Chapter 8 Performance Analysis of Alpha-Beta Pruning

1. Chapter 8 Performance Analysis of Alpha-Beta Pruning

2. Performance analysis of Alpha-Beta Pruning • Since alpha-beta pruning performs a minimax search while pruning much of the tree, its effect is to allow a deeper search with the same amount of computation. • The question: how much does alpha-beta improve performance? • The best way to characterize is asymptotic effective branching factor. • The dth root of the number of nodes (in a search to depth d, in the limit of large d) • number of nodes generated at depth d / number of nodes generated at depth d-1.

3. Performance analysis of Alpha-Beta Pruning • The efficiency of alpha-beta pruning depends upon the order in which nodes are encountered at the search frontier. • Thus, we consider 3 different cases: • worst case - the algorithm doesn’t perform any cutoffs at all • best case • average case

4. MAX MIN 4 4 2 4 8 2 14 4 2 6 8 1 2 12 14 4 5 3 2 6 7 8 9 1 10 2 11 12 13 14 14 Example of alpha-beta worst case • Evaluation from left to right

5. Lower Bound for Minimax Algorithms • We consider a lower bound on the number of leaf nodes that must be examined by any minimax algorithm. • In minimax algorithm, it’s a guaranty to return the minimax value vof the root node of a game tree. • verifying maximum value = v verifying value  v && value  v. • Any correct minimax algorithm must explore: • a strategy for Max • a strategy for Min

6. value  v: doesn’t matter what min does Strategy for max: subtree containing: one child of each Max node all b children of each min node value  v: doesn’t matter what Max does Strategy for min: subtree containing: one child of each Min node all b children of each Max node Strategies for Min and Max

7. Example strategy for Min: strategy for Max: Min strategy mixed Max strategy

8. Strategy for Max d is even  leaf nodes d is odd  leaf nodes Strategy for Min d is even  leaf nodes d is odd  leaf nodes Lower Bound for Minimax Algorithms - Analysis • Assume : • uniform branching factor of b uniform depth of d levels • Max move is at the root.

9. Lower Bound for Minimax Algorithms - Analysis Total number of distinct leaf nodes: • d is odd : • d is even :  note: • there is a single leaf node in common of both strategies.

10. b d/2 + b d/2 -1 = O(bd/2 ) Lower Bound for Minimax Algorithms - Analysis • This is the number of leaf nodes that must be examined by any minimax algorithm. • This is the lower bound of the time complexity.

11. Minimax value of game trees What will be the minimax value in the following cases • Special case • leaf nodes: • are actual terminal positions, • have the exact values of WIN or LOSS. • General case • arbitrary leaf node values In the course we will only deal with the special case

12. WIN-LOSS Trees analytic model • uniform branching factor b • uniform depth d • Max is to move at the root • depth d is even • terminal nodes labeled WIN with probability P0 • terminal nodes labeled LOSS with probability 1 - P0

13. Example: Board Splitting • Two players:verticaland horizontal • square sheet of graph paper, bd/2 squares on each side • each square V with probability P0 and H with probability 1 - P0 • vertical’sturn: divides the board vertically into b equals slices, discarding all but one of them.

14. Board Splitting • horizontal’s turn: divides the board horizontally into b equals slices, discarding all but one of them. • Result: the initial in the only square left indicates the winner.

15. Pn probability that Max force a win, given that Max is to move  2n moves in the tree Qn probability that Max force a win, given that min is to move  2n -1 moves in the tree Pn Qn Pn-1 Max min Complexity of WIN-LOSS Tree

16. WIN-LOSS Trees  This is the probability that a Max node at any higher level in the tree will be a win for Max

17. Min is to move to be a win for Max.  all of its children must be wins for Max probability that all b children of a node are win for Max  Q n = (Pn - 1)b Max is to move to be a loss for Max  all of its children must be losses for Max probability that a node is a loss for Max  1 - probability that it is a win for Max  1 - P n = (1 - Q n) WIN-LOSS Trees

18. WIN-LOSS Trees 1.00 crossover point: determines the probability of a win for Max 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 0.00ב 0.20 0.40 0.60 0.80 1.00 Graph showing iterations of function f(x) = 1 - (1 - x2)2.

19. WIN-LOSS Trees If the probability of a win for MAX at the leaves is grater than crossover point, then the large enough game tree is almost certainly a forced win for MAX!

