Section CS3191 3
Medium Games
Andrea Schalk Department of Computer Science, University of Manchester
Andrea Schalk: CS3191 Section 3 p. 1/24
Medium games
The topic of Section 3 are medium games.
Andrea Schalk: CS3191 Section 3 p. 2/24
Medium games
The topic of Section 3 are medium games. For a game to be medium-sized it has to be larger than smallin other words, it should be so large that it is not feasible to describe it via the Players strategies.
Andrea Schalk: CS3191 Section 3 p. 2/24
Medium games
The topic of Section 3 are medium games. For a game to be medium-sized it has to be larger than smallin other words, it should be so large that it is not feasible to describe it via the Players strategies. It will become clear in the course of this section when a game is, in fact large.
Andrea Schalk: CS3191 Section 3 p. 2/24
Medium games
The topic of Section 3 are medium games. For a game to be medium-sized it has to be larger than smallin other words, it should be so large that it is not feasible to describe it via the Players strategies. It will become clear in the course of this section when a game is, in fact large. The idea for this section is to employ computer power to help us nd good strategies for a game. To some extent we will be building on our theory of games developed in Sections 1 and 2, since the notion of game tree will still be present.
Andrea Schalk: CS3191 Section 3 p. 2/24
Algorithmic point of view
Andrea Schalk: CS3191 Section 3 p. 3/24
Algorithms
In order to solve a small game, the following tasks have to be performed:
Andrea Schalk: CS3191 Section 3 p. 4/24
Algorithms
In order to solve a small game, the following tasks have to be performed:
generate all the strategies for all the players;
Andrea Schalk: CS3191 Section 3 p. 4/24
Algorithms
In order to solve a small game, the following tasks have to be performed:
generate all the strategies for all the players; calculate the pay-offs when playing the various strategies against each other;
Andrea Schalk: CS3191 Section 3 p. 4/24
Algorithms
In order to solve a small game, the following tasks have to be performed:
generate all the strategies for all the players; calculate the pay-offs when playing the various strategies against each other; nd equilibrium points.
Andrea Schalk: CS3191 Section 3 p. 4/24
Algorithms
In order to solve a small game, the following tasks have to be performed:
generate all the strategies for all the players; calculate the pay-offs when playing the various strategies against each other; nd equilibrium points.
Andrea Schalk: CS3191 Section 3 p. 4/24
Algorithms
In order to solve a small game, the following tasks have to be performed:
generate all the strategies for all the players; calculate the pay-offs when playing the various strategies against each other; nd equilibrium points.
Andrea Schalk: CS3191 Section 3 p. 4/24
Algorithms
In order to solve a small game, the following tasks have to be performed:
generate all the strategies for all the players; calculate the pay-offs when playing the various strategies against each other; nd equilibrium points. X
Andrea Schalk: CS3191 Section 3 p. 4/24
Beyond small games
Andrea Schalk: CS3191 Section 3 p. 5/24
How do games grow?
Question. How does the number of positions grow with the size of the game tree?
Andrea Schalk: CS3191 Section 3 p. 6/24
How do games grow?
Question. How does the number of positions grow with the size of the game tree? How does the number of strategies (added up over all players) grow with the size of the game tree?
Andrea Schalk: CS3191 Section 3 p. 6/24
How do games grow?
Question. How does the number of positions grow with the size of the game tree? How does the number of strategies (added up over all players) grow with the size of the game tree?
If the tree branches at least into two at most decision points then for a tree of height m there are about
Andrea Schalk: CS3191 Section 3 p. 6/24
How do games grow?
Question. How does the number of positions grow with the size of the game tree? How does the number of strategies (added up over all players) grow with the size of the game tree?
If the tree branches at least into two at most decision points then for a tree of height m there are about
2m+1 1
decision points, that is their number grows exponentially.
Andrea Schalk: CS3191 Section 3 p. 6/24
How do games grow?
To count the number of strategies, we need to know how many decision points each player has (nodes when it is his turn), and how they relate.
Andrea Schalk: CS3191 Section 3 p. 6/24
How do games grow?
