Instability of Scoring Heuristic In games with value exchange, the heuristics are very bumpy Make smoothing assumptions search for "quiesence"

Transcription

1 More on games Gaming Complications Instability of Scoring Heuristic In games with value exchange, the heuristics are very bumpy Make smoothing assumptions search for "quiesence" The Horizon Effect No matter how deep you search, there may be a loss just beyond the horizon. Use secondary searching on a few final candidates

2 Knowledge Search Tradeoff One of the foundational principles of AI The more you know, the less you have to search. In minimax game playing with a static evaluation function, the "knowledge" estimates how good a position is. If it was "perfect knowledge" it would be equivalent to full unrolling of the game tree possible only for small games Search Random Testing Brute-Force examination Hill Climbing Organized Search Heuristic Search Locally informed methods "Strong" methods Mathematical Solutions Insight and Intuition more knowledge less search

3 Branch and Bound Algorithmic Technique When exploring multiple paths, use knowledge (of value, optimality, cost) of KNOWN paths to "prune" other branches: If you can prove that a branch of a search tree cannot POSSIBLY be better than another, dont search it! Alpha-Beta Pruning A variety of Branch and Bound for searching game trees If we are guaranteed a score by making move A, then don't bother searching responses to move B, once any response to B is less than our guarantee

4 What are Alpha and Beta Greatest Lower Bound on my score (worst you can do to me) Least Upper Bound on your score (Best I can do against you) These are used recursively in a flip-flop fashion A B C D E F G H I J K L

5 = -3 Worst you can do if I choose move B A SAVES 2 NODES!!!!! k@-3 means D is no better B 4-3 C 7 D 9 E F G H I J K L H@-5 means C is worse than B Iterative deepening First search to depth 1 Then search to depth 2 and so on Combined with some caching, iterative deepening is used in many game playing systems. Especially when playing with a clock!

6 Caching Each time you do a search, you are re-calculating a lot of boards and heuristics Find a way to keep some of those around to avoid re-calculation Related to Memoization Canonical Forms many games have symmetries player a or player b Rotation of board Mirror image of board Storing board in a canonical form will lessen memory and computational requirements Can the board be converted to an integer? comparison becomes faster "="

7 Games are an active research topic Initial attempts at Robotic Sports Robotic Olympics, Demolition Derby Video Games and Internet Games Real Time Performance No Robotics Problem Arbitrary sophistication Interactive Knowledge Games Are computers good at trivia? Machine Learning of Games More stuff Game Theory Horizon Effect Search vs Knowledge proof Alpha beta pruning Iterative Deepening Caching states Canonical forms Book Moves

8 Game Theory? AI has not focused on "Game Theory" Mathematical Model of Decisionmaking under adversary with cooperation Von Neumann, 1934 Nash 1950 (Nash Equilibria) Field of Mathematical Economics. What is a two-player game in normal form? Each player chooses a move and receives a payoff based on others zero Sum Win Lose Win Lose In a zero sum game, the sum of payoffs to all players, is zero.

9 Conversion from Extensive (Tree) to Normal (Table) Can be done for small games combinatorial explosion tho Positive Sum Games Possible (Iterated) Prisoner's (Choice is a) Dilemma prisoner dilemma cooperate defect cooperate defect

10 Game Theory Concepts pure strategy vs mixed strategy A pure strategy is to always play one line or column A mixed strategy is to play according to a probability distribution Game Theory Concepts Nash Equilbrium If there is a set of strategies with the property that no player can benefit by changing her strategy while the other players keep their strategies unchanged, then that set of strategies and the corresponding payoffs constitute the Nash Equilibrium.

11 Value of Game Theory Extensive Literature Game of Chicken International relations MAD Nuclear Strategy evolution of cooperation positive sum games Prisoner's dilemma's Evolution (Maynard Smith) Hawks vs Doves Markets, bidding Organization and Team theory