Stat155 Game Theory Lecture 8: Symmetry in two player zero

Outline for today
Stat155
Game Theory
Lecture 8: Symmetry in two player zero-sum games
Solving two player zero-sum games
Recall: definitions, solving 2 × 2, 2 × n and m × 2 games, principle of
indifference
Symmetry
Peter Bartlett
Invariance of payoffs under permutations
Invariance of strategies under permutations
Optimal strategies can be invariant
September 20, 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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Saddle points
Dominated pure strategies
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
Definition
A strategy ei for Player I is dominated in payoff matrix A if there is a
mixture x ∈ ∆m with xi = 0 so that, for all j ∈ {1, . . . , n},
X
xi 0 ai 0 j ≥ aij .
max aij ∗ = ai ∗ j ∗ = min ai ∗ j .
i
j
Theorem: Saddle points are optimal
i
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
Solving 2 × n and m × 2 games
How to solve a 2 × 2 game
1
Check for a saddle point.
2
If no saddle points, find equalizing strategies.
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 .
(Ferguson, 2014)
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(Ferguson, 2014)
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Principle of indifference
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
for all j,
if yj > 0,
m
X
l=1
m
X
l=1
xl alj ≥ V ,
for all i,
xl alj = V ,
if xi > 0,
n
X
l=1
n
X
Solving linear systems
ail yl ≤ V ,
If we know that an optimal strategy x for Player I has certain
components positive, we can solve the corresponding “indifference
equalities” to find y .
ail yl = V .
l=1
If one player is playing optimally, any action that has positive weight
in the other player’s optimal mixed strategy is a suitable response.
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Example
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Example
Rock, Paper, Scissors
Rock, Paper, Scissors
Payoff matrix?
What probabilities should be positive?
What are optimal strategies?
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Outline
Symmetry
Example: Submarine Salvo
Solving two player zero-sum games
Recall: definitions, solving 2 × 2, 2 × n and m × 2 games, principle of
indifference
Symmetry
Invariance of payoffs under permutations
Invariance of strategies under permutations
Optimal strategies can be invariant
(Karlin and Peres, 2016)
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Symmetry
Symmetry
Submarine Salvo payoff matrix
Submarine Salvo payoff matrix
1
2
3
4
5
6
7
8
9
12
1
1
0
0
0
0
0
0
0
23
0
1
1
0
0
0
0
0
0
14
1
0
0
1
0
0
0
0
0
25
0
1
0
0
1
0
0
0
0
36
0
0
1
0
0
1
0
0
0
45
0
0
0
1
1
0
0
0
0
47
0
0
0
1
0
0
1
0
0
56
0
0
0
0
1
1
0
0
0
58
0
0
0
0
1
0
0
1
0
69
0
0
0
0
0
1
0
0
1
78
0
0
0
0
0
0
1
1
0
89
0
0
0
0
0
0
0
1
1
1
2
3
4
5
6
7
8
9
12
1
1
0
0
0
0
0
0
0
23
0
1
1
0
0
0
0
0
0
14
1
0
0
1
0
0
0
0
0
25
0
1
0
0
1
0
0
0
0
36
0
0
1
0
0
1
0
0
0
45
0
0
0
1
1
0
0
0
0
47
0
0
0
1
0
0
1
0
0
56
0
0
0
0
1
1
0
0
0
58
0
0
0
0
1
0
0
1
0
69
0
0
0
0
0
1
0
0
1
78
0
0
0
0
0
0
1
1
0
89
0
0
0
0
0
0
0
1
1
What happens if I flip the board from left to right?
1 ↔ 3, 4 ↔ 6, 7 ↔ 9;
12 ↔ 23, 14 ↔ 36, 45 ↔ 56, 47 ↔ 69, 78 ↔ 89.
What happens if I flip the board from left to right?
1 ↔ 3, 4 ↔ 6, 7 ↔ 9;
12 ↔ 23, 14 ↔ 36, 45 ↔ 56, 47 ↔ 69, 78 ↔ 89.
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Nothing!
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Outline
Symmetry
Definition
A game with payoff matrix A ∈ Rm×n is invariant under a permutation πx
on {1, . . . , m} if there is a permutation πy on {1, . . . , n} such that, for all
i, j, aij = aπx (i),πy (j) .
Solving two player zero-sum games
Recall: definitions, solving 2 × 2, 2 × n and m × 2 games, principle of
indifference
Symmetry
Example: for a left-to-right flip, πx permutes bomb positions, πy
permutes submarine positions.
If A is invariant under permutations π1 and π2 on {1, . . . , m}, then it
is invariant under π1 ◦ π2 .
