Midterm 2
Stat155
Game Theory
Lecture 21: Midterm 2 Review
“This is an open book exam: you can use any printed or written material,
but you cannot use a laptop, tablet, or phone (or any device that can
communicate). There are three questions, each consisting of three parts.
Each part of each question carries equal weight. Answer each question in
the space provided.”
Peter Bartlett
November 8, 2016
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Topics
Series and parallel games
Series and parallel games.
Two-player general-sum games
A series game: Both players first play G1 , then both play G2 .
Value is V1 + V2 .
Payoff matrices, dominant strategies, safety strategies, Nash eq.
Multiplayer general-sum games
Utility functions, Nash equilibria. Nash’s Theorem
Congestion games and potential games
Congestion games: Every congestion game has a pure Nash equilibrium
Potential games: Every congestion game is a potential game
Criticisms of Nash equilibria
Evolutionarily stable strategies
A parallel game: Both players simultaneously decide which game to
play, and an action in that game.
If they choose the same game, they get the payoff from that game.
If they choose different games, the payoff is 0.
Value is 1/(1/V1 + 1/V2 ).
Electrical networks
ESS and Nash equilibria
Multiplayer ESS
Correlated equilibrium
Price of anarchy
Braess’s paradox
The impact of adding edges
Classes of latencies
Pigou networks
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Topics
Cooperative versus noncooperative games
Series and parallel games.
Two-player general-sum games
Payoff matrices, dominant strategies, safety strategies, Nash eq.
Multiplayer general-sum games
Noncooperative games
Utility functions, Nash equilibria. Nash’s Theorem
Congestion games and potential games
Congestion games: Every congestion game has a pure Nash equilibrium
Potential games: Every congestion game is a potential game
Criticisms of Nash equilibria
Evolutionarily stable strategies
Players play their strategies simultaneously.
They might communicate (or see a common signal; e.g., a traffic
signal), but there’s no enforced agreement.
Natural solution concepts:
Nash equilibrium, correlated equilibrium.
No improvement from unilaterally deviating.
ESS and Nash equilibria
Multiplayer ESS
Correlated equilibrium
Price of anarchy
Braess’s paradox
The impact of adding edges
Classes of latencies
Pigou networks
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General-sum games
General-sum games
Dominated pure strategies
Notation
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 .
A two-person general-sum game is specified by two payoff matrices,
A, B ∈ Rm×n .
Simultaneously, Player I chooses i ∈ {1, . . . , m} and the Player II
chooses j ∈ {1, . . . , n}.
Similarly, a pure strategy ej for Player II is dominated by ej 0 in payoff
matrix B if, for all i ∈ {1, . . . , m},
Player I receives payoff aij .
Player II receives payoff bij .
bij ≤ bij 0 .
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General-sum games
General-sum games
Nash equilibria
Safety strategies
A pair (x∗ , y∗ ) ∈ ∆m × ∆n is a Nash equilibrium for payoff matrices
A, B ∈ Rm×n if
A safety strategy for Player I is an x∗ ∈ ∆m that satisfies
max x > Ay∗ = x∗> Ay∗ ,
min x∗> Ay = max min x > Ay .
y ∈∆n
x∈∆m
x∈∆m y ∈∆n
max x∗> By = x∗> By∗ .
x∗ maximizes the worst case expected gain for Player I.
y ∈∆n
Similarly, a safety strategy for Player II is a y∗ ∈ ∆n that satisfies
If Player I plays x∗ and Player II plays y∗ , neither player has an
incentive to unilaterally deviate.
min x > By∗ = max min x > By .
x∈∆m
y ∈∆n x∈∆m
x∗ is a best response to y∗ , y∗ is a best response to x∗ .
y∗ maximizes the worst case expected gain for Player II.
In general-sum games, there might be many Nash equilibria, with
different payoff vectors.
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Comparing two-player general-sum and zero-sum games
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Comparing two-player general-sum and zero-sum games
Zero-sum games
1
A pair of safety strategies is a Nash equilibrium (minimax theorem)
2
Hence, there is always a Nash equilibrium.
