1. Choose A Subgame to Play by Zhichao “Leaf” Li August 1 st, 2003

2. Choose a Subgame to Play Problem Description Decompose a Game Strategies to Choose a Subgame Minimax Strategy Hotstrat Experiments Future Work Reference

3. Problem Description Go endgame usually can be decomposed into several Subgames Play a single Subgame is trivial But, Question: Which Subgame should be chosen to play?

4. Game Decomposed Decompose the game (3 subgames here) Search out Combinatorial Games mapping to Subgames 0 | -1 0 | -2 || -3 5 || 2 | 0 Choose a Subgame (How?)

5. Methods of Choosing a Subgame Minimax Search Hotstrat Thermostrat Sentestrat Others? Subgame 1: 0 | -1 Subgame 2: 0 | -2 || -3 Subgame 3: 5 || 2 | 0 Play Which Subgame?

6. Minimax Strategy Use Minimax to find out the best Subgame Advantage: Best Results Disadvantage: needs Time and Space Improvement: Prune Subgame by Incentive Example: 4 | -1 dominates 1 | 0 because 4 – (4|-1) > 1 – (1|0)

7. Hotstrat Choose Subgame with highest Temperature Subgame 1: 0 | 1Temp = 0.5 Subgame 2: 0 | -2 || -3Temp = 1.25 Subgame 3: 5 || 2 | 0Temp = 2 Subgame 3 is Hottest, Play it Advantage: Fast

8. Hotstrat Disadvantage: often not the Best Result Example: Right to Move Subgame 1: 8 | 6 || 5 | -1Temp = 2 Subgame 2: 4 | -1Temp = 2.5 But play Subgame 1 is better Improvement: Consider Sente position

9. Experiments (2-level) 100 Games 5 Subgames/Game In every Subgame: Format: a | b || c | d d = 0 c = d + random(50) b = c + random(50) a = b + random(50) 56 | 31 || 12 | 0 101 | 63 || 48 | 0 68 | 54 || 17 | 0 60 | 19 || 9 | 0 117 | 79 || 49 | 0

10. Experiments (2-level) Left: MinimaxRight: MinimaxScore: 20408 Left: MinimaxRight: HotstratScore: 20566 Left: HotstratRight: MinimaxScore: 20268 Min vs Hot - Min vs Min = 158 (/41 = 3.85) Min vs Min - Hot vs Min = 140 (/34 = 4.12) Min vs Hot > Min vs Min: 41 Games (out of 100) Min vs Min > Hot vs Min: 34 Games (out of 100)

11. Experiments (3-level) Based of 2-level Games On 2-level Subgame a | b || c | d a+d1 | a-d1 || b+d2 | b-d2 ||| c+d3 | c-d3 || d+d4 | d-d4 di = random(10) 56|31 || 12|0 -> 65|47 || 39|23 ||| 12|12 || 8|-8 101|63 || 48|0 -> 106|96 || 63|63 ||| 56|40 || 1|-1

12. Experiments (3-level) Left: MinimaxRight: HotstratScore: 20579 Left: MinimaxRight: MinimaxScore: 20453 Left: HotstratRight: MinimaxScore: 20307 Min vs Hot - Min vs Min = 126 (/42 = 3.00) Min vs Min - Hot vs Min = 146 (/46 = 3.17) Min vs Hot > Min vs Min: 42 Games (out of 100) Min vs Min > Hot vs Min: 46 Games (out of 100)

13. Future Work How to narrow down bounds of a Subgame (this helps Minimax to prune subgames) How to improve Hotsrat Other Strategy (Thermostrat, Sentestrat?) Repetition in a Subgame (like Ko) Two options in a move Others?

14. Reference M. Müller. Decomposition search: A combinatorial games approach to game tree search, with applications to solving Go endgames. In IJCAI-99, volume 1, pages 578-583, 1999. M. Müller. Global and local game tree search. Information Sciences, 135(3-4):187-206, 2001. E. Berlekamp, J. Conway, and R. Guy. Wining Ways for your Mathematical Plays, Volume 1, 2 nd edition. A K Peters, Natick, 2001 T. Cazenave. Comparative evaluation of strategies based on the values of direct threats. In Board Games in Academia V, Barcelona, 2002.