Uncertainty and insurance in strategic market games

Summary. For perfectly competitive economies under uncertainty, there is a well-known equivalence between a formulation with contingent goods and one with state-specific securities followed by spot markets for goods. In this paper, I examine whether this equivalence carries over to a particular form of imperfect competition. Specifically, I look at three Shapley-Shubik strategic market games: one with contingent commodities, one with Arrow securities traded under imperfect competition and one with Arrow securities traded under perfect competition. First I compare the feasibility constraints of these three games. Then I compare their equilibrium sets. As in Peck and Shell (1989), the only common equilibria between the first and the second game are those which involve no transfer of income across states. However, if the securities markets are competitive, then the set of equilibria of the contingent commodities game and the securities game coincide. Received: June 16, 1997; revised version: April 30, 1998     

The bargaining set in strategic market games

This paper presents a hybrid equilibrium notion that blends together the ‘cooperative’ and the ‘noncooperative’ theories of competition. In particular, the Mas-Colell bargaining set has been modified in order to accommodate the features of strategic market games. In other words, allocations, objections and counter objections of the standard bargaining set theory are described for an economy, where trades among groups of individuals are conducted via the Shapley–Shubik mechanism. In the main part of the paper, it is proved that in atomless economies the allocations resulting from this equilibrium notion are competitive.     

Comparative statics in non-cooperative games via transfinitely iterated play

Tarski's fixed point theorem establishes the existence of a Nash equilibrium when (a) each player's lattice of strategies is complete and (b) the composite best response function is isotone. Suppose the composite best response function is also isotone in one of the underlying parameters. Then our transfinite variant of iterated best response play is shown to “converge” and to produce an unambiguous comparative statics analysis. The efficacy of this approach is demonstrated with three examples.     

Core structure and comparative statics in a hybrid matching market

Flexible firms compete by means of wages in the Assignment market while rigid firms have no flexibility over terms of appointment in the Marriage market. Workers trade with both kinds of firms in the hybrid market.Examples show that standard results that characterize the core of the Marriage market (respectively, Assignment market) are not robust to the entrance of flexible (respectively, rigid) firms to this market. A new algebraic structure provides a different characterization for the core of the hybrid model and reflects a sort of robustness to the exit of rigid (respectively, flexible) firms from this market. Meaningful comparative static results are derived.     

“Very Nice” trivial equilibria in strategic market games

Following Shapley [Theory of Measurement of Economic Externalities, Academic Press, New York, 1976], we study the problem of the existence of a Nash Equilibrium (NE) in which each trading post is either active or “legitimately” inactive, and we call it a Shapley NE. We consider an example of an exchange economy, borrowed from Cordella and Gabszewicz [Games Econ. Behav. 22 (1998) 162–169], which satisfies the assumptions of Dubey and Shubik [J. Econ. Theory 17 (1978) 1–20], and we show that the trivial equilibrium, the unique NE of the associated strategic market game, is not “very nice,” in the sense that it is not “legitimately” trivial. This result has the more general implication that, under the Dubey and Shubik's assumptions, a Shapley NE may fail to exist.     

Retrading in market games

When agents are not price takers, they typically cannot obtain an efficient real location of resources in one round of trade. This paper presents a non-cooperative model of imperfect competition where agents can retrade allocations, consistent with Edgeworth's idea of recontracting. We show (a) there are Pareto optimal allocations, including competitive equilibrium allocations, that can be approximated arbitrarily closely when trade is myopic, i.e., when agents play a static Nash equilibrium at every round of retrading; (b) any converging sequence of allocations generated by myopic retrading can be supported along some retrade-proof subgame perfect equilibrium path when traders anticipate future rounds of trading.     

Comparative statics of monopoly pricing

Summary. When consumers' willingness-to-pay increases by a uniform amount, the change in the resulting monopoly price is generally indeterminate. Our analysis identifies sufficient conditions on the underlying demand curve which predict both the sign and the magnitude of the resulting price change.     

On eliciting beliefs in strategic games

Several recent studies in experimental economics have tried to measure beliefs of subjects engaged in strategic games with other subjects. Using data from one such study we conduct an experiment where our experienced subjects observe early rounds of strategy choices from that study and are given monetary incentives to report forecasts of choices in later rounds. We elicit beliefs using three different scoring rules: linear, logarithmic, and quadratic. We compare forecasts across the scoring rules and compare the forecasts of our trained observers to forecasts of the actual players in the original experiment. We find significant differences across scoring rules. The improper linear scoring rule produces forecasts closer to 0 and 1 than the proper rules, and these forecasts are poorly calibrated. The two proper scoring rules induce significantly different distributions of forecasts. We find that forecasts by observers under both proper scoring rules are significantly different from the forecasts of the actual players, in terms of accuracy, calibration, and the distribution of forecasts. We also find evidence for belief convergence among the observers.     

