Juegos minimax AlfaBeta

Juegos

Nelson Becerra Correa

Fundación FABBECOR-ONG

Bibliografía• Conferencia : Miguel Illescas, Gran Maestro Internacional.

Kasparov, Deep Blue y la mente del ajedrecista.

• http://www.ocf.berkeley.edu/~yosenl/extras/alphabeta/alphabeta

• Jorge Egger M. Desarrollo de la maestría ajedrecística computacional. Utilizando recursos restringidos. Universidad de chile. Facultad de ciencias físicas y matemáticas. Departamento de ciencias de la computación, Santiago de chile, Julio 2003.

• Técnicas de inteligencia artificial en Juego. Nelson Becerra Correa, Fondo de publicaciones Universidad Distrital (FJC), Año 2003.

• w.acm.uiuc.edu/sigart/docs/MinimaxPresentation.ppt

http://www.ocf.berkeley.edu/~yosenl/extras/alphabeta/alphabeta

http://www.ocf.berkeley.edu/~yosenl/extras/alphabeta/alphabeta

TEORIA DE JUEGOS Fué creada para solucionar problemas

económicos. Los pioneros de esta teoría son John Von Newman y Oskar Morgestern.

En la actualidad se aplica a la Psicología, la administarción , las ciencias políticas etc.

Clasificación de los juegos

NUMERO PARTICIPANTES

•Unipersonales•Bipersonales•N-personas N>2

NUMEROESTRATEGIAS

•Finito (estrategias finitas)•Infinitas

DE ACUERDO A LA

INFORMACION

•Información perfecta (Secuenciales)

•Información Imperfecta (Yan ken Po) (Dilema pri)

DE ACUERDO A LA SUMA

•Juegos de suma cero (ajedrez, oso, go,mankala)

•Juegos de suma distinta de cero (F)

Como juegan las maquinas





DEEP-BLUE potencia de calculo

Función de evaluación

Función de evaluación

Posición que un programa sin Tablas de Transposición no puede solucionar.Las Blancas ganan mediante 1.Rb1!! Rb7 2.Rc1!!

1. 30 movimientos sería necesaria, y eso tomaría miles de horas de tiempo de CPU.

2. Con tablas Hash 6 movimientos, puesto que existen 130 diferentes tipos de posiciones obtenibles desde la inicial.

Basic Idea

• Search problem– Searching a tree of the possible moves in order

to find the move that produces the best result– Depth First Search algorithm

• Assume the opponent is also playing optimally– Try to guarantee a win anyway!

Árbol de variantes

1. En los inicios de 1970 la búsqueda en árboles de variantes alcanzaba la cantidad aproximada de 200 posiciones por segundo.

2. En año 2003, DEEP BLUE busca en 2.000.000 de posiciones por segundo.

3. Los mejores programas logran examinar todas las secuencias de movimientos con una profundidad de 8 a 10 movidas en el árbol (4 a 5 jugadas por bando).

Required Pieces for Minimax

• An initial state– The positions of all the pieces– Whose turn it is

• Operators– Legal moves the player can make

• Terminal Test– Determines if a state is a final state

• Utility Function

Utility Function• Gives the utility of a game state

– utility(State)

• Examples– -1, 0, and +1, for Player 1 loses, draw, Player 1 wins,

respectively– Difference between the point totals for the two players– Weighted sum of factors (e.g. Chess)

• utility(S) = w1f

1(S) + w

2f

2(S) + ... + w

nf

n(S)

– f1(S) = (Number of white queens) – (Number of black queens), w

1 = 9

– f2(S) = (Number of white rooks) – (Number of black rooks), w

2 =

5– ...

Two Agents• MAX

– Wants to maximize the result of the utility function

• MIN– Wants to minimize the result of the evaluation

function

… MINIMAXEjemplo de definición de función evaluadora:• Si p no es posición ganadora para ningún ganado, entonces:

– e(p)= (el nº de filas, columnas y diagonales completas que todavía estan libres para MAX) – (nº de filas, columnas, diagonales completas que todavía estan libres para MIN)

• Si p es una posición ganadora para MAX, entonces– e(p)= ∞ (donde ∞ indica un número positivo muy grande)

• Si p es una posición ganadora para MIN, entonces– e(p)= -∞

… MINIMAXOX

Se tiene que e(p)=6 – 4 = 2

http://www.ocf.berkeley.edu/~yosenl/extras/alphabeta/alphabeta.html

Basic Algorithm

ALGORITMOS DE JUEGOS

• Algoritmos de primera generación

– se caracterizan por confiar en la capacidad bruta de las maquinas

• Algoritmos de segunda generación:

– Surge la inquietud de que mientras la maquina examina miles de posiciones su oponente humano solo examina algunas de ellas

ALGORITMOS DE PRIMERA GENERACION

• MINIMAX

• NEGAMAX

• ALFA/BETA

• FALFA/BETA

• LALFA/BETA

• PALFA/BETA

• SCOUT

MINIMAX NEGAMAX

ALFA/BETA FALFA/BETA

LALFA/BETA PALFA/BETA

SCOUT

Max

Min

Max

Min

Starting node and labelsMiniMax Search

Max

Min

Max

Min

Continue expand the search space

Max

Min

Max

Min

Continue expand the search space

Max

Min

Max

Min2 3 5 09 7 4

Expand the search space down3-ply

Max

Min

Max

Min2 3 5 09 7 4

Propagate the leaf values backwardPick Max values at Max Level

9 03 77

Max

Min

Max

Min2 3 5 09 7 4

Continue propagate the values backwardPick Min values at Min Level

9 03 77

3 0

Max

Min

Max

Min2 3 5 09 7 4

Continue propagate the values backwardPick Max values at Max Level

9 03 77

3 0

3

Max

Min

Max

Min2 3 5 09 7 4

The move picked by Max

9 03 77

3 0

3

Example MINIMAX

• Coins game– There is a stack of N coins– In turn, players take 1, 2, or 3 coins from the

stack– The player who takes the last coin loses

Coins Game: Formal Definition

• Initial State:

