Juche in international trade

We show that in trade between two parties, one of whom has much greater endowment that the other, calculated by the Nash bargaining solution, tends to exploit the poorer party, as compared with the competitive equilibrium. This can be interpreted as saying that trade between a developed country and a developing nation, each acting on the basis of self-interest, will result in exploitation of the developing nation. This illustrates one aspect of the idea of Juche. We also show that if trade in two goods among two parties is extended to a third, one of the two parties will be worse off as a result under natural hypotheses. This can be interpreted as explaining why trade between a developing country and a developed country might make some groups in the developing country rich but other groups poor, a second aspect of Juche.     

Coalition-proof equilibria in a voluntary participation game

We examine the coalition-proof equilibria of a participation game in the provision of a (pure) public good. We study which Nash equilibria are achieved through cooperation, and we investigate coalition-proof equilibria under strict and weak domination. We show that under some incentive condition, (i) a profile of strategies is a coalition-proof equilibrium under strict domination if and only if it is a Nash equilibrium that is not strictly Pareto-dominated by any other Nash equilibrium and (ii) every strict Nash equilibrium for non-participants is a coalition-proof equilibrium under weak domination.     

Some interesting properties of maximin strategies

Since the seminal paper of Nash (1950) game theoretic literature has focused mostly on equilibrium and not on maximin (minimax) strategies. We study the properties of these strategies in non-zero-sum strategic games that possess (completely) mixed Nash equilibria. We find that under certain conditions maximin strategies have several interesting properties, some of which extend beyond 2-person strategic games. In particular, for n-person games we specify necessary and sufficient conditions for maximin strategies to yield the same expected payoffs as Nash equilibrium strategies. We also show how maximin strategies may facilitate payoff comparison across Nash equilibria as well as refine some Nash equilibrium strategies.     

Nash equilibrium based fairness

There are several approaches of sharing resources among users. There is a noncooperative approach wherein each user strives to maximize its own utility. The most common optimality notion is then the Nash equilibrium. Nash equilibria are generally Pareto inefficient. On the other hand, we consider a Nash equilibrium to be fair as it is defined in a context of fair competition without coalitions (such as cartels and syndicates). We show a general framework of systems wherein there exists a Pareto optimal allocation that is Pareto superior to an inefficient Nash equilibrium. We consider this Pareto optimum to be ??Nash equilibrium based fair.?? We further define a ??Nash proportionately fair?? Pareto optimum. We then provide conditions for the existence of a Pareto-optimal allocation that is, truly or most closely, proportional to a Nash equilibrium. As examples that fit in the above framework, we consider noncooperative flow-control problems in communication networks, for which we show the conditions on the existence of Nash-proportionately fair Pareto optimal allocations.     

Nash implementation of competitive equilibria in the job-matching market

This paper studies Nash implementation in the job-matching market where each worker works for only one firm and a firm hires as many workers as it wishes. We show that the competitive equilibrium correspondence (CEC) is the smallest Nash implementable correspondence satisfying individual rationality and Pareto indifference. Furthermore, the CEC is the minimal monotonic extension of the worker-optimal and firm-optimal subcorrespondences. We offer two “good” mechanisms that implement this correspondence in Nash equilibrium.     

Customers’ abandonment strategy in an <Emphasis Type="Italic">M</Emphasis> / <Emphasis Type="Italic">G</Emphasis> / 1 queue

We consider an M / G / 1 queue in which the customers, while waiting in line, may renege from it. We show the Nash equilibrium profile among customers and show that it is defined by two sequences of thresholds. For each customer, the decision is based on the observed past (which determines from what sequence the threshold is taken) and the observed queue length (which determines the appropriate element in the chosen sequence). We construct a set of equations that has the Nash equilibrium as its solution and discuss the relationships between the properties of the service time distribution and the properties of the Nash equilibrium, such as uniqueness and finiteness.     

