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

2. Noam Nisan Market Equilibrium x ij - fraction of good j that player i gets p j - price of good j Each player gets his “demand”  Bundle that he prefers most under current prices Market clears  demand=supply for each good goodsBuyers / sellers x ij pjpj

3. Noam Nisan 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?

4. Noam Nisan Simple Market Model for this talk m heterogeneous indivisible items to be allocated among n bidders. Each bidder i has a valuation v i, where v i (S) is his real value for the set S of items. Walrasian equilibrium: prices + allocation 2 items1item 139 2 items1item 127 $4.5

5. Noam Nisan Lack of equilibrium 2 items1item 80 2 items1item 66 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”?

6. Noam Nisan The Market as a Game Every player i makes an offer b ij 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 = v i (S i ) - ∑ j  Si b ij

7. Noam Nisan Single item case 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 V 1 =8 V 2 =6 V 3 =4 b 1 =6 b 2 =6 b 3 =4 An equilibrium A game

8. Noam Nisan 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 2 items1item 139 2 items1item 127 $4.5

9. Noam Nisan 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

10. Noam Nisan Mixed Nash for AND-OR game 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 2 items1item 10 2 items1item vv AND bidder OR bidder (v>0.5( 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)

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

12. Noam Nisan 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: 1. In eq., AND player never bids a total of more than 1. 2.  OR can get utility v-O(lg m/m) by bidding 2/m on random log m items 3.  Pr[OR looses in equilibrium] ≤ O(lg m/√m) 4. Social welfare ≤ v + Pr[AND wins] ≤ O(lg m/√m)

13. Noam Nisan 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.

14. Noam Nisan 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  OPT i bid the median value of the highest other bid for j in EQ. This would win each item with probability ½. Expected value is ≥ v i (OPT i )/2  Uses sub-modularity (or fractional sub-additivity) Expected payment is ≤ ∑ j  OPTi E EQ [price j ]

15. Noam Nisan Proof (cont.) Since the deviation from last slide cannot be profitable: E EQ [value i ] - E EQ [payment i ] ≥ v i (OPT i )/2 - ∑ j  OPTi E EQ [price j ] Summing over all players i: E EQ [SW] - E EQ [Revenue] ≥ SW(OPT)/2 - E EQ [Revenue]

16. Noam Nisan 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

17. Noam Nisan 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

18. Noam Nisan Thank You!