penny auctions
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Introduction Model Fixed-price Finite Discussion
Penny auctions501 presentation
Toomas Hinnosaar
May 21, 2009
IntroductionModelFixed-priceFiniteDiscussion
Toomas Hinnosaar Penny auctions
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Introduction Model Fixed-price Finite Discussion What? Literature
What do we know about penny auctions?Ian Ayres (Yale) at FreakonomicsThis is an example of what auction theorists call an “all-pay” auction,and it’s a game you want to avoid playing if you possibly can.
Wikipedia on Bidding fee schemeBecause the outcome of the auction-like process is uncertain, the ”fee”spent on the bid is actually equivalent to a wager, and the wholeenterprise is actually a deceptive form of gambling.
Tyler Cowen (George Mason) at Marginal RevolutionIf traders are overconfident, as much as the finance literature alleges,there ought to be a way to exploit that tendency. And so there is.
Jeff Atwood (blogger)In short, swoopo is about as close to pure, distilled evil in a business planas I’ve ever seen.
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Introduction Model Fixed-price Finite Discussion What? Literature
What are Penny Auctions?
Best explained by an example from http://www.swoopo.com/
Some extreme outcomes from auctions selling cash:▶ $1,000 sold after 15 bids. Winner: $3.75, total: ≈ $10.40.▶ $80 was sold after one bid. Winner paid $0.15, losers nothing.▶ $1,000, free auction. Winner spent: $805.50, total (23100
bids): ≈ $16,100.▶ $80 Cash!, free auction. Winner spent: $194.25
(“Congratulations, Newstart16! Savings: 0%”), total $950.
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Introduction Model Fixed-price Finite Discussion What? Literature
All-pay auctionsSPAPA Second Price All-Pay Auctions = War of Attrition
▶ Two contestants compete for a prize. While in competition,they incur constant flow cost of time. The one who stays incompetition longer, wins the prize.
▶ Introduced by Smith (1974) to study the evolutionary stabilityof animal conflicts.
▶ Also applied to price wars, bargaining, patent competition.▶ Full characterization of equilibria under full information by
Hendricks, Weiss, and Wilson (1988).FPAPA (First Price) All-Pay Auctions
▶ Widely used to model: Rent-seeking, R&D races, politicalcontests, lobbying, job-promotion tournaments.
▶ Two-player dollar auction Shubik (1971).▶ Full characterization of equilibria under full information by
Baye, Kovenock, and de Vries (1996).▶ Siegel (2008) offers closed-form characterization for players’
equilibrium payoffs in a quite general class of all-pay contests.Toomas Hinnosaar Penny auctions
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Introduction Model Fixed-price Finite Discussion What? Literature
One paper about the same type of auctions
Platt, Price, and Tappen (March 31, 2009) “Pay-to-bid Auctions”:▶ Main focus is empirical.▶ The same data source and get similar stylized facts.▶ Theoretical model simpler: bidders never have to make
simultaneous decisions.▶ They get that their model predicts the outcomes of auctions
reasonably well for 34 of the items,
▶ Exception: video game systems (more aggressive bidding),where they need to add risk-loving preferences to generate theoutcome.
▶ Expected revenue to the seller is always strictly less than v(although not by much).
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Introduction Model Fixed-price Finite Discussion Notation Timing SSSPNE
Model
▶ N + 1 players, i ∈ {0, 1, . . . ,N},▶ Discrete time, rounds t = 0, 1, 2, . . . ,▶ ε = bid increment, 2 cases: ε = 0 and ε > 0
ε = 0 ε > 0C = bid cost c = C c = C
εP0 = starting price p0 = 0 p0 = 0Pt = price at round t pt = Pt − P0 pt = Pt−P0
ε
V = market value v = V − P0 v = V−P0ε
▶ An auction = (N, v , c, 1[ε > 0]).▶ Assumption: v − c > bv − cc and v > c + 1.
