Two-person Nonzero-sum Game Research Articles

Open-loop and closed-loop local and remote stochastic nonzero-sum game with inconsistent information structure

In this paper, the open-loop and closed-loop local and remote stochastic nonzero-sum game (LRSNG) problem is investigated. Different from previous works, the stochastic nonzero-sum game problem under consideration is essentially a special class of two-person nonzero-sum game problem, in which the information sets accessed by the two players are inconsistent. More specifically, both the local player and the remote player are involved in the system dynamics, and the information sets obtained by the two players are different, and each player is designed to minimise its own cost function. For the considered LRSNG problem, both the open-loop and closed-loop Nash equilibrium are derived. The contributions of this paper are given as follows. Firstly, the open-loop optimal Nash equilibrium is derived, which is determined in terms of the solution to the forward and backward stochastic difference equations (FBSDEs). Furthermore, by using the orthogonal decomposition method and the completing square method, the feedback representation of the optimal Nash equilibrium is derived for the first time. Finally, the effectiveness of our results is verified by a numerical example.

A development in the existing nonlinear model and to study its impact on the Nash equilibrium in a two-person non-zero sum game

A non-zero sum two-person game can be formulated as a nonlinear program problem for finding the Nash equilibrium. Although intensive research has been done in developing methods for solving such nonlinear problems, very little work has been done in modifying the existing model in order to generate a more refined Nash solution. The study focuses on modifying the nonlinear programming model by incorporating some additional constraints and analysing if the altered model would generate a better result. The utility of the additional constraints is discussed, in terms of the payoff received by either of the players, for different problem situations.

A development in the existing nonlinear model and to study its impact on the Nash equilibrium in a two-person non-zero sum game

A non-zero sum two-person game can be formulated as a nonlinear program problem for finding the Nash equilibrium. Although intensive research has been done in developing methods for solving such nonlinear problems, very little work has been done in modifying the existing model in order to generate a more refined Nash solution. The study focuses on modifying the nonlinear programming model by incorporating some additional constraints and analysing if the altered model would generate a better result. The utility of the additional constraints is discussed, in terms of the payoff received by either of the players, for different problem situations.

On a strategic model of pollution control

This paper proposes a strategic model of pollution control. A firm, representative of the productive sector of a country, aims at maximizing its profits by expanding its production. Assuming that the output of production is proportional to the level of pollutants’ emissions, the firm increases the level of pollution. The government of the country aims at minimizing the social costs due to the pollution, and introduces regulatory constraints on the emissions’ level, which then effectively cap the output of production. Supposing that the firm and the government face both proportional and fixed costs in order to adopt their policies, we model the previous problem as a stochastic impulse two-person nonzero-sum game. The state variable of the game is the level of the output of production which evolves as a general linearly controlled one-dimensional Ito-diffusion. Following an educated guess, we first construct a pair of candidate equilibrium policies and of corresponding equilibrium values, and we then provide a set of sufficient conditions under which they indeed realize an equilibrium. Our results are complemented by a numerical study when the (uncontrolled) output of production evolves as a geometric Brownian motion, and the firm’s operating profit and the government’s running cost functions are of power type. An analysis of the dependency of the equilibrium policies and values on the model parameters yields interesting new behaviors that we explain as a consequence of the strategic interaction between the firm and the government.

The audience effect and the role of deception in the expression of male mating preferences

Males of several species have been shown to alter their mate preference in the presence of an eavesdropping rival. This evasive tactic has been interpreted as an attempt by the courting male to drive the attention of the rival away from the preferred female. The fitness return of this deceptive strategy will depend on the costs of cheating for the actor (the displayer) and the benefits for the rival (the bystander) of copying the choice of the displayer. We developed a two-person nonzero sum game between two males that compete for mating with one of two receptive females. Males could assess female quality with a varying level of uncertainty, which was modelled using a Bayesian statistical decision theory approach. We explored the actor and bystander payoffs under different levels of uncertainty in mate assessment and difference in quality between females. We found that when being eavesdropped on is costly (i.e. when females differ largely in quality), males are expected to cheat to reduce the amount of public information that is available to the unintended audience. However, under these circumstances, the value of the public information is low and the bystander is not expected to copy the choice of the actor. Our model suggests that deceptive male choice may evolve only under relatively restricted conditions and suggest that other explanations, such as, for example, a reduction in the risk of precopulatory male–male competition may be more likely. Future theoretical and empirical work will be necessary to test alternative interpretations of the audience effects in male mate choice.

