Minimizing Efficiency Loss in Mechanism and Protocol Design

1. Minimizing Efficiency Loss in Mechanism and Protocol Design Tim Roughgarden (Stanford) includes joint work with: Shuchi Chawla (Wisconsin), Ho-Lin Chen (Stanford), Aranyak Mehta (IBM Almaden), Mukund Sundararajan (Stanford), Gregory Valiant (UC Berkeley)

2. Reasons for Efficiency Loss Non-cooperative equilibria: • no control of underlying game, players' actions Auction design: • players have private "valuations" for goods • can use VCG mechanism to maximize efficiency • but suboptimality inevitable if goal includes: • poly-time + hard allocation (combinatorial auctions) • different (e.g. maxmin) objective [Nisan/Ronen 99] • revenue constraints

3. Quantifying Efficiency Loss Early applications: • price of anarchy [Kousoupias/Papadimitriou 99], etc. • approximation mechanisms • both poly-time combinatorial auctions and maxmin objectives This talk: mechanism/protocol design to minimize worst-case efficiency loss. • mechanism design s.t. revenue constraint • protocol design to minimize price of anarchy • full information but implementation constraints

4. Cost-Sharing Problems • general case: set U of players, cost function C defined on U (incurred by mechanism) • special case: fixed-tree-multicast rooted tree T with fixed edge costs c; C(S) = cost of subtree spanning S • [Feigenbaum/Papadimitriou/Shenker 00] • player i has valuation vi for winning Terminology: • surplus of S = v(S) - C(S) [where v(S) = Σi vi]

5. Cost-Sharing Mechanisms • cost-sharing mechanism: collect bids, pick winning set S, determines prices for winners Natural goals: • truthful + "individually rational" • economically efficient (maximizes surplus) • "budget-balance" (revenue covers cost incurred) • VCG fails miserably here • fact: 3 goals mutually incompatible [Green/Laffont, Roberts 70s], [Feigenbaum/Krishnamurthy/Sami/Shenker 03]

6. Shapley Mechanism for Multicast e3 • collects bids (bi for each i) • initialize S = all players • share each edge equally among its users • if bi pi for all i, done. • else drop a player i with bi< pi and iterate e2 e1 Price = c(e1) + c(e2)/3 + c(e3)/4

7. Moulin Mechanisms [Moulin 99] e3 Given: cost fn C(S) on subsets S of U Cost-Sharing Method: for every set S, defines a cost share χ(i,S) for every i in S (“suggested prices”) Defn: χ is ß-budget-balanced (ß-BB) if prices charged within ß of C(S) Moulin mechanism: simulate ascending auction using χ to compute prices at each iteration. e2 e1 Price = c(e1) + c(e2)/3 + c(e3 )/4

8. Moulin Mechanisms: Good News Fact:[Moulin 99] if cost-sharing method χ is monotone (price for each player only increases), then the Moulin mechanism is truthful. • utility = vi- pi if i wins, 0 otherwise • reason: same as a classical ascending auction Also: • groupstrategyproof (form of collusion-resistance) • prices charged cover cost incurred (up to ß factor)

9. Moulin Mechanisms: Bad News Claim: Moulin mechanisms (e.g., the Shapley mechanism) need not maximize surplus. k players with valuations: 1,1/2, 1/3, … , 1/k e1 = 1 + e

10. Moulin Mechanisms: Bad News Claim: Moulin mechanisms (e.g., the Shapley mechanism) need not maximize surplus. • opt surplus (ln k) - 1, Shapley surplus = 0 k players with valuations: 1,1/2, 1/3, … , 1/k e1 = 1 + e

11. Moulin Mechanisms: Bad News Claim: Moulin mechanisms (e.g., the Shapley mechanism) need not maximize surplus. • opt surplus (ln k) - 1, Shapley surplus = 0 Negative result[GL,R,FKSS]: no truthful mechanism gets non-trivial approximation of BB + surplus. k players with valuations: 1,1/2, 1/3, … , 1/k e1 = 1 + e

12. Measuring Surplus Loss Goal: minimize worst-case surplus loss. • surplus of S: v(S) - C(S) Defn: social cost of S: π(S) = C(S) + v(U\S) • U = set of all players • note: social cost = -surplus + v(U) Bad example: opt social cost 1, Shapley social cost  ln k e1 = 1 + e 1,1/2, 1/3, … , 1/k

