Non-price Equilibria in Markets of Discrete goods

1. Non-price Equilibria in Markets of Discrete goods Avinatan Hassidim, Haim Kaplan, Yishay Mansour, Noam Nisan Noam Nisan

2. Market Equilibrium • xij - fraction of good j that player i gets • pj - price of good j • Each player gets his “demand” • Bundle that he prefers most under current prices • Market clears • demand=supply for each good pj xij Buyers / sellers goods Noam Nisan

3. Main Dogma of Economics • Market equilibria exist • (…, Arrow-Debreu, …): theorem if “convexity” • A Market equilibrium gives an efficient allocation • “First welfare theorem” • Convexity is a big assumption • Does not hold for indivisible goods • What happens without it? Noam Nisan

4. Simple Market Model for this talk • m heterogeneous indivisible items to be allocated among n bidders. • Each bidder i has a valuation vi, where vi(S) is his real value for the set S of items. • Walrasian equilibrium: prices + allocation $4.5 $4.5 $4.5 Noam Nisan

5. Lack of equilibrium AND bidder OR bidder • AND Wins?  one of the prices ≤ 4  OR wants it • OR Wins an item?  other item has price 0  OR prefers other item • So what “happens”? Noam Nisan

6. The Market as a Game • Every player i makes an offer bij for each item j • Complete information • Highest bidder on each item wins it and pays his offer • Ties are handled according to a pre-defined tie-breaking rule • Utility of player i = vi(Si) - ∑jSi bij Noam Nisan

7. Single item case An equilibrium A game • Other equilibria exist (e.g. 67 and/or 45) • Assuming that ties are broken in player 1’s favor • Otherwise, no pure equilibrium exists • But (6+, 6, 4) is an -equilibrium b1=6 V1=8 b2=6 V2=6 b3=4 V3=4 Noam Nisan

8. Pure Equilibria Theorem: Pure Nash equilibria of the game correspond to the Walrasian equilibria. Comment: Exactly so for some tie-breaking rule, -Nash for all tie-breaking rules. Corollary: Any pure equilibrium is efficient (PoA=1). Proof: (WN) Everyone bids eq. prices; winner + (NW) with winning prices $4.5 $4.5 $4.5 Noam Nisan

9. Is there always a mixed-Nash equilibrium? • Nash theorem does not apply: Continuum of strategies, discontinuous utilities • For some games, for some tie breaking rules, there is no exact mixed Nash equilibrium. • Simon&Zame ’90 implies that for some (randomized) tie breaking rule a mixed Nash equilibrium exists. • Utility functions are continuous except for ties • Conjecture: -Nash for every tie-breaking rule Noam Nisan

10. Mixed Nash for AND-OR game Bids y for each item, with 0≤y≤1/2 according to cumulative distribution: Pr[bid ≤y]=(v-1/2)/(v-y) (atom at 0: Pr[y = 0] = 1-1/(2v)) Bids x for random item, with 0≤x≤1/2 according to cumulative distribution: Pr[bid≤x]=x/(1-x) Proof (for symmetric deviations): • Any 0≤t≤1/2 is best-reply for OR: • Expected utility = Pr[AND’s-bid ≤ t](v-t) = (v-1/2) = constant • Any 0≤t≤1/2 is best-reply for AND: • Expected utility = Pr[OR’s-bid ≤ t](1-t) – t = 0 = constant AND bidder OR bidder (v>0.5( Noam Nisan

11. Price of Anarchy and Stability • Mixed Nash equilibria are not always efficient • How inefficient? • Previous work about PoA of 2nd price auctions: • Christodoulou, Kovacs & Schapira 2008 • Lucier & Borodin 2010 • Bhawalkar & Roughgarden 2011 Noam Nisan

12. First Non-Welfare Theorem Consider an AND-OR game with m items • AND player has value 1 for the bundle of m items • OR player has value v = 1/√m for any single item Theorem: Any equilibrium has welfare ≤ O(lg m/√m) Proof: • In eq., AND player never bids a total of more than 1. •  OR can get utility v-O(lg m/m) by bidding 2/m on random log m items •  Pr[OR looses in equilibrium] ≤ O(lg m/√m) • Social welfare ≤ v + Pr[AND wins] ≤ O(lg m/√m) Noam Nisan

13. Approximate First Welfare Theorem Theorem: for every game the social welfare of any mixed Nash equilibrium is at least  fraction of the optimum, where: •  ≤ m, in general •  ≤ log m, for sub-additive valuations •  ≤ 2, for sub-modular and XOS valuations • This is also true in the Bayesian setting, for Bayes-Nash equilibria. Noam Nisan

14. Proof of  ≤ 2 for sub-modular case • Let us look at a mixed Nash equilibrium EQ. • Consider the following deviation for player i: for each item jOPTibid the median value of the highest other bid for j in EQ. • This would win each item with probability ½. • Expected value is ≥ vi(OPTi)/2 • Uses sub-modularity (or fractional sub-additivity) • Expected payment is ≤ ∑jOPTi EEQ[pricej] Noam Nisan

15. Proof (cont.) Since the deviation from last slide cannot be profitable: EEQ[valuei] - EEQ[paymenti] ≥ vi (OPTi)/2 - ∑jOPTi EEQ[pricej] Summing over all players i: EEQ[SW] - EEQ[Revenue] ≥ SW(OPT)/2 - EEQ[Revenue] Noam Nisan

16. Conclusions • Looked at Nash equilibria in markets • Pure-Nash corresponds to price-based equilibrium • Mixed-Nash exists even when no price eq. exist • Analyzed mixed-Nash equilibria in some basic cases • Unlike pure equilibria, Mixed equilibria may have an efficiency loss • We can bound the efficiency loss Noam Nisan

17. Further work • Within our model: • Existence of (ε-) mixed equilibrium for all tie breaking rules? • Characterization of all mixed equilibria in our games • PoA lower bounds for sub-classes of valuations • General program • Other market models (budgets, two-sided….) • Other “auction” rules • More on non-complete information models Noam Nisan

18. Thank You! Noam Nisan