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Penny Auctions

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    Penny Auctions Penny Auctions Presentation Transcript

    • Introduction Model Fixed-price Finite Discussion Penny auctions 501 presentation Toomas Hinnosaar May 21, 2009 Introduction Model Fixed-price Finite Discussion 1 Toomas Hinnosaar Penny auctions
    • Introduction Model Fixed-price Finite Discussion What? Literature What do we know about penny auctions? Ian Ayres (Yale) at Freakonomics This 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 scheme Because the outcome of the auction-like process is uncertain, the ”fee” spent on the bid is actually equivalent to a wager, and the whole enterprise is actually a deceptive form of gambling. Tyler Cowen (George Mason) at Marginal Revolution If 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 plan as I’ve ever seen. 2 Toomas Hinnosaar Penny auctions
    • 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. 3 Toomas Hinnosaar Penny auctions
    • Introduction Model Fixed-price Finite Discussion What? Literature All-pay auctions SPAPA 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 in competition longer, wins the prize. Introduced by Smith (1974) to study the evolutionary stability of 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, political contests, 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. 4 Toomas Hinnosaar Penny auctions
    • 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 3 reasonably well for 4 of the items, Exception: video game systems (more aggressive bidding), where they need to add risk-loving preferences to generate the outcome. Expected revenue to the seller is always strictly less than v (although not by much). 5 Toomas Hinnosaar Penny auctions
    • 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>0 C C = bid cost c C c ε P0 = starting price p0 0 p0 0 Pt −P0 Pt = price at round t p t Pt − P0 pt ε V −P0 V = market value v V − P0 v ε An auction = N, v , c, 1 ε > 0 . Assumption: v − c > v − c and v > c 1. 6 Toomas Hinnosaar Penny auctions
    • 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 K 1 ε > 0 , each who submitted a bid pays 1 c and becomes the leader at t 1 with probability K . 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 K 1 ε > 0 , each who submitted a 1 bid pays C and becomes the leader at t 1 with probability K . t ∞ All get −∞. 7 Toomas Hinnosaar Penny auctions
    • 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 full histories irrelevant. When ε > 0: equilibrium is q p p∈{0,1,... } , solved at each p to for symmetric MSNE q p . When ε 0: equilibrium is q0 , q , solved at round t > 0 for symmetric MSNE q and at t 0 for q0 . 8 Toomas Hinnosaar Penny auctions
    • Introduction Model Fixed-price Finite Discussion Result Properties Not SS Main result: Equilibrium in the case ε 0 Theorem 1 In the case ε 0, there exists a unique SSSPNE q0 , q , such that 1. q ∈ 0, 1 is uniquely determined by N c 1−q N q . v 2. If N 1 2, then q0 0, otherwise q0 ∈ 0, 1 is uniquely determined by c 1 − q N N 1 q0 . v N−1 N −1 K N− K 1 1 N q q 1−q . K 0 K K 1 9 Toomas Hinnosaar Penny auctions
    • 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: N−1 N−1 v 1− 1−q v 1−q 0 ⇐⇒ v 0. Leader’s continuation value: v ∗ 1−q Nv. 10 Toomas Hinnosaar Penny auctions
    • Introduction Model Fixed-price Finite Discussion Result Properties Not SS Getting N q # of bids by opponents Pr K E v K 0 1 − q N−1 v∗ 1 ∗ 1 K 1 N − 1 q 1 − q N−2 2v 2v 1 ∗ 2 K 2 N − 1 N − 2 q 2 1 − q N−3 3 v 3v ... ... ... 1 ∗ N−1 K N −1 q N−1 N v N v Since v ∗ 1−q Nv and v 0, expected value from a bid is N−1 N −1 K N− K 1 v∗ N 0 q 1−q −c 1−q N q v −c K 0 K K 1 N c 1−q N q v ⇒ unique q ∈ 0, 1 . 11 Toomas Hinnosaar Penny auctions
    • Introduction Model Fixed-price Finite Discussion Result Properties Not SS Solution for q0 By a similar argument q0 ∈ 0, 1 and N N K 1 0 q 1 − q0 N−K v ∗ − c ⇐⇒ K 0 K 0 K 1 N c N 1−q N 1 q0 1−q N q . v 1 1 N 1 q0 N q ∈ ,1 ⊂ ,1 . N N 1 ⇒ unique q0 ∈ 0, 1 . 12 Toomas Hinnosaar Penny auctions