Correlation and unmediated cheap talk in repeated games with imperfect monitoring

This paper studies n-player \((n\ge 3)\) undiscounted repeated games with imperfect monitoring. We prove that all uniform communication equilibrium payoffs of a repeated game can be obtained as Nash equilibrium payoffs of the game extended by unmediated cheap talk. We also show that all uniform communication equilibrium payoffs of a repeated game can be reached as Nash equilibrium payoffs of the game extended by a pre-play correlation device and a cheap-talk procedure that only involves public messages; furthermore, in the case of imperfect public and deterministic signals, no cheap talk is conducted on the equilibrium path.     

Semidefinite complementarity reformulation for robust Nash equilibrium problems with Euclidean uncertainty sets

Consider the N-person non-cooperative game in which each player’s cost function and the opponents’ strategies are uncertain. For such an incomplete information game, the new solution concept called a robust Nash equilibrium has attracted much attention over the past several years. The robust Nash equilibrium results from each player’s decision-making based on the robust optimization policy. In this paper, we focus on the robust Nash equilibrium problem in which each player’s cost function is quadratic, and the uncertainty sets for the opponents’ strategies and the cost matrices are represented by means of Euclidean and Frobenius norms, respectively. Then, we show that the robust Nash equilibrium problem can be reformulated as a semidefinite complementarity problem (SDCP), by utilizing the semidefinite programming (SDP) reformulation technique in robust optimization. We also give some numerical example to illustrate the behavior of robust Nash equilibria.     

Nonzero-sum non-stationary discounted Markov game model

The goal of this paper is provide a theory of K-person non-stationary Markov games with unbounded rewards, for a countable state space and action spaces. We investigate both the finite and infinite horizon problems. We define the concept of strong Nash equilibrium and present conditions for both problems for which strong Nash or Nash equilibrium strategies exist for all players within the Markov strategies, and show that the rewards in equilibrium satisfy the optimality equations.     

Generalising diagonal strict concavity property for uniqueness of Nash equilibrium

In this paper, we extend the notion of diagonally strictly concave functions and use it to provide a sufficient condition for uniqueness of Nash equilibrium in some concave games. We then provide an alternative proof of the existence and uniqueness of Nash equilibrium for a network resource allocation game arising from the so-called Kelly mechanism by verifying the new sufficient condition. We then establish that the equilibrium resulting from the differential pricing in the Kelly mechanism is related to a normalised Nash equilibrium of a game with coupled strategy space.     

On a game in manufacturing

We analyse a non-zero sum two-person game introduced by Teraoka and Yamada to model the strategic aspects of production development in manufacturing. In particular we investigate how sensitive their solution concept (Nash equilibrium) is to small variations in their assumptions. It is proved that a Nash equilibrium is unique if it exists and that a Nash equilibrium exists when the capital costs of the players are zero or when the players are equal in every respect. However, when the capital costs differ, in general a Nash equilibrium exists only when the players' capital costs are high compared to their profit rates.     

A Strong Anti-Folk Theorem

We study the properties of finitely complex, symmetric, globally stable, and semi-perfect equilibria. We show that: (1) If a strategy satisfies these properties then players play a Nash equilibrium of the stage game in every period; (2) The set of finitely complex, symmetric, globally stable, semi-perfect equilibrium payoffs in the repeated game equals the set of Nash equilibria payoffs in the stage game; and (3) A strategy vector satisfies these properties in a Pareto optimal way if and only if players play some Pareto optimal Nash equilibrium of the stage game in every stage. Our second main result is a strong anti-Folk Theorem, since, in contrast to what is described by the Folk Theorem, the set of equilibrium payoffs does not expand when the game is repeated.This paper is a revised version of Chapter 3 of my Ph.D. thesis, which has circulated under the title “An Interpretation of Nash Equilibrium Based on the Notion of Social Institutions”.     

A best-response approach for equilibrium selection in two-player generalized Nash equilibrium problems

ABSTRACT

In this paper, we propose a best-response approach to select an equilibrium in a two-player generalized Nash equilibrium problem. In our model we solve, at each of a finite number of time steps, two independent optimization problems. We prove that convergence of our Jacobi-type method, for the number of time steps going to infinity, implies the selection of the same equilibrium as in a recently introduced continuous equilibrium selection theory. Thus the presented approach is a different motivation for the existing equilibrium selection theory, and it can also be seen as a numerical method. We show convergence of our numerical scheme for some special cases of generalized Nash equilibrium problems with linear constraints and linear or quadratic cost functions.     

