509). Likewise, in a conventional military context, Davis and Shapiro describe anti-access and area denial as being "cost-imposing strategies," a description these authors suggest is particularly useful in the counterterrorism context. In addition, game theory can help avoid military confrontations altogether, thereby avoiding unnecessary friendly casualties. In this regard, Schofield (1999) emphasizes that, "The inevitability of armed conflict in the classical sense is not a foregone conclusion in a terrorist incident, but hinges on many variables" (p. 8). Finally, as Davis and Shapiro (2003) point out, "Projecting and sustaining power in distant antiaccess and area-denial environments is now one of the Department of Defense's key operational goals of the military transformation" (p. 42). In this environment, identifying more effective approaches to the management of these battlefield scenarios has assumed new relevance and importance.
Scope of Study
The scope of the proposed study will extend to an analysis of relevant resources published within the past 10 years (except for historical references) and in the English language.
Rationale of Study
Because resources are by definition scarce, it is vitally important for policymakers at all levels to make the most of the resources they possess and game theory appears to represent a valuable technique for maximizing the effectiveness of weapon systems today. According to Kreps (1990), "Game theory comprises formal mathematical models of 'games' that are examined deductively" (p. 7). Moreover, Read emphasizes that, "Game theory can be used to analyze the ordinary events of humans" (p. 466). Just as with more traditional economic theory, the advantages of using game theory include the following:
Game theory provides a clear and precise language for communicating insights and notions. In particular, it provides us with general categories of assumptions so that insights and intuitions can be transferred from one context to another and can be cross-checked between different contexts.
Game theory provides the ability to subject particular insights and intuitions to the test of logical consistency.
Game theory helps trace back from observations to underlying assumptions to determine what assumptions are really at the heart of particular conclusions (Kreps, 1990).
These advantages are particularly important in the context of waging war on non-state actors where the tactics involved may be focused on persistent area denial methods. As Davis and Shapiro (2003) emphasize, "When considering the future of rapid strike operations against the full range of terrorist targets, U.S. military planners must assume that adversary adaptations will include uniquely suited forms of antiaccess and area denial" (p. 43). Such uniquely suited forms of antiaccess and area denial tactics will require a more comprehensive approach than sheer firepower alone will provide. As Davis and Shapiro conclude, "These adaptations, together with the demand by senior policymakers to have viable military options against such targets, suggest that new combinations of combat power and high responsiveness may be necessary to deal with such contingencies" (p. 43).
Review of the Literature
Although game theory has received an increasing amount of attention in recent years, the concept actually originated in the 17th century by mathematicians seeking to solve the gambling problems associated with French nobility with too much time on their hands (Kelly, 2004). Originally, game theory was primarily concerned with two-person zero-sum interactions based on its origins in parlor games such as chess and cards (Kelly). According to Flanagan (1998), more recently, "Game theory emerged as a distinct intellectual enterprise in 1944 with the publication of the Theory of Games and Economic Behavior, by John yon Neumann and Oskar Morgenstern. Its maturity was signaled fifty years later by the award of the 1994 Nobel Prize for economics to three eminent scholars in the field" (p. 122). The fundamental stages of development of game theory were as follows:
1928: Von Neumann demonstrates his minimax theory. This demonstration occurs within the framework of a category of two-person zero-sum games in which chance (hazard) plays no part, at least no explicit part, and in which the results depend solely upon the reason of the players, not upon their ability. Such "strategic games" lend themselves naturally to an economic interpretation.
1937: Pursuing his topological work on the application of the fixed-point theorem, Von Neumann discovers the existence of a connection between the minimax problem in game theory and the saddle point problem as an equilibrium in economic theory.
1940: Von Neumann chooses the economist O. Morgenstern to assist him in the composition of what would become the first treatise of game theory. The title of their work is explicit: the theoretical understanding of games is presented as relevant to the analysis of economic behavior (Schmidt, p. 2).
Today, game theory is a popular analytical tool in economics and political science, and to a lesser degree, psychology, sociology, and the other social sciences (Flanagan). In sum, "Game theory is a branch of mathematics involving the creation and study of models of situations in which outcomes are interdependent on choices made by two or more actors" (p. 121). According to Read (2004), typical game theory scenarios present each player with a decision that he or she might, or might not, make, along with their preferences for the possible combinations of their decisions. In any event, the first player in a game has two strategies that can be used (i.e., option 1 or option 2); likewise, the opposing player has four strategies: (a) execute their option 1 regardless of what the first person does; (b) execute their option 2 regardless of what the first person does; - execute their option 1 if the first player chooses their option 1, and their option 2 if the first player chooses their option 2 (tit-for-tat); or (d) execute their option 2 if the first player chooses their option 1, and their option 1 if the first player chooses his or her option 1 (tat-for-tit) (Read, p. 466). The individual preferences are then analyzed to determine if one or the other player has a dominant strategy, in other words, a strategy that is superior for that player irrespective of the action taken by the other player (Read).
Although significant variations exist, a game model typically requires the following elements:
Players. These are assumed to be rational actors weighing costs and benefits as they pursue their own goals. There must be two or more players.
Rules of the game. These define the limits of action - what can and cannot be done in the game.
Strategies. These are the choices that the players can make within the rules of the game. A strategy is a complete set of choices from beginning to end of the game. For example, if a player can make three different decisions, and for each decision there are two alternatives, he has eight different strategies for the whole game.
Payoffs. These are the outcomes that accrue to players depending on the choice of strategies they and their opponents make. Payoffs may be either ordinal or cardinal.
Solutions. A solution is the set of payoffs arising from the strategies that rational players would choose under the rules of the game; in some cases, there are multiple solutions. Indeed, sometimes there are multiple solution concepts, that is, more than one line of reasoning that rational actors might employ (Flanagan).
According to Kreps, game theory is divided into two branches: (a) cooperative and (b) non-cooperative game theory as follows:
In non-cooperative game theory the unit of analysis is the individual participant in the game who is concerned with doing as well for himself as possible subject to clearly defined rules and possibilities. If individuals happen to undertake behavior that in common parlance would be labeled 'co-operation,' then this is done because such co-operative behavior is in the best interests of each individual singly; each fears retaliation from others if co-operation breaks down.
In cooperative game theory, the unit of analysis is most frequently the group or, in the standard jargon, the coalition; when a game is specified, part of the specification is what each group or coalition of players can achieve, without too much reference to how the coalition would effect a particular outcome or result (Kreps).
There is also a relatively recent innovation known as evolutionary game theory (EGT), which is "a formal, mathematical approach within evolutionary economics, which thus far has been mainly applied to economics as a refinement of the Nash equilibrium concept" (Villena & Villena, 2004, p. 585).
According to Kelly (2003), "A two-person zero-sum game is one in which the pay-offs add up to zero. They are strictly competitive in that what one player gains, the other loses. The game obeys a law of conservation of utility value, where utility value is never created or destroyed, only transferred from one player to another" (p. 77). The term "zero-sum game" is used to describe this because the gain achieved by one player is regarded as a loss to another; because the loss and gain cancel each other out, the net result is a sum of zero, therefore the name "zero-sum game"; in…