3
If there are multiple Nash equilibria, they form a convex set, and the
expected payoff is identical within that set.
Thus, any two Nash equilibria give the same payoff.
Zero-sum games
4
If each player has an equalizing mixed strategy
(that is, x > A = v 1> and Ay = v 1),
then this pair of strategies is a Nash equilibrium.
(from the principle of indifference)
General-sum games
General-sum games
4
1
A pair of safety strategies might be unstable.
(Opponent aims to maximize their payoff, not minimize mine.)
2
There is always a Nash equilibrium (Nash’s Theorem).
3
There can be multiple Nash equilibria, with different payoff vectors.
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If each player has an equalizing mixed strategy
for their opponent’s payoff matrix
(that is, x > B = v2 1> and Ay = v1 1),
then this pair of strategies is a Nash equilibrium.
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Topics
Multiplayer general-sum games
Series and parallel games.
Two-player general-sum games
Notation
A k-person general-sum game is specified by k utility functions,
uj : S1 × S2 × · · · × Sk → R.
Payoff matrices, dominant strategies, safety strategies, Nash eq.
Multiplayer general-sum games
Utility functions, Nash equilibria. Nash’s Theorem
Congestion games and potential games
Congestion games: Every congestion game has a pure Nash equilibrium
Potential games: Every congestion game is a potential game
Criticisms of Nash equilibria
Evolutionarily stable strategies
Player j can choose strategies sj ∈ Sj .
Simultaneously, each player chooses a strategy.
Player j receives payoff uj (s1 , . . . , sk ).
k = 2: u1 (i, j) = aij , u2 (i, j) = bij .
ESS and Nash equilibria
Multiplayer ESS
For s = (s1 , . . . , sk ), let s−i denote the strategies without the ith one:
Correlated equilibrium
Price of anarchy
s−i = (s1 , . . . , si−1 , si+1 , . . . , sk ).
Braess’s paradox
The impact of adding edges
Classes of latencies
Pigou networks
And write (si , s−i ) as the full vector.
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Multiplayer general-sum games
Multiplayer general-sum games
Definition
Definition
A vector (s1∗ , . . . , sk∗ ) ∈ S1 × · · · × Sk is a pure Nash equilibrium for utility
functions u1 , . . . , uk if, for each player j ∈ {1, . . . , k},
max uj (sj , s∗−j ) = uj (sj∗ , s∗−j ).
sj ∈Sj
If the players play these sj∗ , nobody has an incentive to unilaterally
deviate: each player’s strategy is a best response to the other players’
strategies.
A sequence (x1∗ , . . . , xk∗ ) ∈ ∆S1 × · · · × ∆Sk (called a strategy profile) is a
Nash equilibrium for utility functions u1 , . . . , uk if, for each player
j ∈ {1, . . . , k},
max uj (xj , x∗−j ) = uj (xj∗ , x∗−j ).
xj ∈∆Sj
Here, we define
uj (x∗ ) = Es1 ∼x1 ,...,sk ∼xk uj (s1 , . . . , sk )
X
=
x1 (s1 ) · · · xk (sk )uj (s1 , . . . , sk ).
s1 ∈S1 ,...,sk ∈Sk
If the players play these mixed strategies xj∗ , nobody has an incentive
to unilaterally deviate: each player’s mixed strategy is a best response
to the other players’ mixed strategies.
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Multiplayer general-sum games
Multiplayer general-sum games
Nash’s Theorem (1951)
Every finite general-sum game has a Nash equilibrium.
Theorem
Proof Idea (two players)
Consider a strategy profile x ∈ ∆S1 × · · · × ∆Sk . Let
Ti = {s ∈ Si : xi (s) > 0}. The following statements are equivalent.
1
2
We find an “improvement” map M(x, y ) = (x̂, ŷ ), so that
x is a Nash equilibrium.
For each i, there is a ci such that
1
2
For all si ∈ Si , ui (si , x−i ) ≤ ci .
