1 Issues on the border of economics and computation נושאים בגבול כלכלה וחישוב Congestion Games, Potential Games and Price of Anarchy Liad Blumrosen ©

Course Outline 1 st part: equilibrium analysis of games, inefficiency of equilibria, dynamics that lead to equilibria. 2 nd part: market design, electronic commerce, algorithmic mechanism design. Book: Algorithmic Game Theory –By Nisan, Roughgarden, Tardos and Vazirani. –Available online:

Today’s Outline Congestion games. –Equilibrium. –Convergence to equilibrium. Potential games. Inefficiency of equilibria: –Price of anarchy –Price of stability –Example: congestion games.

Reminder: Nash Equilibrium Consider a game: –S i is the set of (pure) strategies for player i S = S 1 x S 2 x … x S n –s = (s 1,s 2,…,s n )  S is a vector of strategies –U i : S  R is the payoff function for player i. Notation: given a strategy vector s, let s -i = (s 1,…,s i-1,s i,…,s n ) –The vector i where the i’th item is omitted. s is a Nash equilibrium if for every i, u i )s i,s -i ) ≥ u i (s i ’,s -i ) for every s i ’  S i

Externalities A standard assumption in classic economics assume no externalities –You only care about what you consume. In reality, people care about the consumption of others:

Congestion games A special class of games that model externalities: u i (consuming A) = f( #agents consuming A ) –“Congestion games” (aka as “network externalities). Can model both negative and positive externalities. –Despite the name that hints for negative externalities. Examples: –Congestion on roads, in restaurants. (negative) –Fax, social network, fashion, standards (file formats, etc.). (positive)

Congestion games Definition: congestion games (משחקי גודש) –A set of players 1,…,n –A set of resources M = {1,…,m} –S i is the set of (pure) strategies of player i i.e., s i  S i is a subset of M. –Cost for the players that use resource j  M depends on the number of players using j : c j (n j ) –For s=(s 1,…,s n ), let n j (s) = the number of players using resource j –The total cost c i for player i:

Congestion games Note: –it only matters how many players use resource j. Not their identities. –Cost structure is symmetric, asymmetry is via the S i ’s. –Externalities may be positive, negative or both. –Payoffs: today, we will mostly talk about costs, and players aim to minimize their cost. As opposed to maximizing utility: c() = -u() The models are game-theoretically equivalent. There are differences when we talk about approximation, etc.

Congestion games Why are we interested in congestion games? –Model some interesting real problems. –Have nice equilibrium properties. –Have nice dynamics properties. –Good example for price-of-anarchy and price-of- stability.

Example 1: network cong. game Resources: the edges. Pure strategies: subsets of edges. Travel time on each edges: f(congestion) Player 1 wants to travel A  D –S 1 ={ {AB,BD}, {AC,CD}, {AC,CB,BD} } Player 2 wants to travel A  B –S 1 ={ {AC,CB}, {AB} } A B C D E c(n)=1 c(n)=n/2 c(n)=n 2 c(n)=10 c(n)=4n Consider the strategy profile: s 1 = {AC,CD} s 2 = {AC,CB} c 1 (s)=4+10 c 2 (s)=4+1 c(n)=n

Equilibria in congestion game Structure of Nash equilibria in congestion games: Theorem: In every congestion game there exists a pure Nash equilibrium. –(At least one…) First observed by Rosenthal (1973). “A class of games possessing pure-strategies Nash equilibria”

Pure eq. proof (slide 1 of 2) –Assume that player i deviates from s i to t i : Recall that s i and t i are subsets of resources Let ΔΦ be: Let Δc be:  ΔΦ= Δc. Proof: Consider the following function (potential function): Economic meaning: unclear….

Pure eq. proof (slide 2 of 2) –Now, consider a pure-strategy profile s*  argmin s Φ(s) –From the previous slide, we can conclude that s* is a Nash equilibrium –Why? Proof:

Equilibria in congestion game The proof leads to another conclusion: –Start with some arbitrary strategic behavior of the players; –at each step some player improves its payoff (“better- response” dynamic);  a pure equilibrium will be reached. Why? –Each improvement strictly improves potential. –there is a finite number of strategy profiles. –Potential is increasing  no strategy profile is repeated.  Better response dynamic converges to a pure- Nash equilibrium in any congestion game.

Potential games We saw that congestion games: –Always have a pure Nash equilibrium –Best-response dynamics leads to such an equilibrium. But the proof seems to be more general, it works whenever we have such a potential function. We now define such games: potential games.

