Two-person Zero-sum Game Research Articles

On a dynamic fractional game

Consider a two-person zero-sum game constructed by a dynamic fractional form. We establish the upper value as well as the lower value of a dynamic fractional game, and prove that the dual gap is equal to zero under certain conditions. It is also established that the saddle point function exists in the fractional game system under certain conditions so that the equilibrium point exists in this game system.

On the optimal strategy in a random game

Consider a two-person zero-sum game played on a random $n$ by $n$ matrix where the entries are iid normal random variables. Let $Z$ be the number of rows in the support of the optimal strategy for player I given the realization of the matrix. (The optimal strategy is a.s. unique and $Z$ a.s. coincides with the number of columns of the support of the optimal strategy for player II.) Faris an Maier (see the references) make simulations that suggest that as $n$ gets large $Z$ has a distribution close to binomial with parameters $n$ and 1/2 and prove that $P(Z=n) \lt 2^{-(k-1)}$. In this paper a few more theoretically rigorous steps are taken towards the limiting distribution of $Z$: It is shown that there exists $a \lt 1/2$ (indeed $a \lt 0.4$) such that $P((1/2-a)n \lt Z \lt (1/2+a)n)$ tends to 1 as $n$ increases. It is also shown that the expectation of $Z$ is $(1/2+o(1))n$. We also prove that the value of the game with probability $1-o(1)$ is at most $Cn^{-1/2}$ for some finite $C$ independent of $n$. The proof suggests that an upper bound is in fact given by $f(n)/n$, where $f(n)$ is any sequence tending to infinity as $n$ increases, and it is pointed out that if this is true, then the variance of $Z$ is $o(n^2)$ so that any $a \gt 0$ will do in the bound on $Z$ above.

Hide and seek in Arizona

Laboratory subjects repeatedly played one of two variations of a simple two-person zero-sum game of “hide and seek”. Three puzzling departures from the prescriptions of equilibrium theory are found in the data: an asymmetry related to the player’s role in the game; an asymmetry across the game variations; and positive serial correlation in subjects’ play. Possible explanations for these departures are considered.

Two-person zero-sum markov games: receding horizon approach

We consider a receding horizon approach as an approximate solution to two-person zero-sum Markov games with infinite horizon discounted cost and average cost criteria. We first present error bounds from the optimal equilibrium value of the game when both players take "correlated" receding horizon policies that are based on exact or approximate solutions of receding finite horizon subgames. Motivated by the worst-case optimal control of queueing systems by Altman, we then analyze error bounds when the minimizer plays the (approximate) receding horizon control and the maximizer plays the worst case policy. We finally discuss some state-space size independent methods to compute the value of the subgame approximately for the approximate receding horizon control, along with heuristic receding horizon policies for the minimizer.

Computational techniques for the verification of hybrid systems

Hybrid system theory lies at the intersection of the fields of engineering control theory and computer science verification. It is defined as the modeling, analysis, and control of systems that involve the interaction of both discrete state systems, represented by finite automata, and continuous state dynamics, represented by differential equations. The embedded autopilot of a modern commercial jet is a prime example of a hybrid system: the autopilot modes correspond to the application of different control laws, and the logic of mode switching is determined by the continuous state dynamics of the aircraft, as well as through interaction with the pilot. To understand the behavior of hybrid systems, to simulate, and to control these systems, theoretical advances, analyses, and numerical tools are needed. In this paper, we first present a general model for a hybrid system along with an overview of methods for verifying continuous and hybrid systems. We describe a particular verification technique for hybrid systems, based on two-person zero-sum game theory for automata and continuous dynamical systems. We then outline a numerical implementation of this technique using level set methods, and we demonstrate its use in the design and analysis of aircraft collision avoidance protocols and in verification of autopilot logic.