    • Introduction Model Fixed-price Finite Discussion Result Properties Not SS Properties of auctions with ε 0 Corollary 1 From Theorem 1 we get the following properties of the auctions with 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 . 13 Toomas Hinnosaar Penny auctions
    • Introduction Model Fixed-price Finite Discussion Result Properties Not SS Properties of auctions with ε 0 Observation 1 1. 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 winner gets v − c and the losers pay nothing. 2. When we fix arbitrarily high number R, then there is positive probability that revenue R > R. This is true since there is positive probability of sale and at each round there is positive probability that all non-leaders submit bids. 3. With positive probability we can even get a case where revenue is bigger than R, but the winner paid just c. 14 Toomas Hinnosaar Penny auctions
    • Introduction Model Fixed-price Finite Discussion Result Properties Not SS Properties of auctions with ε 0 Observation 2 1. Details do not affect payoffs much: E R|sale v , payoffs to players always 0. c 2. As v increases, both q and q0 will decrease. As c → 0, the object is never sold. 3. As N increases, since N q is decreasing in N, both q and q0 decrease. 15 Toomas Hinnosaar Penny auctions
    • Introduction Model Fixed-price Finite Discussion Result Properties Not SS Uniqueness comes from Symmetry and Stationarity Remark 1 Without Symmetry and Stationarity, almost anything is possible. 1. i’s favorite equilibrium: i always bids and all the other players always pass. Gives v − c to i and 0 to others. 2. Using this we can construct other equilibria, for example such that No-one bids (if j bids, take i j and go to i’s favorite eq). Players bid by some rule up to v /c an then quit. 3. With suitable randomizations: any revenue from c to v . 16 Toomas Hinnosaar Penny auctions
    • Introduction Model Fixed-price Finite Discussion 2 players N > 2 players Finiteness of the game when ε > 0 Notation: p v − c ≥ 1, γ v − c − v − c > 0. Lemma 1 Fix any equilibrium. None of the players will place bids at prices pt ≥ p. That is, q p 0 for all p ≥ p. Corollary 2 1. 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. 17 Toomas Hinnosaar Penny auctions
    • Introduction Model Fixed-price Finite Discussion 2 players N > 2 players 2-player case: equilibria Proposition 1 Suppose ε > 0 and N 1 2. Then in any SSSPNE the strategies q are such that 0 ∀p ≥ p and ∀p p − 2i > 0, i ∈ , q p 1 ∀p p − 2i 1 > 0, i ∈ , and q 0 is determined for each v , c by one of the following cases. 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 ∈ 0, 1 . v c 1 18 Toomas Hinnosaar Penny auctions
    • Introduction Model Fixed-price Finite Discussion 2 players N > 2 players 2-player case: results Observation 3 Some 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. 19 Toomas Hinnosaar Penny auctions
    • Introduction Model Fixed-price Finite Discussion 2 players N > 2 players N > 2 players: Equilibria Theorem 2 In case ε > 0, there exists a SSSPNE q → 0, 1 , such that q and the corresponding continuation value functions are recursively characterized (C1), (C2), or (C3) at each p < p and q p 0 for all 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. 20 Toomas Hinnosaar Penny auctions
    • Introduction Model Fixed-price Finite Discussion 2 players N > 2 players N > 2 players: Expected revenue Corollary 3 With ε > 0, in any SSSPNE, we can say the following about E 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 . 21 Toomas Hinnosaar Penny auctions
    • Introduction Model Fixed-price Finite Discussion 2 players N > 2 players N > 2 players: Realized prices Lemma 2 With ε > 0, in any SSSPNE, p ∈ {2, . . . , p} st q p−1 q p 0. In particular, q p − 1 > 0. Proposition 2 If ε > 0 and q 0 > 0, then the highest price reached with strictly probability, 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 4 When the object is sold and condition γ < N − 1 c is satisfied, 1. R > v with positive probability, 2. R < v with positive probability 22 Toomas Hinnosaar Penny auctions