For si ∈ Ti , ui (si , x−i ) = ci .
←− No better response outside Ti
←− Indifferent within Ti
Compare with the principle of indifference in the zero-sum case.
1
x̂ > Ay > x > Ay (or x̂ > Ay = x > Ay , if x was a best response to y ),
2
x > B ŷ > x > By (or x > B ŷ = x > By , if y was a best response to x),
3
M is continuous.
It’s easy to find a map like this.
For instance, take a step in the direction of increasing payoff:
x̂ = P∆m (x + ηAy ),
ŷ = P∆n (y + ηB > x).
A Nash equilibrium is a fixed point of M.
The existence of a Nash equilibrium follows from Brouwer’s
fixed-point theorem.
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Nash’s Theorem
Topics
Series and parallel games.
Two-player general-sum games
Payoff matrices, dominant strategies, safety strategies, Nash eq.
Multiplayer general-sum games
Utility functions, Nash equilibria. Nash’s Theorem
Congestion games and potential games
Brouwer’s Fixed-Point Theorem
A continuous map f : K → K from a convex, closed, bounded K ⊆
has a fixed point, that is, an x ∈ K satisfying f (x) = x.
Congestion games: Every congestion game has a pure Nash equilibrium
Potential games: Every congestion game is a potential game
Rd
Criticisms of Nash equilibria
Evolutionarily stable strategies
ESS and Nash equilibria
Multiplayer ESS
Correlated equilibrium
Price of anarchy
Braess’s paradox
The impact of adding edges
Classes of latencies
Pigou networks
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Congestion games: games with a pure Nash equilibrium
Congestion games
Definition
A congestion game has
Example
k players
A
m facilities {1, . . . , m} (e.g., edges)
(1,2,4)
I, II, III
For player i, there is a set Si of strategies that are sets of facilities,
s ⊆ {1, . . . , m} (e.g., paths)
(2,3,5)
S
(2,3,5)
For facility j, there is a cost vector cj ∈ Rk , where cj (n) is the cost of
facility j when it is used by n players.
T
(1,2,6)
For a sequence s = (s1 , . . . , sn ), the utilities of the players are defined by
X
costi (s) = −ui (s) =
cj (nj (s)),
(2,4,8)
B
j∈si
where nj (s) = |{i : j ∈ si }| is the number of players using facility j.
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Congestion games
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Congestion games
Proof
For a sequence s = (s1 , . . . , sn ), the utilities of the players are defined by
X
costi (s) = −ui (s) =
cj (nj (s)),
We define a potential function Φ : S1 × · · · × Sk → R as
Φ(s) =
j∈si
j (s)
m nX
X
cj (l).
j=1 l=1
Fix strategies for the k players s = (s1 , . . . , sk ).
What happens when Player i changes from si to si0 ?
∆costi = costi (si0 , s−i ) − costi (si , s−i )
X
X
=
cj (nj (s) + 1) −
cj (nj (s))
where nj (s) = |{i : j ∈ si }| is the number of players using facility j.
Egalitarian: The utilities depend on how many players use each
facility, and not on which players use it.
j∈(si0 −si )
Theorem
|
Every congestion game has a pure Nash equilibrium.
{z
increased cost
}
= Φ(si0 , s−i ) − Φ(si , s−i )
j∈(si −si0 )
|
{z
decreased cost
}
= ∆Φ.
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The potential Φ reflects how a player’s costs change.
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Congestion games
Potential games
Proof
Definition
If we start at an arbitrary s, and update one player’s choice to
decrease that player’s cost, the potential must decrease.
Consider a multiplayer game G :
k players
Continuing updating other player’s strategies in this way, we must
eventually reach a local minimum (there are only finitely many
strategies).
For player i, there is a set Si of strategies.
Since no player can reduce their cost from there, we have reached a
pure Nash equilibrium.
This gives an algorithm for finding a pure Nash equilibrium: Update
the choice of one player at a time to improve their cost.
For player i, there is costi : S1 × · · · × Sk → R.