Potential games Definition: (exact) potential game A game is an exact potential game if there is a function Φ:S  R such that Definition: (ordinal) potential game The same, but with instead of (*) (*)

Example: prisoners dilemma Consider the prisoners dilemma: CooperateDefect Cooperate -1, -1-5, 0 Defect 0, -5-3,-3 Let’s present it via costs instead of utilities…

Example: prisoners dilemma Consider the prisoners dilemma: CooperateDefect Cooperate 1, 15, 05, 0 Defect 0, 53,33,3 Is this an exact potential game? Goal: assign a number to each entry, such that: Δ potential= Δ utilities

Example: prisoners dilemma Consider the prisoners dilemma: CooperateDefect Cooperate 1, 15, 05, 0 Defect 0, 53,33,3 We can build a graph: –V = strategy profiles –E = moving from one vertex to another is a best response The game is a potential game iff this graph has no cycles. –How can we find the (ordinal) potential function? –No cycles: finite improvement paths.

Example: prisoners dilemma –Cycles in the local improvement graph  no potential function. If Φ exists: Φ(TT) < Φ(HT) < Φ(HH) < Φ(TH) < Φ(TT) -1,11,-1 -1,1 TailHeads Tail Heads

Eq. in potential games Theorem: every (finite) potential game has a pure- strategy equilibrium. Theorem: in every (finite) potential game best- response dynamic converges to an equilibrium. Proof: As before.

Potential games and cong. games What other games have this nice property other than congestion games? Answer: none. Theorem (Monderer & Shapley): every exact potential game is a congestion game. (we already saw the converse)

Outline Congestion games. –Equilibrium. –Convergence to equilibrium. Potential games. Inefficiency of equilibria: –Price of anarchy –Price of stability –Example: congestion games.

Quality of equilibria We saw: congestion games admit pure Nash equilibria Are these equilibria “good” for the society? Approximately good? We will need to: –specify some objective function. –Define “approximation”. –Deal with multiplicity of equilibria.

Price of anarchy/stability Price of anarchy: Price of stability: Cost of worst Nash eq. Optimal cost Cost of best Nash eq. Optimal cost When talking about cost minimization, POA and POS ≥1 Concepts are not restricted to pure equilibria (similar concepts available for other types of equilibria)

Examples Optimization goal: social welfare (=sum of payoffs) Optimal cost: 1+1=2 Cost of worst NE = cost of best NE = 6 –One Nash equilibrium. POA = POS = 3 CooperateDefect Cooperate 1, 15, 05, 0 Defect 0, 53,33,3

Examples Optimization goal: social welfare Two pure equilibria: (Ballet, Ballet), (Football, Football) Optimal cost: 2+1=3 Cost of worst NE 1+4 = 5 –POA=5/3 Cost of best NE 1+2 = 3 –POS=1 BalletFootball Ballet 2, 15, 55, 5 Football 5, 51,41,4

Approximation measurements Several approximation concepts in the design of algorithms: –Approximation ratio (approximation algorithms): what is the price of limited computational resources. –Competitive ratio (online algorithms): what is the price for not knowing the future. –Price of anarchy: the price of lack of coordination –Price of stability: price of selfish decision making with some coordination.

Price of stability in cong. games Meaning: in such games there exists pure Nash equilibria with cost which is at most double the optimal cost. Also known: POA in linear congestion games ≤ 2.5 Theorem: in congestion games with linear cost function, POS ≤ 2 –Objective: cost minimization. –Linear cost: c j (n j )=a j n j +b j for some a j,b j ≥0

Price of stability – proof (1 of 2) Proof: let Φ = potential function from previous slides. Consider a strategy profile s  S. We first compare: Φ(s) and c(S) = Σ i  N c i (s) Φ(s)≤ c(s) ≤ 2Φ(s)

Price of stability – proof (2 of 2) Proof: for every strategy profile s, we have Let s* = argmin s Φ(s). As argued before, s* is a pure Nash equilibrium. Let s opt be the optimal solution, c(s opt ) = min s c(s) Then, c(s*)  POS ≤ c(s*)/c(s opt ) ≤ 2 Φ(s)≤ c(s) ≤ 2Φ(s) ≤ 2Φ(s*)≤ 2Φ(s opt )≤ 2c(s opt )

Summary We discussed a class of games: congestion games. Model environments with externalities. Equivalent to the class of potential games. Admits a pure Nash equilibrium Best-response dynamic convergence to such a Nash equilibrium. We discussed the POA and POS in congestion games.