Risk Aversion and
Optimal Reserve Prices
in First and Second-Price Auctions
Audrey Hu
University of Amsterdam
Stephen A. Matthews
University of Pennsylvania
Liang Zou
University of Amsterdam
The Benchmark Model

Ex ante symmetric buyers



An indivisible object for sale to n
potential risk-neutral buyers
Buyers’ values are independently drawn
ex ante from the same distribution F
Main results (e.g. Myerson 1981)


Revenue (payoff) equivalence
Optimal reserve price is the same in
the standard auctions
Extension to Risk-Averse Buyers

Under the same reserve price:


Higher revenue in FPA than in SPA
(Holt 1980, Maskin and Riley 1984)
Buyers’ preference differs from the
seller (Matthews 1987)
 DARA buyers prefer SPA
 IARA buyers prefer FPA
 CARA buyers are indifferent
The Present Model

Both the seller and the buyers are
risk-averse




Private value v[L,H]
Buyers have the same utility function
uB
The seller has the utility function uS ,
and reservation value v0 ≥ L
Endogenize the reserve price
Symmetric Equilibrium


In SPA, buyers bid up to their value
The bid function in FPA is characterized by:
g (v)u B (v-b)
b1 (v,r)=
G (v)u B ' (v-b)
where G (v)  F n 1 (v); g (v)  G ' (v)

New findings:
1  b1 (v, r )  0 and b2 (v, r )  0.
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Comparative Statics

Let “hat” denote more risk-averse buyers:
b1 (v, r )  bˆ1 (v, r )
b (v, r )  bˆ (v, r )
2
2
The more risk averse the bidders are, the
more rapidly the bid function increases in
a bidder's value, but the more slowly it
increases in the reserve price.
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Seller’s Expected Utility

In the FPA and SPA, respectively,
H
VI (r ) = n  u S (b(v, r ))G (v)dF (v) F (r ) n u S (v0 )
r
v


VII (r )  n  u s (r )G (r )   u s ( y )dG ( y )dF (v)  F (r ) n u S (v0 )
r 
r

H
where the first term comes form the event
of a sale, and the last term from the even
of no sale.
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Marginal Expected Utility

In the FPA and SPA, respectively,
H
VI ' ( r ) =  u S ' (b(v, r ))b2 (v, r )G (v ) dF (v )
(MB I )
r
 nG ( r ) f ( r )[u S ( r )  u S (v0 )]
VII ' ( r )  nG ( r )(1  F ( r ))u S ' ( r )
 nG ( r ) f ( r )[u S ( r )  u S (v0 )]
(MC I )
(MB II )
(MC II )
where the first term is the marginal benefit
of raising the reserve price, and the last
term is the marginal cost at the level of r.
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FPA vs. SPA

The difference between the seller’s
marginal utilities is
VI ' (r )  VII ' (r )
 uS ' (b(v, r )) b2 (v, r )G (v) 
 nG(r )uS ' (r )  
 1dF (v)  0
u S ' (r )
G (r )

r 
H
This is negative because (i) utility function
is concave and (ii) the marginal change in
expected payment as r increases is lower in
the FPA than in the SPA.
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Main Results



The seller's optimal reserve price
decreases in his own risk aversion, and
more so in the FPA.
The seller's optimal reserve price in the
FPA (not in the SPA) decreases in the
buyers' risk aversion.
At the interim stage, FPA is preferred by
all buyer types in a lower interval, as well
as by the seller.
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Efficiency Implications


Both the seller’s and the buyers’ risk
aversion can be a disguised blessing in
terms of ex post efficiency. (Higher risk
aversion leads to higher probability of a
sale.)
The seller and buyers with CARA or IARA
always prefer the FPA to the SPA. At least
some of the DARA buyers (with low values)
also prefer the FPA. In these cases FPA
Pareto dominates SPA.
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