We say Φ : S1 × · · · × Sk → R is a potential for game G if ∆Φ = ∆costi ,
that is, for all i, s and si0 ,
Φ(si0 , s−i ) − Φ(si , s−i ) = costi (si0 , s−i ) − costi (si , s−i ).
We say that G is a potential game if it has a potential.
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Potential games
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Topics
Series and parallel games.
Two-player general-sum games
Payoff matrices, dominant strategies, safety strategies, Nash eq.
Multiplayer general-sum games
Utility functions, Nash equilibria. Nash’s Theorem
In considering congestion games, we proved two things:
Congestion games and potential games
Congestion games: Every congestion game has a pure Nash equilibrium
Potential games: Every congestion game is a potential game
Theorem
1
Every congestion game is a potential game.
2
Every potential game has a pure Nash equilibrium.
Criticisms of Nash equilibria
Evolutionarily stable strategies
ESS and Nash equilibria
Multiplayer ESS
Correlated equilibrium
Price of anarchy
Braess’s paradox
The impact of adding edges
Classes of latencies
Pigou networks
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What’s wrong with Nash equilibria?
Evolutionarily stable strategies
Will all players know everyone’s utilities?
Maximizing expected utility does not (explicitly) model risk aversion.
Will players maximize utility and completely ignore the impact on
other players’ utilities?
There is a population of individuals.
There is a game played between pairs of individuals.
How can the players find a Nash equilibrium?
Each individual has a pure strategy encoded in its genes.
How can the players agree on a Nash equilibrium to play?
The two players are randomly chosen individuals.
Will players actually randomize?
A higher payoff gives higher reproductive success.
This can push the population towards stable mixed strategies.
Alternative equilibrium concepts
Correlated equilibrium
Evolutionary stability
Equilibria in perturbed games.
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Evolutionarily stable strategies
Evolutionarily stable strategies
Suppose that x is invaded by a small population of mutants z:
x is replaced by (1 − )x + z.
Will the mix x survive?
Consider a two-player game with payoff matrices A, B.
Suppose that it is symmetric (A = B > ).
x’s utility:
Consider a mixed strategy x.
z’s utility:
Think of x as the proportion of each pure strategy in the population.
x> A (z + (1 − )x) = x> Az + (1 − )x> Ax
z> A (z + (1 − )x) = z> Az + (1 − )z> Ax
Definition
Mixed strategy x ∈ ∆n is an evolutionarily stable strategy (ESS) if, for any
pure strategy z,
1
2
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z> Ax ≤ x> Ax
If z> Ax = x> Ax then z> Az < x> Az.
←− (x, x) is a Nash equilibrium.
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ESS and Nash equilibria
ESS and Nash equilibria
Theorem
Every ESS is a Nash equilibrium.
Definition
A strategy profile x = (x1∗ , . . . , xk∗ ) ∈ ∆S1 × · · · × ∆Sk is a strict Nash
equilibrium for utility functions u1 , . . . , uk if, for each player
j ∈ {1, . . . , k}, for all xj ∈ ∆Sj that is different from xj∗ ,
This follows from the definition:
Definition
A strategy x ∈ ∆n is an evolutionarily stable strategy (ESS) if, for any
pure strategy z 6= x,
1
2
uj (xj , x∗−j ) < uj (xj∗ , x∗−j ).
z> Ax ≤ x> Ax
If z> Ax = x> Ax then z> Az < x> Az.
A Nash equilibrium has uj (xj , x∗−j ) ≤ uj (xj∗ , x∗−j ).
Proof: If every pure strategy z satisfies z> Ax ≤ x> Ax, then every
mixed strategy z (mixture of pure strategies) must obey the same
inequality. Hence, (x, x) is a Nash equilibrium.
By the principle of indifference, only a pure Nash equilibrium can be a
strict Nash equilibrium.
The converse is true for strict Nash equilibria.
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ESS and Nash equilibria
ESMS: evolutionarily stable against mixed strategies
Pure versus mixed
Theorem
An ESS is a Nash equilibrium (x∗ , x∗ ) satisfying, for all ei 6= x∗ , if
ei> Ax∗ = x∗> Ax∗ , then ei> Aei < x∗> Aei .