20. WIN-LOSS Trees • Let be is the fixed point of the iteration (crossover point). For b = 2, This value is also known as “golden ratio” . • The probability that the root of a game tree is a forced win for Max: Even through wins and losses are chosen randomly at the leaf nodes, we can predict the win-loss status of the root of a sufficiently deep minimax tree with almost certainty, simply by knowing the probability of a win at the leaves!

21. Minimax convergence theorem • We now generalize our result to the case of leaf nodes with arbitrary numerical values. We adopt a following model: • Uniform branching factor b • uniform depth d • leaves are assigned random numeric value, but from a common probability distribution function Fv0(v) = P(v0 v). • v0 is a particular node value chosen from this distribution

22. Minimax convergence theorem • Let’s determine the probability distribution of the minimax value of the root of a tree • in the limit of large depth • as a function of the probability distribution of the leaf values • with the expression of the distribution of the minimax values at 2n levels above the leaves as Fvn(v)

23. Minimax convergence theorem a leaf node is a win for Max its value> v • Minimax value of a max node will be greater than v any one of its children has a minimax value greater than v. For any value of v: • The minimax values propagate up the same as in the win-loss trees. a leaf node is a win for Min its value >v • Minimax value of a min node will be greater than v all of its children has a minimax value greater than v.

24. The theorem: Minimax convergence theorem • In win-loss treePn is the probability that a Max node 2n levels above the leaves is a forced win for Max. • In general game tree

25. Minimax convergence theorem The Meaning: • Probability distribution is zero up to a particular value of v (v*). in a b-ary tree: • beyond v*, the probability distribution function of the minimax value of the root is 1. • The probability density function of the step distribution function Fvn(v) is an impulse at v*. All the probability mass is concentrated at v *.

26. Conclusion for a minimax tree We can predict exactly what the minimax value of the root of the tree will be. • Given: • arbitrary terminal values chosen independently from the same distribution • limit of large depth

27. Example (application) Goal: • a fuse that will burn out after a specific time. problem: • we only have fuses that have same broad distribution of burn-out time. solution: • we connect two fuses in parallel • the burn-out time of the whole circuit will be the maximum of the burn-out time of the individual fuses • the circuit will remain closed until both fused burn-out • The burn-out time of the entire circuit is the minimax value of the burn-out times of the individuals fuses.

28. Average-case time complexity - Win-Loss game tree • We assume the previous model. • Assume that According to the minimax convergence theorem, at sufficiently high levels of the tree, all nodes are losses for Max and wins for Min. Max node  all the children must be examined  all the children will be a loss for Max Min node  only the first node must be examined  it will be a win for Max

29. Average-case time complexity - Win-Loss game tree If we follow any path from root, we will branch: • only one way at the alternating Max levels • b ways at every other level @ effective branching factor of b1/2 The asymptotic number of leaf nodes in the limit of large depthO(bd/2 )

30. Average-case time complexity - Win-Loss game tree • Assume that In this case, at sufficiently high levels of the tree, all nodes are losses for Max and wins for Min. • As the above case, this alse results in effective b.f of b1/2 • Assume that (extremely rare case) • Pearl shows : effective branching factor =

31. Average-case time complexity - Trees with arbitrary terminal values • There are two possibilities for choosing the leaf values: • a continuous distribution - segment of the real number number line. • Minimax values of all nodes will be equal • Alpha-beta pruning will realize its best performance • a discrete distribution - only a finite number of distinct values. • The probability that any node takes an any particular value is zero • Pearl shows: effective branching factor =

32. Chapter 9 Multi player games

33. Introduction • We generalize the 2-player-perfect-information algorithms, to the case of non-cooperative perfect-information -more players games. • No coalitions between players. • Examples: • Chinese Checkers with 6 players. • Othello extended by having different colored pieces for each player.

34. Maxn (maxn) Algorithm Assumption: • the players alternate moves • each player tries to maximize his/her return • and indifferent to returns of others.

35. Maxn Algorithm • At frontier nodes, evaluation function returns an n-tuple of values: (player1, P2, P3, …. Player n) • For example: • Othello - return number of pieces for each player.

