1. Uniform Price Auctions: Equilibria and Efficiency Vangelis Markakis Athens University of Economics & Business (AUEB) 1 Orestis Telelis University of Liverpool

2. Outline Intro to Multi-unit Auctions Uniform Price Auctions Pure Nash Equilibria: Existence, Computation and Efficiency Bayes-Nash Equilibria 2

3. Multi-unit Auctions Auctions for selling multiple identical units of a single good In practice: US Treasury notes, bonds UK electricity auctions (output of generators) Radio spectrum licences Various online sales 3

4. Multi-unit Auctions Online sites offering multi-unit auctions UK −uk.ebid.netuk.ebid.net Greece −www.ricardo.grwww.ricardo.gr Australia −www.quicksales.com.auwww.quicksales.com.au … 4

5. Some Notation n bidders k available units of an indivisible good Bidder i has valuation function v i : [k]  R −v i (j) = value of bidder i for obtaining j units Alternative description with marginal valuations: −m i (j) = v i (j) – v i (j-1) = additional value for obtaining the j-th unit, if already given j-1 units −(m i (1), m i (2),…, m i (k)): vector of marginal values 5

6. (Symmetric) Submodular Valuations 6 In the multi-unit setting, a valuation v i is submodular iff  x ≤ y, v i (x+1) - v i (x) ≥ v i (y+1) – v i (y) Hence: m i (1) ≥ m i (2) ≥ … ≥ m i (k) Discrete analog of concavity     Value # bottles

7. A Bidding Format for Multi-unit Auctions Used in various multi-unit auctions [Krishna ’02, Milgrom ’04] 1.The auctioneer asks each bidder to submit a vector of decreasing marginal bids b i = (b i (1), b i (2),…, b i (k)) b i (1) ≥ b i (2) ≥ … ≥ b i (k) 2.The bids are ranked in decreasing order and the k highest win the units 7

8. Example 8 b 1 = (45, 42, 31, 22, 15) b 2 = (35, 27, 20, 12, 7) b 3 = (40, 33, 24, 14, 9)

9. Example 9 # units … 45 bids 42 40 35 33 31 winning bidslosing bids (45, 42, 31, 22, 15) (35, 27, 20, 12, 7) (40, 33, 24, 14, 9) supply How should we charge the winners? … 0

10. Standard Auction Formats 10 1.Multi-unit Vickrey auction (VCG) [Vickrey ’61] −Each bidder pays the externality he causes to the others −Generalization of single-item 2 nd price auction −Good theoretical properties, strategyproof, but barely used in practice 2.Discriminatory Price Auctions −Bidders pay their bids for the units won −Generalization of 1 st price auction −Not strategyproof, but widely used in practice

11. Standard Auction Formats (cont’d) 11 3.Uniform Price Auctions [Friedman 1960] −Same price for every unit −Price is set so that Supply = Demand (market clears) −Interval of prices to pick from: [highest losing bid, lowest winning bid] −This talk: price = highest losing bid −For 1 unit, same as Vickrey auction −For ≥ 2 units, not strategyproof, but widely used in practice (following the campaign of Miller and Friedman in the 90’s)

12. Example Revisited 12 # units … 45 bids 42 40 35 33 31 winning bidslosing bids (45, 42, 31, 22, 15) (35, 27, 20, 12, 7) (40, 33, 24, 14, 9) supply Interval of candidate prices = [31, 33] Uniform price = 31 … 0

13. Uniform Price Auctions 13 Pros Intuitively the right thing to do: identical goods should cost the same! No complaints arising from price discrimination Cons Not strategyproof Nash equilibria are usually inefficient Debate still going on for treasury auctions: Uniform Price vs Discriminatory?

14. Equilibria in Uniform Price Auctions 14 Q1: Existence? Q2: Computation? Q3: Social Inefficiency – Price of Anarchy? We will focus on Nash equilibria in undominated strategies

15. Equilibria in Uniform Price Auctions 15 Q1: Existence? Theorem: For bidders with submodular valuations, a pure Nash equilibrium (PNE) always exists

16. Properties of Nash Equilibria 16 Lemma 1: It is a weakly dominated strategy to declare a bid b i = (b i (1), b i (2),…, b i (k)) s.t. 1. b i (1) ≠ v i (1) 2. b i (j) > m i (j), for some j  [k] For bidders with submodular valuations:

17. Properties of Nash Equilibria 17 Lemma 2: Let b be a PNE in undominated strategies. There always exists an equilibrium b’ resulting in the same allocation, s.t. 1. b’ i (x) = m i (x),  i and every x ≤ # units won 2. the new price is either 0 or v i (1) for some bidder i For bidders with submodular valuations: Lemma 1: It is a weakly dominated strategy to declare a bid b i = (b i (1), b i (2),…, b i (k)) s.t. 1. b i (1) ≠ v i (1) 2. b i (j) > m i (j), for some j  [k]