To count the number of strategies, we need to know how many decision points each player has (nodes when it is his turn), and how they relate. We look at an example: Let us assume we have a complete binary tree where two players move alternatingly.
Andrea Schalk: CS3191 Section 3 p. 6/24
How do games grow?
To count the number of strategies, we need to know how many decision points each player has (nodes when it is his turn), and how they relate. We look at an example: Let us assume we have a complete binary tree where two players move alternatingly. height 0 no pos. 1 strats for P1 1 strats for P2 1 all strats 2
Andrea Schalk: CS3191 Section 3 p. 6/24
How do games grow?
To count the number of strategies, we need to know how many decision points each player has (nodes when it is his turn), and how they relate. We look at an example: Let us assume we have a complete binary tree where two players move alternatingly. height 0 1 no pos. 1 3 strats for P1 1 2 strats for P2 1 1 all strats 2 3
Andrea Schalk: CS3191 Section 3 p. 6/24
How do games grow?
To count the number of strategies, we need to know how many decision points each player has (nodes when it is his turn), and how they relate. We look at an example: Let us assume we have a complete binary tree where two players move alternatingly. height 0 1 2 no pos. 1 3 7 strats for P1 1 2 2 strats for P2 1 1 4 all strats 2 3 6
Andrea Schalk: CS3191 Section 3 p. 6/24
How do games grow?
To count the number of strategies, we need to know how many decision points each player has (nodes when it is his turn), and how they relate. We look at an example: Let us assume we have a complete binary tree where two players move alternatingly. height 0 1 2 3 4 5 6 7 8 no pos. 1 3 7 15 31 63 127 255 511 strats for P1 1 2 2 8 8 128 128 32768 32768 strats for P2 1 1 4 4 64 64 16384 16384 1073741824 all strats 2 3 6 12 72 192 16512 49152 1073774592
Andrea Schalk: CS3191 Section 3 p. 6/24
What to do?
The fact that the number of strategies in a game typically grows exponentially makes it quite clear that we soon move out of the realms of games that can be solved with the methods from Section 2.
Andrea Schalk: CS3191 Section 3 p. 7/24
What to do?
The fact that the number of strategies in a game typically grows exponentially makes it quite clear that we soon move out of the realms of games that can be solved with the methods from Section 2. The aim for this section is to nd good strategies without creating all strategies rst.
Andrea Schalk: CS3191 Section 3 p. 7/24
What to do?
The fact that the number of strategies in a game typically grows exponentially makes it quite clear that we soon move out of the realms of games that can be solved with the methods from Section 2. The aim for this section is to nd good strategies without creating all strategies rst. For that we assume that the game tree is given in a computer.
Andrea Schalk: CS3191 Section 3 p. 7/24
What to do?
The fact that the number of strategies in a game typically grows exponentially makes it quite clear that we soon move out of the realms of games that can be solved with the methods from Section 2. The aim for this section is to nd good strategies without creating all strategies rst. For that we assume that the game tree is given in a computer. However, our algorithms will not require that the game is present in memory at all timesit is sufcient if it can be created on demand.
Andrea Schalk: CS3191 Section 3 p. 7/24
The minimax algorithm
Andrea Schalk: CS3191 Section 3 p. 8/24
Values
For a game of perfect information we dene the value of a position p for Player X to be the pay-off he can guarantee for himself at the very least.
Andrea Schalk: CS3191 Section 3 p. 9/24
Values
For a game of perfect information we dene the value of a position p for Player X to be the pay-off he can guarantee for himself at the very least. (This harkens back to Scotties and Amelias original analysis, which we will take to its natural conclusion in this section.)
Andrea Schalk: CS3191 Section 3 p. 9/24
Values
For a game of perfect information we dene the value of a position p for Player X to be the pay-off he can guarantee for himself at the very least. Note that the player has to assume that the others will do their worst in order for the guarantee to work.
Andrea Schalk: CS3191 Section 3 p. 9/24
Values
For a game of perfect information we dene the value of a position p for Player X to be the pay-off he can guarantee for himself at the very least.
