lecture5b - CMSC 498T, Game Theory 5b. Zero-Sum...

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Nau: Game Theory 1 Updated 10/10/11 CMSC 498T, Game Theory 5b. Zero-Sum Imperfect-Information Games Dana Nau University of Maryland
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Nau: Game Theory 2 Updated 10/10/11 Examples Most card games Ø Bridge, crazy eights, cribbage, hearts, gin rummy, pinochle, poker, spades, … A few board games Ø battleship, kriegspiel chess All of these games are finite, zero-sum, perfect recall West North East South 6 2 8 Q Q J 6 5 9 7 A K 5 3 A 9
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Nau: Game Theory 3 Updated 10/10/11 Bridge Four players Ø North and South are partners Ø East and West are partners Equipment: Ø deck of 52 playing cards Phases of the game Ø dealing the cards distribute them equally among the four players Ø bidding negotiation to determine what suit is trump Ø playing the cards West North East South
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Nau: Game Theory 4 Updated 10/10/11 Playing the Cards Declarer: person who chose the trump suit Dummy: the declarer’s partner Ø Dummy turns cards face up Ø Declarer plays both declarer’s and dummy’s cards Trick: the basic unit of play Ø one player leads a card Ø others must follow suit if possible Ø trick is won by highest card of the suit that was led, unless someone plays a trump Keep going until all cards have been played Scores based on how many tricks were bid, how many were taken West North East South 6 2 8 Q Q J 6 5 9 7 A K 5 3 A 9
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Nau: Game Theory 5 Updated 10/10/11 Game Tree Search in Bridge Imperfect information in bridge: Ø Don’t know what cards the others have (except the dummy) Ø Many possible card distributions, so many possible moves If we encode the additional moves as additional branches in the game tree, this increases the branching factor b Number of nodes is exponential in b Ø Worst case: about 6x10 44 leaf nodes Ø Average case: about 10 24 leaf nodes A bridge game takes about 1 ½ minutes Ø Not enough time to search the tree b = 2 b = 3 b = 4
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Nau: Game Theory 6 Updated 10/10/11 Monte Carlo Sampling Generate many random hypotheses for how the cards might be distributed Generate and search the game trees Ø Average the results This approach has some theoretical problems Ø The search is incapable of reasoning about actions intended to gather information actions intended to deceive others Ø Despite these problems, it seems to work well in bridge It can divide the size of the game tree by as much as 5.2x10 6 Ø (6x10 44 )/(5.2x10 6 ) = 1.1x10 38 Better, but still quite large Ø Thus this method by itself is not enough Ø It’s usually combined with state aggregation
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Nau: Game Theory 7 Updated 10/10/11 State aggregation Modified version of transposition tables Ø Each hash-table entry represents a set of positions that are considered to be equivalent Ø Example: suppose we have AQ532 View the three small cards as equivalent:
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lecture5b - CMSC 498T, Game Theory 5b. Zero-Sum...

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