1.
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
2.
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
3.
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
4.
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
5.
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
6.
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
7.
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
8.
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
9.
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
10.
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
11.
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
12.
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
13.
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
14.
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
15.
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
16.
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
17.
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
18.
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
19.
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
20.
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
21.
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
22.
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
23.
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
24.
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
25.
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
26.
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
27.
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
28.
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
29.
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
30.
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
31.
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
32.
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
33.
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
34.
Stylized facts A: SSSPNE B: N q C: Example
34
Toomas Hinnosaar Penny auctions
But this cleared everything up for me http://cashinunder24hrs.com
Get Funky with the Funky Shark (LOL)