13. Measuring Surplus Loss Goal: minimize worst-case surplus loss. • surplus of S: v(S) - C(S) Defn: social cost of S: π(S) = C(S) + v(U\S) • U = set of all players • note: social cost = -surplus + v(U) Bad example: opt social cost 1, Shapley social cost  ln k Defn: a mechanism is α-approximate if it is an α-approximation algorithm w.r.t. the social cost objective (in the usual sense). e1 = 1 + e 1,1/2, 1/3, … , 1/k

14. Goal + Main Result High-level goal: subject to reasonable BB, design mechanism with smallest approximation factor. • note: requires both upper + lower bound results • precisely quantifies inevitable surplus loss

15. Goal + Main Result High-level goal: subject to reasonable BB, design mechanism with smallest approximation factor. • note: requires both upper + lower bound results • precisely quantifies inevitable surplus loss Main result: complete soln for Moulin mechanisms. • [Roughgarden/Sundararajan STOC 06], [Chawla+R+S WINE 06], [R+S IPCO 07]

16. Goal + Main Result High-level goal: subject to reasonable BB, design mechanism with smallest approximation factor. • note: requires both upper + lower bound results • precisely quantifies inevitable surplus loss Main result: complete soln for Moulin mechanisms. • [Roughgarden/Sundararajan STOC 06], [Chawla+R+S WINE 06], [R+S IPCO 07] Ex: multicast: Shapley is optimal Moulin mechanism • approximation factor of social cost = Hk • extends to all submodular cost functions

17. More Examples Examples: • uncapacitated facility location: the [Pal-Tardos 03] mechanism = optimal Moulin mechanism • optimal approximation = Θ(log k) • Steiner tree: the [Jain-Vazirani 01] mechanism = optimal Moulin mechanism • optimal approximation factor of social cost = Θ(log2 k) • also extends to Steiner forest mechanism of [Konemann/Leonardi/Schaefer SODA 05] and rent-or buy mechanism of [Gupta/Srinivasan/Tardos 03]

18. Proof Techniques Part I: (problem-independent) • identify parameter of a monotone cost-sharing method that controls approximation factor of Moulin mechanism [upper and lower bounds] • reduces property of mechanism to property of method Part II: (problem-dependent) • prove upper bound on parameter for favorite mechanisms, lower bound for all mechanisms • has flavor of analysis of online algorithms

19. A Natural Lower Bound • consider a cost-sharing method χ for C + corresponding Moulin mechanism M • order the players of U = {1,2,...,k} • let xi = χ(i,{1,2,...,i}) • set vi = xi - e • M outputs Ø, social cost  Σi xi ; OPT is ≤ C(U) Σi χ(i,{1,2,...,i})/C(U) lower bounds approximation factor e1 = 1 + e 1,1/2, 1/3, … , 1/k

20. A Natural Lower Bound • consider a cost-sharing method χ for C + corresponding Moulin mechanism M • order the players of U = {1,2,...,k} • let xi = χ(i,{1,2,...,i}) • set vi = xi - e • M outputs Ø, social cost  Σi xi ; OPT is ≤ C(U) Σi χ(i,{1,2,...,i})/C(U) lower bounds approximation factor Defn: the summability α of χ for C is the largest lower bound arising in this way. e1 = 1 + e 1,1/2, 1/3, … , 1/k

21. A Key Theorem Summary: a Moulin mechanism based on an α-summable cost-sharing method is no better than α-approximate.

22. A Key Theorem Summary: a Moulin mechanism based on an α-summable cost-sharing method is no better than α-approximate. Theorem[Roughgarden/Sundararajan STOC 06]: a Moulin mechanism based on an α-summable, ß-BB cost-sharing method is (α+ß)-approximate. Point: for every O(1)-BB method χ, the parameter α completely characterizes the approximation factor of the corresponding mechanism.

