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lecture4a

# 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: Øඏ Agent 1 gets 2 cookies, agent 2 gets 0 cookies Øඏ They each get 1 cookie Øඏ Agent 1 gets 0 cookies, agent 2 gets 2 cookies Agent 2 chooses to accept or reject the split: Øඏ Accept => they each get their cookies(s) Øඏ 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
Nau: Game Theory 3 Updated 10/5/11 Extensive Form The sharing game is a game in extensive form Øඏ A game representation that makes the temporal structure explicit Øඏ Doesn’t assume agents act simultaneously Extensive form can be converted to normal form Øඏ So previous results carry over Øඏ But there are additional results that depend on the temporal structure In a perfect-information game, the extensive form is a game tree : Øඏ Choice (or nonterminal ) node : place where an agent chooses an action Øඏ Edge : an available action or move Øඏ Terminal node : a final outcome Øඏ 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 Øඏ ρ ( h ) = the player to move at h Øඏ χ ( h ) = {all available actions at h } Øඏ σ ( h,a ) = node produced by action a at node h Øඏ h ’s children or successors = { σ ( h,a ) : a χ ( h )} If h is a node (either terminal or nonterminal), then Øඏ h ’s history = the sequence of actions leading from the root to h Øඏ h’ s descendants = all nodes in the subtree rooted at h The book doesn’t give the nodes names Øඏ 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
Nau: Game Theory 5 Updated 10/5/11 Pure Strategies Pure strategy for agent i in a perfect-information game: Øඏ 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 Øඏ Which action at which node? Øඏ 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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