Two-person Zero-sum Game Research Articles

Patrolling a perimeter

In this paper we study the problem of patrolling a perimeter. The general situation considered here can correspond to different tactical problems and it is studied from the point of view of game theory. To put the ideas in a context we describe it as follows. An intruder seeks to carry out a sabotage on the perimeter of a protected zone. He has to perform the action along n consecutive days and has to position himself each day at one of m strategic points placed on this border. The first day he can take his place at any of the m points, but on successive days he can move only to adjacent points. Furthermore, the perimeter is protected by a patroller, who will select each day one of the m points to inspect. The strategic situation is modeled as a two-person zero-sum game, which is developed on a cyclic set of m points over n time units. We prove some interesting properties of the strategies, solve the game in closed form under certain constraints and obtain bounds for the value of the game in several non-solved cases.

The equivalence of linear programs and zero-sum games

In 1951, Dantzig showed the equivalence of linear programming problems and two-person zero-sum games. However, in the description of his reduction from linear programs to zero-sum games, he noted that there was one case in which the reduction does not work. This also led to incomplete proofs of the relationship between the Minimax Theorem of game theory and the Strong Duality Theorem of linear programming. In this note, we fill these gaps.

CSI Usage over Parallel Fading Channels under Jamming Attacks: A Game Theory Study

Consider a parallel channel with M independent flat-fading subchannels. There exists a smart jammer which has possession of a copy of perfect channel state information (CSI) measured and sent back by a receiver to its transmitter. Under this model, a class of two-person zero-sum games is investigated where either achievable mutual information rate or Chernoff bound is taken as the underlying pay-off function with the strategy space of each player determined by respective power control and hopping functions. More specifically, we have tackled and answered the following three fundamental questions. The first one is about whether the transmitter and jammer should hop or fully use all degrees of freedom over the entire parallel channels given the full CSI available to both of them, i.e. to hop or not to hop. The second question is about the impact of sending back CSI on system performance considering that the smart jammer can exploit CSI to further enhance its interference effects, i.e. to feedback or not to feedback. The last question is about whether the amount of feedback information can be reduced given the mutual restrictions between transmitter and jammer, i.e. when to feedback and when not to.

The MIMO Radar and Jammer Games

The interaction between a smart target and a smart MIMO radar is investigated from a game theory perspective. Since the target and the radar form an adversarial system, their interaction is modeled as a two-person zero-sum game. The mutual information criterion is used in formulating the utility functions. The unilateral, hierarchical, and symmetric games are studied, and the equilibria solutions are derived.

A Fictitious Play Algorithm for Matrix Games with Fuzzy Payoffs

Fuzzy matrix games, specifically two‐person zero‐sum games with fuzzy payoffs, are considered. In view of the parametric fuzzy max order relation, a fictitious play algorithm for finding the value of the game is presented. A numerical example to demonstrate the presented algorithm is also given.

A GameTheoretic Model for Defense of an Oceanic Bastion Against Submarines

Abstract : We develop a game-theoretic model called BASTION to guide the employment of antisubmarine warfare (ASW) platforms such as ships and aircraft that are defending a stationary oceanic bastion from attack by hostile submarines. The model is an example of a two-person zero-sum game with some additional variables (ship locations) that are under the control of the maximizing defender, but known to the minimizing attacker. The attacker knowing the ship locations, but not the locations of other platforms such as aircraft, must select a path to the bastion. The probability of detecting the attacker as it follows this path is the objective shared by the opponents.

In This Issue

In This Issue

On the Saddle-Point Solution and the Large-Coalition Asymptotics of Fingerprinting Games

We study a fingerprinting game in which the number of colluders and the collusion channel are unknown. The encoder embeds fingerprints into a host sequence and provides the decoder with the capability to trace back pirated copies to the colluders. Fingerprinting capacity has recently been derived as the limit value of a sequence of maximin games with mutual information as their payoff functions. However, these games generally do not admit saddle-point solutions and are very hard to solve numerically. Here under the so-called Boneh-Shaw marking assumption, we reformulate the capacity as the value of a single two-person zero-sum game, and show that it is achieved by a saddle-point solution. If the maximal coalition size is k and the fingerprinting alphabet is binary, we show that capacity decays quadratically with k. Furthermore, we prove rigorously that the asymptotic capacity is 1/(k^2 2ln2) and we confirm our earlier conjecture that Tardos' choice of the arcsine distribution asymptotically maximizes the mutual information payoff function while the interleaving attack minimizes it. Along with the asymptotic behavior, numerical solutions to the game for small k are also presented.

