Transcript Game Theory

Game Theory
“If you don’t think the math
matters, then you don’t know the
right math.”
Chris Ferguson
2002 World Series of Poker Champion
Prisoners
You and I have been arrested for a bank
robbery (of which we are guilty!), and
they have evidence to convict us, but not
enough for the maximum sentence.
They decide to separate us and try to get
each of us to confess and implicate the
other one.
A Dilemma
If we both stonewall (i.e. keep our mouths shut)
then we will get 2 years in prison.
If we both confess, we’ll each get 4 years in
prison.
But if only one person finks (confesses), that
person will get 1 year in prison, while the
person who stayed quiet gets 5 years!
Summary of Your
Outcomes
I Stonewall
I Fink
You Stonewall
2 years prison
5 years prison
You Fink
1 year prison
4 years prison
A Dominant Strategy
 No matter what I choose, you’ll be better
off having finked.
 But the same reasoning applies for me!
 So even though we’d both be better off
had we both stonewalled (2 years prison
each instead of 4), unless we can
cooperate (i.e. make binding
agreements), we are both forced to fink.
Summary of Both of
Our Outcomes
I Stonewall
I Fink
You
Stonewall
(-2,-2)
(-5,-1)
You Fink
(-1,-5)
(-4,-4)
Equilibrium
Outcome
A ‘game’ is any situation
in which:
 There are at least two players
 Each player has a number of possible
strategies
 The strategies chosen by each player
determine the outcome of the game
 Associated to each outcome is a
collection of numerical payoffs to each
player
Payoffs in the Prisoner’s
Dilemma:
S
Me
F
S
(-2,-2)
(-5,-1)
F
(-1,-5)
(-4,-4)
You
The first number in each ordered pair indicates the
payoff to the first player, whose strategy options
correspond to the rows of the matrix.
Domination
A strategy A dominates a strategy B if
every outcome in A is at least as good
(for that player) as the corresponding
outcome in strategy B.
Dominance Principle: A rational player
should never play a dominated strategy.
A game of total conflict:
The following table indicates payoffs for a
game. Notice that the payoffs are the
negative of each other, so what is good
for you is bad for me, and vice versa!
A
Me
B
A
(5,-5)
(2,-2)
B
(0,0)
(-4,4)
You
 Strategy A is dominant for You.
 Strategy B is dominant for Me.
 By the Dominance Principle, the outcome
of this game should be AB for rational
players.
A
Me
B
A
(5,-5)
(2,-2)
B
(0,0)
(-4,4)
You
Zero-Sum Games
In the previous game, since we know that
the payoffs for the second player are
negatives of the payoffs for the first
player, we could have described the
outcomes with a matrix where each entry
has payoffs for the first player only.
Me
A
You
B
A
5
2
B
0
-4
For the game below, what
is your rational strategy?
Me
A
B
C
1
3
7
B
6
4
6
C
6
2
1
A
Me
You
A
B
C
A
1
3
7
B
6
4
6
C
6
2
1
The rational outcome is BB.
A Game without domination
You
A
Me
B
A
-1
3
9
-1
B
6
4
8
4
C
8
2
-2
-2
8
4
9
maximum
C
Minimum
maximin
minimax
Goal: Guarantee yourself the least-worst result.
Saddle Points
 An outcome in a matrix game is called a
saddle point if the entry at that outcome is
both less than or equal to any entry in its row,
and greater than or equal to any entry in its
column.
 If the minimax column and maximin row are
equal in payoff, the corresponding outcome is
a saddle point.
 That payoff is called the value of the game.
Saddle Points
-1
3
1
6
4
8
3
2
-2
Expected value
Me
You
A
B
A
1
3
B
4
2
This game does not
have a dominant
strategy or saddle point.
But you happen to
know that I am going to
play strategy A half the
time and Strategy B the
other half of the time.
What should you do?
Expected Value
 If you choose strategy A, your expected
payoff will be
1
1
(1) (3)  2
2
2
 If you choose strategy B, your expected
payoff will be
1
1
( 4) (2)  3
2
2
Mixed Strategy
What happens if I play strategy A one
fourth of the time and strategy B the
remaining three fourths of the time?
Me
You
A
B
A
1
3
B
4
2
Mixed Strategy
 Your expected payoff for playing strategy A
will be:
1
3
(1) (3)  2.5
4
4
 Your expected payoff for playing strategy B
will be:
1
3
(4) (2)  2.5
4
4
Mixed Strategy
 By playing randomly with the correct
weights for each strategy, I can
guarantee that your strategy choice
makes no difference. This is called a
mixed-strategy on my part.
 You can even know exactly what my
mixed strategy is, and it gives you no
advantage!
Computing a Mixed
Strategy Equilibrium
 I want to play strategy A with some weight p,
and strategy B with weight 1-p.
 I want the expected payoffs to you to be
independent of your strategy.
 So I need to solve the equation:
p(1) (1  p)(3)  p(4)  (1  p)(2)
 The solution is p=1/4.
Utility Theory
 Payoffs (also called utilities by
economists) can be selected by ranking
outcomes in order of preference.
 The magnitude of these rankings need
not matter, just the order. Thus the
payoffs are called ordinal utilities.
Utility Theory
 If the magnitudes can be compared, then
the payoffs are called cardinal utilities.
 The most common situation with cardinal
utilities is when the payoffs correspond to
money.
 Only cardinal utilities make sense with
mixed strategies!
Multiple-move games
 It is X’s turn to act in
Tic-Tac-Toe:
 What are X’s
strategies?
O O
A
B
C
X
D
X O
Bibliography
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Game Theory and Strategy, Philip D. Straffin
For All Practical Purposes, COMAP
The Mathematics of Poker, Bill Chen and Jerrod Ankenman
Microeconomic Theory, Walter Nicholson
Theory of Games and Economic Behavior, John von Neumann, et.
al.