Lecture 3 SeqAndSimGames

# Lecture 3 SeqAndSimGames - Economics 101 UCLA Fall 2010...

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Unformatted text preview: Economics 101, UCLA Fall 2010 Jernej Copic Lecture 3, 10/5/10. Game Theory: sequential games and simultaneous games definitions and further examples. Meanwhile, in his downtown o ﬃ ce, No1 was similarly contemplating the meeting with Jeremiah. But he had more of a taste for abstraction. So he started out by writing down what he thought would be a reasonable definition of a sequential game. He pulled out a wrinkled papyrus and started scribbling. He wrote the following. A sequential-moves game is REALLY given by a game tree, but to describe it I can say that: • There is a bunch of players, well, 2 players in this case, denoted by i = 1 and i = 2. • A sequence in which the players move; in this case I move first and Jeremiah moves second. • Each player has some actions available, whenever he moves; player 1 has the set of actions A 1 and player 2 has his set of actions A 2 . For instance, in the case of my meeting with Jeremiah, I will be player 1, and Jeremiah player 2, my actions are A 1 = { Downtown, cab } , and his actions are A 2 = { taxi, UCLA } . I will call a = ( a 1 , a 2 ) the profile of players’ actions, where a 1 ∈ A 1 and a 2 ∈ A 2 . • There are payo ff s, or utilities, for each player, depending on what profile of actions has been played. For instance, here, when a = ( a 1 , a 2 ) = ( Downtown, taxi ), then u 1 ( a ) = 200 and u 2 ( a ) = 100. • Most important! Each player has some strategies. A strategy is di ff erent from action in that it describes what a player will do in every situation (also called a history ). A strategy of player i is denoted by s i , and the set of his strategies is denoted by S i . Well, here then, for player 1 this is no di ff erent then just choosing an action, since he only chooses once at the beginning. So his strategies are S 1 = A 1 1 But player 2 has more strategies! - His strategy must say what he will do in every situ- ation, and a situation is described simply by the actions that player 1 has taken before him. For example, one strategy of player 2 here is ( taxi | Downtown, UCLA | cab ). But I shouldn’t confuse this with an action profile! A strategy of player 2 just describes how he would behave depending on what actions player 1 has taken. So the strategy ( taxi | Downtown, UCLA | cab ) really means that Jeremiah would take a taxi if I stay Downtown, and would stay at UCLA if I take a cab. In my little problem, player 2 then has a total of 4 strategies: S 2 = { ( taxi | Downtown, UCLA | cab ) , ( UCLA | Downtown, UCLA | cab ) , ( taxi | Downtown, taxi | cab ) , ( UCLA | Downtown, taxi | cab ) } ....
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Lecture 3 SeqAndSimGames - Economics 101 UCLA Fall 2010...

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