BACK TO PRISON The only Nash equilibrium in Prisoners dilemma is - - PowerPoint PPT Presentation

ALGOS TRUTH JUSTICE

Game Theory II: Price of Anarchy

Teachers: Ariel Procaccia (this time) and Alex Psomas

BACK TO PRISON

• 1,-1
• 9,0

0,-9

• 6,-6
• 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
• We can make this arbitrarily bad

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)

• bjective 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

EXAMPLE: COST SHARING

• 𝑜 players in weighted directed graph

𝐻

• Player 𝑗 wants to get from 𝑡𝑗 to 𝑢𝑗;

strategy space is 𝑡𝑗 → 𝑢𝑗 paths

• Each edge 𝑓 has cost 𝑑𝑓
• Cost of edge is split between all

players using edge

• Cost of player is sum of costs over

edges on path

𝑡2 𝑡1

𝑢1 𝑢2

10 10 10 1 1 1 1

EXAMPLE: COST SHARING

• With 𝑜 players, the example on the

right has a NE with social cost 𝑜

• Optimal social cost is 1
• It follows that the price of anarchy
• f cost sharing games is at least 𝑜
• It is easy to see that the price of

anarchy of cost sharing games is at most 𝑜 — why?

𝑢 𝑡

𝑜 1

EXAMPLE: COST SHARING

• Think of the 1 edges as cars, and

the 𝑙 edge as mass transit

• Bad Nash equilibrium with cost

𝑜

• Good Nash equilibrium with

cost 𝑙

• Now let’s modify the example…

𝑡1 𝑡2 𝑡𝑜 𝑢

… 1 1 1 𝑙

EXAMPLE: COST SHARING

• OPT = 𝑙 + 1
• Only equilibrium has cost

𝑙 ⋅ 𝐼(𝑜)

• Therefore, the price of

stability of cost sharing games is at least Ω(log 𝑜)

• We will show that the price
• f stability is Θ(log 𝑜)

𝑡1 𝑡2 𝑡𝑜 𝑢

… 𝑙 + 1 𝑙 1 𝑙 2 𝑙 𝑜

POTENTIAL GAMES

• A game is an exact potential game if there

exists a function Φ: ς𝑗=1

𝑜

𝑇𝑗 → ℝ such that for all 𝑗 ∈ 𝑂, for all 𝒕 ∈ ς𝑗=1

𝑜

𝑇𝑗, and for all 𝑡𝑗

′ ∈ 𝑇𝑗,

cost𝑗 𝑡𝑗

′, 𝒕−𝑗 − cost𝑗 𝒕 = Φ 𝑡𝑗 ′, 𝒕−𝑗 − Φ(𝒕)

• The existence of an exact potential function

implies the existence of a pure Nash equilibrium — why?

POTENTIAL GAMES

• Theorem: the cost sharing game is an exact

potential game

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

Φ 𝒕 = ෍

𝑓

෍

𝑙=1 𝑜𝑓(𝒕) 𝑑𝑓

𝑙

• If player changes paths, pays

𝑑𝑓 𝑜𝑓 𝒕 +1 for each new

edge, gets

𝑑𝑓 𝑜𝑓 𝒕 for each old edge, so Δcost𝑗 = ΔΦ ∎

POTENTIAL GAMES

• Theorem: The cost of stability of cost sharing

games is 𝑃(log 𝑜)

• Proof:
• It holds that

cost 𝒕 ≤ Φ 𝒕 ≤ 𝐼 𝑜 ⋅ cost(𝒕)

• Take a strategy profile 𝒕 that minimizes Φ
• 𝒕 is an NE
• cost 𝒕 ≤ Φ 𝒕 ≤ Φ OPT ≤ 𝐼 𝑜 ⋅ cost(OPT) ∎

COST SHARING SUMMARY

• Upper bounds: ∀cost sharing game,
• PoA: ∀NE 𝒕,

cost 𝒕 ≤ 𝑜 ⋅ cost(OPT)

• PoS: ∃NE 𝒕 s.t.

cost 𝒕 ≤ 𝐼 𝑜 ⋅ cost(OPT)

• Lower bounds: ∃cost sharing game s.t.
• PoA: ∃NE 𝒕 s.t.

cost 𝒕 ≥ 𝑜 ⋅ cost(OPT)

• PoS: ∀NE 𝒕,

cost 𝒕 ≥ 𝐼 𝑜 ⋅ cost(OPT)

NETWORK FORMATION GAMES

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

build that touch 𝑤

• Strategy profile 𝒕 induces undirected graph

𝐻(𝒕)