Every strict Nash equilibrium is an ESS.
Proof: A strict Nash equilibrium has z> Ax < x> Ax for z 6= x, so both
conditions defining an ESS are satisfied.
Definition
2
Say that a symmetric strategy (x∗ , x∗ ) is evolutionarily stable against
mixed strategies (ESMS) if it is a Nash equilibrium and, for all mixed
strategies z 6= x∗ , if z > Ax∗ = x∗> Ax∗ , then z > Az < x∗> Az.
(ESS is sometimes defined this way, e.g., Leyton-Brown and Shoham)
A strategy x ∈ ∆n is an evolutionarily stable strategy (ESS) if, for any
pure strategy z 6= x,
1
What about invasion by a mixed strategy?
Clearly, every ESMS strategy is an ESS.
z> Ax ≤ x> Ax
Theorem
If z> Ax = x> Ax then z> Az < x> Az.
For a two-player 2 × 2 symmetric game, every ESS is ESMS.
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Multiplayer evolutionarily stable strategies
Multiplayer evolutionarily stable strategies
Consider a symmetric multiplayer game (that is, unchanged by
relabeling the players).
Definition
(Suppose, for simplicity, that the utility for player i depends on si and on
the set of strategies played by the other players, but is invariant to a
permutation of the other players’ strategies.)
A strategy x ∈ ∆n is an evolutionarily stable strategy (ESS) if, for any
pure strategy z 6= x,
Suppose that a symmetric mixed strategy x is invaded by a small
population of mutants z: x is replaced by (1 − )x + z.
Will the mix x survive?
1
x’s utility:
z’s utility:
u1 (x, z + (1 − )x, z + (1 − )x)
2
= u1 (x, z, x) + u1 (x, x, z) + (1 − 2)u1 (x, x, x) + O(2 )
u1 (z, z, x) + u1 (z, x, z) + (1 − 2)u1 (z, x, x) + O(2 ).
←− x is a Nash equilibrium.
u1 (z, x−1 ) ≤ u1 (x, x−1 )
If u1 (z, x−1 ) = u1 (x, x−1 ) then for all j 6= 1,
u1 (z, z, x−1,−j ) < u1 (x, z, x−1,−j ).
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Topics
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Correlated equilibria: A driving example
Series and parallel games.
Two-player general-sum games
Payoff
Payoff matrices, dominant strategies, safety strategies, Nash eq.
Go
Stop
Multiplayer general-sum games
Utility functions, Nash equilibria. Nash’s Theorem
Congestion games and potential games
Congestion games: Every congestion game has a pure Nash equilibrium
Potential games: Every congestion game is a potential game
Criticisms of Nash equilibria
Evolutionarily stable strategies
ESS and Nash equilibria
Multiplayer ESS
Go
(-100,-100)
(-1,1)
Stop
(1,-1)
(-1,-1)
Nash equilibria
2
99
2
99
(Go, Stop), (Stop, Go), 101
, 101
, 101
, 101
.
(because we want indifference: −100p + 1 − p = −p − (1 − p))
Better solution?
Correlated equilibrium
Price of anarchy
A traffic signal: Pr((Red, Green)) = Pr((Green, Red)) = 1/2, and
both players agree: Red means Stop, Green means Go.
Braess’s paradox
The impact of adding edges
Classes of latencies
Pigou networks
After they both see the traffic signal, the players have no incentive to
deviate from the agreed actions.
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Correlated equilibrium
Correlated equilibrium
Definition
Definition
For a two player game with strategy sets S1 = {1, . . . , m} and
S2 = {1, . . . , n}, a correlated strategy pair is a pair of random variables
(R, C ) with some joint probability distribution over pairs of actions
(i, j) ∈ S1 × S2 .
A correlated strategy pair for a two-player game with payoff matrices A
and B is a correlated equilibrium if
1 for all i, i 0 ∈ S , Pr(R = i) > 0 ⇒ E [a
1
i,C |R = i] ≥ E ai 0 ,C |R = i .
