Penny Auctions

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.
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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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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.
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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
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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).

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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>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.

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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 −∞.

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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 .

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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

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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.

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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 .
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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 .

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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 .

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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



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.

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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 .

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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.

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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

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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.

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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.

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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 .

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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
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Conclusion: Can explain winners’ “savings”

Figure: Distribution of the winner’s savings in diﬀerent types of auctions
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Conclusion: Can’t explain high average profit margin

Figure: Distribution of the profit margin in different types of auctions
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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: 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

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Stylized facts: # of bids

Figure: Distribution of the number of bids submitted in diﬀerent types of
auctions 27


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 .
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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


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 .

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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
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E R|p > 0 8.62 < 9.1 Hinnosaar
Toomas
v. Penny auctions



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

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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

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