

![0_image_0.png](0_image_0.png)

provided by α-β pruning, can be contrasted with α-β pruning itself, which gives a reduction when compared to pure minimax of b → b . 75 if the moves are ordered randomly (Pearl, 1982) and b → b 0.5 if the ordering is optimal.

The method was also applied to full deals of 52 cards, which can be solved while expanding an average of 18,000 nodes per deal. 5  This works out to about a second of CPU
time.

## 3. Monte Carlo Cardplay Algorithms

One way in which we might use our perfect-information cardplay engine to proceed in a realistic situation would be to deal the unseen cards at random, biasing the deal so that it was consistent both with the bidding and with the cards played thus far. We could then analyze the resulting deal double dummy and decide which of our possible plays was the strongest. Averaging over a large number of such Monte Carlo samples would allow us to deal with the imperfect nature of bridge information. This idea was initially suggested by Levy (Levy, 1989), although he does not appear to have realized (see below) that there are problems with it in practice.

Algorithm 3.0.1 (Monte Carlo card selection) To select a move from a candidate set

## M Of Such Moves:
