Cognitive Game Theory Alpha-Beta minimax search Inductive Adversary - PDF document

Cognitive Game Theory Alpha-Beta minimax search Inductive Adversary Modeling Evolutionary Chess Jennifer Novosad, Justin Fox and Jeremie Pouly Our lecture topic is cognitive game. We are interested in this subject because games are a simple representation of reality on which we can test any concept developed in artificial intelligence. For this reason games have always been considered as an attractive framework for new developments. Our talk in divided in three parts: • Jeremie will first give a quick review of the minimax search and present a few improvements including alpha-beta cutoffs, transposition table and move ordering. He will also introduce the two demonstrations of the lecture. • Jennifer • Justin 1

Motivation • Good benchmark – Similar to military or financial domains • Computer can beat humans • Fun • $ 2

Reasoning Techniques for Games Games Statistical Search Inference Hidden Minimax/ Adversary Evolutionary Bayesian … Markov … Alpha-Beta model Algorithms Nets Models 3

Cognitive Game Theory • Alpha/Beta Search – Jeremie • Adversary Modeling – Jennifer • Evolutionary Algorithms – Justin We return to the outline to note that the next section of this talk will now focus on a still small, but more detailed and less abstract example of how evolutionary algorithms may be applied to create chess players. This example can be found in the paper: Kendall and Whitwell. An Evolutionary Approach for the Tuning of a Chess Evaluation Function using Population Dynamics , Proc. 2001 IEEE Congress on Evolutionary Computation. 4

Cognitive Game Theory • Alpha/Beta Search – Minimax search – Evaluation function – Alpha-Beta cutoffs – Other improvements – Demo • Adversary Modeling • Evolutionary Algorithms We return to the outline to note that the next section of this talk will now focus on a still small, but more detailed and less abstract example of how evolutionary algorithms may be applied to create chess players. This example can be found in the paper: Kendall and Whitwell. An Evolutionary Approach for the Tuning of a Chess Evaluation Function using Population Dynamics , Proc. 2001 IEEE Congress on Evolutionary Computation. 5

Adversarial search • Two-person games: Players = Max & Min – Max wants to win – Min wants Max to loose MAX Initial Board Situation New Board MIN Situations : : : Win Loss Draw Loss MAX 1 -1 0 -1 Final Board Situations - End Games 6

Minimax search • Basic Assumption • Strategy: – MAX wants to maximise its payoff – MIN is trying to prevent this. • MiniMax procedure maximises MAX’s moves and minimises MIN’s moves. 7

An example MAX a 1 MIN b c 1 -1 Terminal d e f g States 1 1 -1 0 Best value for MAX is 1 8

Minimax recursive procedure Function MINIMAX (called at each node): • If terminal state then return payoff • Else if MAX node then use MINIMAX on the children and return the maximum of the results. • Otherwise ( MIN node ), use MINIMAX on the children and return the minimum of the results. 9

Problems • Time complexity: O(b m ) b branching factor and m depth of the terminal states (Chess, b=35, m=100 � 35 100 ≈ 10 154 nodes to visit) • Not possible to search the full game tree Cutoff the tree at a certain depth • But payoffs defined only at terminal states 10

Cognitive Game Theory • Alpha/Beta Search – Minimax search – Evaluation function – Alpha-Beta cutoffs – Other improvements – Demo • Adversary Modeling • Evolutionary Algorithms We return to the outline to note that the next section of this talk will now focus on a still small, but more detailed and less abstract example of how evolutionary algorithms may be applied to create chess players. This example can be found in the paper: Kendall and Whitwell. An Evolutionary Approach for the Tuning of a Chess Evaluation Function using Population Dynamics , Proc. 2001 IEEE Congress on Evolutionary Computation. 11

Heuristic evaluation function • Estimate the chance of winning from board configuration. • Important qualities: – Must agree with terminal states – Must be fast to compute – Should be accurate enough • Chess or checkers: Value of all white pieces – Value of all black pieces 12

Heuristic evaluation function Val = (4*1) – (4*1+1*2) = -2 Val ??? 13

Our evaluation function • Normal checker = 100000 • 4 parameters (long): – King value – Bonus central square for kings – Bonus move forward for checkers – Bonus for order of the moves (*depth/2) 14

Our evaluation function • Normal checker = + 3*Bonus 100000 • 4 parameters (long): + 2*Bonus – King value – Bonus central square + 1*Bonus for kings – Bonus move forward for checkers No Bonus – Bonus for order of the moves (*depth/2) 15

Cognitive Game Theory • Alpha/Beta Search – Minimax search – Evaluation function – Alpha-Beta cutoffs – Other improvements – Demo • Adversary Modeling • Evolutionary Algorithms We return to the outline to note that the next section of this talk will now focus on a still small, but more detailed and less abstract example of how evolutionary algorithms may be applied to create chess players. This example can be found in the paper: Kendall and Whitwell. An Evolutionary Approach for the Tuning of a Chess Evaluation Function using Population Dynamics , Proc. 2001 IEEE Congress on Evolutionary Computation. 16

Alpha-Beta pruning • Search deeper in the same amount of time • Basic idea: prune away branches that cannot possibly influence the final decision • Similar to the Branch-and-Bound search (two searches in parallel: MAX and MIN) 17

General case MAX MIN m : MAX MIN n If m is better than n for MAX then n will never get into play because m will always be chosen in preference. 18

Review of Branch-and-Bound root Var A A 1 A 2 A 3 1 0 4 Var B B 1 B 2 B 3 B 1 B 2 B 3 = 4 3 12 8 4 4 6 Best assignment: [A1,B1], value = 3 19

Alpha-Beta procedure • Search game tree keeping track of: – Alpha: Highest value seen so far on maximizing level – Beta: Lowest value seen so far on minimizing level • Pruning: – MAX node : prune parent if node evaluation smaller than Alpha – MIN node : prune parent if node evaluation greater than Beta 20

Branch-and-Bound analogy • MIN: minimize board valuation � minimize constraints in Branch-and-Bound • MAX: maximize board valuation � inverse of Branch-and-Bound (but same idea) • Prune parent instead of current node (stop expanding siblings) 21

Example MIN 3 2 Min Beta not Beta = 3 Beta = 3 define = 4 3 2 Max 3 1 -5 4 1 0 2 Beta: Lowest value seen so far on minimizing level 22

Example MAX 3 10 Max Alpha not Alpha = 3 Alpha = 10 define ≤ 2 3 10 Min 3 11 5 2 14 24 10 Alpha: Highest value seen so far on maximizing level 23

Beta cutoffs MaxValue (Node,a,b ) If CutOff-Test(Node) then return Eval(Node) For each Child of Node do a = Max(a, MinValue(Child,a,b)) if a = b then return b Return a 24

Alpha cutoffs MinValue (Node,a,b ) If CutOff-Test(Node) then return Eval(Node) For each Child of Node do b = Min(b, MinValue(Child,a,b)) if b = a then return a Return b 25

Alpha-Beta gains • Effectiveness depends on nodes ordering • Worse case: no gain (no pruning) � O(b d ) • Best case (best first search) � O(b d/2 ) i.e. allows to double the depth of the search! • Expected complexity: O(b 3d/4 ) 26

Cognitive Game Theory • Alpha/Beta Search – Minimax search – Evaluation function – Alpha-Beta cutoffs – Other improvements – Demo • Adversary Modeling • Evolutionary Algorithms We return to the outline to note that the next section of this talk will now focus on a still small, but more detailed and less abstract example of how evolutionary algorithms may be applied to create chess players. This example can be found in the paper: Kendall and Whitwell. An Evolutionary Approach for the Tuning of a Chess Evaluation Function using Population Dynamics , Proc. 2001 IEEE Congress on Evolutionary Computation. 27

Other improvements • Nodes ordering (heuristic) • Quiescent search (variable depth & stable board) • Transposition tables (reconnect nodes in search tree) 28

Advanced algorithm MaxValue (Node,a,b ) If board already exist in transposition tables then if new path is longer return value in the table Save board in transposition table If CutOff-Test(Node) then if quiescent board then return Eval(Node) Find all the children and order them (best first) For each Child of Node (in order) do a:=Max(a,MinValue(Child,a,b)) if a>=b then return b Return a 29

Statistics: opening Depth Minimax Alpha- + Move Quiesc. Transpo. Beta ordering search tables Number 4 3308 278 271 * 2237 of nodes 6 217537 5026 3204 41219 50688 8 15237252 129183 36753 649760 859184 Search 4 0 0 0 * 0 time (sec.) 6 3 0 0 0 1 8 201 1 0 9 12 30

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Statistics: jumps available Depth Minimax Alpha- + Move Quiesc. Transpo. Beta ordering search tables Number 4 8484 2960 268 * 5855 of nodes 6 695547 99944 2436 170637 172742 8 56902251 2676433 22383 2993949 3488690 Search 4 0 0 0 * 0 time (sec.) 6 9 1 0 2 2 8 739 34 0 38 46 32