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Introduction Model Fixed-price Finite Discussion Notation Timing SSSPNE
Timing
t = 0 At price 0 all bidders simultaneously: Bid / Pass.▶ If K = 0 bids, game ends, seller keeps the object, all bidders
get 0.▶ If K > 0 bids, p1 = K1[ε > 0], each who submitted a bid pays
c and becomes the leader at t = 1 with probability 1K .
t > 0 All non-leaders simultaneously: Bid / Pass.▶ If K = 0 bids, game ends, leader gets v − pt , non-leaders get 0.▶ If K > 0 bids, pt+1 = pt + K1[ε > 0], each who submitted a
bid pays C and becomes the leader at t + 1 with probability 1K .
t =∞ All get −∞.
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Introduction Model Fixed-price Finite Discussion Notation Timing SSSPNE
Solution concept: SSSPNE = S+S+SPNE
▶ Subset of SPNE that satisfy▶ “Symmetry”: being in the same situation, players behave
similarly.▶ “Stationarity”: only directly payoff-relevant characteristics
matter (current price and active bidders), time and fullhistories irrelevant.
▶ When ε > 0: equilibrium is (q(p))p∈{0,1,... }, solved at each pto for symmetric MSNE q(p).
▶ When ε = 0: equilibrium is (q0, q), solved at round t > 0 forsymmetric MSNE q and at t = 0 for q0.
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Introduction Model Fixed-price Finite Discussion Result Properties Not SS
Main result: Equilibrium in the case ε = 0
Theorem 1In the case ε = 0, there exists a unique SSSPNE (q0, q), such that
1. q ∈ (0, 1) is uniquely determined by
(1− q)NΨN(q) =cv .
2. If N + 1 = 2, then q0 = 0, otherwise q0 ∈ (0, 1) is uniquelydetermined by
(1− q)NΨN+1(q0) =cv .
ΨN(q) =N−1∑K=0
(N − 1
K
)qK (1− q)N−(K+1) 1
K + 1 .
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Introduction Model Fixed-price Finite Discussion Result Properties Not SS
Proof: Finding equilibria when N + 1 ≥ 3
▶ Let N + 1 ≥ 3.▶ No equilibria where q = 1 – the game never ends, −∞ to all.▶ No equilibria where q = 0 – could get v − c > 0 with certainty.▶ ⇒ in any equilibrium q ∈ (0, 1).▶ A non-leader is indifferent:
v = [1− (1− q)N−1]v + (1− q)N−10 ⇐⇒ v = 0.
▶ Leader’s continuation value: v∗ = (1− q)Nv .
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Introduction Model Fixed-price Finite Discussion Result Properties Not SS
Getting ΨN(q)
# of bids by opponents Pr(K ) E (v ′)K = 0 (1− q)N−1 v∗K = 1 (N − 1)q(1− q)N−2 1
2 v∗ + 12 v
K = 2 (N − 1)(N − 2)q2(1− q)N−3 13 v∗ + 2
3 v. . . . . . . . .
K = N − 1 qN−1 1N v∗ + N−1
N v
▶ Since v∗ = (1− q)Nv and v = 0, expected value from a bid is
0 =N−1∑K=0
(N − 1
K
)qK (1−q)N−(K+1) v∗
K + 1−c = (1−q)NΨN(q)v−c
▶ (1− q)NΨN(q) = cv ⇒ unique q ∈ (0, 1).
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Introduction Model Fixed-price Finite Discussion Result Properties Not SS
Solution for q0
By a similar argument q0 ∈ (0, 1) and
0 =N∑
K=0
(NK
)qK
0 (1− q0)N−K 1K + 1 v∗ − c ⇐⇒
(1− q)NΨN+1(q0) =cv = (1− q)NΨN(q).
ΨN+1(q0) = ΨN(q) ∈( 1
N , 1)⊂( 1
N + 1 , 1).
⇒ unique q0 ∈ (0, 1).
