A_Course_in_Game_Theory_-_Martin_J._Osborne 30

# A_Course_in_Game_Theory_-_Martin_J._Osborne 30 - essence of...

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Page 15 he chooses , given that every other player j chooses his equilibrium action . Briefly, no player can profitably deviate, given the actions of the other players. The following restatement of the definition is sometimes useful. For any define B i ( a -i ) to be the set of player i 's best actions given a - i : We call the set-valued function B i the best-response function of player i . A Nash equilibrium is a profile a * of actions for which This alternative formulation of the definition points us to a (not necessarily efficient) method of finding Nash equilibria: first calculate the best response function of each player, then find a profile a * of actions for which for all . If the functions B i are singleton-valued then the second step entails solving | N | equations in the | N | unknowns 2.3 Examples The following classical games represent a variety of strategic situations. The games are very simple: in each game there are just two players and each player has only two possible actions. Nevertheless, each game captures the
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Unformatted text preview: essence of a type of strategic interaction that is frequently present in more complex situations. • Example 15.3 (Bach or Stravinsky ? (BoS )) Two people wish to go out together to a concert of music by either Bach or Stravinsky. Their main concern is to go out together, but one person prefers Bach and the other person prefers Stravinsky. Representing the individuals' preferences by payoff functions, we have the game in Figure 16.1. This game is often referred to as the "Battle of the Sexes"; for the standard story behind it see Luce and Raiffa (1957, pp. 90-91). For consistency with this nomenclature we call the game "BoS". BoS models a situation in which players wish to coordinate their behavior, but have conflicting interests. The game has two Nash equilibria: ( Bach, Bach ) and ( Stravinsky, Stravinsky ). That is, there are two steady states: one in which both players always choose Bach and one in which they always choose Stravinsky ....
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