Game Theory

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Game Theory
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Game theory is a mathematical theory that deals
with the general features of competitive
situations. The final outcome depends primarily
upon the combination of strategies selected by
the adversaries.
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Two key Assumptions (a) Both players are
rational (b) Both players choose their strategies
solely to increase their own welfare.
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Payoff Table
Player 2
Strategy
1 2 3
1 2 4 1 0 5 0 1 -1
1 2 3
Player 1
Each entry in the payoff table for player 1
represents the utility to player 1 (or the
negative utility to player 2) of the outcome
resulting from the corresponding strategies used
by the two players.
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A strategy is dominated by a second strategy if
the second strategy is always at least as good
regardless of what the opponent does. A
dominated strategy can be eliminated immediately
from further consideration.
Player 2
Strategy
1 2 3
1 2 4 1 0 5 0 1 -1
1 2 3
Player 1
For player 1, strategy 3 can be eliminated. ( 1
gt 0, 2 gt 1, 4 gt -1)
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1 2 3
1 2 4 1 0 5
1 2
For player 2, strategy 3 can be eliminated. ( 1
lt 4, 1 lt 5 )
1 2
1 2 1 0
1 2
For player 1, strategy 2 can be eliminated. ( 1
1, 2 lt 0 )
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1 2
1 2
1
For player 2, strategy 2 can be eliminated. ( 1
lt 2 )
Consequently, both players should select their
strategy 1. A game that has a value of 0 is said
to be a fair game.
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Minimax criterion To minimize his maximum losses
whenever resulting choice of strategy cannot be
exploited by the opponent to then improve his
position.
Player 2
Strategy
Minimum
1 2 3
-3 0 -4
-3 -2 6 2 0 2 5 -2 -4
1 2 3
Player 1
5 0 6
Maximum
Minimax value
Maximin value
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The value of the game is 0, so this is fair
game Saddle Point A Saddle point is an entry
that is both the maximin and minimax.
Player 2
Strategy
Minimum
1 2 3
-3 0 -4
-3 -2 6 2 0 2 5 -2 -4
1 2 3
Player 1
5 0 6
Maximum
Saddle point
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There is no saddle point. An unstable
solution
Player 2
Strategy
Minimum
1 2 3
-2 -3 -4
0 -2 2 5 4 -3 2 3 -4
1 2 3
Player 1
5 4 2
Maximum
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Mixed Strategies
probability that player 1 will use strategy i
( i 1,2,,m), probability that player 2 will
use strategy j ( j 1,2,,n),
Expected payoff for player 1
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Minimax theorem If mixed strategies are allowed,
the pair of mixed strategies that is optimal
according to the minimax criterion provides a
stable solution with (the
value of the game), so that neither player can do
better by unilaterally changing her or his
strategy.
maximin value minimax value
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Graphical Solution Procedure
Player 2
Probability
Pure Strategy
Probability
1 2 3
0 -2 2 5 4 -3
1 2
Player 1
Expected Payoff
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Expected Payoff
Expected payoff for player 1
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Player 1 wants to maximize the minimum expected
payoff. Player 2 wants to minimize the expected
payoff.
6 5 4 3 2 1 0 -1 -2 -3 -4
Maximin point
Expected payoff
1.0
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The optimal mixed strategy for player 1 is
So the value of the game is
The optimal strategy
(1)
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When player 1 is playing optimally (
), this inequality will be an equality, so that
(2)
Because is a probability distribution,
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because would violate
(2),
Because the ordinate of this line must equal at
, and because it must never exceed
,
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To solve for and , select two values
of (say, 0 and 1),
The optimal mixed strategy for player 2 is
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Solving by Linear Programming
Expected payoff for player 1
The strategy is optimal if
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For each of the strategies
where one and the rest equal 0. Substituting
these values into the inequality yields
Because the are probabilities,
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The two remaining difficulties are (1) is
unknown (2) the linear programming problem has
no objective function. Replacing the
unknown constant by the variable and
then maximizing , so that
automatically will equal at the optimal
solution for the LP problem.
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Player 2
Example
Probability
Pure Strategy
Probability
1 2 3
0 -2 2 5 4 -3
1 2
Player 1
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Player 2
The dual
Probability
Pure Strategy
Probability
1 2 3
0 -2 2 5 4 -3
1 2
Player 1
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Question 1 Consider the game having the following
payoff table.
(a) Formulate the problem of finding optimal
mixed strategies according to the minimax
criterion as a linear programming problem. (b)
Use the simplex method to find these optimal
mixed strategies.