Nash equilibrium strategy for fuzzy non-cooperative games

In this article, two-person zero-sum games are investigated in the fuzzy environment. Several models constructed by Maeda in the symmetrical fuzzy environment are extended to the models in the asymmetric fuzzy environment. The existence of equilibrium strategies for these extended models is proposed in the asymmetric fuzzy environment. However, in some cases, Nash equilibrium strategies may not exist. Therefore, two special cases are presented for which Nash equilibrium strategies do exist. In order to investigate the existence of (weak) Pareto Nash equilibrium strategies for fuzzy matrix games, we introduce the concept of crisp bi-matrix games with parameters. By solving the parametric bi-matrix games, we obtain the (weak) Pareto Nash equilibrium strategies for the fuzzy matrix games.

New optimality conditions for average-payoff continuous-time Markov games in Polish spaces

This paper concerns two-person zero-sum games for a class of average-payoff continuous-time Markov processes in Polish spaces. The underlying processes are determined by transition rates that are allowed to be unbounded, and the payoff function may have neither upper nor lower bounds. We use two optimality inequalities to replace the so-called optimality equation in the previous literature. Under more general conditions, these optimality inequalities yield the existence of the value of the game and of a pair of optimal stationary strategies. Under some additional conditions we further establish the optimality equation itself. Finally, we use several examples to illustrate our results, and also to show the difference between the conditions in this paper and those in the literature. In particular, one of these examples shows that our approach is more general than all of the existing ones because it allows nonergodic Markov processes.

Obtaining a fair arbitration outcome

The ‘minimax theorem’ is the most recognized theorem for determining strategies in a two-person zero-sum game. Other common strategies exist such as the ‘maximax principle’ and ‘minimize the maximum regret principle’. All these strategies follow the Von Neumann and Morgenstern linearity axiom which states that numbers in the game matrix must be cardinal utilities and can be transformed by any positive linear function f (x) = ax +b, a > 0, without changing the information they convey. This paper describes risk-averse strategies for a two-person zero-sum game where the linearity axiom may not hold. With connections to gambling theory, there is evidence to show why it can be optimal for the favourable player to adopt risk-averse strategies. Based on this approach, an arbitration value is obtained in a litigation game, where the amount awarded to the victim is less than expectation and shown to be ‘fairer’ when compared with the amount obtained using the Von Neumann and Morgenstern game theory framework.

A robust von Neumann minimax theorem for zero-sum games under bounded payoff uncertainty

The celebrated von Neumann minimax theorem is a fundamental theorem in two-person zero-sum games. In this paper, we present a generalization of the von Neumann minimax theorem, called robust von Neumann minimax theorem, in the face of data uncertainty in the payoff matrix via robust optimization approach. We establish that the robust von Neumann minimax theorem is guaranteed for various classes of bounded uncertainties, including the matrix 1-norm uncertainty, the rank-1 uncertainty and the columnwise affine parameter uncertainty.

Solving Two-Person Zero-Sum Game by Matlab

In this article we present an overview on two-person zero-sum games, which play a central role in the development of the theory of games. Two-person zero-sum games is a special class of game theory in which one player wins what the other player loses with only two players. It is difficult to solve 2-person matrix game with the order m×n(m≥3,n≥3). The aim of the article is to determine the method on how to solve a 2-person matrix game by linear programming function linprog() in matlab. With linear programming techniques in the Matlab software, we present effective method for solving large zero-sum games problems.

Pure and applied science

Pure and applied science

A foraging problem: Sit-and-wait versus active predation

The literature on foraging shows that some predators use a combination of ambush and active search to locate a prey. Let us suppose that a prey must go every day to some determined places to feed, and to another place, 0, to drink. A predator can stay at zone 0 waiting for the prey (sit-and-wait strategy) or it can move between the different places where the prey will go to eat (search strategy). If predator and prey meet each other in the same place, prey will be caught with a probability depending on the place. We study this problem in different situations, modelling them as two-person zero-sum games. We solve them in closed form, giving optimal strategies for prey and for predator and the value of the games.

