Outline for today
Stat155
Game Theory
Lecture 12: General-Sum Games
Two-player general-sum games
Definitions: payoff matrices, dominant strategies, safety strategies,
Nash equilibrium.
Examples: Prisoners’ dilemma, Stag hunt, Parking tickets
Peter Bartlett
October 6, 2016
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General-sum games
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General-sum games
Notation
A two-person general-sum game is specified by two payoff matrices,
A, B ∈ Rm×n .
Dominated pure strategies
Simultaneously, Player I chooses i ∈ {1, . . . , m} and the Player II
chooses j ∈ {1, . . . , n}.
Player I receives payoff aij .
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 .
Player II receives payoff bij .
Similarly, a pure strategy ej for Player II is dominated by ej 0 in payoff
matrix B if, for all i ∈ {1, . . . , m},
bij ≤ bij 0 .
Zero-sum game: B = −A.
Because it’s easier to view, we will often write a single bimatrix, that
is, a matrix with ordered pair entries (aij , bij ).
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General-sum games
General-sum games
Safety strategies
Nash equilibria
A safety strategy for Player I is an x∗ ∈ ∆m that satisfies
min
y ∈∆n
x∗> Ay
A pair (x∗ , y∗ ) ∈ ∆m × ∆n is a Nash equilibrium for payoff matrices
A, B ∈ Rm×n if
>
= max min x Ay .
x∈∆m y ∈∆n
max x > Ay∗ = x∗> Ay∗ ,
x∗ maximizes the worst case expected gain for Player I.
x∈∆m
max x∗> By = x∗> By∗ .
Similarly, a safety strategy for Player II is a y∗ ∈ ∆n that satisfies
y ∈∆n
min x > By∗ = max min x > By .
x∈∆m
y ∈∆n x∈∆m
If Player I plays x∗ and Player II plays y∗ , neither player has an
incentive to unilaterally deviate.
y∗ maximizes the worst case expected gain for Player II.
Recall that, for zero-sum games, the safety strategy for Player II was
defined using A (because in that case, B = −A):
max x > Ay∗ = min max x > Ay .
x∈∆m
x∗ is a best response to y∗ , y∗ is a best response to x∗ .
In general-sum games, there might be many Nash equilibria, with
different payoffs.
y ∈∆n x∈∆m
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Example: Prisoners’ Dilemma
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Example: Prisoners’ Dilemma
Payoff matrices
Payoff matrices
silent
confess
silent
(-1,-1)
(0,-10)
silent
confess
confess
(-10,0)
(-8,-8)
silent
(-1,-1)
(0,-10)
confess
(-10,0)
(-8,-8)
Dominant strategy: confess.
How can they achieve the better outcome?
Repeated play? Communication? Penalties/payments?
Try it!
(non-cooperative!)
(Karlin and Peres, 2016)
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For now, we’ll consider non-cooperative games:
Communication might or might not occur, but there are no binding
agreements.
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Example: Stag Hunt
Example: Stag Hunt
Dominant strategy?
Safety strategy?
(Recall: for Player I, choose x to maximize miny x > Ay .)
Pure strategy: hare.
(hare, hare) is a pure Nash equilibrium. Payoff (1,1).
Payoff matrices
stag
hare
stag
(4,4)
(2,0)
(stag, stag) is a pure Nash equilibrium. Payoff (4,4).
hare
(0,2)
(1,1)
Mixed Nash equilibrium:
For (x, y ) to be a Nash equilibrium, the players don’t want to shift to
a different mixture.
4y = 2y + 1 − y ,
4x = 2x + 1 − x.
So (1/3, 1/3) is a Nash equilibrium. Payoff (4/3,4/3).
(Karlin and Peres, 2016)
What are the expected payoffs in each case?
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Example: Parking Tickets
Example: Parking Tickets
Dominant strategy?
Safety strategy?
Pure strategies: legal, inspect.
Payoff: (0,-1). Not stable!
Pure Nash equilibrium? None.
Payoff matrices
legal
illegal
don’t inspect
(0,0)
(10,-10)
inspect
(0,-1)
(-90,-6)
Mixed Nash equilibrium:
For (x, y ) to be a Nash equilibrium, the players don’t want to shift to
a different mixture.
0 = 10y − 90(1 − y ),
meridiancity.org
−10(1 − x) = −x − 6(1 − x).
So (0.8, 0.9) is a Nash equilibrium.
Expected payoff? (0,-2).
(c.f. the payoff for the (unstable) safety strategies)
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Outline
Two-player general-sum games
Definitions: payoff matrices, dominant strategies, safety strategies,
Nash equilibrium.
Examples: Prisoners’ dilemma, Stag hunt, Parking tickets
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