2 for all j, j 0 ∈ S , Pr(C = j) > 0 ⇒ E [b
2
R,j |C = j] ≥ E bR,j 0 |C = j .
Example
c.f. a Nash equilibrium
In the traffic signal example, Pr(Stop, Go) = Pr(Go, Stop) = 1/2.
A strategy profile (x, y) ∈ ∆Sm × ∆Sn is a Nash equilibrium iff
1 for all i, i 0 ∈ S , Pr(R = i) > 0 ⇒ E [a
1
i,C ] ≥ E ai 0 ,C .
2 for all j, j 0 ∈ S , Pr(C = j) > 0 ⇒ E [b
2
R,j ] ≥ E bR,j 0 .
c.f. a pair of mixed strategies
If we have x ∈ ∆Sm and y ∈ ∆Sn , then choosing the two actions (R, C )
independently gives Pr(R = i, C = j) = xi yj .
In the traffic signal example, we cannot have Pr(Stop, Go) > 0 and
Pr(Go, Stop) > 0 without Pr(Go, Go) > 0.
When R and C are independent, these expectations and the conditional
expectations are identical.
Thus, a Nash equilibrium is a correlated equilibrium.
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Correlated equilibria
Correlated equilibria
CE vs NE
Nash’s Theorem implies there is always a correlated equilibrium.
They are easy to find, via linear programming.
It is not unusual for correlated equilibria to achieve better solutions
for both players than Nash equilibria.
Implementation
We can think of a correlated equilibrium being implemented in two
equivalent ways:
1
2
There is a random draw of a correlated strategy pair with a known
distribution, and the players see their strategy only.
There is a draw of a random variable (‘external event’) with a known
probability distribution, and a private signal is communicated to the
players about the value of the random variable. Each player chooses a
mixed strategy that depends on this private signal (and the dependence
is common knowledge).
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Combinations of correlated equilibria
Given any two correlated equilibria, you can combine them to obtain
another: Imagine a public random variable that determines which of
the correlated equilibria will be played. Knowing which correlated
equilibrium is being played, the players have no incentive to deviate.
The payoffs are convex combinations of the payoffs of the two CEs.
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Topics
Braess’s paradox
Series and parallel games.
Two-player general-sum games
Example: before
Payoff matrices, dominant strategies, safety strategies, Nash eq.
Multiplayer general-sum games
Utility functions, Nash equilibria. Nash’s Theorem
Congestion games and potential games
Congestion games: Every congestion game has a pure Nash equilibrium
Potential games: Every congestion game is a potential game
Criticisms of Nash equilibria
Evolutionarily stable strategies
Example: after
ESS and Nash equilibria
Multiplayer ESS
Correlated equilibrium
Price of anarchy
Braess’s paradox
The impact of adding edges
Classes of latencies
Pigou networks
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(Karlin and Peres, 2016)
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Price of anarchy
Price of anarchy
Definition
For a routing problem, define
price of anarchy =
Example: Nash equals socially optimal
average travel time in worst Nash equilibrium
.
minimal average travel time
The minimum is over all flows.
The flow minimizing average travel time is the socially optimal flow.
The price of anarchy reflects how much average travel time can
decrease in going from a Nash equilibrium flow (where all individuals
choose a path to minimize their travel time) to a prescribed flow.
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Price of anarchy
Price of anarchy
Example: price of anarchy = 4/3
Theorem
1
For linear latency functions, the price of anarchy is 1.
2
For affine latency functions, the price of anarchy is no more than 4/3.
(Karlin and Peres, 2016)
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Price of anarchy
Nash equilibrium flows exist (and they’re easy to find)
Traffic flow
Theorem
A flow from source s to destination t in a directed graph is a mixture
of paths from s to t, with mixture weight fP for path P.
We write the flow on an edge e as
X
Fe =
fP .