Question. Why do we insist on having a game of perfect information here?
Andrea Schalk: CS3191 Section 3 p. 9/24
Values
For a game of perfect information we dene the value of a position p for Player X to be the pay-off he can guarantee for himself at the very least.
Question. Why do we insist on having a game of perfect information here?
We can deal with elements of chance if we stick to the expected pay-off as before.
Andrea Schalk: CS3191 Section 3 p. 9/24
Calculating the value
The following proposition shows us how to calculate the value of a position for a given player.
Andrea Schalk: CS3191 Section 3 p. 10/24
Calculating the value
The following proposition shows us how to calculate the value of a position for a given player.
p
p1
p2
pn
Andrea Schalk: CS3191 Section 3 p. 10/24
Calculating the value
The following proposition shows us how to calculate the value of a position for a given player.
p
p1
p2
pn
Proposition 3.1 Let p be a position with p1 , p2 , . . . , pm the positions which can be reached from p with one move (see the gure). Then the value for player X of p, vX (p), is
p is nal pX (p) max X to move at p 1im vX (pi ) vX (p) = Y = X to move at p min1im vX (pi ) q v (p ) + q v (p ) + + q v (p ) chance move at p; 1 X 1 2 X 2 n X n qi : probability for pi
Andrea Schalk: CS3191 Section 3 p. 10/24
Proof
p is nal pX (p) max X to move at p 1im vX (pi ) vX (p) = Y = X to move at p min1im vX (pi ) q v (p ) + q v (p ) + + q v (p ) chance move at p; 1 X 1 2 X 2 n X n qi : probability for pi
Andrea Schalk: CS3191 Section 3 p. 11/24
Proof
p is nal pX (p) max X to move at p 1im vX (pi ) vX (p) = Y = X to move at p min1im vX (pi ) q v (p ) + q v (p ) + + q v (p ) chance move at p; 1 X 1 2 X 2 n X n qi : probability for pi
This is really obvious from the denition:
Andrea Schalk: CS3191 Section 3 p. 11/24
Proof
p is nal pX (p) max X to move at p 1im vX (pi ) vX (p) = Y = X to move at p min1im vX (pi ) q v (p ) + q v (p ) + + q v (p ) chance move at p; 1 X 1 2 X 2 n X n qi : probability for pi
If it is Player X s turn at p then he can choose the position with the maximum value and from there use a strategy to achieve that value.
Andrea Schalk: CS3191 Section 3 p. 11/24
Proof
p is nal pX (p) max X to move at p 1im vX (pi ) vX (p) = Y = X to move at p min1im vX (pi ) q v (p ) + q v (p ) + + q v (p ) chance move at p; 1 X 1 2 X 2 n X n qi : probability for pi
If it is Player X s turn at p then he can choose the position with the maximum value and from there use a strategy to achieve that value. If it is some other players turn he cannot do better than guarantee the value that he might get in the worst case, which is the minimum of the values that can be reached from p with one move.
Andrea Schalk: CS3191 Section 3 p. 11/24
Proof
p is nal pX (p) max X to move at p 1im vX (pi ) vX (p) = Y = X to move at p min1im vX (pi ) q v (p ) + q v (p ) + + q v (p ) chance move at p; 1 X 1 2 X 2 n X n qi : probability for pi
If it is Player X s turn at p then he can choose the position with the maximum value and from there use a strategy to achieve that value. If it is some other players turn he cannot do better than guarantee the value that he might get in the worst case, which is the minimum of the values that can be reached from p with one move. If it is a chance move then the minimal expected pay-off from p is
q1 vX (p1 ) + q2 vX (p2 ) + + qm vX (pm ).
Andrea Schalk: CS3191 Section 3 p. 11/24
The values of a game
Note that unless we have a 2-person zero-sum game there will not be a unique value which we can take as relevant for all the players.
Andrea Schalk: CS3191 Section 3 p. 12/24
The values of a game
Note that unless we have a 2-person zero-sum game there will not be a unique value which we can take as relevant for all the players. The various values for the root for each player give the pay-off they can guarantee for themselves in the worst case. Of course, they might do better when the game is played!