First-order algorithm with $${\mathcal{O}({\rm ln}(1{/}\epsilon))}$$ convergence for $${\epsilon}$$ -equilibrium in two-person zero-sum games

We propose an iterated version of Nesterov’s first-order smoothing method for the two-person zero-sum game equilibrium problem $$\min_{x \in Q_1}\max_{y \in Q_2} {x}^{\rm T}{Ay} = \max_{y \in Q_2} \min_{x \in Q_1} {x}^{\rm T}{Ay}.$$ This formulation applies to matrix games as well as sequential games. Our new algorithmic scheme computes an $${\epsilon}$$ -equilibrium to this min-max problem in $${\mathcal {O}\left(\frac{\|A\|}{\delta(A)} \, {\rm ln}(1{/}\epsilon)\right)}$$ first-order iterations, where δ(A) is a certain condition measure of the matrix A. This improves upon the previous first-order methods which required $${\mathcal {O}(1{/}\epsilon)}$$ iterations, and it matches the iteration complexity bound of interior-point methods in terms of the algorithm’s dependence on $${\epsilon}$$. Unlike interior-point methods that are inapplicable to large games due to their memory requirements, our algorithm retains the small memory requirements of prior first-order methods. Our scheme supplements Nesterov’s method with an outer loop that lowers the target $${\epsilon}$$ between iterations (this target affects the amount of smoothing in the inner loop). Computational experiments both in matrix games and sequential games show that a significant speed improvement is obtained in practice as well, and the relative speed improvement increases with the desired accuracy (as suggested by the complexity bounds).

Solutions for fuzzy matrix games

In this paper, we deal with two-person zero-sum games with fuzzy payoffs and fuzzy goals. We have presented two models for studying two-person zero-sum matrix games with fuzzy payoffs and fuzzy goals. We assume that each player has a fuzzy goal for each of the payoffs. We obtained that the fuzzy relation approach and the max–min solution are equivalent.

Repeated games with public uncertain duration process

We consider repeated games where the number of repetitions θ is unknown. The information about the uncertain duration can change during the play of the game. This is described by an uncertain duration process Θ that defines the probability law of the signals that players receive at each stage about the duration. To each repeated game Γ and uncertain duration process Θ is associated the Θ-repeated game ΓΘ. A public uncertain duration process is one where the uncertainty about the duration is the same for all players. We establish a recursive formula for the value VΘ of a repeated two-person zero-sum game ΓΘ with a public uncertain duration process Θ. We study asymptotic properties of the normalized value vΘ = VΘ/E(θ) as the expected duration E (θ) goes to infinity. We extend and unify several asymptotic results on the existence of lim vn and lim vλ and their equality to lim vΘ. This analysis applies in particular to stochastic games and repeated games of incomplete information.

Data-based Optimal Control for Discrete-time Zero-sum Games of 2-D Systems Using Adaptive Critic Designs

In this paper, an iterative adaptive critic design (ACD) algorithm is proposed to solve a class of discrete-time two-person zero-sum games for Roesser type 2-D system. The idea is to use adaptive critic technique to obtain the optimal control pair iteratively to make the performance index function reach the saddle point of the zero-sum games. The proposed iterative ACD algorithm can be implemented based on the input and state data without the system model. Stability analysis of the 2-D system is presented and the convergence property of the performance index function is also proved. Neural networks are used to approximate the performance index function and compute the optimal control policies, respectively, for facilitating the implementation of the iterative ACD algorithm. The optimal control scheme of the air drying process is given to illustrate the performance of the proposed method.

Complexity and effective prediction

Let G = 〈 I , J , g 〉 be a two-person zero-sum game. We examine the two-person zero-sum repeated game G ( k , m ) in which players 1 and 2 place down finite state automata with k , m states respectively and the payoff is the average per-stage payoff when the two automata face off. We are interested in the cases in which player 1 is “smart” in the sense that k is large but player 2 is “much smarter” in the sense that m ≫ k . Let S ( g ) be the value of G where the second player is clairvoyant, i.e., would know player 1's move in advance. The threshold for clairvoyance is shown to occur for m near min ( | I | , | J | ) k . For m of roughly that size, in the exponential scale, the value is close to S ( g ) . For m significantly smaller (for some stage payoffs g) the value does not approach S ( g ) .