1. Machine Learning of Bridge Bidding By Dan Emmons Computer Systems Laboratory 2008-2009

2. Bridge Bidding is Hard Both cooperative agents and opposing agents must be dealt with Only partial information is available to each player Effectiveness of all bids cannot be evaluated until the end of the entire bidding sequence Multiplicity of meanings for each bid Some hands can be readily handled with multiple bids while other hands can be readily handled by no bids

3. Three Necessary Parts A way to select bids that overcomes the limitation of partial information A way to evaluate a bidding scenario by counting tricks that can be earned in play A way to improve partnership bidding agreements inductively to improve overall bidding through learning

4. Monte Carlo Sampling C: QJT94 D: 732 H: AQJ S: KT ?? ? C: 73 D: AK984 H: K4 S: 96 ? ? ? C: 6 D: QT85 H: 73 S: AQ9742 ? ? ? C: AQ83 D: QJ32 H: A32 S: Q5 ?? ? C: J93 D: A43 H: AKQT84 S: T ?? ? C: T6 D: Q72 H: AQ62 S: 8732 ?? ? C: AJ8 D: K94 H: KJT5 S: K85 ?? ?

5. The Bid Decision Hierarchy Root Node Constraints: None Actions: Pass Constraints: 15-17 HCP, Balanced Actions: 1NT Constraints: 5+ Hearts Actions: 1H Constraints: 4+ Diamonds Actions: 1D Constraints: 13+ HCP Actions: 1C, 1D, 1H, 1S, 1NT High Priority Low Priority

6. Double-Dummy Solver Implementation MTD(f) is used with a transposition table Two pruning extra pruning techniques: o Only check one of adjacent cards in the same hand o Assume the player does not want to lose with a higher card than necessary Hash values are computed so as to hash equivalent hand positions to the same value: Clubs: A K J Diamonds: 9 7 2 Hearts: 6 5 4 3 2 Spades: K 9 After the club queen has been played Clubs: K Q J Diamonds: 9 7 2 Hearts: 6 5 4 3 2 Spades: K 9 After the club ace has been played

7. Sample Output of Implemented Solver North: Clubs: T 7 5 3 2 Diamonds: J Hearts: A Q J T Spades: T 9 7 West:East: Clubs: 6Clubs: A J 8 Diamonds: A K T 7 5Diamonds: Q 9 8 Hearts: 9 8 4Hearts: 5 3 Spades: Q J 6 2Spades: A K 8 5 4 South: Clubs: K Q 9 4 Diamonds: 6 4 3 2 Hearts: K 7 6 2 Spades: 3 Trick Counts for Each Declarer (North, South, East, West): Clubs: 9 9 3 3 Diamonds: 2 2 11 11 Hearts: 7 7 3 3 Spades: 0 0 11 11 No Trump: 2 2 8 8

8. Current Bidding Performance Dealer: West Vulnerable: All North Clubs: A 8 4 Diamonds: Q J 8 Hearts: A T Spades: A T 8 6 3 WestEast Clubs: Q 5 2Clubs: J 6 3 Diamonds: 6 4 3 2Diamonds: A T 9 Hearts: 9 7 4Hearts: Q J 8 5 Spades: J 7 2Spades: K Q 9 South Clubs: K T 9 7 Diamonds: K 7 5 Hearts: K 6 3 2 Spades: 5 4 SouthWestNorthEast 4DPass4H XPassPass4S XPassPassPass 4SX Vul – East Down 7 Score: -2000 Dealer: West Vulnerable: None North Clubs: A K 7 6 Diamonds: J T 8 4 Hearts: Q T 8 3 Spades: 2 WestEast Clubs: 9 8 5 4Clubs: J 2 Diamonds: 9 7 6Diamonds: A Q 2 Hearts: J 2Hearts: A K 9 7 6 4 Spades: 8 7 6 3Spades: K 9 South Clubs: Q T 3 Diamonds: K 5 3 Hearts: 5 Spades: A Q J T 5 4 SouthWestNorthEast PassPassPass 2SPass3HPass 3SX4CPass 4SPass4NTPass 5CPass5HPass 5SXPassPass Pass 5SX Nonvul - South Making Exact Score: 650

9. Third Quarter Improvements Give bidding agents a more rigid framework of rules and constraints as a basic system Teach agents to refine their bidding system inductively, reducing the average branching factor of the bidding look-ahead and giving the partner agent more information per bid Hold IMP-scored games between refined and unrefined bidders to verify improvement Test a computer bidding pair against human opponents