A globalized Newton method for the computation of normalized Nash equilibria

The generalized Nash equilibrium is a Nash game, where not only the players’ cost functions, but also the constraints of a player depend on the rival players decisions. We present a globally convergent algorithm that is suited for the computation of a normalized Nash equilibrium in the generalized Nash game with jointly convex constraints. The main tool is the regularized Nikaido–Isoda function as a basis for a locally convergent nonsmooth Newton method and, in another way, for the definition of a merit function for globalization. We conclude with some numerical results.     

Smoothing approach to Nash equilibrium formulations for a class of equilibrium problems with shared complementarity constraints

The equilibrium problem with equilibrium constraints (EPEC) can be looked on as a generalization of Nash equilibrium problem (NEP) and the mathematical program with equilibrium constraints (MPEC) whose constraints contain a parametric variational inequality or complementarity system. In this paper, we particularly consider a special class of EPECs where a common parametric P-matrix linear complementarity system is contained in all players?? strategy sets. After reformulating the EPEC as an equivalent nonsmooth NEP, we use a smoothing method to construct a sequence of smoothed NEPs that approximate the original problem. We consider two solution concepts, global Nash equilibrium and stationary Nash equilibrium, and establish some results about the convergence of approximate Nash equilibria. Moreover we show some illustrative numerical examples.     

集值映射的Nash平衡点的存在定理

本文引入了集值映射的Nash平衡点的概念，它以通常的Nash平衡点及Loose Nash平衡点为特例，并在紧和非紧的假设下，得到集值映射的Nash平衡点的存在定理，其中在非紧的情况下使用escaping序列的定义．     

相位阻尼信道条件下量子斗鸡博弈模型均衡解分析

利用量子博弈的相关理论,以噪音强度和记忆强度为参量,建立了相位阻尼信道条件下的量子斗鸡博弈模型,求出了模型的量子纳什均衡解,讨论了两参量对均衡解稳定性的影响,得出在无记忆相位阻尼信道条件下,当噪音强度小于阈值0.24时,纳什均衡仍然为帕累托最优解,当噪音强度大于0.24时,均衡解演变为另2个均衡解,不再是帕累托最优;在有记忆相位阻尼信道条件下,当噪音强度小于0.24,且记忆强度大于0.5时,均衡解是稳定的.特殊地,当信道是完全记忆时,均衡解的稳定性与噪音强度无关.     

Loss aversion equilibrium

The Nash equilibrium solution concept for games is based on the assumption of expected utility maximization. Reference dependent utility functions (in which utility is determined not only by an outcome, but also by the relationship of the outcome to a reference point) are a better predictor of behavior than expected utility. In particular, loss aversion is an important element of such utility functions.  We extend games to include loss aversion characteristics of the players. We define two types of loss-aversion equilibrium, a solution concept endogenizing reference points. The two types reflect different procedures of updating reference points during the game. Reference points emerge as expressions of anticipation which are fulfilled.  We show existence of myopic loss-aversion equilibrium for any extended game, and compare it to Nash equilibrium. Comparative statics show that an increase in loss aversion of one player can affect her and other players' payoffs in different directions. Received August 1998/Revised version February 2000     

Determining all Nash equilibria in a (bi-linear) inspection game

This paper addresses a “game” between an inspection agency and multiple inspectees that are subject to random inspections by that agency. We provide explicit (easily computable) expressions for all possible Nash equilibria and verify that none is left out. In particular, our results characterize situations when there exists a unique Nash equilibrium. We also explore special features of the Nash equilibria and the solution of the problem the inspection agency faces in a non-strategic environment.     

Uncoupled automata and pure Nash equilibria

We study the problem of reaching a pure Nash equilibrium in multi-person games that are repeatedly played, under the assumption of uncoupledness: EVERY player knows only his own payoff function. We consider strategies that can be implemented by finite-state automata, and characterize the minimal number of states needed in order to guarantee that a pure Nash equilibrium is reached in every game where such an equilibrium exists.