1. Temperature Discovery Martin M ü ller, Markus Enzenberger and Jonathan Schaeffer zIntroduction: local and global search yLocal search algorithms yTemperature zEnvironments and coupon stacks zTemperature discovery search zFirst results

2. Local and Global Search Local search Partition game into sum of subgames Local analysis Problem: how to evaluate local results? Central question: which sums of games are wins? Global search Single, monolithic game state Full board evaluation Single game tree, minimax backup Central question: what is the minimax score?

3. Why Local Search? zGlobal Alpha-beta: Search time exponential in size of full problem zLocal search: time exponential in size of subproblems

5. Results of Local Searches z1. Exact: combinatorial game value (Winning Ways, my Ph.D. thesis on Go endgames)  2. Inexact, but “ very good ” : temperatures, thermographs (Go: Berlekamp, Spight, Fraser, M ü ller, Amazons: Theo Tegos) z3. Even less exact: heuristic search to estimate the temperature (This work, with Markus and Jonathan)

6. 1. Decomposition Search zUsual: global game tree search zDS: Divide-and-conquer approach zIdea: yDivide game into sub-games yDo a local search zCombine local results: Combinatorial game theory

7. 2. Temperatures, Thermographs

8. 3. Temperature Discovery  Problem: Thermographs computed “ bottom-up ” zNeeds complete local game tree zSometimes too expensive zHeuristic evaluation works well in global search zIdea: use it in local search to estimate temperature

9. Temperature Discovery zA different way to compute temperatures (Berlekamp):  Play local game + “ Coupon stack ”  Choose between play on the board and “ coupon ” (move of known value) zTemperature of coupon of value t is t. So can estimate temp of board!

10. Example zCoupon stack 3,2,1,0,-1 zAmazons board zSearch depth 4 z1. B: Coupon(3) 2. W: C8-C7xC8 3. B: Coupon(2) z4. W: Coupon(1) 9.. X. 8.. W. 7 X.. B 6. X.. A B C D

11. Example (cont ’ d) zUses heuristic evaluation of board zDepth-limited search zResult: ywhen does it change from taking coupons to board? yEstimate for the temperature

12. Experiments (1) zRun temp. discovery search on small areas  Compare estimated t against exact t from Theo Tegos ’ Databases zPlot real t vs estimated t zWorks OK, but still some problems/bugs?

13. Experiments (2) zSample starting positions with 2, 4 and 6 subgames zSubgame size 4x4, 5x5 zTemperature discovery in each local game  Simple ‘ hotstrat ’ player zPlay 2x200 games against Arrow (full board search)

14.  ‘ Coupon player ’ vs Arrow yAbout 10 sec./move Two, four, six 4x4 subgames

16. Two and Four 5x5 subgames

17.  13.25 average over 200 pairs of games (stdDev 11.5) 5x5 subgames

18. zArrow(10sec) vs Arrow on four 4x4 zDifferent time limits for opponent Control experiment 5s1s30s10s

19. Sample 4x5x5 Game

20. zMore experiments, e.g. 6x5x5, 6x6,... zTry on real games zBetter sum game algorithm zTune, fix temperature discovery search zOptimal solver? (Needs global search too) zThe real goal - apply to Go! To Do...

21. Summary zLocal search algorithm zTry to discover temperature by minimax search zApplications: Amazons, future: Go zFirst results: it works... zStill lots of open questions