lecture5b

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

This preview shows pages 1–8. Sign up to view the full content.

Nau: Game Theory 1 Updated 10/10/11 CMSC 498T, Game Theory 5b. Zero-Sum Imperfect-Information Games Dana Nau University of Maryland

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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:

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 30

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

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online