Pure equilibria in a simple dynamic model of strategic market game

We present a discrete model of two-person constant-sum dynamic strategic market game. We show that for every value of discount factor the game with discounted rewards possesses a pure stationary strategy equilibrium. Optimal strategies have some useful properties, such as Lipschitz property and symmetry. We also show value of the game to be nondecreasing both in state and discount factor. Further, for some values of discount factor, exact form of optimal strategies is found. For β less than , there is an equilibrium such that players make large bids. For β close to 1, there is an equilibrium with small bids. Similar result is obtained for the long run average reward game.     

Linear-quadratic,two-person,zero-sum differential games: Necessary and sufficient conditions

We consider linear-quadratic, two-person, zero-sum perfect information differential games, possibly with a linear target set. We show a necessary and sufficient condition for the existence of a saddle point, within a wide class of causal strategies (including, but not restricted to, pure state feedbacks). The main result is that, when they exist, the optimal strategies are pure feedbacks, given by the classical formulas suitably extended, and that existence may be obtained even in the presence of a conjugate point within the time interval, provided it is of a special type that we calleven.The partial support of the Trieste Unit of the GNAS, Italian CNR, is gratefully acknowledged.     

Uniqueness of Radial Solutions for the Fractional Laplacian

We prove general uniqueness results for radial solutions of linear and nonlinear equations involving the fractional Laplacian (?Δ)^s with s ? (0,1) for any space dimensions N ≥ 1. By extending a monotonicity formula found by Cabré and Sire , we show that the linear equation has at most one radial and bounded solution vanishing at infinity, provided that the potential V is radial and nondecreasing. In particular, this result implies that all radial eigenvalues of the corresponding fractional Schrödinger operator H = (?Δ)^s + V are simple. Furthermore, by combining these findings on linear equations with topological bounds for a related problem on the upper half‐space , we show uniqueness and nondegeneracy of ground state solutions for the nonlinear equation for arbitrary space dimensions N ≥ 1 and all admissible exponents α > 0. This generalizes the nondegeneracy and uniqueness result for dimension N = 1 recently obtained by the first two authors and, in particular, the uniqueness result for solitary waves of the Benjamin‐Ono equation found by Amick and Toland .© 2016 Wiley Periodicals, Inc.     

Correlated equilibria in some classes of two-person games

A correlated equilibrium in a two-person game is “good” if for everyNash equilibrium there is a player who prefers the correlated equilibrium to theNash equilibrium. If a game is “best-response equivalent” to a two-person zero-sum game, then it has no good correlated equilibria. But games which are “almost strictly competitive” or “order equivalent” to a two-person zero-sum game may have good correlated equilibria.     

Denumerable state stochastic games with limiting average payoff

We study stochastic games with countable state space, compact action spaces, and limiting average payoff. ForN-person games, the existence of an equilibrium in stationary strategies is established under a certain Liapunov stability condition. For two-person zero-sum games, the existence of a value and optimal strategies for both players are established under the same stability condition.The authors wish to thank Prof. T. Parthasarathy for pointing out an error in an earlier version of this paper. M. K. Ghosh wishes to thank Prof. A. Arapostathis and Prof. S. I. Marcus for their hospitality and support.     

A unique equilibrium efficient frontier with fixed-sum outputs in data envelopment analysis

Based on the equilibrium efficient frontier data envelopment analysis (EEFDEA) approach, Fang (J Oper Res Soc 67:412–420, 2015a) developed an equivalent linear programming model to improve and strengthen the EEFDEA approach. Furthermore, Fang (2015a) indicated that his secondary goal approach can achieve a unique equilibrium efficient frontier. However, through a simple counterexample we demonstrate that Fang’s secondary goal approach cannot always achieve uniqueness of the equilibrium efficient frontier. In this paper, we propose an algorithm based on the secondary goal approach to address the problem. The proposed algorithm is proven mathematically to be an effective approach to guaranteeing the uniqueness of the equilibrium efficient frontier.     

On equilibrium points of diagonaln-person games

In this paper, we compute explicitly the equilibrium points of diagonaln-person games when all of them have the same number of strategies. This number is arbitrary. A wide generalization of two-person games is immediately obtained.The author is grateful to Professor Joel Cohen who visited IMASL during the winter of 1987 and commented on the paper.     

