1. Maximizing the Chance of Winning in Searching Go Game Trees Presenter: Ling Zhao March 16, 2005 Author: Keh-Hsun Chen Accepted by Information Sciences

2. Motivation Traditional approach in Go: maximize territory Traditional approach in Go: maximize territory Should it better to maximize the probability of winning? Should it better to maximize the probability of winning?

3. Expected territory vs. chance of winning k groups, prob p i to be value A i, and prob 1– p i to be –A i k groups, prob p i to be value A i, and prob 1– p i to be –A i (q i, A ’ i ) either (p i, A i ) or (1-p i, -A i ) (q i, A ’ i ) either (p i, A i ) or (1-p i, -A i ) 2^n combinations 2^n combinations

4. Case study: All groups are safe Territory score is a good predication of the outcome of the game in end games. Territory score is a good predication of the outcome of the game in end games. Less reliable in opening or middle game. Less reliable in opening or middle game. Major difficulty: measuring no man’s lands Major difficulty: measuring no man’s lands

5. Frontier space points of a block 1. Must be adjacent empty points of the block 2. Must have an adjacent point which is not the same color of the block 3. Can be used to measure openness of the boundary. openness of the boundary.

6. Frontier space points Usually the total number of frontier space points (F) is 0 at the beginning, increases until to its peak (about 60) in the middle game, then decreases to 0 in the end. M is the move number. Usually the total number of frontier space points (F) is 0 at the beginning, increases until to its peak (about 60) in the middle game, then decreases to 0 in the end. M is the move number. if (M < 100) if (M < 100) if (E A > 60+(100-M)/4) E W = 1; if (E A > 60+(100-M)/4) E W = 1; else if (E A < -60-(100-M)/4) E W = 0; else E W = 0.5+ 0.5 * E A /(60+(100-M)/4); else if (E A > F) E W = 1; else if (E A < -F) E W = 0; else E W = 0.5+0.5*E A /F;

7. Case study: Existence of unsafe groups k groups, the first k 1 groups are safe, and the rest are unsafe. k groups, the first k 1 groups are safe, and the rest are unsafe. Pessimistic evaluation: Pessimistic evaluation: Optimistic evaluation: Optimistic evaluation:

8. Battles An unsafe group’s transitive closure of adjacent unsafe groups forms a battle. An unsafe group’s transitive closure of adjacent unsafe groups forms a battle. Evaluation of one battle Evaluation of one battle Probability p 1, p 2,……p n with sum of 1. E W = E W =

9. Multi-battle situation Combinatorial game model: Combinatorial game model: G 1 = {A | B} G 1 = {A | B} G = G 1 + G 2 + …+ G n Probabilistic combinatorial game (PCG) model: Probabilistic combinatorial game (PCG) model: G 1 = {A 1, p 1, A 2, p 2 | B 1, q 1, B 2, q 2 } G = G 1 + G 2 + …+ G n

10. Solution Mini-max based on winning percentage Mini-max based on winning percentage Terminal nodes: no branching in the game Terminal nodes: no branching in the game

11. Experimental results Experimental Go intellect is slightly inferior to the regular version. Experimental Go intellect is slightly inferior to the regular version. Reasons: Reasons: Probability and the correspondent outcome score are difficult to estimate when there are one or more battles. Solutions: more thorough knowledge engineering and implementation. Solutions: more thorough knowledge engineering and implementation.

12. Lessons learned Dynamic modification of weights on some move generators. For example, reduce weight for attacking moves when far ahead. Dynamic modification of weights on some move generators. For example, reduce weight for attacking moves when far ahead. Adjust territory evaluation by the probability of winning. For example, if the winning percentage is 99%, add 10 points to territory score. Adjust territory evaluation by the probability of winning. For example, if the winning percentage is 99%, add 10 points to territory score. Incremental increase of performance found from experiments from the above two techniques. Incremental increase of performance found from experiments from the above two techniques.

13. Conclusions Right direction for Go. Right direction for Go. More concrete experimental results. More concrete experimental results. Interesting problem in itself and possible applications in other games like Amazon. Interesting problem in itself and possible applications in other games like Amazon. Need better implementation for computing winning probability. Need better implementation for computing winning probability.