Andrea Schalk: CS3191 Section 3 p. 12/24
The values of a game
Note that unless we have a 2-person zero-sum game there will not be a unique value which we can take as relevant for all the players. The various values for the root for each player give the pay-off they can guarantee for themselves in the worst case. Of course, they might do better when the game is played! We have given a recursive algorithmin order to nd the value (for a player) of some node we rst have to know the value (for that player) of all its children.
Andrea Schalk: CS3191 Section 3 p. 12/24
How to nd the valuebottom-up
max P1 min ? P2
-1
max
4
?
?
2
-2
P1
1
2
-1
2
1
Andrea Schalk: CS3191 Section 3 p. 13/24
How to nd the valuebottom-up
max
P1
min
?
-1
-2
P2
max
4
2
2
2
-2
P1
1
2
-1
2
1
Andrea Schalk: CS3191 Section 3 p. 13/24
How to nd the valuebottom-up
max ?
P1
min
2
-1
-2
P2
max
4
2
2
2
-2
P1
1
2
-1
2
1
Andrea Schalk: CS3191 Section 3 p. 13/24
How to nd the valuebottom-up
max 2 P1 min 2 -2 P2
-1
max
4
2
2
2
-2
P1
1
2
-1
2
1
Andrea Schalk: CS3191 Section 3 p. 13/24
Alternatives?
Question. How does a program implementing this algorithm traverse the tree?
Andrea Schalk: CS3191 Section 3 p. 14/24
Alternatives?
Question. How does a program implementing this algorithm traverse the tree?
This would have to be implemented using breadth-rst traversal. This is usually not a very efcient way of traversing a tree, in particular one that is being created as it is being searched!
Andrea Schalk: CS3191 Section 3 p. 14/24
Alternatives?
Question. How does a program implementing this algorithm traverse the tree?
This would have to be implemented using breadth-rst traversal. This is usually not a very efcient way of traversing a tree, in particular one that is being created as it is being searched!
Question. If you had to write a program for nding the value of a game, would you do it in this way?
Andrea Schalk: CS3191 Section 3 p. 14/24
Alternatives?
Question. How does a program implementing this algorithm traverse the tree?
This would have to be implemented using breadth-rst traversal. This is usually not a very efcient way of traversing a tree, in particular one that is being created as it is being searched!
Question. If you had to write a program for nding the value of a game, would you do it in this way? Can you think of alternatives?
Andrea Schalk: CS3191 Section 3 p. 14/24
Alternatives?
Question. How does a program implementing this algorithm traverse the tree?
This would have to be implemented using breadth-rst traversal. This is usually not a very efcient way of traversing a tree, in particular one that is being created as it is being searched!
Question. If you had to write a program for nding the value of a game, would you do it in this way? Can you think of alternatives?
It is much better to traverse the tree depth-rst. We will study this idea next.
Andrea Schalk: CS3191 Section 3 p. 14/24
Depth-rst algorithm
P1 ?
P2
-1
P1
4
2
-2
1
2
-1
2
1
Andrea Schalk: CS3191 Section 3 p. 15/24
Depth-rst algorithm
P1 ?
P2
?
-1
P1
4
2
-2
1
2
-1
2
1
Andrea Schalk: CS3191 Section 3 p. 15/24
Depth-rst algorithm
P1 ?
P2
?
-1
P1
4
?
2
-2
1
2
-1
2
1
Andrea Schalk: CS3191 Section 3 p. 15/24
Depth-rst algorithm
P1 ?
P2
?
-1
P1
4
?
2
-2
1
2
-1
2
1
Andrea Schalk: CS3191 Section 3 p. 15/24
Depth-rst algorithm
P1 ?
P2
?
-1
P1
4
2
2
-2
1
2
-1
2
1
Andrea Schalk: CS3191 Section 3 p. 15/24
Depth-rst algorithm
P1 ?
P2
?
-1
P1
4
2
?
2
-2
1
2
-1
2
1
Andrea Schalk: CS3191 Section 3 p. 15/24
Depth-rst algorithm
P1 ?