Commitments to threat strategies in Nash bargaining

Nash's two-person bargaining model consists of two stages: threat strategies and utility demands are chosen in the first and second stages respectively. Here players commit to play the threat strategies chosen in the first stage for the case where disagreement occurs in the second stage. Whether a player commits or not to play a threat strategy, however, is voluntary in principle. This leads to some possible extensions of Nash's model so that players' commitment choices are taken into account. In this paper, we consider three extensions. In the two person case, these three extensions give essentially the same result as that given by Nash. This is not the case for more than two players; the result depends upon an extension. In one extension, Nash's result always holds for more than two players. In the other two extensions, however, we give a three person example where not all players choose commitments in equilibrium.     

Structure of equilibria inN-person non-cooperative games

Here we study the structure of Nash equilibrium points forN-person games. For two-person games we observe that exchangeability and convexity of the set of equilibrium points are synonymous. This is shown to be false even for three-person games. For completely mixed games we get the necessary inequality constraints on the number of pure strategies for the players. Whereas the equilibrium point is unique for completely mixed two-person games, we show that it is not true for three-person completely mixed game without some side conditions such as convexity on the equilibrium set. It is a curious fact that for the special three-person completely mixed game with two pure strategies for each player, the equilibrium point is unique; the proof of this involves some combinatorial arguments.     

Take-and-guess games

This paper studies two classes of two-person zero-sum games in which the strategies of both players are of a special type. Each strategy can be split into two parts, a taking and a guessing part. In these games two types of asymmetry between the players can occur. In the first place, the number of objects available for taking does not need to be the same for both players. In the second place, the players can be guessing sequentially instead of simultaneously; the result is asymmetric information. The paper studies the value and equilibria of these games, for all possible numbers of objects available to the players, for the case with simultaneous guessing as well as for the variant with sequential guessing.      

Value and perfection in stochastic games

A stochastic game isvalued if for every playerk there is a functionr ^k:S→R from the state spaceS to the real numbers such that for every ε>0 there is an ε equilibrium such that with probability at least 1−ε no states is reached where the future expected payoff for any playerk differs fromr ^k(s) by more than ε. We call a stochastic gamenormal if the state space is at most countable, there are finitely many players, at every state every player has only finitely many actions, and the payoffs are uniformly bounded and Borel measurable as functions on the histories of play. We demonstrate an example of a recursive two-person non-zero-sum normal stochastic game with only three non-absorbing states and limit average payoffs that is not valued (but does have ε equilibria for every positive ε). In this respect two-person non-zero-sum stochastic games are very different from their zero-sum varieties. N. Vieille proved that all such non-zero-sum games with finitely many states have an ε equilibrium for every positive ε, and our example shows that any proof of this result must be qualitatively different from the existence proofs for zero-sum games. To show that our example is not valued we need that the existence of ε equilibria for all positive ε implies a “perfection” property. Should there exist a normal stochastic game without an ε equilibrium for some ε>0, this perfection property may be useful for demonstrating this fact. Furthermore, our example sews some doubt concerning the existence of ε equilibria for two-person non-zero-sum recursive normal stochastic games with countably many states. This research was supported financially by the German Science Foundation (Deutsche Forschungsgemeinschaft) and the Center for High Performance Computing (Technical University, Dresden). The author thanks Ulrich Krengel and Heinrich Hering for their support of his habilitation at the University of Goettingen, of which this paper is a part.     

Intuitive and noncompetitive equilibria in weakly efficient auctions with entry costs

I study weakly efficient auctions with entry costs, under the IPV assumption, following Tan and Yilankaya [Tan, G., Yilankaya, O., 2006. Equilibria in second price auctions with participation costs. Journal of Economic Theory 130, 205–219]. First, I generalize their Proposition 4 to what I call (generalized) intuitive equilibrium. By such I prove that if bidders’ valuation distributions are ordered in a (weak) first order domination ranking, then there exists an equilibrium in cutoff strategies where cutoffs are (weakly) increasingly ordered with respect to the domination ranking. Stronger bidders are thus ex ante more likely to participate. A second result states a necessary and sufficient condition for the existence of a noncompetitive cutoff equilibrium, in which only one bidder (if any) takes part in the auction. Neither the uniform distribution nor any distribution first order stochastically dominated by the uniform may ever satisfy that condition. If both intuitive and nonintuitive equilibria exist, I conjecture that intuitive equilibria tend to yield higher ex ante efficiency, while nonintuitive ones might yield higher expected revenues.     