Zero-sum games for continuous-time Markov chains with unbounded transition and average payoff rates

This paper is a first study of two-person zero-sum games for denumerable continuous-time Markov chains determined by given transition rates, with an average payoff criterion. The transition rates are allowed to be unbounded, and the payoff rates may have neither upper nor lower bounds. In the spirit of the ‘drift and monotonicity’ conditions for continuous-time Markov processes, we give conditions on the controlled system's primitive data under which the existence of the value of the game and a pair of strong optimal stationary strategies is ensured by using the Shapley equations. Also, we present a ‘martingale characterization’ of a pair of strong optimal stationary strategies. Our results are illustrated with a birth-and-death game.

Zero-sum games for continuous-time Markov chains with unbounded transition and average payoff rates

This paper is a first study of two-person zero-sum games for denumerable continuous-time Markov chains determined by given transition rates, with an average payoff criterion. The transition rates are allowed to be unbounded, and the payoff rates may have neither upper nor lower bounds. In the spirit of the ‘drift and monotonicity’ conditions for continuous-time Markov processes, we give conditions on the controlled system's primitive data under which the existence of the value of the game and a pair of strong optimal stationary strategies is ensured by using the Shapley equations. Also, we present a ‘martingale characterization’ of a pair of strong optimal stationary strategies. Our results are illustrated with a birth-and-death game.

On finite strategy sets for finitely repeated zero-sum games

We study finitely repeated two-person zero-sum games in which Player 1 is restricted to mixing over a fixed number of pure strategies while Player 2 is unrestricted. We describe an optimal set of pure strategies for Player 1 along with an optimal mixed strategy. We show that the entropy of this mixed strategy appears as a factor in an exact formula for the value of the game and thus is seen to have a direct numerical effect on the game's value. We develop upper and lower bounds on the value of these games that are within an additive constant and discuss how our results are related to the work of Neyman and Okada on strategic entropy (Neyman and Okada, 1999, Games Econ. Behav. 29, 191–223). Finally, we use these results to bound the value of repeated games in which one of the players uses a computer with a bounded memory and is further restricted to using a constant amount of time at each stage.

Raid games across a set with cyclic order

In this paper we shall deal with two-person zero-sum games where the strategies of the players are intervals on a set with cyclic order, and the payoff function is a function depending on the intersection of the strategies for the players. These games correspond to similar tactical problems. We give a general method to obtain a solution of the discrete games of this kind. This method is applied to solve the games with monotonic payoff function and a game with no monotonic one. We also solve a continuous game.

A policy-improvement type algorithm for solving zero-sum two-person stochastic games of perfect information

We give a policy-improvement type algorithm to locate an optimal pure stationary strategy for discounted stochastic games with perfect information. A graph theoretic motivation for our algorithm is presented as well.

The game for the speed of convergence in repeated games of incomplete information

We consider an infinitely repeated two-person zero-sum game with incomplete information on one side, in which the maximizer is the (more) in- formed player. Such games have value vyð pÞ for all 0 a p a 1. The informed player can guarantee that all along the game the average payoper stage will be greater than or equal to vyðpÞ (and will converge from above to vyðpÞ if the minimizer plays optimally). Thus there is a conflict of interest between the two players as to the speed of convergence of the average payos-to the value vyð pÞ. In the context of such repeated games, we define a game for the speed of convergence, denoted SGyðpÞ, and a value for this game. We prove that the value exists for games with the highest error term, i.e., games in which vnðpÞ� vyðpÞ is of the order of magnitude of 1ffiffi

Notes on a stochastic game with information structure

The game described below appeared in conversations after a talk given by Lloyd Shapley at the Second International Workshop in Game Theory in Berkeley, Summer 1970. This was one of the problems circulating around the conference to which several participants contributed. The general problem is to study two-person zero-sum games with stochastic movement among subgames in which the subgame being played is not precisely known, but in which information about this aspect may be gathered or partially revealed by the strategy choices of the players, the actual game being played, and chance. Without stochastic movement among the subgames, such a game would be a repeated game of incomplete information defined and studied by the Mathematica Inc. group at Princeton 1966–1968; see the book of Aumann and Maschler, ‘‘Repeated Games with Incomplete Information’’, MIT Press, 1995. Granted that the problem is interesting, one searches for the simplest non-trivial example of the class and is led to the following game. Subgames. There are two basic subgames, G1 with matrix ð0 1Þ, and G2