P2
?
-1
P1
4
2
2
2
-2
1
2
-1
2
1
Andrea Schalk: CS3191 Section 3 p. 15/24
Depth-rst algorithm
P1 ?
P2
2
-1
P1
4
2
2
2
-2
1
2
-1
2
1
Andrea Schalk: CS3191 Section 3 p. 15/24
Depth-rst algorithm
P1 ?
P2
2
-1
?
P1
4
2
2
2
-2
1
2
-1
2
1
Andrea Schalk: CS3191 Section 3 p. 15/24
Depth-rst algorithm
P1 ?
P2
2
-1
-2
P1
4
2
2
2
-2
1
2
-1
2
1
Andrea Schalk: CS3191 Section 3 p. 15/24
Depth-rst algorithm
P1 2
P2
2
-1
-2
P1
4
2
2
2
-2
1
2
-1
2
1
Andrea Schalk: CS3191 Section 3 p. 15/24
Depth-rst algorithm
P1 2
P2
2
-1
-2
P1
4
2
2
2
-2
1
2
-1
2
1
This is known as the minimax-algorithm.
Andrea Schalk: CS3191 Section 3 p. 15/24
Minimax on 2-person zero-sum games
If we apply the minimax algorithm to a 2-person zero-sum game of perfect information, then
Andrea Schalk: CS3191 Section 3 p. 16/24
Minimax on 2-person zero-sum games
If we apply the minimax algorithm to a 2-person zero-sum game of perfect information, then
the value calculated for Player 1 is the value of the game in the sense of Section 2;
Andrea Schalk: CS3191 Section 3 p. 16/24
Minimax on 2-person zero-sum we games
If apply the minimax algorithm to a 2-person zero-sum game of perfect information, then
the value calculated for Player 1 is the value of the game in the sense of Section 2; the value calculated for Player 2 is the negative of that calculated for Player 1.
Andrea Schalk: CS3191 Section 3 p. 16/24
Minimaxwhat does it do?
Clearly, the minimax algorithm cannot help at all in dectecting mixed-strategy equilibrium points.
Andrea Schalk: CS3191 Section 3 p. 17/24
Minimaxwhat does it do?
Clearly, the minimax algorithm cannot help at all in dectecting mixed-strategy equilibrium points. It will only calculate the actual value of a 2-person zero-sum game if a pure strategy equilibrium point exists;
Andrea Schalk: CS3191 Section 3 p. 17/24
Minimaxwhat does it do?
Clearly, the minimax algorithm cannot help at all in dectecting mixed-strategy equilibrium points. It will only calculate the actual value of a 2-person zero-sum game if a pure strategy equilibrium point exists; we can tell that this is the case when the values for the two players are the negative of each other.
Andrea Schalk: CS3191 Section 3 p. 17/24
Minimaxwhat does it do?
Clearly, the minimax algorithm cannot help at all in dectecting mixed-strategy equilibrium points. It will only calculate the actual value of a 2-person zero-sum game if a pure strategy equilibrium point exists; we can tell that this is the case when the values for the two players are the negative of each other. If the algorithm remembers how to achieve the calculated value it calculates a strategy for guaranteeing it.
Andrea Schalk: CS3191 Section 3 p. 17/24
Minimax on non zero-sum games
What does the value for a player mean when the game is not zero-sum?
Andrea Schalk: CS3191 Section 3 p. 18/24
Minimax on non zero-sum games
What does the value for a player mean when the game is not zero-sum? It is a minimal pay-off that each player can guarantee for himself.
Andrea Schalk: CS3191 Section 3 p. 18/24
Minimax on non zero-sum games
What does the value for a player mean when the game is not zero-sum? It is a minimal pay-off that each player can guarantee for himself. This makes the assumption that the other players do their worst from our players point of viewclearly that is not realistic.
Andrea Schalk: CS3191 Section 3 p. 18/24
Minimax on non zero-sum games
What does the value for a player mean when the game is not zero-sum? It is a minimal pay-off that each player can guarantee for himself. This makes the assumption that the other players do their worst from our players point of viewclearly that is not realistic. Note that the minimax algorithm cannot work on games of imperfect information.
