lecture4a - CMSC 498T, Game Theory 4a. Extensive-Form Games...

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Nau: Game Theory 1 Updated 10/5/11 CMSC 498T, Game Theory 4a. Extensive-Form Games Dana Nau University of Maryland
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Nau: Game Theory 2 Updated 10/5/11 The Sharing Game Suppose agents 1 and 2 are two children Someone offers them two cookies, but only if they can agree how to share them Agent 1 chooses one of the following options: Ø Agent 1 gets 2 cookies, agent 2 gets 0 cookies Ø They each get 1 cookie Ø Agent 1 gets 0 cookies, agent 2 gets 2 cookies Agent 2 chooses to accept or reject the split: Ø Accept => they each get their cookies(s) Ø Otherwise, neither gets any 2-0 1-1 0-2 no yes no yes no yes (0,0) (2,0) (0,0) (1,1) (0,0) (0,2) 2’s move 2’s move 2’s move 1’s move
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Nau: Game Theory 3 Updated 10/5/11 Extensive Form The sharing game is a game in extensive form Ø A game representation that makes the temporal structure explicit Ø Doesn’t assume agents act simultaneously Extensive form can be converted to normal form Ø So previous results carry over Ø But there are additional results that depend on the temporal structure In a perfect-information game, the extensive form is a game tree : Ø Choice (or nonterminal ) node : place where an agent chooses an action Ø Edge : an available action or move Ø Terminal node : a final outcome Ø At each terminal node h , each agent i has a utility u i ( h ) 2-0 1-1 0-2 no yes no yes no yes (0,0) (2,0) (0,0) (1,1) (0,0) (0,2) 2’s move 2’s move 2’s move 1’s move
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Nau: Game Theory 4 Updated 10/5/11 Notation from the Book (Section 4.1) H = {nonterminal nodes} Z = {terminal nodes} If h is a nonterminal node, then Ø ρ ( h ) = the player to move at h Ø χ ( h ) = {all available actions at h } Ø σ ( h,a ) = node produced by action a at node h Ø h ’s children or successors = { ( h,a ) : a χ ( h )} If h is a node (either terminal or nonterminal), then Ø h ’s history = the sequence of actions leading from the root to h Ø h’ s descendants = all nodes in the subtree rooted at h The book doesn’t give the nodes names Ø The labels tell which agent makes the next move 2-0 1-1 0-2 no yes no yes no yes (0,0) (2,0) (0,0) (1,1) (0,0) (0,2) 2 2 2 1 Different notation than I’m used to, so I might not always remember it
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Nau: Game Theory 5 Updated 10/5/11 Pure Strategies Pure strategy for agent i in a perfect-information game: Ø Function telling what action to take at every node where it’s i ’s choice i.e., every node h at which ρ ( h ) = i The book specifies pure strategies as lists of actions Ø Which action at which node? Ø Either assume a canonical ordering on the nodes, or use different action names at different nodes Sharing game: Agent 1 has 3 pure strategies: S 1 = { 2-0, 1-1, 0-2 } Agent 2 has 8 pure strategies: S 2 = { (yes, yes, yes), (yes, yes, no), (yes, no, yes), (yes, no, no), (no, yes, yes), (no, yes, no), (no, no, yes), (no, no, no) } 2-0 1-1 0-2 no yes no yes no yes (0,0) (2,0) (0,0) (1,1) (0,0) (0,2) 2 2 2 1
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This note was uploaded on 01/13/2012 for the course CMSC 498T taught by Professor Staff during the Fall '11 term at Maryland.

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lecture4a - CMSC 498T, Game Theory 4a. Extensive-Form Games...

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