On the observational equivalence of random matching

Random matching is often used in economic models as a means of introducing uncertainty in sequential decision problems. We show that random matching processes that satisfy standard proportionality laws are not unique. We give conditions on the payoffs and transition functions of sequential decision models that insure that economic models are robust to the nonuniqueness of the matching process. Under these conditions, the information contained in the proportionality laws is all that is needed to know about the matching process to formulate the model.     

Search theory,competitive equilibrium,and the Nash bargaining solution

We investigate a canonical search-theoretic model without entry. Two agents are randomly matched with a long side being rationed. The matched agents face a pair of randomly drawn non-transferable payoffs, and then choose whether or not to form a partnership subject to a small probability of exogenous break down. As this probability and friction vanish, the Nash bargaining solution emerges as the unique undominated strategy equilibrium outcome if the mass of each party is the same. If the size of one party is larger than the other, the short side extracts the entire surplus, a sharp contrast to Rubinstein and Wolinsky (1985) [16].     

A three-player dynamic majoritarian bargaining game

We analyze an infinitely repeated divide-the-dollar bargaining game with an endogenous reversion point. In each period a new dollar is divided among three legislators according to the proposal of a randomly recognized member—if a majority prefer so—or according to previous period's allocation otherwise. Although current existence theorems for Markovian equilibria do not apply for this dynamic game, we fully characterize a Markov equilibrium. The equilibrium is such that irrespective of the discount factor or the initial division of the dollar, the proposer eventually extracts the whole dollar in all periods. We also show that proposal strategies are weakly continuous in the status quo that equilibrium expected utility is not quasi-concave, and the correspondence of voters’ acceptance set (the set of allocations weakly preferred over the status quo) fails lower hemicontinuity.     

The role of private information in dynamic matching and bargaining: Can it be good for efficiency?

We consider a private information replica of the dynamic matching and bargaining model of Mortensen and Wright (2002). We find that private information typically deters entry. But, the welfare can actually be higher under private information.     

Delays and partial agreements in multi-issue bargaining

We model a situation in which two players bargain over two pies, one of which can only be consumed starting at a future date. Suppose the players value the pies asymmetrically: one player values the existing pie more than the future one, while the other player has the opposite valuation. We show that players may consume only a fraction of the existing pie in the first period, and then consume the remainder of it, along with the second pie, at the date at which the second pie becomes available. Thus, our model features a special form of bargaining delay, in which agreements take place in multiple stages. Such partial agreements arise when players are patient enough, when they expect the second pie to become available soon, and when the asymmetry in their valuations is large enough.     

Sorting versus screening: Search frictions and competing mechanisms

In a market where sellers compete by posting trading mechanisms, we allow for a general search technology and show that its features crucially affect the equilibrium mechanism. Price posting prevails when meetings are rival, i.e., when a meeting by one buyer reduces another buyer's meeting probability. Under price posting buyers reveal their type by sorting ex-ante. Only if the meeting technology is sufficiently non-rival, price posting is not an equilibrium. Multiple buyer types then visit the same sellers who screen ex-post through auctions.     

A theory of political cycles

We study how the proximity of elections affects policy choices in a model in which policymakers want to improve their reputation to increase their reelection chances. Policymakers' equilibrium decisions depend on both their reputation and the proximity of the next election. Typically, incentives to influence election results are stronger closer to the election (for a given reputation level), as argued in the political cycles literature, and these political cycles are less important when the policymaker's reputation is better. Our analysis sheds light on other agency relationships in which part of the compensation is decided upon infrequently.     

Bayesian learning and convergence to Nash equilibria without common priors

Summary. Consider an infinitely repeated game where each player is characterized by a “type” which may be unknown to the other players in the game. Suppose further that each player's belief about others is independent of that player's type. Impose an absolute continuity condition on the ex ante beliefs of players (weaker than mutual absolute continuity). Then any limit point of beliefs of players about the future of the game conditional on the past lies in the set of Nash or Subjective equilibria. Our assumption does not require common priors so is weaker than Jordan (1991); however our conclusion is weaker, we obtain convergence to subjective and not necessarily Nash equilibria. Our model is a generalization of the Kalai and Lehrer (1993) model. Our assumption is weaker than theirs. However, our conclusion is also weaker, and shows that limit points of beliefs, and not actual play, are subjective equilibria. Received: March 3, 1995; revised version: February 17, 1997     

Efficient delays in a stochastic model of bargaining

Summary. We consider a k-player sequential bargaining model in which both the cake size and the identity of the proposer are determined by a stochastic process. For the case where the cake is a simplex (of random size) and the players share a common discount factor, we establish the existence of a unique stationary subgame perfect payoff which is efficient and characterize the conditions under which agreement is delayed. We also investigate how the equilibrium payoffs depend on the order in which the players move and on the correlation between the identity of the proposer and the cake size.Received: November 5, 1996; revised version: December 31, 1996This revised version was published online in February 2005 with corrections to the cover date.     

Symmetry-based compromise and the Nash solution to convex bargaining problems

A new axiom, symmetry-based compromise, is introduced to study solutions to convex bargaining problems. It is shown that any efficient solution that is also scale invariant and satisfies symmetry-based compromise must be Nash.     

