14 Game Theory - Game Theory In previous chapters, we have...

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± ± ± ± ± ± ± ± ± ± ± Game Theory In previous chapters, we have encountered many situations in which a single decision maker chooses an optimal decision without reference to the effect that the decision has on other deci- sion makers (and without reference to the effect that the decisions of others have on him or her). In many business situations, however, two or more decision makers simultaneously choose an ac- tion, and the action chosen by each player affects the rewards earned by the other players. For ex- ample, each fast-food company must determine an advertising and pricing policy for its product, and each company’s decision will affect the revenues and pro±ts of other fast-food companies. Game theory is useful for making decisions in cases where two or more decision makers have con²icting interests. Most of our study of game theory deals with situations where there are only two decision makers (or players), but we brie²y study n -person (where n ± 2) game theory also. We begin our study of game theory with a discussion of two-player games in which the players have no common interest. 14.1 Two-Person Zero-Sum and Constant-Sum Games: Saddle Points Characteristics of Two-Person Zero-Sum Games 1 There are two players (called the row player and the column player). 2 The row player must choose 1 of m strategies. Simultaneously, the column player must choose 1 of n strategies. 3 If the row player chooses his i th strategy and the column player chooses his j th strat- egy, then the row player receives a reward of a ij and the column player loses an amount a ij . Thus, we may think of the row player’s reward of a ij as coming from the column player. Such a game is called a two-person zero-sum game, which is represented by the ma- trix in Table 1 (the game’s reward matrix ). As previously stated, a ij is the row player’s TABLE 1 Example of Two-Person Zero-Sum Game Row Player’s Column Player’s Strategy Strategy Column 1 Column 2 ±±± Column n Row 1 a 11 a 12 ²²² a 1 n Row 2 a 21 a 22 ²²² a 2 n ² ²² ² Row m a m 1 a m 2 ²²² a mn
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804 CHAPTER 14 Game Theory reward (and the column player’s loss) if the row player chooses his i th strategy and the column player chooses his j th column strategy. For example, in the two-person zero-sum game in Table 2, the row player would re- ceive two units (and the column player would lose two units) if the row player chose his second strategy and the column player chose his first strategy. A two-person zero-sum game has the property that for any choice of strategies, the sum of the rewards to the players is zero. In a zero-sum game, every dollar that one player wins comes out of the other player’s pocket, so the two players have totally conflicting in- terests. Thus, cooperation between the two players would not occur.
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14 Game Theory - Game Theory In previous chapters, we have...

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