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Two-Person Zero-Sum Games

Alan Washburn
Publication Date: 
Number of Pages: 
International Series in Operations Research and Management Science
[Reviewed by
Charles Ashbacher
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A two-person zero-sum (TPZS) game is one where the consequence of each action (assuming it results in some change) is that what one player gains is what the other player looses. They are generally represented as a matrix of payoffs based on the various available moves. For convenience of analysis, we assume one player wants to maximize their holdings or, failing that, to minimize their losses, while the other wants to do the reverse. Hence, the appropriate use of the terms maxmin and minmax in describing “strategies.” As Washburn points out, the term “strategy” is misapplied to a situation where there are rules with such specific results, yet by long-standing convention we are stuck with it.

At first glance it would seem that TPZS games are a very restrictive subset of the broad application of game theory. While that is true in the strictest interpretation, the strength of this book is the reality that to a first approximation many more complex situations can be effectively modeled by a TPZS game.

The first example given by Washburn is that of a submarine attacking a convoy of ships protected by a ring of destroyers. While the attack could be by a so-called wolf pack of submarines and the destroyers would cooperate in the defense of the convoy, locally the fight comes down to one sub against one ship, at least in the short term. In many, if not most, cases that is the critical timeframe, for if the sub evades the first responder, it is likely to get safely away. This situation can be effectively modeled as a TPZS game of action versus reaction with one player making the first move.

While the military applications are significant, the most currently relevant material is the chapter on network interdiction, where the goal of one player is to maximize the flow while the goal of the other is the reverse. If one player is a software-driven communications network and the other is a software-driven malicious entity, then the complexity of of the situation is reduced to a TPZS game. Given the enormous value of modern communication networks, estimates are that $2.2 quadrillion were electronically transferred in 2008, the importance of this is obvious.

An excellent and properly expansive look at TPZS games, this book is a suitable textbook for classes in game theory. There are exercises at the end of the chapters with solutions to all included in an appendix. 

Charles Ashbacher splits his time between consulting with industry in projects involving math and computers, teaching college classes and co-editing The Journal of Recreational Mathematics. In his spare time, he reads about these things and helps his daughter in her lawn care business.

Single Person Background

Maxmin Versus Minmax

Matrix Games

Markov (Multistage) Games

Games with a Continuum of Strategies

Blotto Games

Network Interdiction

Search Games

Miscellaneous Games.