Resale in Second Price Auctions with Costly Participation
Okan Y¬lankaya (Koç University)
(joint with Görkem Çelik, ESSEC and THEMA)
1
Introduction
Auctions are important
Fixed number of bidders
– Stochastic
Endogenous entry
– Information acquisition
Participation costs
– Purchase bid documents, register for the auction, travel to the auction site
– Arrange …nancing, secure bid or performance bonds in order to be able to bid in the auction
– Pre-qualify for the auction
Preparing a “bid” may be complicated and costly when a bid is not only a dollar amount, but also a detailed business plan with documentation
Possible ex-post ine¢ ciency
– “Participants” vs. “bidders”
– Even in a symmetric environment
Opportunity for resale
Active resale/secondary markets
– Real estate, treasury bills, artwork...
Literature
Participation Costs: Green and La¤ont (1984), Samuelson (1985), Stege-man (1996), Tan and Yilankaya (2006), Celik and Yilankaya (2009)
Resale: Gupta and Lebrun (1999), Haile (1999, 2003), Garratt and Tröger (2006), Hafalir and Krishna (2008)
2
The Environment
Standard symmetric IPV environment:
– The valuation of the risk-neutral seller is zero
– 2 risk-neutral (potential) bidders (n in the paper)
– vi 2 [0;1]; F(:) continuously di¤erentiable, f(:) the density
– (Generalized) monotone hazard rate condition:
For any x 2 (0;1], F(x)f(v)F(v) is decreasing in v for all v 2 [0; x]
(Slightly less standard) Participation is costly: c 2 (0;1)
Payo¤ of non-participation is zero
Bidders independently decide (interim)
– whether to participate or not
3
The Benchmark: No Resale
Tan and Yilankaya (2006)
When c = 0 bidding your valuation is weakly dominant
With c > 0, not quite, but
– If participate, then cannot do better than bidding your valuation
Expected payo¤ is strictly increasing in valuation
Cuto¤ strategies
b1(v1) =
(
N o if v1 a
v1 if v1 > a ; b2(v2) = (
N o if v2 b v2 if v2 > b
with b a
Let ~L (a; b) = aF (b) and ~H (a; b) = bF(a) + Rab(b v)dF(v)
(Bayesian-Nash) Equilibrium (cuto¤s) a and b
~L (a ; b ) c 0, with equality if a > 0 ~H (a ; b ) c 0, with equality if b < 1
Unique symmetric equilibrium: a = b = vs 2 (c; 1), where
vsF(vs) c = 0
Proposition 1 a) If F(:) is concave, then no asymmetric equilibrium exists (for any c and n).
Proof (by a picture)
n > 2
Local condition
Welfare:
~
S (a; b) =
Z b
a vF (b)dF (v) +
Z 1
b vdF (v) 2
Proposition 2 If F(:) is strictly convex, then the asymmetric equilibrium gen-erates a higher surplus than the symmetric one.
For all c 2 (0;1)
Proof idea:
@S~(a; b)
@a = f (a) [~L (a; b) c] @S~(a; b)
@b = f (b) [~H (a; b) c]
Welfare is decreasing in a and increasing in b for the set of points where
4
Entry with Resale
Allocative ine¢ ciency in the asymmetric equilibrium
Opportunity for resale
Timing: SPA (possibly) followed by an optimal resale auction by the winner
Same type of an equilibrium at the SPA:
– Cuto¤s a and b
4.1
Optimal Resale Auction
Re-seller (bidder 1 with v 2 [a; b]) facing a bidder whose value is distrib-uted on [0; b] with FF(b)(:)
Standard optimal auction (Myerson, 1981)
– BUT, the re-seller’s valuation is NOT commonly known
– Informed principal’s problem
– Yilankaya (1999)
Makes a di¤erence even in the IPV environment
SPA with r is optimal for re-seller with v
r = v + F (b) F (r)
f (r)
Independent of n > 2
r (v) 2 (v; b], r0 (v) 2 (0;1), r (b) = b
Expected payment of bidder w facing r at the resale auction (conditional on winning)
4.2
Equilibrium Bids in the Original Auction
Bid “adjusted values”/gross expected payo¤s inclusive of resale
Opportunity for resale exists i¤ bidder 1 with v 2 [a; b] wins
Calculate expected payo¤ directly (“seller in the optimal auction”)
(v) = b
Z b
v
F(r(x))
F(b) dx
Indirectly (Revenue Equivalence Theorem)
– Probability of keeping the object: FF(r(v))(b)
– Derivative of the expected payo¤
4.3
Equilibrium Participation Decisions
Resale a¤ects not only bids in an auction, but also who participates
Again, just “indi¤erence at cuto¤s”
– More work now: Payo¤ of nonparticipating is not constant (resale)
Bidder 1 with value a
– Out: 0
– In: With probability F(b) get (a)
– Payo¤ di¤erential (gross of the participation cost):
L (a; b) = (a) F (b)
Bidder 2 with value b
– Out: Buy at the resale auction for r
b
Z
a
[b r(v)]dF (v)
– In:
bF (a) +
b
Z
a
[b (v)] dF (v)
– Payo¤ di¤erential:
H (a; b) = bF (a) + b
Z
a
Equilibrium cuto¤s a and b
L (a ; b ) c, with equality if a > 0 H (a ; b ) c, with equality if b < 1
One solution: a = b = vs
– No resale on the equilibrium path
Proposition 3 Suppose that there exists an asymmetric no-resale equilibrium with b > a . There exists an asymmetric resale equilibrium with a < a
Proof idea:
For b > a > 0
L (a; b) = (a) F (b) > aF (b) = ~L(a; b)
H (a; b) = bF (a) + b
Z
a
[r(v) (v)]dF (v)
< bF(a) +
Z b
a (b v)dF(v) = ~H (a; b)
Welfare:
S (a; b) =
Z b
a [F(r (v))v +
Z b
r(v) wdF (w)]dF (v) +
Z 1
b vdF (v) 2
(1 F (a))c (1 F (b))c
Proposition 4 Suppose that there exists an asymmetric no-resale equilibrium with b > a . Suppose further that vfF(v)(v) is weakly increasing. There exists an asymmetric resale equilibrium which generates a higher social surplus than does this asymmetric no-resale equilibrium.
Remark 1 a) vfF(v)(v) is weakly increasing for power functions
5
Uniform Example
No-resale benchmark:
– Unique equilibrium (a = b = pc)
– Our results do not apply
With resale:
If c < 16 : a = 0; b = p6c
If 16 c 14 : a = 0; b = 1
If c > 14 : a = 2pc 1; b = 1
8c: S (a ; b ) S~(a ; b ) > 0
