Two-Player Zero-Sum Games and Duality
Battle of Wits
Is there an equilibrium?
primal
dual
max cx
s.t.
Ax b
x⇥0
min yb
s.t. yA c
y 0
Duality Theorem
Primal has an optimal solution if and only if dual does.
If x* and y* are optimal then cx* = y*b
e62: lecture 16
1
20161129
CTF-150 versus the Somali Pirates
north
coast
west
coast
area
intercept
probability
north
1/3
west
1/2
south
3/4
south
coast
e62: lecture 16
2
20161129
Strategies
pirates
north
•
•
•
intercept
probability
1/3
west
1/2
south
3/4
CTF-150
area
success
probability
north west south
north
2/3
1
1
west
1
1/2
1
south
1
1
1/4
payoff
matrix
=P
Pirates assume CTF to be “clairvoyant”
Pirate strategy A: north
•
•
CTF-150 strategy: north
Success probability: 2/3
•
•
CTF-150 strategy: south
Success probability: 3/4
Pirate strategy B: equiprobable actions
e62: lecture 16
3
Can the pirates do better?
20161129
Optimal Strategy against a Clairvoyant Opponent
•
Possible strategies for pirates and coast guard
X =
•
•
(
x ⇥ ⇤3 x
0,
3
X
)
Y=
xn = 1
n=1
(
y ⇥ ⇤3 y
0,
3
X
n=1
)
yn = 1
Pirate’s probability of success = yP x
Pirate selects a strategy (assuming optimal
countermeasure)
✓
◆
max min yP x
x X
•
y Y
What if coast guard strategizes similarly?
◆
min max yP x
y Y
✓
x X
This is an equilibrium!
e62: lecture 16
4
20161129
General Two-Player Zero-Sum Game
•
•
•
Actions
•
•
Payoff matrix: P
⇥M
N
Strategies
•
•
•
Player 1: {1, 2, 3, ..., N}
Player 2: {1, 2, 3, ..., M}
⇥
Player 1: X = x ⇥ ⇤N x
0,
N
⌅
Player 2: Y = y ⇥ ⇤M y
0,
M
⌅
m=1
M
⇤
ym = 1
N
Expected payoff: yP x =
e62: lecture 16
xn = 1
n=1
⇥
⇤
xn ym Pmn
m=1 n=1
5
20161129
Conservative Strategies via Linear Programming
Observation: for a clairvoyant opponent, a pure counterstrategy suffices
•
Player 1
max min yP x
x X
y Y
u
•
Player 2
min max yP x
y Y x X
v
e62: lecture 16
6
max u
s.t.
ue
Px
N
n=1 xn = 1
x⇥0
min v
s.t. ve
yP
M
m=1 ym = 1
y 0
20161129
Equilibrium
•
•
•
Def. An equilibrium is a pair of strategies (x*, y*) such that
y* is optimal for player 2 if player 1 uses x*
x* is optimal for player 1 if player 2 uses y*
•
•
No incentives to deviate:
y Px
y Px
yP x
⇤x ⇥ X , y ⇥ Y
Conservative strategies attain an equilibrium
Minimax Theorem
max min yP x = min max yP x
x X
y Px
y Y
min yP x = max y P x
y⇥Y
y Px
e62: lecture 16
y Y x X
x⇥X
y Px
7
yP x
y Px
⇤x ⇥ X , y ⇥ Y
20161129
Proof of Minimax Theorem
max min yP x
x X
y Y
max u
s.t.
ue
Px
N
n=1 xn = 1
x⇥0
e62: lecture 16
min max yP x
y Y x X
duals
u =v
8
min v
s.t. ve
yP
M
m=1 ym = 1
y 0
20161129
The Art of Decision Framing
e62: lecture 16
9
20161129