For a DAG with latency functions `e that are continuous, non-decreasing,
and non-negative, if there is a path from source to destination, there is a
Nash equilibrium unit flow.
Proof idea
This is the non-atomic version of a congestion game.
P3e
For the atomic version (finite number of players), we showed that
there is a pure Nash equilibrium that can be found by descending a
potential function.
Latency on an edge e is a non-decreasing function of Fe , written
`e (Fe ).
P
Latency on a path P is the total latency, LP (f ) = e∈P `e (Fe ).
P
P
Average latency is L(f ) = P fP LP (f ) = e Fe `e (Fe ).
The same approach works here. The potential function is
X Z Fe
Φ(f ) =
`e (x) dx.
A flow f is a Nash equilibrium if, for all P and P 0 , if fP > 0,
LP (f ) ≤ LP 0 (f ).
e
0
If f is not a Nash equilibrium flow, then Φ(f ) is not minimal.
Φ is convex, on a convex, compact set, so it has a minimum.
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Price of anarchy
The impact of adding edges
Theorem
For linear latencies, that is, `e (x) = ae x, with ae ≥ 0, if f is a Nash
equilibrium flow and f ∗ is a socially optimal flow (that is, L(f ∗ ) is
minimal), then
L(f ) = L(f ∗ ).
Theorem
Consider a network G with a Nash equilibrium flow fG and average latency
LG (fG ), and a network H with additional roads added. Suppose that the
price of anarchy in H is no more than α. Then any Nash equilibrium flow
fH has average latency LH (fH ) ≤ αLG (fG ).
Proof
Theorem
For affine latencies, that is, `e (x) = ae x + be , with ae , be ≥ 0, if f is a
Nash equilibrium flow and f ∗ is a socially optimal flow (that is, L(f ∗ ) is
minimal), then
4
L(f ) ≤ L(f ∗ ).
3
LH (fH ) ≤ αLH (fH∗ ) ≤ αLH (fG∗ ) = αLG (fG∗ ) ≤ αLG (fG ).
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Classes of latency functions
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Pigou networks and price of anarchy
Price of anarchy
A Pigou network
Suppose we allow latency functions from some class L.
For example, we have considered
L = {x 7→ ax : a ≥ 0} ,
What about
L = {x 7→ ax + b : a, b ≥ 0} ,
L=
(
x 7→
X
d
)
ad x d : ad ≥ 0 ?
We’ll insist that latency functions are non-negative and
non-decreasing.
It turns out that the price of anarchy in an arbitrary network with
latency functions chosen from L is at most the price of anarchy in a
certain small network with these latency functions: a Pigou network.
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(Karlin and Peres, 2016)
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Pigou networks and price of anarchy
Nonlinear latency functions
Example: nonlinear latency `e (x) = x d
Theorem
d
Define the Pigou price of anarchy as the price of anarchy for this network
with latency function ` and total flow r :
αr (`) =
r `(r )
.
minx≥0 (x`(x) + (r − x)`(r ))
(why x ≥ 0?)
For any network with latency functions from L and total flow 1, the price
of anarchy is no more than
Price of anarchy
Nash equilibrium flow: all through top edge. L(f ) = 1.
Socially optimal flow:
max max αr (`).
0≤r ≤1 `∈L
L(f ∗ ) = min(1 − x + x d+1 ) = 1 − d(d + 1)−(d+1)/d .
x
Price of anarchy:
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Topics
Series and parallel games.
Two-player general-sum games
Payoff matrices, dominant strategies, safety strategies, Nash eq.
Multiplayer general-sum games
Utility functions, Nash equilibria. Nash’s Theorem
Congestion games and potential games
Congestion games: Every congestion game has a pure Nash equilibrium
Potential games: Every congestion game is a potential game
Criticisms of Nash equilibria
Evolutionarily stable strategies
ESS and Nash equilibria
Multiplayer ESS
Correlated equilibrium
Price of anarchy
Braess’s paradox
The impact of adding edges
Classes of latencies
Pigou networks
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1
d
.
−(d+1)/d
ln d
1 − d(d + 1)
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