Andrea Schalk: CS3191 Section 3 p. 18/24
Minimax on non zero-sum games
What does the value for a player mean when the game is not zero-sum? It is a minimal pay-off that each player can guarantee for himself. This makes the assumption that the other players do their worst from our players point of viewclearly that is not realistic. Note that the minimax algorithm cannot work on games of imperfect information. Once again, we get a less than desirable solution to our problem, but it is still the best we can do.
Andrea Schalk: CS3191 Section 3 p. 18/24
Minimax on non zero-sum games
What does the value for a player mean when the game is not zero-sum? It is a minimal pay-off that each player can guarantee for himself. This makes the assumption that the other players do their worst from our players point of viewclearly that is not realistic. Note that the minimax algorithm cannot work on games of imperfect information. Once again, we get a less than desirable solution to our problem, but it is still the best we can do. The problem with the idea of trying to take into account what other players might really do is that when this is done by all the players then this results in a process which usually goes around in circles.
Andrea Schalk: CS3191 Section 3 p. 18/24
More improvements?
The minimax algorithm visits each leaf once. It visits other nodes slightly more often (depending on how many leaves they have).
Andrea Schalk: CS3191 Section 3 p. 19/24
More improvements?
The minimax algorithm visits each leaf once. It visits other nodes slightly more often (depending on how many leaves they have). One could write a minimax algorithm that visits every node at most twice if the tree data structure used knows about siblings. However, the improvement in efency usually isnt worth worrying about.
Andrea Schalk: CS3191 Section 3 p. 19/24
More improvements?
The minimax algorithm visits each leaf once. It visits other nodes slightly more often (depending on how many leaves they have). One could write a minimax algorithm that visits every node at most twice if the tree data structure used knows about siblings. However, the improvement in efency usually isnt worth worrying about. Note that once the program has found the value of a node, it is ne if it forgets everything below that node.
Andrea Schalk: CS3191 Section 3 p. 19/24
More improvements?
The minimax algorithm visits each leaf once. It visits other nodes slightly more often (depending on how many leaves they have). One could write a minimax algorithm that visits every node at most twice if the tree data structure used knows about siblings. However, the improvement in efency usually isnt worth worrying about. Note that once the program has found the value of a node, it is ne if it forgets everything below that node. It seems that visiting every node just once (or maybe twice) is the best we can possibly do to determine the value of a node in this way.
Andrea Schalk: CS3191 Section 3 p. 19/24
More improvements?
The minimax algorithm visits each leaf once. It visits other nodes slightly more often (depending on how many leaves they have). One could write a minimax algorithm that visits every node at most twice if the tree data structure used knows about siblings. However, the improvement in efency usually isnt worth worrying about. Note that once the program has found the value of a node, it is ne if it forgets everything below that node. It seems that visiting every node just once (or maybe twice) is the best we can possibly do to determine the value of a node in this way. This turns out to be
wrong.
Andrea Schalk: CS3191 Section 3 p. 19/24
Alpha-beta pruning
Andrea Schalk: CS3191 Section 3 p. 20/24
Alpha-beta pruning
The following algorithm works by passing along information about what is known so far.
Andrea Schalk: CS3191 Section 3 p. 21/24
Alpha-beta pruning
The following algorithm works by passing along information about what is known so far.
max
(, )
min
max
4
-2
-4
Andrea Schalk: CS3191 Section 3 p. 21/24
Alpha-beta pruning
The following algorithm works by passing along information about what is known so far.
max
(, )
min
(, )
max
4
-2
4
Andrea Schalk: CS3191 Section 3 p. 21/24
Alpha-beta pruning
The following algorithm works by passing along information about what is known so far.
max
(, )
min
(, )
max
4
-2
4
Andrea Schalk: CS3191 Section 3 p. 21/24
Alpha-beta pruning
The following algorithm works by passing along information about what is known so far.
