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

T RUTH J USTICE A LGOS Game Theory II: Price of Anarchy Teachers: Ariel Procaccia (this time) and Alex Psomas

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 • We can make this arbitrarily bad -1,-1 -9,0 0,-9 -6,-6

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

EXAMPLE: COST SHARING • 𝑜 players in weighted directed graph 𝐻 𝑡 1 𝑡 2 1 • Player 𝑗 wants to get from 𝑡 𝑗 to 𝑢 𝑗 ; 1 strategy space is 𝑡 𝑗 → 𝑢 𝑗 paths • Each edge 𝑓 has cost 𝑑 𝑓 10 10 10 • Cost of edge is split between all players using edge 1 1 • Cost of player is sum of costs over 𝑢 1 𝑢 2 edges on path

EXAMPLE: COST SHARING • With 𝑜 players, the example on the right has a NE with social cost 𝑜 𝑡 • Optimal social cost is 1 • It follows that the price of anarchy 𝑜 1 of cost sharing games is at least 𝑜 • It is easy to see that the price of anarchy of cost sharing games is at 𝑢 most 𝑜 — why?

EXAMPLE: COST SHARING • Think of the 1 edges as cars, and the 𝑙 edge as mass transit • Bad Nash equilibrium with cost 0 0 0 𝑜 … 𝑡 1 𝑡 2 𝑡 𝑜 • Good Nash equilibrium with 𝑙 cost 𝑙 1 1 1 • Now let’s modify the example… 𝑢

EXAMPLE: COST SHARING • OPT = 𝑙 + 1 • Only equilibrium has cost 𝑙 ⋅ 𝐼(𝑜) 0 0 0 • Therefore, the price of … 𝑡 1 𝑡 2 𝑡 𝑜 𝑙 + 1 stability of cost sharing games is at least Ω(log 𝑜) 𝑙 𝑙 𝑙 • We will show that the price 1 2 𝑜 of stability is Θ(log 𝑜) 𝑢

POTENTIAL GAMES • A game is an exact potential game if there 𝑜 exists a function Φ: ς 𝑗=1 𝑇 𝑗 → ℝ such that 𝑜 for all 𝑗 ∈ 𝑂 , for all 𝒕 ∈ ς 𝑗=1 𝑇 𝑗 , and for all ′ ∈ 𝑇 𝑗 , 𝑡 𝑗 ′ , 𝒕 −𝑗 − cost 𝑗 𝒕 = Φ 𝑡 𝑗 ′ , 𝒕 −𝑗 − Φ(𝒕) cost 𝑗 𝑡 𝑗 • 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 𝑜 𝑓 𝒕 be the number of players using 𝑓 under 𝒕 ◦ Define the potential function 𝑜 𝑓 (𝒕) 𝑑 𝑓 Φ 𝒕 = ෍ ෍ 𝑙 𝑓 𝑙=1 𝑑 𝑓 ◦ If player changes paths, pays 𝑜 𝑓 𝒕 +1 for each new 𝑑 𝑓 edge, gets 𝑜 𝑓 𝒕 for each old edge, so Δcost 𝑗 = 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 • cost 𝑤 𝒕 = 𝛽𝑜 𝑤 𝒕 + σ 𝑣 𝑒(𝑣, 𝑤) , where 𝑜 𝑤 = # edges bought by 𝑤 , 𝑒 is shortest path in # edges • cost 𝒕 = σ 𝑣≠𝑤 𝑒 𝑣, 𝑤 + 𝛽|𝐹|

EXAMPLE: NETWORK FORMATION NE with 𝛽 = 3 Suboptimal Optimal

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

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 = 4𝑜 2 − 6𝑜 + 2 < 4 ∎ 3𝑜 2 − 3𝑜 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 + 𝑜 2 ◦ Buying a connected graph costs at least 𝑜 − 1 𝛽 ◦ There are Ω 𝑜 2 distances • Distance costs ≤ 𝑒𝑜 2 ⇒ 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 𝑣 is the shortest path from 𝑣 to 𝑣 ′ ⇒ 𝑄 ◦ 𝑄 𝑣 ≤ 𝑒 𝑄 𝑤 ≤ 𝑒 − 1 as 𝑄 𝑤 ∪ {𝑓} is shortest path from 𝑣 to ◦ 𝑤 ′ ∎ 𝑊 𝑓 𝑤 𝑤′ 𝑄 𝑤 𝑓 𝑓′ 𝑄 𝑣 𝑣 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(𝑜 2 𝑒) 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 = 𝑙 2 • If 2 𝑒 𝑣, 𝑤 𝛽 < 𝑙 2 ≤ ⇒ 𝑒 𝑣, 𝑤 > 2 𝛽 2 then it is beneficial to add edge — contradiction ∎