Competitive bargaining equilibrium

In a simple exchange economy we propose a bargaining procedure that leads to a Walrasian outcome as the agents become increasingly patient. The competitive outcome therefore obtains even if agents have market power and are not price-takers. Moreover, where in other bargaining protocols the final outcome depends on bargaining power or relative impatience, the outcome here is determinate and depends only on preferences and endowments. Our bargaining procedure involves bargaining over prices and maximum quantity constraints, and it guarantees convergence to a Walrasian outcome for any standard exchange economy. In contrast, without quantity constraints we show that equilibrium is generically inefficient.     

Matching and bargaining models of markets: approximating small markets by large markets

Summary. We show that the equilibrium of a matching and bargaining model of a market in which there is a finite number of agents at each date need not be near the equilibrium of a market with a continuum of agents, although matching probabilities are the same in both markets. Holding the matching process fixed, as the finite market becomes large its equilibrium approaches the equilibrium of its continuum limit.Received: January 22, 1996; revised version: September 24, 1996This revised version was published online in February 2005 with corrections to the cover date.     

Simple Markovian equilibria in dynamic spatial legislative bargaining

The paper proves, by construction, the existence of Markovian equilibria in a dynamic spatial legislative bargaining model. Players bargain over policies in an infinite horizon. In each period, a sequential protocol of proposal-making and voting, with random proposer recognitions and a simple majority, produces a policy that becomes the next period's status-quo; the status-quo is endogenous. The construction relies on simple strategies determined by strategic bliss points computed by the algorithm we provide. A strategic bliss point, the dynamic utility ideal, is a moderate policy relative to a bliss point, the static utility ideal. Moderation is strategic and germane to the dynamic environment; players moderate in order to constrain the future proposals of opponents. Moderation is a strategic substitute; when a player's opponents do moderate, she does not, and when they do not moderate, she does. We provide conditions under which the simple strategies induced by the strategic bliss points computed by the algorithm deliver a Stationary Markov Perfect equilibrium, and we prove its existence in generic games with impatient players and in symmetric games. Because the algorithm constructs all equilibria in simple strategies, we provide their general characterization, and we show their generic uniqueness.     

Monotonic and implementable solutions in generalized matching problems

We consider one-to-one matching problems when preferences are weak orders and search for (Maskin-)monotonic and implementable solutions. We show that for any coalition structure , the strong -core is monotonic if and only if it is the individually rational solution. We introduce a solution, which we call the “union strong -core”, and show that this solution is the minimal way to expand the strong -core recovering monotonicity. We also show that the union strong -core is implementable.     

Convergence to perfect competition of a dynamic matching and bargaining market with two-sided incomplete information and exogenous exit rate

Consider a decentralized, dynamic market with an infinite horizon and incomplete information in which buyers and sellers' values for the traded good are private and independently drawn. Time is discrete, each period has length δ, and each unit of time a large number of new buyers and sellers enter the market. Within a period each buyer is matched with a seller and each seller is matched with zero, one, or more buyers. Every seller runs a first price auction with a reservation price and, if trade occurs, the seller and winning buyer exit with their realized utility. Traders who fail to trade either continue in the market to be rematched or exit at an exogenous rate. We show that in all steady state, perfect Bayesian equilibria, as δ approaches zero, equilibrium prices converge to the Walrasian price and realized allocations converge to the competitive allocation.     

Von Neumann-Morgenstern stable sets, discounting, and Nash bargaining

We establish a link between von Neumann-Morgenstern stable set and the Nash solution in a general n-player utility set. The stable set-solution is defined with respect to a dominance relation: payoff vector u dominates v if one player prefers u even with one period delay. We show that a stable set exists and, if the utility set has a smooth surface, any stable set converges to the Nash bargaining solution when the length of the period goes to zero.     

On the convergence of reinforcement learning

This paper examines the convergence of payoffs and strategies in Erev and Roth's model of reinforcement learning. When all players use this rule it eliminates iteratively dominated strategies and in two-person constant-sum games average payoffs converge to the value of the game. Strategies converge in constant-sum games with unique equilibria if they are pure or if they are mixed and the game is 2×2. The long-run behaviour of the learning rule is governed by equations related to Maynard Smith's version of the replicator dynamic. Properties of the learning rule against general opponents are also studied.     

Reputation and exogenous private learning

Short-run competitors in the chain store game receive noisy signals of the long-run incumbent firm's type. The history of signals, which in the limit is fully revealing, is observable to the competitors but possibly not to the incumbent. As long as there is sufficient noise in the signals, then in any equilibrium a patient weak incumbent obtains a payoff strictly higher than her minmax payoff.     

Investment timing and learning externalities

We study a duopoly model of investment, in which each player learns about the quality of a common value project by observing some public background information, and possibly the experience of his rival. Investment costs are private information, and the background signal takes the form of a Poisson process conditional on the quality of the project being low. The resulting attrition game has a unique, symmetric equilibrium, which depends on initial public beliefs. We determine the impact of changes in the cost and signal distributions on investment timing, and how equilibrium is affected when a first-mover advantage is introduced.