Mark Stegeman

Existence and Uniqueness of Maximal Reductions Under Iterated Strict Dominance

Iterated elimination of strictly dominated strategies is an order dependent procedure. It can also generate spurious Nash equilibria, fail to converge in countable steps, or converge to empty strategy sets. If best replies are well-defined, then spurious Nash equilibria cannot appear; if strategy spaces are compact and payoff functions are uppersemicontinuous in own strategies, then order does not matter; if strategy sets are compact and payoff functions are continuous in all strategies, then a unique and nonempty maximal reduction exists. These positive results extend neither to the better-reply secure games for which Reny has established the existence of a Nash equilibrium, nor to games in which (under iterated eliminations) any dominated strategy has an undominated dominator. Copyright The Econometric Society 2002.

Non-Nash equilibria of Darwinian dynamics with applications to duopoly

Abstract Consider a symmetric, differentiated duopoly. If firms’ strategy choices, in the repeated game, follow a stochastic Darwinian process, then they cluster around a strategy profile that is typically not a one-shot Nash equilibrium. This profile is invariant under a broad class of transformations of the strategy space (e.g. Bertrand vs. Cournot); this implies that mixing imitative and rational decision-makers can produce purely imitative outcomes. The evolution of objectives consistently distorts behavior toward revenue maximization, and the distortion increases in ‘good times’ of high demand and low costs. We generalize the results beyond duopoly to symmetric, two-player games.

Sequential Auctions of Endogenously Valued Objects

Two completely informed but possibly asymmetric bidders buy or sell identical “claims” in sequential auctions. They subsequently receive monetary prizes that depend upon the final allocation of claims. Iterated elimination of weakly dominated strategies leaves a unique Nash equilibrium. For any prize schedule, prices weakly decline as the auctions progress, and points of strict decline have a simple characterization. For one class of prize schedules, which arises naturally if duopolists bid for a scarce input, the equilibrium is completely characterized; many initial allocations generate the same final (unequal) division of claims, which may be interpreted as the natural market structure. Journal of Economic Literature Classification Numbers: D43, D44, L11.

Sequential Procurement With Subcontracting

Two symmetric sellers are approached sequentially by fragmented buyers. Each buyer conducts a second-price auction and purchases from the seller who offers the lower price. Winning an auction affects bidding for future contracts because the sellers have nonconstant marginal costs. We assume that the sellers are completely informed, and we study the unique equilibrium that survives iterated elimination of weakly dominated strategies. If subcontracting between the sellers is impossible, the final allocation of contracts is generally inefficient. If postauction subcontracting is possible, the sellers can be worse off, ex ante, than when subcontracting is impossible.

Stochastic Darwinian equilibria in small and large populations

Abstract We establish necessary and sufficient conditions for the stability of stochastic Darwinian dynamics in quadratic games. Each players strategy adjusts through mutation and selection shocks, and stability is independent of the rates at which these shocks arrive. Given stability, we characterize the midpoint of the nondegenerate ergodic distribution. In small populations, some equilibria correspond to relative payoff maximization, but others are unanticipated by existing static concepts. In the large population limit of a finite population, the set of stable Nash equilibria strictly includes all equilibria stable under myopic best reply, but some strict Nash equilibria are highly unstable. The stability result shows, for the first time, that large finite populations converge to Nash play even if they do not understand the game and strategies are so numerous that most are never played. The large population stability condition is related to risk dominance and, separately, to the static CSS condition.

Rigid monopoly prices

A consumer in the real world typically must visit (e.g. by phone) a monopolist to observe its price, even though the consumer may have correct expectations about that price. This causes monopoly prices to be higher and stickier than is predicted by the textbook model. If visiting costs are small, and consumers do not observe the firms costs, then prices conform to the textbook model when costs are high but are downwardly rigid when costs drop below a threshold. Price flexibility increases as the fraction of ignorant consumers, who observe their idiosyncratic valuations of the product only after visiting the firm, increases. In a repeated game, a ‘ratchet’ equilibrium, in which price increases are permanent and price decreases temporary within a certain band, supports equilibria for which price is rigid on the equilibrium path but which Pareto dominate the fluctuating one-shot equilibrium. The ratchet equilibrium has the advantages that price is a continuous function of past prices near the equilibrium path and the price coordination problem is solved in a natural way: the equilibrium price is the lowest price that can be sustained by such a ratchet. This equilibrium price exceeds but is close to the average prediction of the textbook model. Additional results are derived for the case in which costs are fixed.

Sufficient Conditions for Inessentiality

Three theorems state conditions sufficient for the inessentiality of equilibrium in a pure exchange, sequence economy. The agents have uncommon priors, state-contingent utility functions, and asymmetric information in every trading period, and they trade different sets of event-contingent claims in different periods. The theorems provide alternative interpretations of the concept of market completeness, reveal two fundamentally different ways to obtain inessentiality, and shed light on the conditions permitting speculation and the role of price-contingent trading. None of the theorems requires ex ante Pareto optimality or the absence of arbitrage opportunities. Copyright 1993 by The Econometric Society.