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Introduction Model Fixed-price Finite Discussion Result Properties Not SS
Properties of auctions with ε = 0
Corollary 1From Theorem 1 we get the following properties of the auctionswith ε = 0:
1. q0 < q.2. If N + 1 > 2, then the probability of selling the object is
1− (1− q0)N+1 > 0. If N + 1 = 2, the seller keeps the object.3. Expected ex-ante value to the players is 0.4. Expected revenue to the seller, conditional on sale, is v .
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Introduction Model Fixed-price Finite Discussion Result Properties Not SS
Properties of auctions with ε = 0
Observation 11. With probability (N + 1)(1− q0)N q0(1− q)N > 0 the seller
sells the object after just one bid and gets R = c. The winnergets v − c and the losers pay nothing.
2. When we fix arbitrarily high number R, then there is positiveprobability that revenue R > R. This is true since there ispositive probability of sale and at each round there is positiveprobability that all non-leaders submit bids.
3. With positive probability we can even get a case whererevenue is bigger than R, but the winner paid just c.
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Introduction Model Fixed-price Finite Discussion Result Properties Not SS
Properties of auctions with ε = 0
Observation 21. Details do not affect payoffs much: E (R|sale) = v , payoffs to
players always 0.2. As c
v increases, both q and q0 will decrease.As c → 0, the object is never sold.
3. As N increases, since ΨN(q) is decreasing in N, both q and q0decrease.
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Introduction Model Fixed-price Finite Discussion Result Properties Not SS
Uniqueness comes from Symmetry and Stationarity
Remark 1Without Symmetry and Stationarity, almost anything is possible.
1. i ’s favorite equilibrium: i always bids and all the other playersalways pass. Gives v − c to i and 0 to others.
2. Using this we can construct other equilibria, for example suchthat
▶ No-one bids (if j bids, take i 6= j and go to i ’s favorite eq).▶ Players bid by some rule up to bv/cc an then quit.
3. With suitable randomizations: any revenue from c to v .
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Introduction Model Fixed-price Finite Discussion 2 players N > 2 players
Finiteness of the game when ε > 0Notation:
▶ p = bv − cc ≥ 1,▶ γ = (v − c)− bv − cc > 0.
Lemma 1Fix any equilibrium. None of the players will place bids at pricespt ≥ p. That is, q(p) = 0 for all p ≥ p.
Corollary 21. max{p − 1 + N,N + 1} is an upper bound of the support of
realized prices.2. The game has ended by time τ ≤ p + N with certainty.3. We can use backwards induction to find any SPNE.
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Introduction Model Fixed-price Finite Discussion 2 players N > 2 players
2-player case: equilibria
Proposition 1Suppose ε > 0 and N + 1 = 2. Then in any SSSPNE the strategiesq are such that
q(p) =
{0 ∀p ≥ p and ∀p = p − 2i > 0, i ∈ ℕ,1 ∀p = p − (2i + 1) > 0, i ∈ ℕ,
and q(0) is determined for each (v , c) by one of the followingcases.
1. If p is an even integer, then q(0) = 0.2. If p is odd integer and v ≥ 3(c + 1), then q(0) = 1.3. If p is odd integer and v < 3(c + 1), then
q(0) = 2 v−(c+1)v+(c+1) ∈ (0, 1).
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Introduction Model Fixed-price Finite Discussion 2 players N > 2 players
2-player case: results
Observation 3Some observations regarding the equilibria in the two-player case.
1. Sensitive to “irrelevant” detail — is p even or odd.2. When v ≥ 3(c + 1), the equilibrium collapses
E (R|p > 0) = 3(c + 1), in general � v .3. One very specific case: p is an odd and v < 3(c + 1)
▶ P(p > 0) ∈ (0, 1),▶ E (R|p > 0) = v , expected payoff to players is 0.▶ P(0) > 0,P(1) > 0,P(2) = 0,P(3) > 0,P(p) = 0,∀p ≥ 4.
