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Potential Functions and the Inefficiency of Equilibria

Potential Functions and the Inefficiency of Equilibria. Tim Roughgarden Stanford University. Pigou's Example. Example: one unit of traffic wants to go from s to t Question: what will selfish network users do? assume everyone wants smallest-possible cost [Pigou 1920].

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Potential Functions and the Inefficiency of Equilibria

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  1. Potential Functions and the Inefficiency of Equilibria Tim Roughgarden Stanford University

  2. Pigou's Example Example:one unit of traffic wants to go from s to t Question:what will selfish network users do? • assume everyone wants smallest-possible cost • [Pigou 1920] cost depends on congestion c(x)=x s t c(x)=1 no congestion effects

  3. Motivating Example Claim:all traffic will take the top link. Reason: • Є > 0  traffic on bottom is envious • Є = 0  equilibrium • all traffic incurs one unit of cost Flow = 1-Є c(x)=x s t c(x)=1 this flow is envious! Flow = Є

  4. Can We Do Better? Consider instead:traffic split equally Improvement: • half of traffic has cost 1 (same as before) • half of traffic has cost ½ (much improved!) Flow = ½ c(x)=x s t c(x)=1 Flow = ½

  5. ½ ½ x 1 s t ½ ½ x 1 Braess’s Paradox Initial Network: Cost = 1.5

  6. ½ ½ x 1 s t ½ ½ x 1 Braess’s Paradox Initial Network: Augmented Network: ½ ½ x 1 0 s t ½ ½ x 1 Cost = 1.5 Now what?

  7. ½ ½ x 1 s t ½ ½ x 1 Braess’s Paradox Initial Network: Augmented Network: x 1 0 s t x 1 Cost = 1.5 Cost = 2

  8. ½ ½ x 1 s t ½ ½ x 1 Braess’s Paradox Initial Network: Augmented Network: All traffic incurs more cost![Braess 68] • also has physical analogs [Cohen/Horowitz 91] x 1 0 s t x 1 Cost = 1.5 Cost = 2

  9. High-Level Overview Motivation: equilibria of noncooperative network games typically inefficient • e.g., Pigou's example + Braess's Paradox • don't optimize natural objective functions Price of anarchy: quantify inefficiency w.r.t some objective function Our goal: when is the price of anarchy small? • when does competition approximate cooperation? • benefit of centralized control is small

  10. Selfish Routing Games • directed graph G = (V,E) • source-destination pairs (s1,t1), …, (sk,tk) • ri = amount of traffic going from si to ti • for each edge e, a cost function ce(•) • assumed continuous and nondecreasing Examples: (r,k=1) c(x)=x ½ c(x)=1 c(x)=x ½ c(x)=0 s1 t1 s1 t1 c(x)=1 ½ ½ c(x)=x c(x)=1

  11. s t Outcomes = Network Flows Possible outcomes of a selfish routing game: • fP = amount of traffic choosing si-ti path P • outcomes of game flow vectors f • flow vector: nonnegative and total flow fP on si-ti paths equals traffic rate ri (for all i)

  12. s t Outcomes = Network Flows Possible outcomes of a selfish routing game: • fP = amount of traffic choosing si-ti path P • outcomes of game flow vectors f • flow vector: nonnegative and total flow fP on si-ti paths equals traffic rate ri (for all i) Question: What are the equilibria (natural selfish outcomes) of this game?

  13. Nash Flows Def: [Wardrop 52]A flow is at Nash equilibrium (or is a Nash flow) if no one can switch to a path of smaller cost. I.e., all flow is routed on min-cost paths. [given current edge congestion] Examples: 1 ½ x x s t s t 1 1 ½ ½ x 1 x 1 0 0 1 s t s t x x 1 1 ½

  14. s t Our Objective Function Definition of social cost: total cost C(f) incurred by the traffic in a flow f. Formally: ifcP(f) = sum of costs of edges of P (w.r.t. flow f), then: C(f) = P fP •cP(f)

  15. s t Our Objective Function Definition of social cost: total cost C(f) incurred by the traffic in a flow f. Formally: ifcP(f) = sum of costs of edges of P (w.r.t. flow f), then: C(f) = P fP •cP(f) Example: x ½ s t Cost = ½•½ +½•1 = ¾ ½ 1

  16. The Price of Anarchy price of anarchy of a game Defn: • definition from [Koutsoupias/Papadimitriou 99] obj fn value of selfish outcome = optimal obj fn value Example: POA = 4/3 in Pigou's example 1 ½ x x s t s t 1 1 ½ Cost = 3/4 Cost = 1

  17. xd 1 1-Є s t 1 0 Є A Nonlinear Pigou Network Bad Example: (d large) equilibrium has cost 1, min cost0

  18. xd 1 1-Є s t 1 0 Є A Nonlinear Pigou Network Bad Example: (d large) equilibrium has cost 1, min cost0 price of anarchy unbounded as d -> infinity Goal: weakest-possible conditions under which P.O.A. is small.

