Multilateral negotiations and formation of coalitions

This paper analyses properties of games modeling multilateral negotiations leading to the formation of coalitions in an environment with widespread externalities. The payoff generated by each coalition is determined by an exogenous partition function (the parameter space). We show that in almost all games, except in a set of measure zero of the parameter space, the Markov perfect equilibrium value of coalitions and the state transition probability that describe the path of coalition formation is locally unique and stable. Therefore, comparative statics analysis are well-defined and can be performed using standard calculus tools. Global uniqueness does not hold in general, but the number of equilibria is finite and odd. In addition, a sufficient condition for global uniqueness is derived, and using this sufficient condition we show that there is a globally unique equilibrium in three-player superadditive games.     

Simple collective equilibria in stopping games

At each moment in time, an alternative from a finite set is selected by a stochastic process. Players observe the selected alternative and sequentially cast a yes or a no vote. If the set of players casting a yes vote is decisive for the selected alternative, it is accepted and the game ends. Otherwise the next period begins. We refer to this class of problems as stopping games. Collective choice games, quitting games, and coalition formation games are particular examples. When the core of a stopping game is non-empty, a subgame perfect equilibrium in pure stationary strategies is shown to exist. But in general, even subgame perfect equilibria in mixed stationary strategies may not exist. We show that aggregate voting behavior can be summarized by a collective strategy. We insist on pure strategies, allow for simple forms of punishment, and provide a constructive proof to show that so-called two-step simple collective equilibria always exist. This implies the existence of a pure strategy subgame perfect equilibrium. We apply our approach to the case with three alternatives exhibiting a Condorcet cycle and to a model of redistributive politics.     

Ramsey equilibrium with liberal borrowing

This paper considers a multi-agent one-sector Ramsey equilibrium growth model with borrowing constraints. The extreme borrowing constraint used in the classical version of the model, surveyed in Becker (2006), and the limited form of borrowing constraint examined in Borissov and Dubey (2015) are relaxed to allow more liberal borrowing by the households. A perfect foresight equilibrium is shown to exist in this economy. We describe the steady state equilibria for the liberal borrowing regime and show that as the borrowing regime is progressively liberalized, the steady state wealth inequality increases. Unlike the case of a limited borrowing regime, an equilibrium path need not converge in the case of liberal borrowing regime. We show through an example that a two period cyclic equilibrium exists when agents are allowed to borrow against their two period future wage income. This result is similar to the possibility of non-convergent equilibrium capital stock sequences in the model with no borrowing.     

A dynamic equilibrium model of search, production, and exchange

We study a general equilibrium model where agents search for production and trading opportunities, that generalizes the existing literature by considering a large number of differentiated commodities and agents with idiosyncratic tastes. Thus, agents must choose nontrivial exchange as well as production strategies. We consider decreasing, constant, and increasing returns to scale in the matching technology, and characterize the circumstances under which there exist multiple steady state equilibria, or multiple dynamic equilibria for given initial conditions. We also characterize the existence of dynamic equilibria that are limit cycles. Equilibria are not generally optimal, and when multiple equilibria coexist they may be ranked. We analyze comparative statics and find that certain intuitive results do not necessarily hold without restrictions on the stochastic structure.     

The stochastic lake game: A numerical solution

In this paper, we numerically solve a stochastic dynamic programming problem for the solution of a stochastic dynamic game for which there is a potential function. The players select a mean level of control. The state transition dynamics is a function of the current state of the system and a multiplicative noise factor on the control variables of the players. The particular application is for lake water usage. The control variables are the levels of phosphorus discharged (typically by farmers) into the watershed of the lake, and the random shock is the rainfall that washes the phosphorus into the lake. The state of the system is the accumulated level of phosphorus in the lake. The system dynamics are sufficiently nonlinear so that there can be two Nash equilibria. A Skiba-like point can be present in the optimal control solution.We analyze (numerically) how the dynamics and the Skiba-like point change as the variance of the noise (the rain) increases. The numerical analysis uses a result of Dechert (1978. Optimal control problems from second order difference equations. Journal of Economic Theory 19, 50–63) to construct a potential function for the dynamic game. This greatly reduces the computational burden in finding Nash equilibria solutions for the dynamic game.     

Vertical Integration and Exclusive Behavior of Insurers and Hospitals

We examine vertical integration and exclusive behavior in health care markets in which insurers and hospitals bilaterally bargain over contracts. We employ a bargaining model of two hospitals and two health insurers competing on premiums. We show that asymmetric equilibria exist in which one insurer contracts one hospital whereas the other insurer contracts both hospitals, even if all players are equally efficient in their production. Asymmetric equilibria arise if hospitals are sufficiently differentiated. In these cases, total industry profits increase and consumer welfare decreases in comparison to the case in which both insurers have contracts with both hospitals. Vertical integration makes these equilibria possible for a wider range of parameters.     

Revealed preferences of individual players in sequential games

This paper studies rational choice behavior of a player in sequential games of perfect and complete information without an assumption that the other players who join the same games are rational. The model of individually rational choice is defined through a decomposition of the behavioral norm assumed in the subgame perfect equilibria, and we propose a set of axioms on collective choice behavior that characterize the individual rationality obtained as such. As the choice of subgame perfect equilibrium paths is a special case where all players involved in the choice environment are each individually rational, the paper offers testable characterizations of both individual rationality and collective rationality in sequential games.     

Oligopoly, Financial Structure, and Resolution of Uncertainty

We characterize equilibria of a multistage game in which competing duopolists may acquire and share information in advance of choosing their financial structure which, in turn, precedes production. Given sufficient uncertainty, equilibria exist in which the efficiency and, possibly, coordination gains to acquiring and sharing perfect information are sufficient to break Brander and Lewis's (1986) result wherein both firms issue debt to their mutual disadvantage. However, more interesting may be the robustness of that result when uncertainty is low or when information is imperfect. The key insight is that the consequences of issuing debt are invariant to the level of uncertainty, given that firms can recalibrate the terms of debt to achieve the Stackelberg solution.     