Distribution Design of Two Rival Decenteralized Supply Chains: a Two-person Nonzero Sum Game Theory Approach

We consider competition between two decentralized supply chains network under demand uncertainty. Each chain consists of one risk-averse manufacturer and a group of risk-averse retailers. These two chains present substitutable products to the geographical dispensed markets. The markets’ demands are contingent upon prices, service levels, and advertising efforts of two supply chains. We formulate the distribution design problem of two rival chains as a two-person nonzero sum game. Since strategic decisions of distribution designs often have priority over tactical ones, we first calculate the equilibrium of tactical decisions for each pair of distribution design scenarios. Then, according to the presented methodology, we find Nash equilibrium solutions of distribution network scenarios for two rival chains. Eventually, the research has concluded with a numerical example in order to illustrate the real applications of the methodology.

A Game Theoretic Model for Trading Pollution Discharge Permits in River Systems

In this paper, by combining a two-person nonzero-sum game, a multi-objective genetic algorithm and a cooperative game, we present a new game theoretic methodology for trading pollution discharge permits in rivers. A trade-off curve between the average treatment level of dischargers and fuzzy risk of low water quality is gained using the optimization model. Then, by using the two-person nonzero-sum game, the best non-dominated solution is chosen from the trade-off curve. The treatment costs of dischargers corresponding to the selected solution are reallocated among dischargers participating in a coalition and side payments are calculated. The proposed model is applied to the Zarjub River in Iran to illustrate its efficiency and applicability.

Infinitesimal characterization of a feedback nash equilibrium in a differential game

A two-person nonzero-sum game is considered. An infinitesimal condition on the set of payoffs in the situation of the Nash equilibrium is proposed. This condition is formulated in terms of the derivative of a set-valued map along a set of directions.

Nondominated equilibrium solutions of a multiobjective two-person nonzero-sum game in extensive form and corresponding mathematical programming problem

In most of studies on multiobjective noncooperative games, games are represented in normal form and a solution concept of Pareto equilibrium solutions which is an extension of Nash equilibrium solu...

Nondominated Equilibrium Solutions of a Multiobjective Two-Person Nonzero-Sum Game and Corresponding Mathematical Programming Problem

With reference to a multiobjective two-person nonzero-sum game, we define nondominated equilibrium solutions and provide a necessary and sufficient condition for a pair of mixed strategies to be a nondominated equilibrium solution. Using the necessary and sufficient condition, we formulate a mathematical programming problem yielding nondominated equilibrium solutions. We give a numerical example and demonstrate that nondominated equilibrium solutions can be obtained by solving the formulated mathematical programming problem.

Extra-proximal methods for solving two-person nonzero-sum games

We consider two-person nonzero-sum game, both in the classical form and in the form of a game with coupled variables. An extra-proximal approach for finding the game’s solutions is suggested and justified. We provide our algorithm with an analysis of its convergence.

Software Release Games

The software release game developed in Ref. 1 is reconsidered in the framework of a two-person nonzero-sum game of timing. More precisely, noisy-type software release strategies are derived in closed form under two different criteria as well as an alternative silent-type software release strategy. Our method overcomes the fatal problem in Ref. 1 and has an advantage on computational tractability. Also, the method can be extended directly to obtain noisy-type strategies.

A game theoretical analysis of the quantity discount problem with perfect and incomplete information about the buyer's cost structure

In this paper, we analyze the quantity discount problem by considering the competitive nature of the problem and the informational structure regarding the buyer's cost structure. We formulate the problem as two-person nonzero-sum game and analyze the seller's optimal quantity discount schedule and the buyer's optimal order quantity by using Stackelberg equilibrium. We show that it is always possible for the seller and the buyer to gain from quantity discount. However, a quantity discount schedule under which the buyer orders more than his EOQ at the discounted price is necessary for the seller and the buyer to gain. The optimal quantity discount schedule when the seller knows the buyer's cost parameters is given by a single break point. When the seller does not know the buyer's cost parameters, an optimal quantity discount schedule may not exist. Two approaches have been developed for the seller to offer quantity discount in this case. The application of our analysis is discussed. Our results can be especially useful when the seller has many buyers.

