1. GAME THEORY:
MAXIMIN AND
MINIMAX STRATEGIES
Presented by Group 9
ANKAN MALLA
ANUSHKA PURBAGOLA
KUMARI JYOTI
NIKHIL TIWARI
PRIYA AGARWAL
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2. Flow Of Presentation
Basic
revision
Types of
Strategy
Assumptions
in Game
Theory
Decision of
Game- 4
Strategies
Drawbacks
of Pure
Strategy
4 Strategies
Using Excel
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3. Game theory: It is the study of decision-making models in situations where strategic interactions among two or more rational
opponents are involved under condition of competition and conflicting interests
Rules of Game. Every game is played according to the set of rules i.e., specific rules which governs the behavior of the players.
Strategy: It is the pre-determined rule by which each player decides his course of action from his list available to him.
In game theory, a player's strategy is any of the options which they choose in a setting where the outcome depends not only on
their own actions but on the actions of others. Some examples of "games" include chess, bridge and poker.
In studying game theory, economists enlist a more rational lens in analyzing decisions rather than the psychological or sociological
perspectives taken when analyzing relationships between decisions of two or more parties in different disciplines.
Strategy set: A player's strategy set defines what strategies are available for them to play. A strategy profile is a list of strategy sets,
ordered from most to least desirable.
Key Concepts
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4. Types of strategy
• Pure Strategy: It is the predetermined course of action to be
employed by the player. It is an unconditional, defined choice
that the players make. They know it in advance. It is usually
represented by a number with which the course of action is
associated.
For example, in the game of Rock-Paper-Scissors , if a player would
choose to only play scissors for each independent trial, regardless
of the other player's strategy, choosing scissors would be the
player's pure strategy.
• Mixed Strategy: In mixed strategy the player decides his
course of action in accordance with some fixed probability
distribution. Probability are associated with each course of
action and the selection is done as per these probabilities.
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5. Assumptions in game theory
• There are finite numbers of competitors (players).
• The players act reasonably.
• Every player strives to maximize gains and minimize losses.
• Each player has finite number of possible courses of action.
• The choices are assumed to be made simultaneously, so that no player
knows his opponent's choice until he has decided his own course of
action.
• The pay-off is fixed and predetermined.
• The pay-offs must represent utilities
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6. Decision of a game: 4
strategies
In Game theory, best strategy for each player is determined
based on some rule. Since both the players are expected to
be rational in their approach this is known as the criteria of
optimality.
Each player lists the possible outcomes from his action and
selects the best action to achieve his objectives. This criteria
of optimality is expressed as Maximax and maximin for the
maximizing player and Minimax and minimin for the
minimizing player.
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7. Decision of a game: 4 strategies
Maximin Criteria: The maximizing player (placed on the left side of the matrix)
lists his minimum gains from each strategy and selects the strategy which
gives the maximum out of these minimum gains.
Minimax Criteria: The minimizing player (Placed on the top side of the matrix)
lists his maximum loss from each strategy and selects the strategy which gives
him the minimum loss out of these maximum losses.
Maximax Criteria: The maximizing player (placed on the left side of the
matrix) lists his maximum gains from each strategy and selects the strategy
which gives the maximum out of these maximum gains.
Minimin Criteria: The minimizing player (Placed on the top side of the matrix)
lists his minimum loss from each strategy and selects the strategy which gives
him the minimum loss out of these minimum losses.
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8. Drawbacks of game theory
• Infinite strategies: In a game theory we assume that there is finite number of possible
courses of action available to each player. But in practice a player may have infinite number
of strategies or courses of action.
• Knowledge about strategy: Game theory assumes that each player as the knowledge of
strategies available to his opponent. But sometimes knowledge about strategy about the
opponent is not available to players. This leads to the wrong conclusions.
• Zero sum games: We have assumed that gain of one person is the loss of another person.
But in practice gain of one person may not be equal to the loss of another person i.e.,
opponent.
• Risk and uncertainty: Game theory does not take into consideration the concept of
probability. So, game theory usually ignores the presence of risk and uncertainty.
• Finite number of competitors: There are finite number of competitors as has been
assumed in the game theory. But in real practice there can be more than the expected
number of players.
• Certainty of Pay off: Game theory assume that payoff is always known in advance. But
sometimes it is impossible to know the pay off in advance. The decision situation in fact
becomes multidimensional with large number of variables.
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9. Sum 1: Maximin and Minimax
* Maximin
*
Minimax
X is the Maximizing Player trying to maximize its gains. Its strategies
are X1, X2, X3, X4. The Maximizing player opts for the Maximin
approach i.e. it chooses the strategy which gives it the maximum
gain from among its minimum gains from each strategy. So G3:G6
are the row minima for each of the strategic alternatives of X.