Invariance of payoffs under permutations
Invariance of strategies under permutations
Optimal strategies can be invariant
Example: In addition to left-to-right flips, we could add
top-to-bottom flips. We have invariance under consecutive flips.
If A is invariant under some set S of permutations, then it is invariant
under the group G of permutations generated by S (that is,
compositions and inverses).
Example: All permutations involving sequences of flips.
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Outline
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Symmetry
Definition
A mixed strategy x ∈ ∆m is invariant under a permutation πx on
{1, . . . , m} if for all i, xi = xπx (i) .
Solving two player zero-sum games
Recall: definitions, solving 2 × 2, 2 × n and m × 2 games, principle of
indifference
Symmetry
Example: x is invariant for the permutation corresponding to a
left-to-right flip if x1 = x3 , x4 = x6 and x7 = x8 .
An orbit of a group G of permutations is a set Oi = {π(i) : π ∈ G }.
Invariance of payoffs under permutations
Invariance of strategies under permutations
Optimal strategies can be invariant
Example: for the group generated by horizontal, vertical, and diagonal
flips (NB: all symmetries!), the orbits are
O1 = {1, 3, 7, 9},
O2 = {2, 4, 6, 8},
O5 = {5}.
If a mixed strategy x is invariant under a group G of permutations,
then for every orbit, x is constant on the orbit.
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Symmetry
Symmetry
Submarine Salvo on orbits
Theorem
If A is invariant under a group G of permutations, then there are optimal
strategies that are invariant under G .
edge
center
0
mid-edge
1
4
1
4
center
0
1
corner
Proof
We take an optimal strategy x, and define the invariant optimal strategy x̄
that, for each orbit O and action i ∈ O, has
x̄i =
1 X
xi 0 .
|O| 0
1
4
Each entry is a uniform average over orbits of the original payoffs.
For Bomber, corner is dominated by mid-edge.
Then for Submarine, center is dominated by edge.
i ∈O
Optimal strategies: Bomber plays mid-edge, Submarine plays edge.
It’s easy to check that this strategy achieves exactly the same value as x.
Bomber puts weight 1/4 on each of 2, 4, 6, 8.
Submarine puts weight 1/8 on each of 12, 23, 14, 36, 47, 69, 78, 89.
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Symmetry
Symmetry
Example: Blotto Games
Two players (let’s call them Clinton and Trump) must decide how to
allocate their resources on advertising campaigns in Iowa and New
Hampshire.
Rock, Paper, Scissors
What are the invariances?
Permutation: Rock → Paper → Scissors → Rock.
Suppose that the player who allocates more resources to a state wins,
and that their gain is 1.
Orbit: {Rock, Paper, Scissors}
Invariant strategies?
(1/3, 1/3, 1/3).
If both allocate the same resources, the gain is 0 (they have equal
chance of winning).
Suppose that Clinton has the budget for 4 campaigns in total, and
Trump 3.
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Symmetry
Symmetry
Blotto Game Payoff Matrix
(3,0)
(2,1)
(1,2)
(0,3)
(4,0)
1
0
0
0
(3,1)
1
1
0
0
(2,2)
(1,3)
(0,4)
0
0
0
1
0
0
1
1
0
0
1
1
Blotto Game Orbits Payoff Matrix
{4, 0}
{3, 1}
(2, 2)
{3, 0}
0.5
0.5
0
{2, 1}
0
0.5
1
These entries are average payoffs on orbits.
The {3, 1} row dominates the {4, 0} row.
Invariances
Then the {3, 0} column dominates the {2, 1} column.
(Let’s ignore the dominated actions)
Thus, Clinton should play {3, 1} and Trump should play {3, 0}.
The states are interchangeable.
Back in the original game:
x ∗ = (0, 1/2, 0, 1/2, 0), y ∗ = (1/2, 0, 0, 1/2).
Orbits for Clinton: {(4, 0), (0, 4)}, {(3, 1), (1, 3)}, {(2, 2)}.
Orbits for Trump: {(3, 0), (0, 3)}, {(2, 1), (1, 2)}.
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Outline
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Outline
Solving two player zero-sum games
Solving two player zero-sum games
Recall: definitions, solving 2 × 2, 2 × n and m × 2 games, principle of
indifference
Symmetry
Recall: definitions, solving 2 × 2, 2 × n and m × 2 games, principle of
indifference
Symmetry
Invariance of payoffs under permutations
Invariance of strategies under permutations
Optimal strategies can be invariant
Invariance of payoffs under permutations
Invariance of strategies under permutations
Optimal strategies can be invariant
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