Applying game theory in libraries: review and preview.

Author:Zhong, Ying


Game theory is the study of how people behave in strategic situations. It is concerned with "how rational individuals make decisions when they are mutually interdependent" (Romp, 1997). "Game" in this context means "sport of any kind," and therefore implies two or more players and a set of rules. The outcome of a game depends on the player him or herself, as well as other players. The root of game theory can be traced to philosophical and political works such as Plato's Republic, in which Socrates worries that "a soldier is better off running away regardless of who is going to win the battle" and "if all of the soldiers reason this way--then this will certainly bring about the outcome in which the battle is lost" (Ross, 2006).

The development of game theory is well summarized as "a conjunction of indirections: ideas, both in mathematics and in economics whose implications and fruitfulness were not understood, dramatizations of concepts for the wrong reasons, and fruits in applications not originally considered" (Arrow, 2003). Von Neumann is considered the founder of game theory (Von Neumann and Morgenstern 1944), including the "minimax" theorem, which is a way of " minimizing the maximum possible loss" (Minimax 2008). It was John Nash's definition and proof of existence for the equilibrium point, however, that exerts the direct impact of game theory on economics.

Strategic games commonly involve three important aspects: players, strategies, and payoffs. The players are "the individuals who make the relevant decisions"; strategies refer to "a complete description of how a player could play a game"; and the pay-offs are "what a player will receive at the end of the game, contingent upon the actions of all the players in the game" (Romp, 1997). As a result, the player will choose a strategy that yields the maximum payoff. Several assumptions are required to understand game theory: individualism, rationality, and mutual interdependence.

The simplest form of game theory is a two-person game. Since each player can choose one of two strategies, the two players will reach four possible decisions as a joint effort. There are three prototypes of two-person game: Prisoner's dilemma, Chicken, and Assurance. Prisoner's dilemma is also known as a game of cooperation, because each player needs to cooperate with the other in order to yield the "dominant" strategy (i.e., the efficient outcome). The following is an explanation of prisoner's dilemma.

Prisoner's dilemma: Two suspects, say Bonnie and Clyde, are arrested for a crime. Lacking sufficient evidence to charge both, the police need at least one of the suspects to confess. To facilitate questioning and possibly get a confession, the police separate the suspects. Each of them is given two choices: confess or not confess. Both are made aware of the consequences of the two choices. Consequences for each prisoner depend on what the other does (confess or not). Consequences are displayed in Table 1.

If Bonnie and Clyde both confess, then each is sentenced to 10 months in prison (the first number in brackets is Bonnie's sentence while the second number is that of Clyde). If one confesses and the other does not, the confessor is released while the non-confessor receives the maximum sentence (20 months). If neither Bonnie nor Clyde confess, then they are held on a technicality for one month and later released. This cell (Do Not Confess, Do Not Confess) is an efficient outcome since it leads to a situation where the two suspects have the lowest combined sentence. If, however, Bonnie confesses and Clyde is made aware of it, then it is in Clyde's best interest also to confess, because he would only get 10 months instead of 20.

In fact, regardless of what Bonnie chooses to do, it is in Clyde 's best interest to confess. "Confess" is a dominant strategy for Clyde. The same holds true for Bonnie. When all is said and done, i.e., in equilibrium, both suspects will choose to confess. Nash equilibrium is a weaker condition of a dominant strategy. John Nash is a mathematician who theorized that in non-cooperative finite games with multiple players, an equilibrium, later known as Nash equilibrium, does exist, such that given the other player's action, one player would not deviate from their strategy choice. Hence the outcome, with cooperation, that would benefit both agents (Do Not Confess, Do Not Confess) is dominated by the outcome (Confess, Confess).

Chicken is a game where "each player prefers not to yield to the other," but "the outcome where neither player yields is the worst possible one for both players" (Chicken [game] 2008). The name refers to the use of "chicken" to mean "cowardly." It is a game of differentiation, since both players must choose a different strategy to maximize pay-off. An assurance game, on the other hand, requires players to choose the same strategy to maximize payoff. The assurance game is also known as the game of coordination or "Battle of the Sexes."

A more complicated form of game theory involves more than two persons. In a multi-person game or n-person game, the number of players choosing a certain strategy will influence the payoff for each player. As a result, more than one solution should be adopted to address individual cases. Another method of categorizing game theory is whether game players will "collude" or "cooperate". In a "non-cooperative" game, each person maximizes his or her own rewards...

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