The number of coins in the stack• Operators:

1. Remove one coin

2. Remove two coins

3. Remove three coins

• Terminal Test:

There are no coins left on the stack• Utility Function: F(S)

– F(S) = 1 if MAX wins, 0 if MIN wins

N = 4K =

N = 3K =

N = 2K =

N = 1K =

N = 0K =

N = 1K =

N = 2K =

N = 0K =

N = 0K =

N = 1K =

N = 0K =

N = 1K =

N = 0K =

N = 0K =

12

3

3 21 2 1 1

1211

N = 0K =

1 F(S)=0

F(S)=0

F(S)=0

F(S)=1

F(S)=1

F(S)=1

F(S)=1

MAXMIN

N = 4K = 1

N = 3K = 0

N = 2K = 0

N = 1K = 1

N = 0K = 1

N = 1K = 0

N = 2K = 1

N = 0K = 1

N = 0K = 1

N = 1K = 0

N = 0K = 0

N = 1K = 1

N = 0K = 0

N = 0K = 0

12

3

3 21

2 1 1

1211

N = 0K = 1

1 F(S)=0F(S)=0 F(S)=0

F(S)=1

F(S)=1

FFS(51791527S)=1

F(S)=1

MAXMIN

Solution

F(S)=1

Analysis• Max Depth: 5• Branch factor: 3• Number of nodes: 15• Even with this trivial example, you can see that

these trees can get very big– Generally, there are O(bd) nodes to search for

• Branch factor b: maximum number of moves from each node• Depth d: maximum depth of the tree

– Exponential time to run the algorithm!– How can we make it faster?

Alpha-Beta Pruning

• Main idea: Avoid processing subtrees that have no effect on the result

• Two new parameters– α: The best value for MAX seen so far– β: The best value for MIN seen so far

• α is used in MIN nodes, and is assigned in MAX nodes

• β is used in MAX nodes, and is assigned in MIN nodes

Alpha-Beta Pruning

• MAX (Not at level 0)– If a subtree is found with a value k greater than

the value of β, then we do not need to continue searching subtrees• MAX can do at least as good as k in this node, so MIN

would never choose to go here!

• MIN– If a subtree is found with a value k less than the

value of α, then we do not need to continue searching subtrees• MIN can do at least as good as k in this node, so MAX

would never choose to go here!

Max

Min

Max

Min

Starting node and labelsAlpha-Beta Prune

Max

Min

Max

Min

Perform a DFS

Max

Min

Max

Min

Continue the DFS

Max

Min

Max

Min2 3

Until reach the depth we want, i.e., 3-ply in this case

Max

Min

Max

Min2 3

3

Propagate leaf values backward

Max

Min

Max

Min2 3

3

3

Max

Min

Max

Min2 3

3

3

3

Max

Min

Max

Min2 3

3

3

3 Alpha Node

Beta Node

Max

Min

Max

Min2 3

3

3

3 Alpha

Beta

5

New DFS path Ended with 5

Max

Min

Max

Min2 3

3

3

3 Alpha

Beta

5

5

Propagate backward

Max

Min

Max

Min2 3

3

3

3 Alpha

Beta

5

5

Because 3 < 5

The branch is pruned.

Beta-Prune

Max

Min

Max

Min2 3

3

3

3 Alpha

Beta

5

5

New 3-ply DFSEnded with 0

0

Max

Min

Max

Min2 3

3

3

3 Alpha

Beta

5

5

0

0

Propagate backward

Max

Min

Max

Min2 3

3

3

3 Alpha

Beta

5

5

0

0

Propagate backward

0

Max

Min

Max

Min2 3

3

3

3 Alpha

Beta

5

5

0

0

The branch is cut offBecause 3 > 0

Alpha Prune

0

Algorithm

N = 4K =

N = 3K =

N = 2K =

N = 1K =

N = 0K =

N = 1K =

N = 2K =

N = 0K =

N = 0K =

N = 1K =

N = 3K =

N = 1K =

N = 0K =

N = 0K =

12

3

3 21

2 1 1

1211

N = 0K =

1 F(S)=0F(S)=0 F(S)=0

F(S)=1

F(S)=1

F(S)=1 F(S)=1

MAXMIN

α =β =

α =β =

α =β =

α =β =

α =β =

α =β =

α =β =

α =β =

α =β =

α =β =

α =β =

α =β =

α =β =

α =β =

α =β =

N = 4K = 0 1

N = 3K = 1 0

N = 2K = 1 0

N = 1K = 1

N = 0K = 1

N = 1K = 0

N = 2K = 1

N = 0K = 1

N = 0K = 1

N = 1K =

N = 3K = 0

N = 1K = 1

N = 0K =

N = 0K = 0

12

3

3 21

2 1 1

1211

N = 0K = 1

1 F(S)=0F(S)=0 F(S)=0

F(S)=1

F(S)=1

F(S)=1 F(S)=1

MAXMIN

α =β = 1 0

α = 0β =

α = 0β = 1 0

α =β = 0

α = 1β =

α = 0β = 0

α = 0β = 0

α = 0β = 1

α = 0β =

α = 0β = 0

α = 0β =

α = 1β = 0

α =β =

α = 1β = 0

α = 0β = 1

Conclusion

• Minimax finds optimal play for deterministic, fully observable, two-player games

• Alpha-Beta reduction makes it faster

Entertainment & Humor

Juegos minimax AlfaBeta