• Cost of building any edge is 𝛽
• cost𝑤 𝒕 = 𝛽𝑜𝑤 𝒕 + σ𝑣 𝑒(𝑣, 𝑤), where

𝑜𝑤 = #edges bought by 𝑤, 𝑒 is shortest path in #edges

• cost 𝒕 = σ𝑣≠𝑤 𝑒 𝑣, 𝑤 + 𝛽|𝐹|

NE with 𝛽 = 3

Suboptimal Optimal

EXAMPLE: NETWORK FORMATION

EXAMPLE: NETWORK FORMATION

• Lemma: If 𝛽 ≥ 2 then any star is optimal, and

if 𝛽 ≤ 2 then a complete graph is optimal

• Proof:
• Suppose 𝛽 ≤ 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 𝛽

• Suppose 𝛽 ≥ 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 𝛽 ∎

EXAMPLE: NETWORK FORMATION

• Theorem:
• 1. If 𝛽 ≥ 2 or 𝛽 ≤ 1, PoS = 1
• 2. For 1 < 𝛽 < 2, PoS ≤ 4/3

For which values of 𝛽 is any star a NE, and for which is any complete graph a NE?

• 1. 𝛽 ≥ 1, 𝛽 ≤ 1
• 3. 𝛽 ≥ 1, none
• 2. 𝛽 ≥ 2, 𝛽 ≤ 1
• 4. 𝛽 ≥ 2, none

Poll 1

?

PROOF OF THEOREM

• Part 1 is immediate from the lemma and

poll

• For 1 < 𝛽 < 2, the star is a NE, while OPT is

a complete graph

• Worst case ratio when 𝛽 → 1:

2𝑜 𝑜 − 1 − 2 𝑜 − 1 + (𝑜 − 1) 𝑜 𝑜 − 1 + 𝑜(𝑜 − 1)/2 = 4𝑜2 − 6𝑜 + 2 3𝑜2 − 3𝑜 < 4 3 ∎

EXAMPLE: NETWORK CREATION

• Theorem [Fabrikant et al. 2003]: The

price of anarcy of network creation games is 𝑃( 𝛽)

• Lemma: If 𝒕 is a Nash equilibrium that

induces a graph of diameter 𝑒, then cost(𝒕) ≤ 𝑃 𝑒 ⋅ OPT

PROOF OF LEMMA

• OPT = Ω 𝛽𝑜 + 𝑜2
• Buying a connected graph costs at least

𝑜 − 1 𝛽

• There are Ω 𝑜2 distances
• Distance costs ≤ 𝑒𝑜2 ⇒ focus on edge

costs

• There are at most 𝑜 − 1 cut edges ⇒

focus on noncut edges

PROOF OF LEMMA

• Claim: Let 𝑓 = (𝑣, 𝑤) be a noncut edge, then the

distance 𝑒(𝑣, 𝑤) with 𝑓 deleted ≤ 2𝑒

• 𝑊

𝑓 = set of nodes s.t. the shortest path from 𝑣 uses 𝑓

• Figure shows shortest path avoiding 𝑓, 𝑓′ = (𝑣′, 𝑤′)

is the edge on the path entering 𝑊

𝑓

• 𝑄

𝑣 is the shortest path from 𝑣 to 𝑣′ ⇒ 𝑄 𝑣 ≤ 𝑒

• 𝑄

𝑤 ≤ 𝑒 − 1 as 𝑄 𝑤 ∪ {𝑓} is shortest path from 𝑣 to

𝑤′ ∎

𝑤 𝑤′ 𝑣 𝑣′

𝑓 𝑓′ 𝑊

𝑓

𝑄

𝑤

𝑄

𝑣

PROOF OF LEMMA

• Claim: There are 𝑃(𝑜𝑒/𝛽) noncut

edges paid for by any vertex 𝑣

• Let 𝑓 = (𝑣, 𝑤) be an edge paid for by 𝑣
• By previous claim, deleting 𝑓 increases

distances from 𝑣 by at most 2𝑒|𝑊

𝑓|

• 𝐻 is an equilibrium ⇒ 𝛽 ≤ 2𝑒 𝑊

𝑓 ⇒

𝑊

𝑓 ≥ 𝛽/2𝑒

• 𝑜 vertices overall ⇒ can’t be more than

2𝑜𝑒/𝛽 sets 𝑊

𝑓 ∎

PROOF OF LEMMA

• 𝑃(𝑜𝑒/𝛽) noncut edges per vertex
• 𝑃(𝑜𝑒) total payment for these per

vertex

• 𝑃(𝑜2𝑒) overall ∎

PROOF OF THEOREM

• By lemma, it is enough to show that the diameter

at a NE ≤ 2 𝛽

• Suppose 𝑒 𝑣, 𝑤 ≥ 2𝑙 for some 𝑙
• By adding the edge (𝑣, 𝑤), 𝑣 pays 𝛽 and improves

distance to second half of the 𝑣 → 𝑤 shortest path by 2𝑙 − 1 + 2𝑙 − 3 + ⋯ + 1 = 𝑙2

• If

𝛽 < 𝑙2 ≤ 𝑒 𝑣, 𝑤 2

2

⇒ 𝑒 𝑣, 𝑤 > 2 𝛽 then it is beneficial to add edge — contradiction∎