CMU 15-896 Noncooperative games 3: Price of anarchy Teacher: - - PowerPoint PPT Presentation

CMU 15-896

Noncooperative games 3: Price of anarchy

Teacher: Ariel Procaccia

15896 Spring 2016: Lecture 19

Back to prison

• The only Nash

equilibrium in Prisoner’s dilemma is bad; but how bad is it?

• Objective function: social

cost sum of costs

• NE is six times worse

than the optimum

2

Cooperate Defect Cooperate

• 1,-1
• 9,0

Defect

0,-9

• 6,-6

15896 Spring 2016: Lecture 19

Anarchy and stability

• Fix a class of games, an objective function, and

an equilibrium concept

• The price of anarchy (stability) is the worst-case

ratio between the worst (best) objective function value of an equilibrium of the game, and that of the optimal solution

• In this lecture:
• Objective function social cost
• Equilibrium concept Nash equilibrium

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15896 Spring 2016: Lecture 19

Example: Cost sharing

• players in weighted directed

graph

• Player wants to get from to ;

strategy space is

paths

• Each edge

has cost

• Cost of edge is split between all

players using edge

• Cost of player is sum of costs over

edges on path

4

• 10

10 10 1 1 1 1

15896 Spring 2016: Lecture 19

Example: Cost sharing

• With

players, the example

• n the right has an NE with

social cost

• Optimal social cost is
• Price of anarchy

5

• 1

Prove that the price of anarchy is at most

15896 Spring 2016: Lecture 19

Example: Cost sharing

• Think of the

edges as cars, and the edge as mass transit

• Bad Nash equilibrium with

cost

• Good Nash equilibrium with

cost

• Now let’s modify the

example…

6

• …

0 0 1 1 1

15896 Spring 2016: Lecture 19

Example: Cost sharing

• OPT
• Only equilibrium has cost
• price of stability is at

least

• We will show that the price
• f stability is

7

• …

0 0 1

• 1
• 2

15896 Spring 2016: Lecture 19

Potential games

• A game is an exact potential game if there

exists a function

• such that

for all , for all

• , and for all
• ,
• 8

Why does the existence of an exact potential function imply the existence

• f a pure Nash equilibrium?

15896 Spring 2016: Lecture 19

Potential games

• Theorem: the cost sharing game is an exact

potential game

• Proof:
• Let be the number of players using under
• Define the potential function

Φ

• If player changes paths, pays
• for each new

edge, gets

• for each old edge, so Δcost ΔΦ ∎

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15896 Spring 2016: Lecture 19

Potential games

• Theorem: The cost of stability of cost sharing

games is

• Proof:
• It holds that

cost

• Take a strategy profile

that minimizes

• is an NE
• cost

OPT

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15896 Spring 2016: Lecture 19

Cost sharing summary

• In every cost sharing game
• NE ,
• NE

such that

• There exist cost sharing games s.t.
• NE

such that

• NE ,

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15896 Spring 2016: Lecture 19

Congestion games

• Generalization of cost sharing games
• players and

resources

• Each player chooses a set of resources (e.g., a

path) from collection of allowable sets of resources (e.g., paths from to )

• Cost of resource is a function
• f the

number of players using it

• Cost of player is the sum over used resources

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15896 Spring 2016: Lecture 19

Congestion games

• Theorem [Rosenthal 1973]: Every

congestion game is an exact potential game

• Proof: The exact potential function is

Φ

• Theorem [Monderer and Shapley 1996]:

Every potential game is isomorphic to a congestion game

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15896 Spring 2016: Lecture 19

Network formation games

• Each player is a vertex
• Strategy of : set of undirected edges to build

that touch

• Strategy profile

induces undirected graph

• Cost of building any edge is
• , where

edges bought by , is shortest path in edges

• 14

15896 Spring 2016: Lecture 19

• NE with

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Suboptimal Optimal

Example: Network formation

15896 Spring 2016: Lecture 19

Example: Network formation

• Lemma: If

then any star is optimal, and if then a complete graph is optimal

• Proof:
• Suppose 2, and consider any graph that is not

complete

• Adding an edge will decrease the sum of distances by

at least 2, and costs only

• Suppose 2 and the graph contains a star, so the

diameter is at most 2; deleting a non-star edge increases the sum of distances by at most 2, and saves ∎

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15896 Spring 2016: Lecture 19

Example: Network formation

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Poll: For which values of is any star a NE, and for which is the complete graph a NE? none none

15896 Spring 2016: Lecture 19

Example: Network formation

• Theorem:

1.

If 2 or 1, PoS 1

2.

For 1 2, PoS 4/3

• Proof:
• Part 1 is immediate from the lemma and poll
• For 1 2, the star is a NE, while OPT is a

complete graph

• Worst case ratio when → 1:

2 1 1 1 1/2 4 6 2 3 3 4 3 ∎

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15896 Spring 2016: Lecture 19

Example: Network creation

• Theorem [Fabrikant et al. 2003]: The price
• f anarcy of network creation games is
• Lemma: If

is a Nash equilibrium that induces a graph of diameter , then

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15896 Spring 2016: Lecture 19

Proof of lemma

• Buying a connected graph costs at least
• There are

distances

• Distance costs
• focus on edge

costs

• There are at most

cut edges focus

• n noncut edges

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15896 Spring 2016: Lecture 19

Proof of lemma

• Claim: Let , be a noncut edge, then the distance

, with deleted 2

• set of nodes s.t. the shortest path from uses
• Figure shows shortest path avoiding , , is

the edge on the path entering

• is the shortest path from to ⇒
• 1 as

∪ is shortest path from to ∎

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• ′

′

• ′

′

• ′

15896 Spring 2016: Lecture 19

Proof of lemma

• Claim: There are

noncut edges paid for by any vertex

• Let

be an edge paid for by

• By previous claim, deleting

increases distances from by at most

• is an equilibrium
• vertices overall

can’t be more than sets

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15896 Spring 2016: Lecture 19

Proof of lemma

• noncut edges per vertex
• total payment for these per vertex
• verall

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15896 Spring 2016: Lecture 19

Proof of theorem

• By lemma, it is enough to show that the

diameter at a NE

• Suppose

for some

• By adding the edge

, pays and improves distance to second half of the shortest path by

• If

, it is beneficial to add edge

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