Unique solvability of the Dirichlet problem¶for weakly harmonic maps

We prove existence and uniqueness of weakly harmonic maps from the unit ball in ℝ ⁿ (with n≥ 3) to a smooth compact target manifold within the class of maps with small scaled energy for suitable boundary data. Received: 9 June 2000 / Revised version: 17 April 2001     

On the uniqueness of time-periodic solutions to the Navier–Stokes equations in unbounded domains

We present a uniqueness theorem for time-periodic solutions to the Navier–Stokes equations in unbounded domains. Thus far, results on the uniqueness of time-periodic solutions to the Navier–Stokes equations in unbounded domain, roughly speaking, have only found that a small time-periodic L ⁿ -solution is unique within the class of solutions which have sufficiently small L ^∞(L ⁿ )-norm. In this paper, we show that a small time-periodic L ⁿ -solution is unique within the class of all time-periodic L ⁿ -solutions, which contains large solutions. We also consider the uniqueness of solutions in weak-L ⁿ space. The proof of the present uniqueness theorem is based on the method of dual equations.      

Cournot equilibrium uniqueness via demi-concavity

A family of oligopolies that possess a unique equilibrium was identified in the second authors doctoral dissertation. For such a family, it is therein specified a class of functions-economically related to the price function of a Cournot oligopoly – that satisfy a particular type of quasi-concavity. The first part of the present article (i) conceptualizes that type of quasi-concavity by introducing the notion of demi-concavity, (ii) considers two possible variants and (iii) provides some calculus properties. The second part, by relying on the results on demi-concavity, proves a Cournot equilibrium uniqueness theorem which is new for the journal literature and subsumes various results thereof. A third part shows an example that illustrates the novelty of the result.     

Equilibrium structures of two supply chains with price and displayed-quantity competition

This paper studies the equilibrium structure of two competing supply chains, each of which consists of one manufacturer and one retailer who faces the demand influenced by price and displayed quantity. Each chain has two structure options: integration or decentralization. Under linear demand, we present the optimal pricing/displayed quantity of all members in the two chains under possible structures: two integrated chains (II), two decentralized chains (DD), and one integrated chain and one decentralized chain (ID or DI). We then analyse the impact of the intensities of price and displayed-quantity competition on the equilibrium structure of two supply chains. The results show that both price and displayed-quantity competition intensities influence significantly the equilibrium structure. Moreover, under certain specific conditions, both price and displayed-quantity competition can have the two chains fall into the prisoner’s dilemma and play a game of chicken as well.     

The golden rule when preferences are time inconsistent

We investigate the classical Ramsey problem of economic growth when the planner uses non-constant discounting. It is well-known that this leads to time inconsistency, so that optimal strategies are no longer implementable. We then define equilibrium strategies to be such that unilateral deviations occurring during a small time interval are penalized. Non-equilibrium strategies are not implementable, so only equilibrium strategies should be considered by a rational planner. We show that there exists such strategies which are (a) smooth, and (b) lead to stationary growth, as in the classical Ramsey model. Finally, we prove an existence and multiplicity result: for logarithmic utility and quasi-exponential discount, there is an interval I such that, for every k in I, there is an equilibrium strategy converging to k. We conclude by giving an example where the planner is led to non-constant discount rates by considerations of intergenerational equity.     

Explicit formula for scalar non-linear conservation laws with boundary condition

We prove an uniqueness and existence theorem for the entropy weak solution of non-linear hyperbolic conservation laws of the form , with initial data and boundary condition. The scalar function u = u(x, t), x > 0, t > 0, is the unknown; the function f = f(u) is assumed to be strictly convex. We also study the weighted Burgers' equation: α ? ? . We give an explicit formula, which generalizes a result of Lax. In particular, a free boundary problem for the flux f(u(.,.)) at the boundary is solved by introducing a variational inequality. The uniqueness result is obtained by extending a semigroup property due to Keyfitz.