by a computer program, and is an "action" system, working hard to make the opponents' bidding as difficult as possible.

With regard to defensive cardplay, the key elements of high level defense are to make it hard for partner to make a mistake while making it easy for declarer to do so. Providing GIB
with these abilities will involve an extra level of recursion in the cardplay, as each element of the Monte Carlo sample must now be considered from other players' points of view, as they generate and then analyze their own samples. These ideas have been implemented but currently lead to small performance degradations (approximately 0.05 IMPs/deal) because the computational cost of the recursive analyses require reducing the size of the Monte Carlo sample substantially. As processor speeds increase, it is reasonable to expect these ideas to bear significant fruit.

In 1997, Martel, a computer scientist himself, suggested that he expected GIB to be the best bridge player in the world in approximately 2003. 1 The work appears to be roughly on schedule.

## 8.3 Other Games

I have left essentially untouched the question of to what extent the basic techniques we have discussed could be applied to games of imperfect information other than bridge.

The ideas that we have presented are likely to be the most applicable in games where the perfect information variant is tractable but computationally challenging, and the assumption that one's opponents are playing with perfect information is a reasonable one.

This suggests that games like hearts and other trick-taking games will be amenable to our techniques, while games like poker (where it is essential to realize and exploit the fact that the opponents also have imperfect information) are likely to need other approaches.

## Acknowledgments

A great many people have contributed to the GIB project over the years. In the technical community, I would like to thank Jonathan Schaeffer, Rich Korf, David Etherington, Bart Massey and the other members of CIRL. In the bridge community, I have received invaluable assistance from Chip Martel, Rod Ludwig, Zia Mahmood, Andrew Robson, Alan Jaffray, Hans Kuijf, Fred Gitelman, Bob Hamman, Eric Rodwell, Jeff Goldsmith, Thomas Andrews and the members of the rec.games.bridge community. The work itself has been supported by Just Write, Inc., by DARPA/Rome Labs under contracts F30602-95-1-0023 and F3060297-1-0294, and by the Boeing Company under contract AHQ569.  To everyone who has contributed, whether named above or not, I owe my deepest appreciation.

## Appendix A. A Summary Of The Rules Of Bridge

We give here a very brief summary of the rules of bridge. Readers wanting a more complete description are referred to any of the many excellent texts available (Sheinwold, 1996).

Bridge is a card game for four players, who are split into two pairs.  Members of a single pair sit opposite one another, so that North-South form one pair and East-West the other.