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Strategic Games:Social Optima and Nash Equilibria
Krzysztof R. AptCWI
& University of Amsterdam
Strategic Games:Social Optima and Nash Equilibria– p. 1/29
Basic Concepts
Strategic games.
Nash equilibrium.
Social optimum.
Price of anarchy.
Price of stability.
Strategic Games:Social Optima and Nash Equilibria– p. 2/29
Strategic Games
Strategic game for |N| ≥ 2 players:
G := (N,{Si}i∈N ,{pi}i∈N).
For each player i
(possibly infinite) set Si of strategies,
payoff function pi : S1× . . .×Sn →R.
Strategic Games:Social Optima and Nash Equilibria– p. 3/29
Basic assumptions
Players choose their strategies simultaneously,
each player is rational: his objective is to maximize hispayoff,
players have common knowledge of the game and ofeach others’ rationality.
Strategic Games:Social Optima and Nash Equilibria– p. 4/29
Three Examples (1)
The Battle of the SexesF B
F 2,1 0,0B 0,0 1,2
Matching PenniesH T
H 1,−1 −1, 1T −1, 1 1,−1
Prisoner’s DilemmaC D
C 2,2 0,3D 3,0 1,1
Strategic Games: Social Optima and Nash Equilibria– p. 5/29
Main Concepts
Notation: si,s′i ∈ Si,s,s′,(si,s−i) ∈ S1× . . .×Sn.
s is a Nash equilibrium if
∀i ∈ {1, . . .,n} ∀s′i ∈ Si pi(si,s−i) ≥ pi(s′i,s−i).
Social welfare of s:
SW (s) :=n
∑j=1
p j(s).
s is a social optimum if SW (s) is maximal.
Strategic Games: Social Optima and Nash Equilibria– p. 6/29
Intuitions
Nash equilibrium:Every player is ‘happy’(played his best response).
Social optimum:The desired state of affairs for the society.
Main problem:Social optima may not be Nash equilibria.
Strategic Games: Social Optima and Nash Equilibria– p. 7/29
Three Examples (2)
The Battle of the Sexes: Two Nash equilibria.F B
F 2,1 0,0B 0,0 1,2
Matching Pennies: No Nash equilibrium.
H TH 1,−1 −1, 1T −1, 1 1,−1
Prisoner’s Dilemma: One Nash equilibrium.
C DC 2,2 0,3D 3,0 1,1
Strategic Games: Social Optima and Nash Equilibria– p. 8/29
Prisoner’s Dilemma in Practice
Strategic Games: Social Optima and Nash Equilibria– p. 9/29
Price of Anarchy and of Stability
Price of Anarchy (Koutsoupias, Papadimitriou, 1999):
SW of social optimumSW of the worst Nash equilibrium
Price of Stability (Schulz, Moses, 2003):
SW of social optimumSW of the best Nash equilibrium
Strategic Games: Social Optima and Nash Equilibria– p. 10/29
Examples
A 3×3 gameL M R
T 2,2 4,1 1,0C 1,4 3,3 1,0B 0,1 0,1 1,1
PoA = 62 = 3.
PoS = 64 = 1.5.
Prisoner’s DilemmaC D
C 2,2 0,3D 3,0 1,1
PoA = PoS = 2.
Strategic Games: Social Optima and Nash Equilibria– p. 11/29
Congestion Games: ExampleAssumptions:
4000 drivers drive from A to B.
Each driver has 2 possibilities (strategies).
T/100
T/100
45
U
R
B
45
A
Problem: Find a Nash equilibrium (T = number of drivers).
Strategic Games: Social Optima and Nash Equilibria– p. 12/29
Nash Equilibrium
T/100
T/100
45
U
R
B
45
A
Answer: 2000/2000.
Travel time: 2000/100 + 45 = 45 + 2000/100 = 65.
Strategic Games: Social Optima and Nash Equilibria– p. 13/29
Braess ParadoxAdd a fast road from U to R.
Each drives has now 3 possibilities (strategies):A - U - B,A - R - B,A - U - R - B.
T/100
T/100
45
U
R
B
45
A 0
Problem: Find a Nash equilibrium.Strategic Games: Social Optima and Nash Equilibria– p. 14/29
Nash Equilibrium
T/100
T/100
45
U
R
B
45
A 0
Answer: Each driver will choose the road A - U - R - B.
Why?: The road A - U - R - B is always a best response.
Strategic Games: Social Optima and Nash Equilibria– p. 15/29
Bad News
T/100
T/100
45
U
R
B
45
A 0
Travel time: 4000/100 + 4000/100 = 80!
PoA (and PoS) went up from 1 to 80/65.
