models have been successfully solved using algorithms inspired by tâtonnement processes .
This paper stresses the idea that a model builder has more freedom to set up the process than a theoretician, for whom only one assignment of prices to excess demand is reasonable; for computational purposes, any assignment is acceptable. The diagonal dominance criterion provides guidelines to set up such an assignment. It is used to show that even intractable examples of global instability can easily be solved .     

Market allocation through search: Equilibrium adjustment and price dispersion

This essay discusses market allocation under uncertainty in a market for a homogeneous good. Market participants contact one another at random times to buy or sell single units of the good. Transactions are carried out at different prices simultaneously. Optimal search rules are employed to describe individuals behavior. Such models provide the framework of a market structure within which equilibrium adjustment processes are analyzed. These decentralized, nontâtonnement processes of price and quantity adjustment are utilized to examine equilibrium. Price dispersion, which characterizes equilibrium, is essentially due to finite lifetimes of market participants.     

Markets for contracts: experiments exploring the compatibility of games and markets for games

Summmary. The research explores the relationship between games and the economic environment in which the games might be embedded. The focus is on a market institution in which agents buy and sell rights to participate in a follow-on stage of strategic interaction. The central question posed concerns how two different types of processes, the game and the market, interact. The market converges to a competitive equilibrium that is consistent with the Nash equilibrium that obtains in the game, and the convergence of the market to a competitive equilibrium lags the convergence of behaviors in the game to a Nash equilibrium.     

Learning behavior and mixed-strategy Nash equilibria

The paper considers whether an adaptive justification, like those commonly available for non-interactive optimization models, can be found for the mixed-strategy Nash equilibrium. Although it is known that such a justification is frequently available for pure-strategy equilibria, it is shown that all members of a wide class of behaviorally plausible learning mechanisms must fail to converge in ‘almost all’ games in which the equilibrium involves mixed strategies. An exact formal analogy is developed, which allows these learning mechanisms to be viewed as discrete ta?tonnement processes in properly chosen competitive exchange economies. In the analogy, the instability derives from the independence of excess demand functions (except at the boundaries) from ‘own’ prices. This independence arises because of the linearity in probabilities of von Neumann-Morgenstern risk preferences, and therefore does not extend to pure-strategy equilibria in general. Thus, assuming that agents will play equilibrium strategies implicitly assumes more sophistication when the equilibrium involves mixed strategies than when it involves only pure strategies.     

Average Monotonic Cooperative Games

A subclass of monotonic transferable utility (T.U.) games is studied: average monotonic games. These games are totally balanced. We prove that the core coincides with both the bargaining set à la Davis and Maschler and the bargaining set à la Mas-Colell. To obtain this result a technique based on reduced games is used. Journal of Economic Literature Classification Number: C71     

Local coordination and market equilibria

We reformulate the local stability analysis of market equilibria in a competitive market as a local coordination problem in a market game, where the map associating market prices to best-responses of all traders is common knowledge and well-defined both in and out of equilibrium. Initial expectations over market variables differ from their equilibrium values and are not common knowledge. This results in a coordination problem as traders use the structure of the market game to converge back to equilibrium. We analyse a simultaneous move and a sequential move version of the market game and explore the link with local rationalizability.     

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.     

EQUILIBRIUM AND THE CORE IN ALONSO'S DISCRETE POPULATION MODEL OF LAND USE*

Conventional wisdom tells us that with no market failure and local nonsatiation of preferences, the core is at least as large as the collection of competitive equilibrium allocations. We confirm this for a standard model featuring private ownership of land. Next we consider the public land ownership version of the model. If the role of land ownership and rent distribution is assumed by a government that ploughs back rent (at least in excess of its agricultural value) to its citizens, the equilibrium allocation remains efficient, but no longer need be in the core.     

Mixed equilibria are unstable in games of strategic complements

In games with strict strategic complementarities, properly mixed Nash equilibria—equilibria that are not in pure strategies—are unstable for a broad class of learning dynamics.     

Risky arbitrage,asset prices,and externalities

We introduce a no-risky-arbitrage price condition (NRAP) for asset market models allowing both unbounded short sales and externalities such as trading volume. We then demonstrate that NRAP is sufficient for the existence of competitive equilibrium in the presence of externalities. Moreover, we show that if all risky arbitrages are utility increasing, then NRAP characterizes competitive equilibrium in the presence of externalities. We are indebted to an anonymous referee for helpful comments on an earlier version of this paper. Page and Wooders are especially grateful to CERMSEM and EUREQua for their support and hospitality which made possible our collaboration.