SCIT1003
Chapter 4: Minimax Equilibrium
in Zero Sum Game

Prof. Tsang
1
Maximin & Minimax Equilibrium in
a zero-sum game
• Minimax - minimizing the maximum loss
(loss-ceiling, defensive)
• Maximin - maximizing the minimum gain
(gain-floor, offensive)
• Minimax = Maximin
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The Minimax Theorem
“Every finite, two-person, zero-sum game
has a rational solution in the form
of a pure or mixed strategy.”
John Von Neumann, 1926
For every two-person, zero-sum game with finite strategies, there
exists a value V and a mixed strategy for each player, such that (a)
Given player 2's strategy, the best payoff possible for player 1 is V,
and (b) Given player 1's strategy, the best payoff possible for
player 2 is −V.
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Pure strategy game: Saddle point
Is this a Nash
Equilibrium?
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3 MaxiMin
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A zero-sum game with a saddle
3 MiniMax
point.
4
Pure & mixed strategies
A pure strategy provides a complete definition of how
a player will play a game. It determines the move a
player will make for any situation they could face.
A mixed strategy is an assignment of a probability to
each pure strategy. This allows for a player to randomly
select a pure strategy.
In a pure strategy a player chooses an action for sure,
whereas in a mixed strategy, he chooses a probability
distribution over the set of actions available to him.
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All you need to know about
Probability
If E is an outcome of action, then P(E) denotes the
probability that E will occur, with the following
properties:
1. 0  P(E)  1 such that:
If E can never occur, then P(E) = 0
If E is certain to occur, then P(E) = 1
2. The probabilities of all the possible outcomes
must sum to 1
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Mixed strategy
• In some zero-sum game, there is no pure
strategy solution (no Saddle point)
• Play’s best way to win is mixing all
possible moves together in a random
(unpredictable) fashion.
• E.g. Rock-Paper-Scissors
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Mixed strategies
Some games, such as Rock-Paper-Scissors, do not
have a pure strategy equilibrium.
In this game, if Player 1 chooses R, Player 2 should choose p, but if Player 2
chooses p, Player 1 should choose S. This continues with Player 2 choosing r
in response to the choice S by Player 1, and so forth.
In games like Rock-Paper-Scissors, a player will want
to randomize over several actions, e.g. he/she can
choose R, P & S in equal probabilities.
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A soccer penalty shot at 12-yard
left or right?
p.145 payoffs are winning probability
Goalie
Left
Left
Kicker
Right
42
58
Right
5
95
7
93
30
70
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A penalty shot at 12-yard
left or right?
If you are the kicker, which side you use?
The best chance you have is 95%. So you kick left.
But the goalie anticipates that because he knows that’s
your best chance. So his anticipation reduces your chance
to 58%.
What if you anticipate that he anticipates … so you kick
right & that increase your chance to 93%.
What if he anticipates that you anticipate that he
anticipates …
If you use a pure strategy, he always has a way to reduce
you chance to win.
10
A penalty shot at 12-yard
left or right?
• To end this circular reasoning, you do
something that the goalie cannot anticipate.
• What if you mix the 2 choices randomly with
50-50 chance?
• Your chance of winning is
(58+93)/2 if the goalie moves to left
(93+70)/2 if the goalie moves to right
Is this better?
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Kicker’s mixture
p.166 graphical solution
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Goalie’s mixture
p.168 graphical solution
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If the goalie improves his skill at saving kicks to the Right side
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A Parking meter game (p.164)
If you pay for the parking, it cause you $1.
If you don’t pay for the parking and you are caught
by the enforcer, the penalty is $50.
Should you take the risk of not paying for the
parking?
How often the enforcer should patrol to keep the
car drivers honest (to pay the parking fee)?
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Parking meter game
p.164
Car driver
Pay
Enforce
Enforcer
Not pay
-1
1
Not
enforce
-50
50
-1
1
0
0
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Mixed strategies
x=probability to take
action R
y=probability to take
action S
no
x
y
no
1-x-y
1-x-y=probability to take
action P
No Nash equilibrium for pure strategy
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They have to be equal if expected payoff
independent of action of player 2
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Janken step game (Japanese RSP)
p.171
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Two-Person, Zero-Sum Games: Summary
•
•
•
•
Represent outcomes as payoffs to row player
Find any dominating equilibrium
Evaluate row minima and column maxima
If maximin=minimax, players adopt pure strategy
corresponding to saddle point; choices are in stable
equilibrium -- secrecy not required
• If maximin minimax, find optimal mixed strategy;
secrecy essential
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Summary: Ch. 4
•
•
•
•
Look for any equilibrium
Dominating Equilibrium
Minimax Equilibrium
Nash Equilibrium
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Assignment 4.1
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Assignment 4.1
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