  19. x s t 1 When Is the Price of Anarchy Bounded? Examples so far: Hope: imposing additional structure on the cost functions helps • worry: bad things happen in larger networks xd x 1 s t 0 s t 1 x 1

  20. x s t 1 Polynomial Cost Functions Def:linear cost fn is of form ce(x)=aex+be Theorem:[Roughgarden/Tardos 00] for every network with linear cost functions: ≤4/3 × cost of Nash flow cost of opt flow

  21. x s t 1 Polynomial Cost Functions Def:linear cost fn is of form ce(x)=aex+be Theorem:[Roughgarden/Tardos 00] for every network with linear cost functions: ≤4/3 × Bounded-deg polys: (w/nonneg coeffs) replace 4/3 by Θ(d/log d) cost of Nash flow cost of opt flow xd tight example s t 1

  22. xd tight example s t 1 A General Theorem Thm:[Roughgarden 02], [Correa/Schulz/Stier Moses 03]fix any set of cost fns. Then, a Pigou-like example 2 nodes, 2 links, 1 link w/constant cost fn) achieves worst POA

  23. xd tight example s t 1 Interpretation Bad news: inefficiency of selfish routing grows as cost functions become "more nonlinear". • think of "nonlinear" as "heavily congested" • recall nonlinear Pigou's example Good news: inefficiency does not grow with network size or # of source-destination pairs. • in lightly loaded networks, no matter how large, selfish routing is nearly optimal

  24. Benefit of Overprovisioning Suppose: network is overprovisioned by β > 0 (β fraction of each edge unused). Then: Price of anarchy is at most ½(1+1/√β). • arbitrary network size/topology, traffic matrix Moral: Even modest (10%) over-provisioning sufficient for near-optimal routing.

  25. Potential Functions • potential games: equilibria are actually optima of a related optimization problem • has immediate consequences for existence, uniqueness, and inefficiency of equilibria • see [Beckmann/McGuire/Winsten 56], [Rosenthal 73], [Monderer/Shapley 96], for original references • see [Roughgarden ICM 06] for survey

  26. The Potential Function Key fact:[BMV 56] Nash flows minimize “potential function”e ∫fce(x)dx (over all flows). ce(fe) 0 e 0 fe 0

  27. The Potential Function Key fact:[BMV 56] Nash flows minimize “potential function”e ∫fce(x)dx (over all flows). Lemma 1: locally optimal solutions are precisely the Nash flows (derivative test). Lemma 2: all locally optimal solutions are also globally optimal (convexity). Corollary: Nash flows exist, are unique. ce(fe) 0 e 0 fe 0

  28. Consequences for the Price of Anarchy Example: linear cost functions. Compare cost + potential function: C(f) = e fe •ce(fe) = e [ae fe + be fe] PF(f) = e ∫fce(x)dx = e [(ae fe)/2 + be fe] 2 2 e 0

  29. Consequences for the Price of Anarchy Example: linear cost functions. Compare cost + potential function: C(f) = e fe •ce(fe) = e [ae fe + be fe] PF(f) = e ∫fce(x)dx = e [(ae fe)/2 + be fe] • cost, potential fn differ by factor of ≤ 2 • gives upper bound of 2 on price on anarchy • C(f) ≤ 2×PF(f) ≤ 2×PF(f*) ≤ 2×C(f*) 2 2 e 0

  30. Better Bounds? Similarly: proves bound of d+1 for degree-d polynomials (w/nonnegative coefficients). • not tight, but qualitatively accurate • e.g., price of anarchy goes to infinity with degree bound, but only linearly • to get tight bounds, need "variational inequalities" • see my ICM survey for details

  31. Variational Inequality Claim: • if f is a Nash flow and f* is feasible, then e fe •ce(fe) ≤e f*•ce(fe) • proof: use that Nash flow routes flow on shortest paths (w.r.t. costs ce(fe)) e

  32. Pigou Bound Recall goal: want to show Pigou-like examples are always worst cases. Pigou bound: given set of cost functions (e.g., degree-d polys), largest POA in a network: • two nodes, two links • one function in given set • one constant function • constant = cost of fully congested top edge xd s t 1

  33. Pigou Bound (Formally) Let S = a set of cost functions. • e.g., polynomials with degree at most d, nonnegative coefficients Definition: the Pigou bound α(S) for S is: max • max is over all choices of cost fns c in S, traffic rate r  0, flow y  0 r • c(r) xd s t y • c(y) + (r-y) • c(r) 1

