Computing uniformly optimal strategies in two-player stochastic games

We provide a computable algorithm to calculate uniform ε-optimal strategies in two-player zero-sum stochastic games. Our approach can be used to construct algorithms that calculate uniform ε-equilibria and uniform correlated ε-equilibria in various classes of multi-player non-zero-sum stochastic games.     

The equilibrium set of two-player games with complementarities is a sublattice

Summary. I prove that the equilibrium set in a two-player game with complementarities, and totally ordered strategy spaces, is a sublattice of the joint strategy space. Received: May 31, 2001; revised version: October 4, 2002     

Experimental study of cursed equilibrium in a signaling game

The laboratory experiment described in this paper provides evidence on play in signaling games in the context of electoral competition. In this game, voters must infer the preferred policy of each candidate from the candidate’s choice of whether to announce (truthfully) his preferred policy or to take no position. Bayesian voters would put high probability on a candidate having an extreme policy preference after observing him take no position, but cursed voters would not fully appreciate the informational content of the decision to take no position. Stated beliefs reveal substantial uncertainty about other players’ strategies. Based on estimates of a structural model of cursed equilibrium allowing for heterogeneity in the degree of cursedness, 32% of choices between candidates are consistent with Bayesian updating, 32% imply no inferences about others’ types after observing their actions, and the remainder indicate partial updating. Though the experiment also includes treatments with subjects in both roles, these estimates are based on interactions with programmed candidates, implying that uncertainty about others’ rationality and strategic sophistication is not driving the result. We also find that the quantal response error structure in which errors depend on payoff differences cannot explain the pattern of errors that subjects make.     

Multiple equilibrium overnight rates in a dynamic interbank market game

We analyse a two-period model of the interbank market, i.e. the market where banks trade liquidity. We assume that banks do not take the interbank interest rate as given, but instead negotiate on interest rates and transaction volumes with each other. The solution concept applied is the Shapley value. We show that there are a multiplicity of average equilibrium interest rates of the first period so that the average interest rate in this period does not convey any information on the expected liquidity situation on the interbank market. As the banks control not only the transaction volumes, but also the interest rates, they can leave the interest rates constant and adjust the transaction volumes when, for example, a liquidity deficit becomes more likely.     

Reputation, control rights and game equilibrium

Using the reputation model of Kreps (1982), Vickers (1986) and Barro(1986), we develop a dynamic game model with incomplete information to examine the relations between the managers of state-owned enterprises(SOEs) in China and the government as the enterprise’s owner. Employing the model, we show that even a noncoopertive manager will not intrude the owner’s interests until the last period of his term in order to maximize his long term utility. The paper also discusses some phenomenona in state-owned enterprises in China, such as “insiders’ control”, “59 phenomenon” and excess on-the-job consumption.      

Computing Cournot equilibrium through maximization over prices

This paper presents a characterization of internal Cournot equilibrium based on first-order conditions corresponding to profit maximization over prices. This characterization may yield significant computational advantage as demand functions need not be inverted and simple first-order conditions are obtained.     

Computing equilibrium bond prices in the Vayanos-Vila model

We develop tools for computing equilibrium bond prices for the discrete-time version of the Vayanos–Vila (2009) model. With the maturity structure included in pricing factors, factor loadings for equilibrium bond yields depends critically on parameters describing maturity structure dynamics and other model parameters. An illustrative example shows that the effect on the yield curve of a supply shock originating in a given maturity, although hump-shaped around the originating maturity, is to change yields broadly across all maturities.     

Nash equilibrium vs. maximin : A comparative game statics analysis

This paper derives comparative game statics results for both maximin and Nash strategies, thereby allowing a more systematic comparison between these two approaches.     

On the generic finiteness of equilibrium outcome distributions in bimatrix game forms

We provide an example of an outcome game form with two players for which there is an open set of utilities for both players such that, in each of the associated games, the set of Nash equilibria induces a continuum of outcome distributions.     

