Outline for today
Stat155
Game Theory
Lecture 7: two player zero-sum games
Zero sum games
Recall: payoff matrices, mixed strategies, von Neumann’s minimax
theorem
Solving two player zero-sum games
Recall: Saddle points, dominated pure strategies, equalizing strategies
Solving 2 × 2 games
Solving 2 × n and m × 2 games
Dominated strategies
Principle of indifference
Peter Bartlett
September 15, 2016
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Recall: Two-player zero-sum games
Recall: Two-player zero-sum games
Definitions
Von Neumann’s Minimax Theorem
For any two-person zero-sum game with payoff matrix A ∈ Rm×n ,
The payoff matrix A ∈ Rm×n represents the payoff to Player I:


a11 a12 · · · a1n
 a21 a22 · · · a2n 


A= .
.. 
..
 ..
.
. 
am1 am2 · · ·
max min x > Ay = min max x > Ay .
x∈∆m y ∈∆n
amn
y ∈∆n x∈∆m
We call the optimal expected payoff the value of the game,
V := max min x > Ay = min max x > Ay .
The expected payoff to Player I when Player I plays mixed strategy
x ∈ ∆m and Player II plays mixed strategy y ∈ ∆n is x > Ay .
x∈∆m y ∈∆n
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y ∈∆n x∈∆m
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Outline
Saddle points
Definition
A pair (i ∗ , j ∗ ) ∈ {1, . . . , m} × {1, . . . , n} is a saddle point (or pure Nash
equilibrium) for a payoff matrix A ∈ Rm×n if
Zero sum games
Recall: payoff matrices, mixed strategies, von Neumann’s minimax
theorem
max aij ∗ = ai ∗ j ∗ = min ai ∗ j .
i
Solving two player zero-sum games
Recall: Saddle points, dominated pure strategies, equalizing strategies
Solving 2 × 2 games
Solving 2 × n and m × 2 games
Dominated strategies
Principle of indifference
j
Theorem
If (i ∗ , j ∗ ) is a saddle point for a payoff matrix A ∈ Rm×n , then
ei ∗ is an optimal strategy for Player I,
ej ∗ is an optimal strategy for Player II, and
the value of the game is ai ∗ j ∗ .
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2 × 2 games
Dominated pure strategies
How to solve a 2 × 2 game
1
Check for a saddle point.
(Is the max of row mins = min of column maxes?)
2
If there are no saddle points, find equalizing strategies.
Definition
A pure strategy ei for Player I is dominated by ei 0 in payoff matrix A if, for
all j ∈ {1, . . . , n},
aij ≤ ai 0 j .
Equalizing strategies are such that, whatever the other player plays, the
expected payoff is the same:
x1 a11 + (1 − x1 )a21 = x1 a12 + (1 − x1 )a22 ,
y1 a11 + (1 − y1 )a12 = y1 a21 + (1 − y1 )a22 .
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Solving 2 × 2 games
Examples
Outline
0
1
3
0
3
0
2
3
2
1
1
2
Zero sum games
Recall: payoff matrices, mixed strategies, von Neumann’s minimax
theorem
Solving two player zero-sum games
Recall: Saddle points, dominated pure strategies, equalizing strategies
Solving 2 × 2 games
Solving 2 × n and m × 2 games
Dominated strategies
Principle of indifference
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Solving 2 × n games
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Dominated pure strategies
Payoff matrix
2 3 1 5
4 1 6 0
Definition
The maximum occurs at the
intersection of the lines
corresponding to columns 2 and 3.
The optimal strategy for Player I is
x = (5/7, 2/7).
Then Player II is indifferent between
columns 2 and 3.
(Ferguson, 2014)
A pure strategy ej for Player II is dominated by columns ej1 , . . . , ejk in
payoff matrix A if there is a convex combination y ∈ ∆n with
{l : yl 6= 0} = {j1 , . . . , jk } such that, for all i ∈ {1, . . . , m},
aij ≥
n
X
ail yl .
l=1
In that case, we can eliminate column j: there is an optimal strategy
for Player II that sets yj = 0.
Column 1 plays no role:
It is dominated by a combination of
columns 2 and 3.
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Solving m × 2 games
Outline
Payoff matrix