On strategic stability in discontinuous games

We identify a class of discontinuous normal-form games whose members possess strategically stable sets, defined according to an infinite-game extension of Kohlberg and Mertens’s (1986) equilibrium concept, and show that, generically, a set is stable if and only if it contains a single Nash equilibrium.     

On Stackelberg games in a homogeneous product market

In a homogeneous product duopoly with concave demand and strictly convex costs we bring together all the standard results of quantity Stackelberg games, provide some new results with price Stackelberg games and compare the equilibrium configuration of the quantity games with the price games. In the price Stackelberg game we show there is a unique SPNE where the leader chooses a lower price than the follower, but both get equal payoffs. We prove that generally quantity Stackelberg games are less competitive than price Stackelberg games. However, we also demonstrate the possibility of a reversal of this result.     

Mixed equilibria in games of strategic complementarities

Summary. The literature on games of strategic complementarities (GSC) has focused on pure strategies. I introduce mixed strategies and show that, when strategy spaces are one-dimensional, the complementarities framework extends to mixed strategies ordered by first-order stochastic dominance. In particular, the mixed extension of a GSC is a GSC, the full set of equilibria is a complete lattice and the extremal equilibria (smallest and largest) are in pure strategies. The framework does not extend when strategy spaces are multi-dimensional. I also update learning results for GSC using stochastic fictitious play. Received: October 16, 2000; revised version: March 7, 2002 RID="*" ID="*" I am very grateful to Robert Anderson, David Blackwell, Aaron Edlin, Peter De Marzo, Ted O'Donoghue, Matthew Rabin, Ilya Segal, Chris Shannon, Clara Wang and Federico Weinschelbaum for comments and advise.     

Comparative statics of degenerate critical points

The correspondence principle is shown to apply to extremum problems whose Hessian matrix is degenerate, yielding, in principle, additional refutable propositions about economic relationships.     

Learning in games with strategic complementarities revisited

Fictitious play is a classical learning process for games, and games with strategic complementarities are an important class including many economic applications. Knowledge about convergence properties of fictitious play in this class of games is scarce, however. Beyond games with a unique equilibrium, global convergence has only been claimed for games with diminishing returns [V. Krishna, Learning in games with strategic complementarities, HBS Working Paper 92-073, Harvard University, 1992]. This result remained unpublished, and it relies on a specific tie-breaking rule. Here we prove an extension of it by showing that the ordinal version of strategic complementarities suffices. The proof does not rely on tie-breaking rules and provides some intuition for the result.     

Tax schemes in a class of differential games

This paper evaluates tax schemes in a class of differential games. The results indicate that there are many tax schemes that support efficient resource usage, but each may fail to implement the targeted resource because of the multiplicity of equilibria. Since all of the equilibria are subgame perfect, it is difficult to predict which specific one arises. Care must then be taken in using a tax scheme as a remedy for the “tragedy of the commons.” The advantages of other policy instruments (including command-and-control regulation and a tradable permit system) are also discussed. I am indebted to Gerhard Sorger, Koji Shimomura, and two anonymous referees for their helpful comments. The detailed suggestions of one of the referees in particular have markedly improved the paper. Any remaining errors are mine. Partial financial support from MEXT KAKENHI(11730017, 18078004) is also gratefully acknowledged.     

Monotone equilibria in Bayesian games of strategic complementarities

For Bayesian games of strategic complementarities, we provide a constructive proof of the existence of a greatest and a least Bayesian Nash equilibrium, each one in strategies that are monotone in type. Our main assumptions, besides strategic complementarities, are that each player's payoff displays increasing differences in own action and the profile of types and that each player's interim beliefs are increasing in type with respect to first-order stochastic dominance (e.g., types are affiliated). The result holds for general action and type spaces (single-, multi-, or infinite-dimensional; continuous or discrete) and no prior is assumed. We also provide the following comparative statics result: the greatest and least equilibria are higher if there is a first-order stochastic dominant shift in the interim beliefs. We apply this result to strategic information revelation in games of voluntary disclosure.     

Finding all equilibria in games of strategic complements

I present a simple and fast algorithm that finds all the pure-strategy Nash equilibria in games with strategic complementarities. This is the first non-trivial algorithm for finding all pure-strategy Nash equilibria.     

Comparative statics in the large based on a fixed point algorithm

The purpose of this paper is to present a theorem on comparative statics in the large by making use of theorems on a fixed point algorithm, and also to show its economic applications focused on Hicksian laws. We provide a unified approach to comparative statics in the large, and obtain laws of demand globally valid for generalized gross-substitute systems.