An "Informationally Robust Equilibrium" for Two-Person Nonzero-Sum Games

This paper considers the implications of explicit slight uncertainty about the strategic sequence of moves, that is, about the informational structure, for a two-person nonzero-sum game. It is shown that the set of associated informationally robust equilibria must be nonempty and must be a subset of the set of Nash equilibria. Although the present approach is conceptually reminiscent of that underlying perfectness, the two concepts are shown to be logically unrelated. There may exist no equilibrium which is both perfect and informationally robust. Journal of Economic Literature Classification Number: C72.

Markov stopping games with random priority

In the paper a construction of Nash equilibria for a random priority finite horizon two-person non-zero sum game with stopping of Markov process is given. The method is used to solve the two-person non-zero-sum game version of the secretary problem. Each player can choose only one applicant. If both players would like to select the same one, then the lottery chooses the player. The aim of the players is to choose the best candidate. An analysis of the solutions for different lotteries is given. Some lotteries admit equilibria with equal Nash values for the players.

An algorithm for the mixed-integer nonlinear bilevel programming problem

The bilevel programming problem (BLPP) is a two-person nonzero sum game in which play is sequential and cooperation is not permitted. In this paper, we examine a class of BLPPs where the leader controls a set of continuous and discrete variables and tries to minimize a convex nonlinear objective function. The follower's objective function is a convex quadratic in a continuous decision space. All constraints are assumed to be linear. A branch and bound algorithm is developed that finds global optima. The main purpose of this paper is to identify efficient branching rules, and to determine the computational burden of the numeric procedures. Extensive test results are reported. We close by showing that it is not readily possible to extend the algorithm to the more general case involving integer follower variables.

A Game of Timing which Shifts Stochastically from a Noisy Version, to a Silent Version

Abstract We consider a two-person non-zero sum game of timing in which the state of bullets possessed by both players shifts stochastically from noisy to silent at random time T in [0, 1] with cdf K(t). The problem is a continuation of the previous paper where the state of the bullets shifted stochastically from silent to noisy. It is shown that the change of information structure available to both the players yields interesting equilibrium strategies. Furthermore, we give remarks on a case where the switching time from a noisy to silent state is not a random variable but a deterministic time in [0, 1], though we cannot derive an equilibrium point except in special cases.

An Optimal Facility Planning for Containerized Shipping under Multi-carrier Competition

Sea-borne container carriers are competing too hard each other on major container lines like the line between Far East and North America. They are seeking the effective planning method for the scale of transportation facilities such as ships, container terminals and containers. There are a few studies on planning these facilities under a given cargo which a carrier should transport without competitors.In this study, we suggest a facility planning model with multi-carrier in order to determine the strategy of carriers in actual competition. In this model, the optimal service level and scale of facilities are determined by the two-person nonzero-sum game with payoff matrix given by two sub-models:(a) shipper's behavior sub-model, (b) carrier's facility planning sub-model.Some case studies show the effectiveness of this model.

A Game of Timing in whichthePlayers Improve their Information Patterns

Abstract We consider a two-person nonzero sum game of timing in which the stales of bullets possessed by both players shift stochastically from silent to noisy at random time T in [0, 1] with cdf H(t). The model is evidently an exlension of the classical games of timing to a nonzero sum version under an uncertain information structure. It is shown that the shift of information structure available to the both players yields interesting equilibrium strategies.

A game theoretic model for two types of customers competing for service

Two types of customers arrive at a single server station and demand service. If a customer finds the server busy upon arrival (or retrial) he immediately departs and conducts a retrial after an exponential period of time and persists this way until he gets served. Both types of customers face linear costs for waiting and conducting retrials and wish to find optimal retrial rates which will minimize these costs. This problem is analysed as a two-person nonzero sum game. Both noncooperative strategies are studied.