36. Maxn Algorithm the entire n-tuple of the child for which the ith component is maximum. evaluation function in each interior node where player i is to move =

37. (7,3,6) 1 (1,7,2) (7,3,6) 2 2 (6,5,4) (1,7,2) (7,3,6) 3 3 3 3 (3,1,8) (2,8,1) (3,1,8) (1,7,2) (5,6,3) (6,5,4) (8,5,4) (7,3,6) (4,2,7) Maxn Algorithm - Example

38. Maxn Algorithm • Formal notations: • M(x) - static heuristic value of node x • M(x,p) - backed-up maxn value of node x by player p. • Mi(x,p) - component of M(x,p) corresponds to the return for player i. • M(xi,p’) = maxMp(xi,p’) over children of node x, • p’ is player that follows player p • tie breaking in favor of leftmost node. • Recursive definition of the maxn node:

39. Maxn Algorithm • Minimax can be viewed as a special case of maxn, when • n = 2, • evaluation function: (x, -x). • Luckhardt & Irani observed: • at nodes where player i is to move, only the ith component of the children need be evaluated. • It may be no less expensive to compute all components. • Without assumptions on values of components, pruning of branches is impossible (with more than 2 players).

40. Paranoid Algorithm • Here we see the other players as one big opponent. • There is my own player. The Max player. • There are all the others. The Min players. • The paranoid algorithm evaluates the tree as follows. • When it is my turn to play – take the maximum of my utility. • When it is not my turn (it is one of them) I take the minimum of my utility.

41. Paranoid Algorithm • With max^n we assumed that each of the other palyers chooses to take its maximum value. • With paranoid algorithm we assume that each of the other players chooses to take my minimum. • Paranoid algorithm is actually MIN^(n-1) MAX. • Paranoid better generalizes minimax because it also gives an insurance of the value of the root node. (see chapter 8)

42. Paranoid Vs. Maxn • Paranoid runs faster as we can see it as a regular minimax algorithm with only two players. The max player (myself) and the min player (them, n-1 players). • In the min phase of minimax, we can span n-1 levels at once without the need to evaluate internal nodes. • The two algorithms are not necessarily comparable since they require different assumptions of the intentions of the players. • Maxn assume that every player wants its own highest score. • Paranoid assumes that the other players want to kill me.

43. Alpha-Beta in multi-Player Games • Tree pruning is possible when : • there is an upper bound on the sum of all components of a tuple • a lower bound on the values of each component. • For example: • Othello - no player can have less than zero, and total number of pieces on the board is equal for all nodes at same level.

44. Immediate Pruning • Player I is to move, and in one child the ith component equals the upper bound of sum on all components. • Obvious that any other child can be pruned. • This is equivalent to situations in the two-player case when a child of a Max node has value of , or a child of a Min node has value of -, indication a won position for the corresponding player.

45. (3,3,3) (3, 6, 6) 1 (3,3,3) 2 (2, 7, 2) (3, 6, 3) 2 2 3 3 3 3 3 (3,3,3) (1,6,2) (4,2,3) (3,1,5) (1,7,1) Shallow Pruning

46. Shallow algorithm Shallow(Node, Player, Bound) IF Node is terminal, RETURN static value Best = Shallow(first Child, next Player, Sum) FOR each remaining Child IF Best[Player] >= Bound, RETURN Best Current = Shallow(Child, next Player, Sum - Best[Player] ) IF Current[Player] > Best[Player], Best = Current RETURN Best

47. Failure of deep pruning • In a 2-player game, alpha-beta allows deep pruning - pruning a node based on bounds inherited from its great-grandparents, or other distant ancestor. Deep pruning does not generalize to more than 2 players!

48. (5, 5, 4) 1 2 2 (5,2,2) (4, 4, 5) 3 3 (6,1,2) 1 1 (2,2,5) (2,3,4) or (3,0,6) Failure of Deep Pruning -Example

49. Optimality of shallow pruning • Theorem: • Every directional algorithm that computes the maxn value of a game tree with more than 2 players must evaluate every terminal node evaluated by shallow pruning under the same order. • Steps of proof: • The formal proof is by induction on the height of the tree and generalizes the result to an arbitrary number of players greater than 2.

50. Optimality of shallow pruning • Try to see a “zipper” effect in the sense that the original order of the “teeth” (nodes) at the bottom determines the order of the teeth at the top, even though no individual tooth can move very far.