On characterization of equilibrium strategy of two-person zero-sum games with fuzzy payoffs

In this paper, we consider fuzzy matrix games, namely, two-person zero-sum games with fuzzy payoffs. Based on fuzzy max order, for such games, we define three kinds of concepts of minimax equilibrium strategies and investigate their properties. First, we shall show that these equilibrium strategies are characterized as Nash equilibrium strategies of a family of parametric bi-matrix games with crisp payoffs. Second, we investigate properties of values of fuzzy matrix games by means of possibility and necessity measures. In addition, we give a numerical example to illustrate utility of our approaches.

Supermodular games and potential games

Potential games and supermodular games are attractive games, especially because under certain conditions they possess pure Nash equilibria. Subclasses of games with a potential are considered which are also strategically equivalent to supermodular games. The focus is on two-person zero-sum games and two-person Cournot games.

Which four-part partition games are isomorphic to Silverman's game?

A four-part partition game on a rectangle is a two-person zero-sum game where the pure strategy sets are intervals and the resulting rectangle is partitioned by three curves into four regions, on each of which the payoff is constant. Such a game is essentially a generalization of Silverman's game, which has been studied extensively by the author and others. It has been shown previously that if two of the partitioning curves of suitable type are given, and the four payoffs are suitably related, there is a unique choice of the third curve for which the game is isomorphic to Silverman's game. In the present note this third curve is expressed explicitly in terms of the first two, provided that one of the two given curves is the middle one of the three.

Being in the Right Place at the Right Time

Being in the Right Place at the Right Time

DISCRETE SEARCH ALLOCATION GAME WITH ENERGY CONSTRAINTS

This paper deals with a search game. In a search space, a target wants to avoid a searcher by selecting his path. The searcher has superior mobility and makes effort to detect the target by distributing divisible searching effort anywhere he wants. The target might move diffusively from the starting points but his motion is restricted by some constraints on his maximum speed and energy consumption. The searcher also has limits on the total amount of searching effort. A payoff of the game is assumed to be the detection probability of the target, which is represented by an exponential function of the cumulative searching effort weighted by the probability distribution of the target. Regardless of the payoff function, we name the game the search allocation game with energy constraints. We formulate it as a two-person zero-sum game and propose a linear programming method to solve it. Our formulation and method have the flexibility to be applied to other search models by a small modification.

On One Feature of Multicriteria Differential Games

Differential two-person zero-sum games with a vector payoff function are considered. A counterexample states that a payoff function component convolution into a linear convolution and further finding saddle point results in the interior instability of a set of such solutions. It is found that such saddle points are Geoffrion saddle points for an initial multicriteria game.

Games solved: Now and in the future

In this article we present an overview on the state of the art in games solved in the domain of two-person zero-sum games with perfect information. The results are summarized and some predictions for the near future are given. The aim of the article is to determine which game characteristics are predominant when the solution of a game is the main target. First, it is concluded that decision complexity is more important than state-space complexity as a determining factor. Second, we conclude that there is a trade-off between knowledge-based methods and brute-force methods. It is shown that knowledge-based methods are more appropriate for solving games with a low decision complexity, while brute-force methods are more appropriate for solving games with a low state-space complexity. Third, we found that there is a clear correlation between the first-player's initiative and the necessary effort to solve a game. In particular, threat-space-based search methods are sometimes able to exploit the initiative to prove a win. Finally, the most important results of the research involved, the development of new intelligent search methods, are described.

Some game theory anecdotes

After 1944, economists learned about Johnny von Neumann’s two-person zero-sum game theory and Nash’s non-cooperative game solution. The present discussion links the Nash analysis back to much earlier 1838 Cournot duopoly theory. And it links the von Neumann breakthrough to earlier and independent perceptions of how randomization provides existent minimax solution to a two-person zero-sum game.