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Introduction Model Fixed-price Finite Discussion 2 players N > 2 players
N > 2 players: Equilibria
Theorem 2In case ε > 0, there exists a SSSPNE q : ℕ→ [0, 1], such that qand the corresponding continuation value functions are recursivelycharacterized (C1), (C2), or (C3) at each p < p and q(p) = 0 forall p ≥ p. The equilibrium is not in general unique.
▶ (C1) = conditions for q(p) = 1 being NE in the stage-game.▶ (C2) = conditions for q(p) = 0 being NE in the stage-game.▶ (C3) = conditions for q(p) ∈ (0, 1) being NE in the
stage-game.▶ Existence by Nash (1951) and construction.▶ Non-uniqueness by example.
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Introduction Model Fixed-price Finite Discussion 2 players N > 2 players
N > 2 players: Expected revenue
Corollary 3With ε > 0, in any SSSPNE, we can say the following aboutE (R|sale).
1. E (R|sale) ≤ v .2. If q(p) < 1,∀p, then E (R|sale) = v .3. In some games in some equilibria E (R|sale) < v .
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Introduction Model Fixed-price Finite Discussion 2 players N > 2 players
N > 2 players: Realized pricesLemma 2With ε > 0, in any SSSPNE, ∄p ∈ {2, . . . , p} stq(p − 1) = q(p) = 0. In particular, q(p − 1) > 0.
Proposition 2If ε > 0 and q(0) > 0, then the highest price reached with strictlyprobability, p∗, satisfies
1. p ≤ p∗ ≤ max{p + N − 1,N + 1},2. If γ < (N − 1)c, then p∗ = max{p + N − 1,N + 1}.
Corollary 4When the object is sold and condition γ < (N − 1)c is satisfied,
1. R > v with positive probability,2. R < v with positive probability
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Introduction Model Fixed-price Finite Discussion
Conclusion: Can explain winners’ “savings”
Figure: Distribution of the winner’s savings in different types of auctions
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Introduction Model Fixed-price Finite Discussion
Conclusion: Can’t explain high average profit margin
Figure: Distribution of the profit margin in different types of auctions
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Introduction Model Fixed-price Finite Discussion
Extensions: how to get E (R|sale) > v
▶ Value to the bidders is bigger than value to the seller.▶ “Entertainment shopping” or “gambling value”▶ Different considerations of cost: c is partly sunk at the
decision points.▶ Incorrect understanding of game.▶ Reputation and Bid butlers.
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Stylized facts A: SSSPNE B: ΨN (q) C: Example
Stylized facts: Averages
Type Obs V P v c p (# of bids)Regular 41760 166.9 46.7 1044 5 242.9Penny 7355 773.3 25.1 75919.2 75 1098.1
Fixed price 1634 967 64.9 6290.7 5 2007.2Free 3295 184.5 0 1222 5 558.5
Nailbiter 924 211.5 8.3 1394.1 5 580.1Beginner 6185 214.5 45.8 1358.5 5 301.6
All auctions 61153 267.6 41.4 10236.3 13.4 420.9
Table: Some statistics about the auctions
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Stylized facts A: SSSPNE B: ΨN (q) C: Example
Stylized facts: # of bids
Figure: Distribution of the number of bids submitted in different types ofauctions
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Stylized facts A: SSSPNE B: ΨN (q) C: Example
Appendix A: SSSPNESome notation:
▶ Histories: ht = (b0, l0, b1, l1, . . . , bt−1, l t−1) ∈ H▶ Pure strategies: bi : H → {0, 1}▶ Mixed (behavioral) strategies: σi : H → [0, 1]
Definition 1σ is Symmetric if ∀t, i , i , ht , if ht is ht with i and i swapped, thenσi (ht) = σi (ht).