18. Equilibria in Uniform Price Auctions 18 Q2: Computation? Theorem: A PNE in undominated strategies satisfying the properties of Lemma 2 can be computed in time poly(n, k) Idea: Iterative ascending process starting with b i (1) = v i (1) Initial price set to 0 or highest losing v i (1) At each step: careful adjustment of price and allocation based on currently least winning bid and current demand

19. Equilibria in Uniform Price Auctions 19 Q3: Social Inefficiency – Price of Anarchy? Let b be a pure Nash equilibrium satisfying the properties of Lemma 2 Resulting allocation: x : = x(b) = (x 1,…, x n ) Social Welfare: SW(b) =  v i (x i ) Let the optimal allocation be y = (y 1,…, y n ) Optimal Welfare: OPT =  v i (y i )

20. Equilibria in Uniform Price Auctions 20 Equilibria of uniform price auctions are usually inefficient due to demand reduction [Ausubel-Cramton ’96] Bidders may have incentives to lower their demand (to avoid paying a high price) PoA = sup OPT/SW(b)

21. Example of Demand Reduction 21 OPT = 3, SW(b) = 13/6 Revealing true profile for bidder 1 results in a price that is too high for him (1, 1, 1) (2/3, 0, 0) (1/2, 0, 0) Real profile (1, 0, 0) (2/3, 0, 0) (1/2, 0, 0) Equilibrium profile

22. Equilibria in Uniform Price Auctions 22 Q3: Social Inefficiency – Price of Anarchy? Theorem: For submodular valuations, PoA ≤ e/e-1 Can demand reduction create a huge loss of efficiency?

23. Proof Sketch 23 W := W(x) = set of winners under b W(y) = winners of optimal allocation Decomposition of W: W 0 = {i  W(y): x i ≥ y i } W 1 = {i  W(y): x i < y i } W 2 = W \ W 0  W 1 Note: All winners of W(y) belong to W (because b i (1) = v i (1)) W = W 0  W 1  W 2 Source of Inefficiency is W 1

24. Proof Sketch 24 Let β j := β j (b) = j-th lowest winning bid of b Because of no-overbidding: Every unit “lost” by some i  W 1 is won by a bidder in W 0  W 2 Units lost by i: r i = y i – x i The sum of winning bids for these units should be ≥ It suffices to find a lower bound α such that:

25. Proof Sketch 25  Consider the deviations of each i  W 1, for obtaining j additional units, for j = 1, 2,…, r i Lemma 1 + 2  new price after each deviation would be β j Since b is a Nash equilibrium no such deviation is profitable 

26. Proof Sketch 26 Manipulation of harmonic terms + Properties of submodular functions

27. Tight Examples 27 Theorem: For any k ≥ 9,  PNE in undominated strategies that recovers at most 1-1/e + 2/k of the optimal welfare Even for 2 bidders with submodular valuations For k=2: PoA = 4/3 Real profile: v 1 = (1, 1), v 2 = (1/2, 0), OPT = 2 Equilibrium: b 1 = (1, 0), b 2 = (1/2, 0), SW(b) = 3/2 For k=3: PoA = 18/13

28. Bayes-Nash Equilibria 28 Incomplete information game Every bidder knows his own valuation and the distribution of valuations for the other bidders V i : domain of bidder i Valuation v i drawn from known probability distribution π i : V i  [0,1] Independent of other bidders’ distributions π =  i π i = product distribution Bidding strategy for i: b i (v i ) A profile b is a Bayes-Nash equilibrium (BNE) if  v i

29. Bayes-Nash Equilibria 29 Let x v = optimal allocation for profile v = (v 1,…,v n ), where v ~ π E[OPT] = E v~π [SW(x v )] PoA for BNE: sup b E[OPT] / E[SW(b)] supremum over all BNE b, and all distributions π.

30. Bayes-Nash Equilibria 30 Proof inspired by the PoA analysis for item-bidding [Christodoulou, Kovacs, Schapira ’08] [Bhawalkar, Roughgarden ’11] Theorem: For the domain of submodular valuations and for any product distribution, the Bayesian PoA is O(logk)

31. Beyond Submodular Valuations 31 Very little known for non-submodular bidders Theorem: For subadditive valuations and 1.Pure Nash equilibria, 2  PoA  4 2.Bayes-Nash equilibria, PoA = O(logk) Subadditive valuations: v(x+y)  v(x) + v(y) Valuation compression is needed for such bidders

32. Open Questions - Future Work Tighten the gap in the Bayesian case Better understanding for non-submodular bidders – Valuation compression (analogous compression happens in the item-bidding format) [Christodoulou, Kovacs, Schapira ’08] [Bhawalkar, Roughgarden ’11] – Other bidding formats? Analysis of Discriminatory price auctions 32

33. Open Questions - Future Work Do you like living in Athens? We are hiring phd students! More info: markakis@gmail.com 33

34. Open Questions - Future Work Do you like living in Athens? We are hiring phd students! 34

35. Thank You! 35