Lecture 11 - Dynamic Game Theory

Lecture 11 - Dynamic Game Theory - Dynamic Game Theory 1...

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1 Dynamic Game Theory 1. Dynamic Games in Extensive Form In dynamic or “sequential move” games players take turns making decisions or “moves” and the payoffs are determined by the sequence of moves after the game ends. Dynamic games are similar to static games, but in dynamic games players move sequentially so the order in which players move matters! To capture the order in which players move, we usually represent dynamic games using game trees or what’s called the extensiveform of a game, such as: Every game tree consists of branches and nodes . A decisionnode , usually represented by a dot or a circle indicates which player’s turn it is to move. Every branch represents a move available to a player at one of their decision nodes. The initial node , usually represented by an open circle, is the starting point of the game. The game ends when a terminalnode is reached at the bottom of the game tree and the payoffs to each player depend on the sequence of moves, or which terminal node is reached ሺexampleሻ. 1 2 2 L1 R1 L2 R2 L2 R2 (4, 1) (0, 2) (1, 3) (5, 2)
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2 Although we usually use game trees to represent dynamic games, we can also put static games into extensive form using “information sets.” For example the extensive form of matching pennies is: The dashed line is used to indicate that both of Player 2’s decision nodes are in the same information set, or that Player 2 has the exact same information about what happened earlier in the game at either decision node. If Player 2 has the same information at either node, then Player 2 can’t know whether Player 1 chose Heads or Tails when it’s his turn to move. Also Player 1 doesn’t know whether Player 2 will choose Heads or Tails because Player 1 is moving first. Therefore, both players are simultaneously unaware of each others’ moves, and the game is ሺby definitionሻ a static or simultaneous move game! Notice that we could reverse the order of moves and payoffs and get the exact same game. The normal form of every game is unique, but there’s more than one way to put every simultaneous move game into extensive form! 1 2 Heads Tails Heads Tails Heads Tails (1, -1) (-1, 1) (-1, 1) (1, -1)
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3 Every node that is not connected to another by a dashed line is assumed to be its own information set! For example, in the dynamic game: Both of Player 2’s decision nodes are in their own information sets. Since Player 2’s decision nodes are in different information sets, Player 2 can have different information about what happened earlier in the game at each decision node. So if Player 2’s first decision node is reached Player 2 will know that Player 1 played L1 and if Player 2’s second decision node is reached Player 2 will know that Player 1 played R1.
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  • Winter '07
  • JUZWIAK,WILLIAM
  • Game Theory, player, Nash