Strategic Games: Social Optima and Nash Equilibria– p. 16/29
Does it Happen?From Wikipedia (‘Braess Paradox’):
In Seoul, South Korea, a speeding-up in traffic aroundthe city was seen when a motorway was removed aspart of the Cheonggyecheon restoration project.
In Stuttgart, Germany after investments into the roadnetwork in 1969, the traffic situation did not improveuntil a section of newly-built road was closed for trafficagain.
In 1990 the closing of 42nd street in New York Cityreduced the amount of congestion in the area.
In 2008 Youn, Gastner and Jeong demonstratedspecific routes in Boston, New York City and Londonwhere this might actually occur and pointed out roadsthat could be closed to reduce predicted travel times.
Strategic Games: Social Optima and Nash Equilibria– p. 17/29
General Model
Congestion games
Each player chooses some set of resources.
Each resource has a delay function associated with it.
Each player pays for each resource used.
The price for the use of the resource depends on thenumber of users.
Theorem (Anshelevich et al., 2004)If the delay functions are linear, then PoA ≤ 4
3.
Strategic Games: Social Optima and Nash Equilibria– p. 18/29
More Concepts
Altruistic games.
Selfishness level.(Based onSelfishness level of strategic games,K.R. Apt and G. Schäfer)
Strategic Games: Social Optima and Nash Equilibria– p. 19/29
Altruistic Games
Given G := (N,{Si}i∈N,{pi}i∈N) and α ≥ 0.
G(α) := (N,{Si}i∈N,{ri}i∈N), where
ri(s) := pi(s)+αSW (s).
When α > 0 the payoff of each player in G(α) dependson the social welfare of the players.
G(α) is an altruistic version of G.
Strategic Games: Social Optima and Nash Equilibria– p. 20/29
Selfishness Level
G is α-selfish if a Nash equilibrium of G(α) is a socialoptimum of G(α).
If for no α ≥ 0, G is α-selfish, thenits selfishness level is ∞.
Suppose G is finite.If for some α ≥ 0, G is α-selfish, then
minα∈R+
(G is α-selfish)
is the selfishness level of G.
Strategic Games: Social Optima and Nash Equilibria– p. 21/29
Three Examples (1)
The Battle of the SexesF B
F 2,1 0,0B 0,0 1,2
Matching PenniesH T
H 1,−1 −1, 1T −1, 1 1,−1
Prisoner’s DilemmaC D
C 2,2 0,3D 3,0 1,1
Strategic Games: Social Optima and Nash Equilibria– p. 22/29
Three Examples (2)
The Battle of the Sexes: selfishness level is 0.F B
F 2,1 0,0B 0,0 1,2
Matching Pennies: selfishness level is ∞.
H TH 1,−1 −1, 1T −1, 1 1,−1
Prisoner’s Dilemma: selfishness level is 1.C D
C 2,2 0,3D 3,0 1,1
C DC 6,6 3,6D 6,3 3,3
Strategic Games: Social Optima and Nash Equilibria– p. 23/29
Selfishness Level vs Price of Stability
NoteSelfishness level of a finite game is 0 iff price ofstability is 1.
TheoremFor every finite α > 0 and β > 1 there is a finite gamewith selfishness level α and price of stability β .
Strategic Games: Social Optima and Nash Equilibria– p. 24/29
Example: Prisoner’s Dilemma
Prisoner’s Dilemma for n players
Each Si = {0,1},
pi(s) := 1− si +2∑j 6=i
s j.
Proposition Selfishness level is 12n−3.
Strategic Games: Social Optima and Nash Equilibria– p. 25/29
Example: Traveler’s Dilemma
Two players, Si = {2, . . .,100},
pi(s) :=
si if si = s−i
si +2 if si < s−i
s−i −2 otherwise.
Problem: Find a Nash equilibrium.
Proposition Selfishness level is 12.
Strategic Games: Social Optima and Nash Equilibria– p. 26/29
Take Home Message
Price of anarchy and price of stability are descriptiveconcepts.
Selfishness level is a normative concept.
Strategic Games: Social Optima and Nash Equilibria– p. 27/29
Some Quotations
Dalai Lama:
The intelligent way to be selfish is towork for the welfare of others.
Microeconomics: Behavior, Institutions, and Evolution,S. Bowles ’04.
An excellent way to promote cooperationin a society is to teach people to careabout the welfare of others.
The Evolution of Cooperation, R. Axelrod, ’84.
Strategic Games: Social Optima and Nash Equilibria– p. 28/29
THANK YOU
Strategic Games: Social Optima and Nash Equilibria– p. 29/29