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1Dynamic Game Theory1.Dynamic Games in Extensive FormIn dynamic or “sequential move” games players take turns makingdecisions or “moves” and the payoffs are determined by the sequenceof moves after the game ends.Dynamic games are similar to static games, but in dynamic gamesplayers move sequentially so the order in which players move matters!To capture the order in which players move, we usually representdynamic games using game trees or what’s called theextensiveformofa game, such as:Every game tree consists ofbranchesandnodes.Adecisionnode, usually represented by a dot or a circle indicateswhich player’s turn it is to move.Every branch represents a move available to a player at one of theirdecision nodes.Theinitial node, usually represented by an open circle, is thestarting point of the game.The game ends when aterminalnodeis reached at the bottom ofthe game tree and the payoffs to each player depend on thesequence 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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2Although we usually use game trees to represent dynamic games, wecan 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 decisionnodes are in the same information set, or that Player 2 has the exactsame information about what happened earlier in the game at eitherdecision node.If Player 2 has the same information at either node, then Player 2can’t know whether Player 1 chose Heads or Tails when it’s his turnto move.Also Player 1 doesn’t know whether Player 2 will choose Heads orTails 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 simultaneousmove game!Notice that we could reverse the order of moves and payoffs and getthe exact same game.The normal form of every game is unique, but there’s more than oneway 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)
3Every node that is not connected to another by a dashed line isassumed 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 happenedearlier in the game at each decision node.So if Player 2’s first decision node is reached Player 2 will know thatPlayer 1 played L1 and if Player 2’s second decision node is reachedPlayer 2 will know that Player 1 played R1.