1. Sum of Combinatorial Games by Leaf Wednesday, October 12th, 2004

2. Past Work Games with More than One Option Amazons Endgames Amazons Endgames vs. Random Games Games with Different Temperatures Future Work

3. Problem Description Given: Subgames in a Game Question: Play on which Subgame? Subgame 1: (left edge) 0 | -1 Subgame 2: (up edge) 0 | -2 || -3 Subgame 3: (bottom edge) 5 || 2 | 0

4. Methods of Choosing a Subgame Minimax Search Hotstrat: Choose the Subgame with highest temperature Hotstrat+: Consider Sente Thermostrat: Choose Subgame from Thermograph Depth-bounded αβ Search Temperature-bounded αβ Serach

5. Random Data (one option) 100 Games 5 Subgames/Game Only One Option in a Move 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

6. Random Data (many options) Based on Games with Only One Option For every move A|B, add one more option A, { A+Random(Δ) | A+Random(Δ) } | B, { B+Random(Δ) | B+Random(Δ) } We let Δ = 10 right now 50,{51|47}|12,{14|11}||8,{15|4}|0,{4|0} 58,{67|52}|23,{29|18}||8,{10|5}|0,{6|-6} 106,{114|99}|86,{88|81}||43,{46|39}|0,{1|-2} 50,{50|46}|38,{42|30}||1,{3|1}|0,{7|-4} 90,{99|85}|83,{92|81}||40,{40|33}|0,{0|-5}

7. Amazons Endgames In Combinatorial Game Suite, (CGSuite) Generate all Amazons Endgame with 3 arrows on 3x3 board Randomly pick up these endgames as subgames Temperatures are low, but with many options and depths {3|1,{2|-1,+-1}} {1/2,{1|1/2}|-2,{0|-2}} {1,{2|1/2,{1|*}}|-3} +-3 {3|-1/2*}

8. Minimax plays Left Minimax plays Right Hotstrat 205.2531.25 Thermostrat 226.519.25 Depth-bounded αβ 3-ply 138124.75 Temperature-bounded αβ 170.586.25 Match Results on Amazons Endgames Play 100 Games, with 20 Subgames in a Game Minimax vs. Minimax: 131.75

9. Match Results on Random Games Play 100 Games, with 5 2-level 2-option Subgames in a Game Difference from Minimax vs. Minimax (Minimax plays Left) Δ246102050 Hotstrat0245387174493 Thermostrat02065111230595 Depth-bounded αβ 3-ply0182161162 Temperature-bounded αβ061743140387

10. Match Results on Random Games Play 100 Games, with 5 2-level 2-option Subgames in a Game Difference from Minimax vs. Minimax (Minimax plays Right) Δ246102050 Hotstrat0315196186431 Thermostrat24186136236536 Depth-bounded αβ 3-ply0472761161 Temperature-bounded αβ0123072152404

11. Some Guess from Results In Amazons Endgames, Depth-bounded αβ plays much better than others Temperature-bounded αβ is also better than Hotstrat and Thermostrat in Amazons Endgames In Random Games, the performance is similar to that in Amazons Endgames When Δ gets better, the difference between Strategies gets bigger

12. Future Work/Questions How to generate Endgames “look like” Amazons or Go Endgames? Why Depth-bounded αβ plays much better than Hotstrat? Others?