The Maximum of these minima is -2 and is marked as the maximin
value.
Y is the Minimizing Player trying to minimize its losses. Its strategies
are Y1,Y2,Y3,Y4. The Minimizing player opts for the Minimax
approach i.e., it chooses the strategy which gives it the minimum
loss from among its maximum losses from each strategy. So C7, D7,
E7, F7 are the column maxima for each of the strategic alternatives
of Y.
The minimum of these maxima is -2 and is marked as the maximin
value.
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10. Sum 1: Maximax and Minimin
* Maximax
*
Minimin
X is the Maximizing Player trying to maximize its gains. Its strategies
are X1, X2, X3, X4. The Maximizing player opts for the Maximax
approach i.e., it chooses the strategy which gives it the maximum
gain from among its maximum gains from each strategy. So G3:G6
are the row minima for each of the strategic alternatives of X.
The Maximum of these maxima is 40 and is marked as the maximax
value.
Y is the Minimizing Player trying to minimize its losses. Its strategies
are Y1,Y2,Y3,Y4. The Minimizing player opts for the Minimin
approach i.e., it chooses the strategy which gives it the minimum
loss from among its minimum losses from each strategy. So C7, D7,
E7, F7 are the column minima for each of the strategic alternatives
of Y.
The minimum of these minima is -6 and is marked as the minimin
value.
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11. Minimax is used in zero-sum games to denote minimizing the opponent's
maximum payoff. In a zero-sum game, this is identical to minimizing one's
own maximum loss, and to maximizing one's own minimum gain. In other
words, a player considers all of the best opponent responses to his
strategies, and selects the strategy such that the opponent's best strategy
gives a payoff as large as possible. Minimax in zero-sum games is the
same as that of Nash equilibrium. Minimax is also used in repeated
games.
Why And When Are These Strategies Applied
"Maximin" is a term commonly used for non-zero-sum games to
describe the strategy which maximizes one's own minimum
payoff. This strategy is applied when an individual seeks out
where the greatest benefit can be found.
12. How Are These Strategies Applied
MAXIMIN:
• Pessimistic Approach
• Determines the best payoff
from the worst payoffs.
• Guaranteed minimum.
• Best minimum
MAXIMAX:
• Optimistic Approach
• Determines the best payoff
from the best payoffs.
• Based on go-for-it strategy
• Best maximum
MINIMIN:
• Determines the worst payoff
from the best regret payoff.
MINIMAX:
• Determines the best payoff
from the worst regret payoffs.
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14. Pseudocode
For Minimax
Theorem
function minimax(node,depth)
if node is a terminal node or depth = 0
return the heuristic value of node
if the adversary is to play at node
let beta := +∞
for each child of node
beta:=min(beta,minimax(child,depth+1))
return beta
else {we are to play at node}
let alpha := -∞
for each child of node
alpha:=max(alpha,minimax(child,depth+1))
return alpha
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16. Pseudocode for Tic
Tac Toe
Finding Best Move: We shall be introducing a new
function called findBestMove(). This function
evaluates all the available moves
using minimax() and then returns the best move the
maximizer can make. The pseudocode for the same
is as follows:function findBestMove(board):
bestMove = NULL
for each move in board :
if current move is better than bestMove
bestMove = current move
return bestMove
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17. Pseudocode for Tic Tac Toe( contd.)
Minimax : To check whether or not the current move is better than the best
move we take the help of minimax() function which will consider all the possible
ways the game can go and returns the best value for that move, assuming the
opponent also plays optimally. The code for the maximizer and minimizer in
the minimax() function is similar to findBestMove(), the only difference is, instead
of returning a move, it will return a value. The pseudocode is as follows:function
minimax(board, depth, isMaximizingPlayer):
else :
bestVal = +INFINITY
for each move in board :
value = minimax(board,
depth+1, true)
bestVal = min( bestVal,
value)
return bestVal
if isMaximizingPlayer :
bestVal = -INFINITY
for each move in board :
value = minimax(board,
depth+1, false)
bestVal = max( bestVal,
value)
return bestVal
if current board state is a
terminal state :
return value of the
board
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18. Pseudocode for Tic Tac Toe(
contd.)
Checking for GameOver state : To check whether the
game is over and to make sure there are no moves left
we use isMovesLeft() function. It is a simple
straightforward function which checks whether a move is
available or not and returns true or false respectively.
Pseudocode is as follows :
function isMovesLeft(board):
for each cell in board:
if current cell is empty:
return true
return false
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