23. Beyond Moulin Mechanisms Question: why obsessed with Moulin mechanisms? • only general technique to achieve truthful + BB • strong lower bounds for approximation for some problems [Immorlica/Mahdian/Mirrokni SODA 05] • non-trivial to design (e.g., for UFL)

24. Beyond Moulin Mechanisms Question: why obsessed with Moulin mechanisms? • only general technique to achieve truthful + BB • strong lower bounds for approximation for some problems [Immorlica/Mahdian/Mirrokni SODA 05] • non-trivial to design (e.g., for UFL) Acyclic Mechanisms[Mehta/Roughgarden/Sundararajan EC 07]: generalizes Moulin mechanisms. • idea: order offers within iteration of ascending auction • most "off-the-shelf" primal-dual algorithms work as is • exponentially better BB + efficiency for e.g. Set Cover

25. Shapley Network Design Games Given: G= (V,E), fixed costs ce • k players = vertex pairs (si,ti) • each picks an si-ti path Shapley cost sharing: • cost of each edge of formed network split equally among users • [Anshelevich et al FOCS 04] • full-information noncooperative game

26. t t 1 1 1 1 1 2 3 k k-1 . . . 1+ k 1+ = = 0 0 0 0 0 s Inefficiency under Shapley Recall:with Shapley cost sharing, • POA = k, even in undirected graphs • POS = Hkin directed graphs • (unknown in undirected graphs)

27. t t 1 1 1 1 1 2 3 k k-1 . . . 1+ k 1+ = = 0 0 0 0 0 s Inefficiency under Shapley Recall:with Shapley cost sharing, • POA = k, even in undirected graphs • POS = Hkin directed graphs • (unknown in undirected graphs) Question #1: can we do better? Question #2: subject to what?

28. In Defense of Shapley Essential properties: (non-negotiable) • "budget-balanced" (total cost shares = cost) • "separable" (cost shares defined edge-by-edge) • pure-strategy Nash equilibria exist Bonus good properties: (negotiable) • "uniform" (same definition for all networks) • "fair" (characterizes Shapley)

29. Key Question The Problem: design edge cost-sharing methods to minimize worst-case POA and/or POS. • directed vs. undirected • uniform vs. non-uniform • single-sink vs. terminal pairs • [Chen/Roughgarden/Valiant 07] Related work: coordination mechanisms [Christodoulou/Koutsoupias/Nanavati ICALP 04], [Immorlica/Li/Mirrokni/Schulz 05], [Azar et al 07] • resource allocation [Johari/Tsitsiklis 07]

30. Directed Graphs Negative result: worst-case POA = k for every cost-sharing method, even non-uniform.

31. Directed Graphs Negative result: worst-case POA = k for every cost-sharing method, even non-uniform. Theorem: Shapley is the optimal uniform cost-sharing method! For every method, either: (1) there is a network game s.t. POS Hk OR (2) there is a network game with no Nash eq.

32. Directed Graphs Negative result: worst-case POA = k for every cost-sharing method, even non-uniform. Theorem: Shapley is the optimal uniform cost-sharing method! For every method, either: (1) there is a network game s.t. POS Hk OR (2) there is a network game with no Nash eq. • Shapley can be justified on efficiency grounds, not just usual fairness/simplicity reasons • open: what's up with non-uniform methods?

33. Undirected Graphs: Uniform Theorem: in undirected graphs, can reduce the worst-case POA to polylogarithmic! • simple uniform priority-based scheme • POA = O(log k) in with single sink, O(log2 k) for pairs (follows from [IW 91], [AA96])

34. Undirected Graphs: Uniform Theorem: in undirected graphs, can reduce the worst-case POA to polylogarithmic! • simple uniform priority-based scheme • POA = O(log k) in with single sink, O(log2 k) for pairs (follows from [IW 91], [AA96]) Theorem: For every unform cost-sharing method, worst-case POA = Ω(log k). [even single-sink] • follows from complete characterization of uniform cost-sharing methods that always admit PNE

35. Undirected: Non-Uniform Theorem: Can reduce POA to 2 in single-sink networks via non-uniform method. • idea: use Prim MST to define priority scheme • easy: matching lower bound Theorem: For every non-uniform method, worst-case POA is general networks is Ω(log k). • extremal graph construction • lower bounds for "oblivious network design"

36. Open Questions Cost-Sharing Mechanism Design: • lower bounds for non-Moulin mechanisms • more applications of acyclic mechanisms • profit-maximization Optimal Protocol Design: • non-uniform methods in directed graphs • lower bounds for scheduling mechanisms • new applications (selfish routing, fair queuing)