max
(, )
min
(, 4]
max
4
-2
-4
Andrea Schalk: CS3191 Section 3 p. 21/24
Alpha-beta pruning
The following algorithm works by passing along information about what is known so far.
max
(, )
min
(, 4]
max
4
(, 4]
-2
-4
Andrea Schalk: CS3191 Section 3 p. 21/24
Alpha-beta pruning
The following algorithm works by passing along information about what is known so far.
max
(, )
min
(, 4]
max
4
[2, 4]
-2
-4
Andrea Schalk: CS3191 Section 3 p. 21/24
Alpha-beta pruning
The following algorithm works by passing along information about what is known so far.
max
(, )
min
(, 4]
max
4
[2, 4] [2, 4]
-2
-4
Andrea Schalk: CS3191 Section 3 p. 21/24
Alpha-beta pruning
The following algorithm works by passing along information about what is known so far.
max
(, )
min
(, 4]
max
4
[2, 4] [2, 4]
-2
-4
Andrea Schalk: CS3191 Section 3 p. 21/24
Alpha-beta pruning
The following algorithm works by passing along information about what is known so far.
max
(, )
min
(, 4]
max
4
2
-2
irr.
-4
Andrea Schalk: CS3191 Section 3 p. 21/24
Alpha-beta pruning
The following algorithm works by passing along information about what is known so far.
max
(, )
min
(, 2]
max
4
2
-2
irr.
-4
Andrea Schalk: CS3191 Section 3 p. 21/24
Alpha-beta pruning
The following algorithm works by passing along information about what is known so far.
max
(, )
min
(, 2]
max
4
-2
(, 2]
irr. 1
-2
-4
Andrea Schalk: CS3191 Section 3 p. 21/24
Alpha-beta pruning
The following algorithm works by passing along information about what is known so far.
max
(, )
min
(, 2]
max
4
-2
(, 2]
irr. 1
-2
-4
Andrea Schalk: CS3191 Section 3 p. 21/24
Alpha-beta pruning
The following algorithm works by passing along information about what is known so far.
max
(, )
min
-2
max
4
-2
irr.
-2
irr.
1
-4
Andrea Schalk: CS3191 Section 3 p. 21/24
Alpha-beta pruning
The following algorithm works by passing along information about what is known so far.
max
[2, )
min
-2
1
max
4
-2
irr.
-2
irr.
1
-4
Andrea Schalk: CS3191 Section 3 p. 21/24
Alpha-beta pruning
The following algorithm works by passing along information about what is known so far.
max
[2, )
min
-2
1
max
4
-2
irr.
-2
irr.
1
-4
Andrea Schalk: CS3191 Section 3 p. 21/24
Alpha-beta pruning
The following algorithm works by passing along information about what is known so far.
max
[1, )
min
-2
1
max
4
-2
irr.
-2
irr.
1
-4
Andrea Schalk: CS3191 Section 3 p. 21/24
Alpha-beta pruning
The following algorithm works by passing along information about what is known so far.
max
[1, )
min
-2
1
[1, )
max
4
-2
irr.
-2
irr.
1
-4
Andrea Schalk: CS3191 Section 3 p. 21/24
Alpha-beta pruning
The following algorithm works by passing along information about what is known so far.
max
[1, )
min
-2
1
[1, )
max
4
-2
irr.
-1
-2
irr.
1
-4
Andrea Schalk: CS3191 Section 3 p. 21/24
Alpha-beta pruning
The following algorithm works by passing along information about what is known so far.
max 1
min
-2
1
irr.
max
4
-2
irr.
-1
-2
irr.
1
-4
Andrea Schalk: CS3191 Section 3 p. 21/24
Alpha-beta pruning
The following algorithm works by passing along information about what is known so far.
max 1
min
-2
1
irr.
max
4
-2
irr.
-1
-2
irr.
1
-4
There are parts of the tree we have not searched at all!
Andrea Schalk: CS3191 Section 3 p. 21/24
How does this work?
On each recursive call of the procedure that determines the value, two parameters
Andrea Schalk: CS3191 Section 3 p. 22/24
How does this work?
On each recursive call of the procedure that determine...