  34. Pigou Bound (Example) Let S = { c : c(x) = ax +b } [linear functions] Recall: the Pigou bound α(S) for S is: max • max is over all choices of cost fns c in S, traffic rate r  0, flow y  0 • choose c(x) = x; r = 1; y = 1/2  get 4/3 • calculus: α(S) = 4/3 [d/ln d for deg-d polynomials] r • c(r) x s t y • c(y) + (r-y) • c(r) 1

  35. Main Theorem (Formally) Theorem:[Roughgarden 02, Correa/Schulz/Stier Moses 03]: For every set S, for every selfish routing network G with cost functions in C, the POA in G is at most α(S). • POA always maximized by Pigou-like examples That is, if f and f* are Nash + optimal flows in G, then C(f)/C(f*) ≤ α(S). • example: POA ≤ 4/3 if G has affine cost fns

  36. Proof of General Thm Let f and f* are Nash + optimal flows in G.

  37. Proof of General Thm Let f and f* are Nash + optimal flows in G. Step 1: for each e, invoke Pigou bound with c = ce, y = f*, r = fe: α(S)  fe •ce(fe)/[f*•ce(f*) + (fe -f*)•ce(fe)] e e e e

  38. Proof of General Thm Let f and f* are Nash + optimal flows in G. Step 1: for each e, invoke Pigou bound with c = ce, y = f*, r = fe: α(S)  fe •ce(fe)/[f*•ce(f*) + (fe -f*)•ce(fe)] Step 2: rearrange and sum over e: C(f*) = e f*•ce(f*) e e e e e e

  39. Proof of General Thm Let f and f* are Nash + optimal flows in G. Step 1: for each e, invoke Pigou bound with c = ce, y = f*, r = fe: α(S)  fe •ce(fe)/[f*•ce(f*) + (fe -f*)•ce(fe)] Step 2: rearrange and sum over e: C(f*) = e f*•ce(f*)  [e fe •ce(fe)]/α(S) + [e (f* - fe)•ce(fe)] e e e e e e e

  40. Proof of General Thm Let f and f* are Nash + optimal flows in G. Step 1: for each e, invoke Pigou bound with c = ce, y = f*, r = fe: α(S)  fe •ce(fe)/[f*•ce(f*) + (fe -f*)•ce(fe)] Step 2: rearrange and sum over e: C(f*) = e f*•ce(f*)  [e fe •ce(fe)]/α(S) + [e (f* - fe)•ce(fe)] Step 3: apply VI e e e e e e e  0

  41. Proof of General Thm Let f and f* are Nash + optimal flows in G. Step 1: for each e, invoke Pigou bound with c = ce, y = f*, r = fe: α(S)  fe •ce(fe)/[f*•ce(f*) + (fe -f*)•ce(fe)] Step 2: rearrange and sum over e: C(f*) = e f*•ce(f*)  [e fe •ce(fe)]/α(S) Step 3: apply VI, done! e e e e e e =C(f)

  42. Recap • selfish routing: simple, basic routing game • inefficient equilibria: Pigou + Braess examples • price of anarchy: ratio of objective fn values of selfish + optimal outcomes • potential functions: equilibria actually solving a related optimization problem • immediate consequence for existence, uniqueness, and inefficiency of equilibria

  43. Recap • variational inequality: inequality based on "first-order condition" satisfied by equilibria • Pigou bound: given a set of cost functions, largest POA in a Pigou-like example • main result: for every set of cost fns, Pigou bound is tight (all multicommodity networks) • POA depends only on complexity of cost functions, not on complexity of network structure

  44. Outline Part I: The Price of Anarchy in Selfish Routing Games Part II: The Price of Stability in Network Connectivity Games

  45. Selfish Network Design Given: G= (V,E), fixed costs ce for all e є E, k vertex pairs (si,ti) Each player wants to build a network in which its nodes are connected. Player strategy: select a path connecting sito ti. • [Anshelevich et al 04]

  46. Shapley Cost Sharing How should multiple players on a single edge split costs? Natural choice is fair sharing, or Shapley cost sharing: Players using e pay for it evenly: ci(P) = Σ ce/ke Each player tries to minimize its cost. e є P

  47. Comparison to Selfish Routing Note:like selfish routing, except: • finite number of outcomes • in selfish routing, outcomes = fractional flows • positive (not negative) externalities • cost function (per player) = ce/ke Objective:C = Σi ci(Pi) = Σ ce • where S = union of Pi's e є S

  48. What's the POA? Example: t1, t2, … tk t 1+ k s s1, s2, … sk

  49. What's the POA? Example: t1, t2, … tk t t 1+ 1+ k k s s s1, s2, … sk OPT (also Nash eq)

  50. What's the POA? Example: t1, t2, … tk t t t 1+ 1+ 1+ k k k s s s s1, s2, … sk OPT (also Nash eq) another Nash eq

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