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Uniform Price Auctions: Equilibria and Efficiency
Vangelis Markakis
Athens University of
Economics & Business
(AUEB)
1
Orestis Telelis University of Liverpool
Outline
• Intro to Multi-unit Auctions
• Uniform Price Auctions• Pure Nash Equilibria: Existence, Computation
and Efficiency• Bayes-Nash Equilibria
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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
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Multi-unit Auctions
Online sites offering multi-unit auctions
• UK− uk.ebid.net
• Greece− www.ricardo.gr
• Australia− www.quicksales.com.au
• …
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Some Notation• n bidders
• k available units of an indivisible good
• Bidder i has valuation function vi : [k] R−vi(j) = value of bidder i for obtaining j units
• Alternative description with marginal valuations:−mi(j) = vi(j) – vi(j-1) = additional value for obtaining the
j-th unit, if already given j-1 units
− (mi(1), mi(2),…, mi(k)): vector of marginal values
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(Symmetric) Submodular Valuations
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• In the multi-unit setting, a valuation vi is submodular iff
x ≤ y, vi(x+1) - vi(x) ≥ vi(y+1) – vi(y)
• Hence: mi(1) ≥ mi(2) ≥ … ≥ mi(k)
Discrete analog of concavity
Value
# bottles
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
• bi = (bi(1), bi(2),…, bi(k))
• bi(1) ≥ bi(2) ≥ … ≥ bi(k)
• The bids are ranked in decreasing order and the k highest win the units
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Example
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b1 = (45, 42, 31, 22, 15)
b2 = (35, 27, 20, 12, 7)
b3 = (40, 33, 24, 14, 9)
Example
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# units
…
45
bids
42
40
35
33
31
winning bids losing bids
(45, 42, 31, 22, 15)
(35, 27, 20, 12, 7)
(40, 33, 24, 14, 9)
supply
How should we charge the winners?
…
0
Standard Auction Formats
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1. Multi-unit Vickrey auction (VCG) [Vickrey ’61]− Each bidder pays the externality he causes to the others
− Generalization of single-item 2nd 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 1st price auction
− Not strategyproof, but widely used in practice
Standard Auction Formats (cont’d)
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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)
Example Revisited
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# units
…
45
bids
42
40
35
33
31
winning bids losing 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
Uniform Price Auctions
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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?
Equilibria in Uniform Price Auctions
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Q1: Existence?
Q2: Computation?
Q3: Social Inefficiency – Price of Anarchy?
We will focus on Nash equilibria in undominated strategies
Equilibria in Uniform Price Auctions
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Q1: Existence?
Theorem: For bidders with submodular valuations, a pure Nash equilibrium (PNE) always exists
Properties of Nash Equilibria
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Lemma 1: It is a weakly dominated strategy to declare a bid bi = (bi(1), bi(2),…, bi(k)) s.t.
• bi(1) ≠ vi(1)• bi(j) > mi(j), for some j [k]
For bidders with submodular valuations:
Properties of Nash Equilibria
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Lemma 2: Let b be a PNE in undominated strategies. There always exists an equilibrium b’ resulting in the same allocation, s.t.
• b’i(x) = mi(x), i and every x ≤ # units won 1. the new price is either 0 or vi(1) for some bidder i
For bidders with submodular valuations:
Lemma 1: It is a weakly dominated strategy to declare a bid bi = (bi(1), bi(2),…, bi(k)) s.t.
• bi(1) ≠ vi(1)• bi(j) > mi(j), for some j [k]
Equilibria in Uniform Price Auctions
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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 bi(1) = vi(1)
• Initial price set to 0 or highest losing vi(1)• At each step: careful adjustment of price and allocation
based on currently least winning bid and current demand
Equilibria in Uniform Price Auctions
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Q3: Social Inefficiency – Price of Anarchy?
Let b be a pure Nash equilibrium satisfying the properties of Lemma 2
Resulting allocation: x := x(b) = (x1,…, xn)
Social Welfare: SW(b) = vi(xi)
Let the optimal allocation be y = (y1,…, yn)
Optimal Welfare: OPT = vi(yi)
Equilibria in Uniform Price Auctions
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• 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)
Example of Demand Reduction
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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
Equilibria in Uniform Price Auctions
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Q3: Social Inefficiency – Price of Anarchy?
Theorem: For submodular valuations,PoA ≤ e/e-1
Can demand reduction create a huge loss of efficiency?
Proof Sketch
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W := W(x) = set of winners under b
W(y) = winners of optimal allocation
Decomposition of W:
• W0 = {i W(y): xi ≥ yi}
• W1 = {i W(y): xi < yi}
• W2 = W \ W0 W1
Note: All winners of W(y) belong to W (because bi(1) = vi(1))
W = W0 W1 W2
Source of Inefficiency is W1
Proof Sketch
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Let βj := βj(b) = j-th lowest winning bid of b
• Because of no-overbidding:
• Every unit “lost” by some i W1 is won by a bidder in W0 W2
• Units lost by i: ri = yi – xi • The sum of winning bids for these units should be ≥
β j
j=1
ri
∑
• It suffices to find a lower bound α such that:
Proof Sketch
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• Consider the deviations of each i W1, for obtaining j additional units, for j = 1, 2,…, ri
• Lemma 1 + 2 new price after each deviation would be βj
• Since b is a Nash equilibrium no such deviation is profitable
Proof Sketch
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Manipulation of harmonic terms+
Properties of submodular functions
Tight Examples
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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: v1 = (1, 1), v2 = (1/2, 0), OPT = 2
Equilibrium: b1 = (1, 0), b2 = (1/2, 0), SW(b) = 3/2
For k=3: PoA = 18/13
Bayes-Nash Equilibria
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Incomplete information game• Every bidder knows his own valuation and the distribution of valuations for the other bidders
• Vi: domain of bidder i
• Valuation vi drawn from known probability distributionπi: Vi [0,1]
• Independent of other bidders’ distributions• π = i πi = product distribution
• Bidding strategy for i: bi(vi)A profile b is a Bayes-Nash equilibrium (BNE) if vi
€
Ev− i ~π − iui b( )[ ] ≥ Ev− i ~π − i
ui c i,b−i( )[ ]
Bayes-Nash Equilibria
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• Let xv = optimal allocation for profile v = (v1,…,vn), where v ~ π
E[OPT] = Ev~π [SW(xv)]PoA for BNE:
supb E[OPT] / E[SW(b)]
• supremum over all BNE b, and all distributions π.
Bayes-Nash Equilibria
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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)
Beyond Submodular Valuations
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• Very little known for non-submodular bidders
Theorem: For subadditive valuations and1. 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
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
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Open Questions - Future Work
• Do you like living in Athens?• We are hiring phd students!• More info: [email protected]
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Open Questions - Future Work
• Do you like living in Athens?• We are hiring phd students!
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Thank You!
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