    • Introduction Model Fixed-price Finite Discussion Conclusion: Can explain winners’ “savings” Figure: Distribution of the winner’s savings in different types of auctions 23 Toomas Hinnosaar Penny auctions
    • 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 24 Toomas Hinnosaar Penny auctions
    • 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. 25 Toomas Hinnosaar Penny auctions
    • Stylized facts A: SSSPNE B: N q C: Example Stylized facts: Averages Type Obs V P v c p (# of bids) Regular 41760 166.9 46.7 1044 5 242.9 Penny 7355 773.3 25.1 75919.2 75 1098.1 Fixed price 1634 967 64.9 6290.7 5 2007.2 Free 3295 184.5 0 1222 5 558.5 Nailbiter 924 211.5 8.3 1394.1 5 580.1 Beginner 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 26 Toomas Hinnosaar Penny auctions
    • Stylized facts A: SSSPNE B: N q C: Example Stylized facts: # of bids Figure: Distribution of the number of bids submitted in different types of auctions 27 Toomas Hinnosaar Penny auctions
    • Stylized facts A: SSSPNE B: N q C: Example Appendix A: SSSPNE Some notation: Histories: ht b 0 , l 0 , b 1 , l 1 , . . . , b t−1 , l t−1 ∈ H Pure strategies: bi H → {0, 1} Mixed (behavioral) strategies: σi H → 0, 1 Definition 1 σ is Symmetric if ∀t, i, i, ht , if ht is ht with i and i swapped, then σi h t σi h t . 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 2 σ is Stationary if ∀i, ht , h, if Li ht Li ht and S ht S ht , then σi ht σi ht . 28 Toomas Hinnosaar Penny auctions
    • Stylized facts A: SSSPNE B: N q C: Example Appendix A: SSSPNE Lemma 3 A strategy profile σ is Symmetric and Stationary if and only if it can be represented by q S → 0, 1 , where q s is the probability bidder i bids at state s ∈ S for each non-leader i ∈ {0, . . . , N}. Lemma 4 With ε > 0, a strategy profile σ is SSSPNE if and only if it can be represented by q S → 0, 1 where q s is the Nash equilibrium in the stage-game at state s, taking into account the continuation values implied by transitions S. Lemma 5 With ε 0, a strategy profile σ is SSSPNE if and only if it can be represented by q S → 0, 1 where q s is the Nash equilibrium in the stage-game at state s, taking into account the continuation values implied by transitions S. 29 Toomas Hinnosaar Penny auctions
    • Stylized facts A: SSSPNE B: N q C: Example Appendix B: Properties of N q N−1 N −1 K N−1−K 1 N q q 1−q . K 0 K K 1 Lemma 6 Let N ≥ 2. Then 1. N q is strictly decreasing in q ∈ 0, 1 . 1 2. limq→0 N q 1, limq→1 N q N. 3. N q > N 1 q for all q ∈ 0, 1 . 30 Toomas Hinnosaar Penny auctions
    • Stylized facts A: SSSPNE B: N q C: Example Appendix C: Auction with 3 equililbria Let 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 0.1183 1 0 8.1 0 0.3681 0.4175 2 1 0 0 0 0 3 0.6996 0.5504 0 0.0119 0.0135 4 0 5.1 0 0.4371 0.4958 5 0.4287 1.3381 0 0.0211 0.0239 6 0.0645 2.7129 0 0.0277 0.0314 7 0 2.1 0 0.0157 0.0178 8 0 1.1 0 0.0001 0.0001 9 0 0.1 0 0 0 Table: Equilibrium with q 2 1 q 0 ∈ 0, 1 , q 1 0, but q 2 1 and 31 E R|p > 0 8.62 < 9.1 Hinnosaar Toomas v. Penny auctions
    • Stylized facts A: SSSPNE B: N 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 0.1061 1 0 8.1 0 0.354 0.3961 2 0.7249 0.5371 0 0.0298 0.0333 3 0.6996 0.5504 0 0.0273 0.0306 4 0 5.1 0 0.3344 0.3741 5 0.4287 1.3381 0 0.0484 0.0542 6 0.0645 2.7129 0 0.0636 0.0711 7 0 2.1 0 0.036 0.0403 8 0 1.1 0 0.0003 0.0003 9 0 0.1 0 0 0 Table: Equilibrium with q 2 0.7249 ∈ 0, 1 32 Toomas Hinnosaar Penny auctions
    • Stylized facts A: SSSPNE B: N 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 1 1 0.7473 0.5174 0 0 2 0 7.1 0 0 3 0.6996 0.5504 0 0 4 0 5.1 0 0 5 0.4287 1.3381 0 0 6 0.0645 2.7129 0 0 7 0 2.1 0 0 8 0 1.1 0 0 9 0 0.1 0 0 Table: Equilibrium with q 2 0 33 Toomas Hinnosaar Penny auctions
    • Stylized facts A: SSSPNE B: N q C: Example 34 Toomas Hinnosaar Penny auctions