Enumeration of Nash equilibria for two-player games

This paper describes algorithms for finding all Nash equilibria of a two-player game in strategic form. We present two algorithms that extend earlier work. Our presentation is self-contained, and explains the two methods in a unified framework using faces of best-response polyhedra. The first method lrsnash is based on the known vertex enumeration program lrs, for “lexicographic reverse search”. It enumerates the vertices of only one best-response polytope, and the vertices of the complementary faces that correspond to these vertices (if they are not empty) in the other polytope. The second method is a modification of the known EEE algorithm, for “enumeration of extreme equilibria”. We also describe a second, as yet not implemented, variant that is space efficient. We discuss details of implementations of lrsnash and EEE, and report on computational experiments that compare the two algorithms, which show that both have their strengths and weaknesses.     

An elasticity approach to equilibrium and preference concentration in the Hotelling game

We consider the standard Hotelling price-and-location game with two firms, and show how equilibrium is affected by the shape of the consumers’ density. By describing the latter in terms of elasticity (Esteban in Int Econ Rev 27:439–444, 1986), we develop a framework which enables us to prove a sufficient condition for the existence of multiple asymmetric equilibria when the density is symmetric, in terms of a lower bound on the Gini coefficient of the distribution of preferences.      

Computing minimal state space recursive equilibrium in OLG models with stochastic production

Using order-theoretic methods, we derive sufficient conditions for the existence, characterization, and computation of minimal state space recursive equilibrium (RE), as well as Stationary Markov equilibrium (SME) for various classes of stochastic overlapping generations models. In contrast to previous work, our methods focus on constructive methods. Our existence results are obtained for models that include public policy (e.g., social security policies, transfers, taxes, etc), production nonconvexities, elastic labor supply, non-monotone income processes, and long-lived agents. We distinguish conditions under which there exist various subclasses of minimal state space RE, including bounded, monotone, non-monotone, semicontinuous, Lipschitz continuous RE. Finally, we provide monotone equilibrium comparative statics results on the space of economies for some RE.     

Testing theories of behavior for extensive-form two-player two-stage games

We examine choices in two-player extensive-form games that give subjects opportunities for individualistic as well as other-regarding behavior, and where each subject makes choices in a variety of games. Following an extensive search over models, where we estimate a single parameter vector for all the games rather than different parameter vectors for each game, we find that (1) the level-n model organizes the data well, (2) other-regarding behaviors in these games appear to consist primarily of egalitarian and utilitarian behaviors, and (3) there is no significant evidence for reciprocal behavior. Electronic Supplementary Material  The online version of this article () contains supplementary material, which is available to authorized users.     

Uniqueness of the index for Nash equilibria of two-player games

Summary. Given a map whose roots are the Nash equilibria of a game, each component of the equilibrium set has an associated index, defined as the local degree of the map. This note shows that for a two-player game, every map with the same roots induces the same index. Moreover, this index agrees with the Shapley index constructed from the Lemke-Howson algorithm. Received: May 30, 1996; revised version June 25, 1996     

Asymptotic expected number of Nash equilibria of two-player normal form games

The formula given by McLennan [The mean number of real roots of a multihomogeneous system of polynomial equations, Amer. J. Math. 124 (2002) 49–73] is applied to the mean number of Nash equilibria of random two-player normal form games in which the two players have M and N pure strategies respectively. Holding M fixed while N→∞, the expected number of Nash equilibria is approximately . Letting M=N→∞, the expected number of Nash equilibria is , where is a constant, and almost all equilibria have each player assigning positive probability to approximately 31.5915 percent of her pure strategies.     

On a master-slave Bertrand game model

A master-slave Bertrand game model is proposed for upstream and downstream monopolies owned by different parties, in which the upstream monopolist's output is used as the main factor of production by the downstream monopolist who is a small purchaser of the upstream monopolist's output. The bifurcation of the Bertrand-Nash equilibrium is analyzed with Schwarzian derivative. Numerical simulations are employed to show the model's complex dynamics by means of the largest Lyapunov exponents (LLEs), bifurcation, time series diagrams and phase portraits. With the modified straight-line stabilization method, chaos control is used to improve the aggregate profits of the two oligopolists. Lastly the welfare impacts of price fluctuations and chaos controls are briefly discussed.