▶ Li (ht) = 1[i = l t ] (= is i the leader at ht)▶ S = {N + 1,N} if ε = 0 and S = {0, 1, 2, . . . } if ε > 0▶ S : H → S in logical way
Definition 2σ is Stationary if ∀i , ht , h, if Li (ht) = Li (ht) and S(ht) = S(ht),then σi (ht) = σi (ht).
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Stylized facts A: SSSPNE B: ΨN (q) C: Example
Appendix A: SSSPNELemma 3A strategy profile σ is Symmetric and Stationary if and only if itcan be represented by q : S → [0, 1], where q(s) is the probabilitybidder i bids at state s ∈ S for each non-leader i ∈ {0, . . . ,N}.
Lemma 4With ε > 0, a strategy profile σ is SSSPNE if and only if it can berepresented by q : S → [0, 1] where q(s) is the Nash equilibrium inthe stage-game at state s, taking into account the continuationvalues implied by transitions S.
Lemma 5With ε = 0, a strategy profile σ is SSSPNE if and only if it can berepresented by q : S → [0, 1] where q(s) is the Nash equilibrium inthe stage-game at state s, taking into account the continuationvalues implied by transitions S.
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Stylized facts A: SSSPNE B: ΨN (q) C: Example
Appendix B: Properties of ΨN(q)
ΨN(q) =N−1∑K=0
(N − 1
K
)qK (1− q)N−1−K 1
K + 1 .
Lemma 6Let N ≥ 2. Then
1. ΨN(q) is strictly decreasing in q ∈ (0, 1).
2. limq→0 ΨN(q) = 1, limq→1 ΨN(q) = 1N .
3. ΨN(q) > ΨN+1(q) for all q ∈ (0, 1).
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Stylized facts A: SSSPNE B: ΨN (q) C: Example
Appendix C: Auction with 3 equililbriaLet N + 1 = 3, v = 9.1, c = 2, ε > 0.
p q(p) v∗(p) v(p) P(p) P(p|p > 0)
0 0.509 0 0.11831 0 8.1 0 0.3681 0.41752 1 0 0 0 03 0.6996 0.5504 0 0.0119 0.01354 0 5.1 0 0.4371 0.49585 0.4287 1.3381 0 0.0211 0.02396 0.0645 2.7129 0 0.0277 0.03147 0 2.1 0 0.0157 0.01788 0 1.1 0 0.0001 0.00019 0 0.1 0 0 0
Table: Equilibrium with q(2) = 1
q(0) ∈ (0, 1), q(1) = 0, but q(2) = 1 andE (R|p > 0) = 8.62 < 9.1 = v .Toomas Hinnosaar Penny auctions
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Stylized facts A: SSSPNE B: ΨN (q) C: Example
Appendix C: Auction with 3 equililbria
p q(p) v∗(p) v(p) P(p) P(p|p > 0)
0 0.5266 0 0.10611 0 8.1 0 0.354 0.39612 0.7249 0.5371 0 0.0298 0.03333 0.6996 0.5504 0 0.0273 0.03064 0 5.1 0 0.3344 0.37415 0.4287 1.3381 0 0.0484 0.05426 0.0645 2.7129 0 0.0636 0.07117 0 2.1 0 0.036 0.04038 0 1.1 0 0.0003 0.00039 0 0.1 0 0 0
Table: Equilibrium with q(2) = 0.7249 ∈ (0, 1)
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Stylized facts A: SSSPNE B: ΨN (q) C: Example
Appendix C: Auction with 3 equililbria
p q(p) v∗(p) v(p) P(p) P(p|p > 0)
0 0 0 11 0.7473 0.5174 0 02 0 7.1 0 03 0.6996 0.5504 0 04 0 5.1 0 05 0.4287 1.3381 0 06 0.0645 2.7129 0 07 0 2.1 0 08 0 1.1 0 09 0 0.1 0 0
Table: Equilibrium with q(2) = 0
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Stylized facts A: SSSPNE B: ΨN (q) C: Example
Toomas Hinnosaar Penny auctions