Implementing the nash extension bargaining solution for non-convex problems

Conley and Wilkie (1993) introduced an axiomatization at the Nash extension bargaining solution defined on a domain of comprehensive but not necessarily convex problems. In this paper we present a non-cooperative game which implements the Nash extension solution in subgame perfect equilibria in the limit as the discount rate applied between rounds of play vanishes.     

On essential, (strictly) perfect equilibria

It is known that generic games within certain collections of infinite-action normal-form games have only essential equilibria. We point to a difficulty in showing that essential equilibria in generic games are (strictly) perfect, and we identify collections of games whose generic members have only essential and (strictly) perfect equilibria.     

Recursive Nash bargaining over a productive asset

We propose an equilibrium concept (the recursive Nash bargaining solution) that describes the outcome of repeated negotiations between two rational agents under the assumptions that the state of the economic system under consideration changes according to the actions of the players and that neither party can make binding commitments to future behavior. This equilibrium is dynamically consistent but typically not Pareto-efficient. As an application, we compute the recursive Nash bargaining solution in a model of two heterogeneous agents bargaining over the use of a productive asset with constant gross rate of return and study how the time-preference rates and the elasticities of substitution affect the solution.     

A limit theorem for systems of social interactions

In this paper, we establish a convergence result for equilibria in systems of social interactions with many locally and globally interacting players. Assuming spacial homogeneity and that interactions between different agents are not too strong, we show that equilibria of systems with finitely many players converge to the unique equilibrium of a benchmark system with infinitely many agents. We prove convergence of individual actions and of average behavior. Our results also apply to a class of interaction games.     

A noncooperative foundation of the asymmetric Nash bargaining solution

We consider a noncooperative multilateral bargaining model with heterogeneous time preferences in which the first rejector of a proposal in the current round becomes the proposer in the next round. We show the existence of a stationary subgame perfect equilibrium (SSPE), characterize SSPEs and show the efficiency of SSPEs. We show that any sequence of SSPE payoff profiles converges to the asymmetric Nash bargaining solution weighted by the inverses of discount rates as the bargaining friction vanishes.     

Coalition formation in general NTU games

A general nontransferable utility (NTU) game is interpreted as a collection of pure bargaining games that can be played by individual coalitions. The threatpoints or claims points respectively, in these pure bargaining games reflect the players' opportunities outside a given coalition. We develop a solution concept for general NTU games that is consistent in the sense that the players' outside opportunities are determined by the solution to a suitably defined reduced game. For any general NTU game the solution predicts which coalitions are formed and how the payoffs are distributed among the players.     

Your failure is my opportunity—Effects of elimination in contests

In this paper, we consider a sequence of multi-prize all-pay auctions, where low-performing players are eliminated along the way. We provide sufficient conditions such that equilibrium behavior is unaffected by the prospects of winning prizes in later rounds. When these conditions are not met, we identify a novel externality effect and derive equilibria in a model with two rounds. We compare our results to knockout tournaments in which players are partitioned into subcontests in each round.     

Equilibrium and arbitrage in incomplete asset markets with fixed prices

At arbitrary prices of commodities and assets, fix-price equilibria exist under weak assumptions: endowments need not satisfy an interiority condition, utility functions need only satisfy a very weak monotonicity requirement, and the asset return matrix allows for redundant assets. Prices of assets may permit arbitrage. At equilibrium, though restricted through endogenously determined trading constraints, arbitrage possibilities may persist; in an example, an individual holds an arbitrage portfolio.     

Determinacy of equilibrium in outcome game forms

We show the generic finiteness of the number of probability distributions on outcomes induced by Nash equilibria for two-person game forms such that either (i) one of the players has no more than two strategies or (ii) both of the players have three strategies, and (iii) for outcome game forms with three players, each with at most two strategies. Finally, we exhibit an example of a game form with three outcomes and three players for which the Nash equilibria of the associated game induce a continuum of payoffs for an open non-empty set of utility profiles.     

Functional sunspot equilibria

Consider a one step forward looking model where agents believe that the equilibrium values of the state variable are determined by a function whose domain is the current value of the state variable and whose range is the value for the subsequent period. An agent’s forecast for the subsequent period uses the belief, where the function that is chosen is allowed to depend on the current realization of an extrinsic random process, and is made with knowledge of the past values of the state variable but not the current value. The paper provides (and characterizes) the conditions for the existence of sunspot equilibria for the model described.     

Communication equilibria with partially verifiable types

This paper studies the set of equilibria that can be achieved by adding general communication systems to Bayesian games in which some information can be certified or, equivalently, in which players’ types are partially verifiable. Certifiability of information is formalized by a set of available reports for each player that varies with the true state of the world. Given these state-dependent sets of reports, we characterize canonical equilibria for which generalized versions of the revelation principle are valid. Communication equilibria and associated canonical representations are obtained as special cases when no information can be certified.     

Investments in education and welfare in a two-sector,random matching economy

We consider a random matching model where heterogeneous agents choose optimally to invest time and real resources in education. Generically, there is a steady state equilibrium where some agents, but not all of them, invest. Regular steady state equilibria are constrained inefficient in a strong sense. The Hosios (1990) condition is neither necessary nor sufficient for constrained efficiency. We also provide restrictions on the fundamentals sufficient to guarantee that equilibria are characterized by overeducation (or undereducation), present some results on their comparative statics properties, and discuss the nature of welfare improving policies.