1 5
4 4
6 2
Zero sum games
Recall: payoff matrices, mixed strategies, von Neumann’s minimax
theorem
Solving two player zero-sum games
Recall: Saddle points, dominated pure strategies, equalizing strategies
Solving 2 × 2 games
Solving 2 × n and m × 2 games
Dominated strategies
Principle of indifference
The minimum occurs for
y1 ∈ [1/4, 1/2].
This is Player II’s optimal strategy.
Player I plays the pure optimal
strategy e2 .
(Ferguson, 2014)
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Principle of indifference
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Principle of indifference
Theorem
We’ve seen several examples where the optimal mixed strategy for
one player leads to a best response from the other that is indifferent
between actions.
This is a general principle.
Suppose a game with payoff matrix A ∈ Rm×n has value V . If x ∈ ∆m
and y ∈ ∆n are optimal strategies for Players I and II, then
for all j,
m
X
l=1
if yj > 0,
m
X
l=1
(Karlin and Peres, 2016)
(Karlin and Peres, 2016)
(Ferguson, 2014)
(Ferguson, 2014)
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xl alj ≥ V ,
xl alj = V ,
for all i,
n
X
l=1
if xi > 0,
n
X
ail yl ≤ V ,
ail yl = V .
l=1
This means that if one player is playing optimally, any action that has
positive weight in the other player’s optimal mixed strategy is a
suitable response.
It implies that any mixture of these “active actions” is also a suitable
response.
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Principle of indifference: Proof
Using principle of indifference
Theorem
Suppose a game with payoff matrix A ∈ Rm×n has value V . If x ∈ ∆m
and y ∈ ∆n are optimal strategies for Players I and II, then
m
n
X
X
for all j,
xl alj ≥ V ,
for all i,
ail yl ≤ V ,
if yj > 0,
l=1
m
X
xl alj = V ,
if xi > 0,
l=1
l=1
n
X
Solving linear systems
Suppose that we have a payoff matrix A and we suspect that an
optimal strategy for Player I has certain components positive, say
x1 > 0, x3 > 0.
ail yl = V .
Then we can solve the corresponding “indifference equalities” to find
y , say
l=1
Proof
1
The inequalities define optimality of x and y . They imply
X
X X
(A) X X
(B) X
V =
xi aij yj =
xi V = V .
Vyj ≤
xi
aij yj ≤
j
2
3
n
X
j
i
i
l=1
a1l yl = V ,
n
X
a3l yl = V .
l=1
i
j
x > Ay
So these are all equalities, and they show that
= V.
Pm
Pm
Suppose yj > 0 and V 6= l=1 xl alj . Then Vyj < l=1 xl alj yj .
Substituting in (B) leads to V < V . Contradiction!
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Example
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Example
Recall: Plus One
Use principle of indifference
We suspect x1 > 0, x2 > 0, x3 > 0.
 
V

Solve Ay = V :
V
 
1/4

y = 1/2, V = 0.
1/4
Reduced payoff matrix
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Example
Example
Diagonal payoff matrix
Rock, Paper, Scissors


a11 0
0
A =  0 a22 0 
0
0 a33
The aii are all positive, so we suspect that all xi , yi > 0 for the
optimal strategies.
Solve
x >A = V
V
V :
x =y =
V =
V
a11
V
a22
V
a33
>
,
1
.
1/a11 + 1/a22 + 1/a33
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Example
Outline
Zero sum games
Recall: payoff matrices, mixed strategies, von Neumann’s minimax
theorem
Rock, Paper, Scissors
Payoff matrix?
Solving two player zero-sum games
What probabilities should be positive?
Recall: Saddle points, dominated pure strategies, equalizing strategies
Solving 2 × 2 games
Solving 2 × n and m × 2 games
Dominated strategies
Principle of indifference
What are optimal strategies?
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