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Optimal Strategy Selection in Combinatorial Games: Exploring Game Endings and Strategies

Explore the world of combinatorial games by delving into strategic endgames and different gameplay temperatures. Discover insights from past work on games with multiple options and future considerations on optimizing gameplay strategies. Compare minimax, hotstrat, and thermostrat approaches in various game scenarios and delve into the analysis of different gameplay depths and temperatures. Uncover the impact of varying strategies on game outcomes and delve into procedural questions for future research.

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Optimal Strategy Selection in Combinatorial Games: Exploring Game Endings and Strategies

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  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. 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 Methods of Choosing a Subgame

  5. 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 Random Data (one option)

  6. 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} Random Data (many options)

  7. 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*} Amazons Endgames

  8. 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)

  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)

  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?

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