Efficient Auctions of Risky Asset to
Heterogeneous Risk Averse (Preferring)
Buyers
By Audrey Hu and Liang Zou
University of Amsterdam
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Introduction
• Suppose a troubled bank (asset) is put forth for
sale through an auction.
▫ Valuation of the asset is difficult, involving
substantial risks in future payoffs.
▫ Potential buyers are heterogeneous in risk
preferences and possess private information.
• How to sell the asset efficiently?
• How varying degrees of risk or buyers' risk
preferences may affect an auction's
performance?
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The VCG Mechanism
• The Vickrey (1961), Clarke (1971), and Groves (1973)
mechanisms have two distinct advantages:
▫ Ex post (or robust) equilibrium
▫ Ex post (Pareto) efficiency
• In other auctions contexts, efficiency may also be
achieved through judiciously designed mechanisms. E.g.,
Cremer and McLean (1985, 1988), Maskin (1992),
Dasgupta and Maskin (2000), Perry and Reny (2002).
• All these studies assume risk neutrality in income.
Consequently, little can be said in these models about
the effects of risk and/or buyers’ attitudes toward risk.
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Existing Studies of Risk Averse Buyers
• Existing studies in auctions typically assume that the
buyers have the same utility function and the auctioned
object carries no ex post risk. E.g., Holt (1981), Riley and
Samuelson (1981), Maskin and Riley (1984), Hu,
Matthews, and Zou (2009).
• Cox, Smith, and Walker (1982, 1988) considered
heterogeneous bidder risk aversion in a private values
model without ensuing ex post risk.
• Eso and White (2004) studied ex post risk in a general
symmetric model of Milgrom and Weber (1982),
assuming buyers have the same utility function (DARA).
This Model
• We extend the VCG mechanisms to a more general
context in which asymmetric buyers have
▫
▫
▫
▫
Private types (risk averse, neutral, or preferring)
Interdependent values
Limited common knowledge
Indirect utility as follows
~
~
U i ( w; hi )  Eu(v (hi , hi )  x  w; hi )
i
where v (h)  v (h) whenever h j  hi
j
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i
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Ordering by Risk Tolerance
• We say that a set of utility functions are ordered
by risk tolerance (ORT) if their marginal utilities
for income satisfy strict log-supermodularity:
u : R  H  R where H is some totally ordered set;
(i) u1 ( x, y )  0 for all ( x, y )  R  H
(ii) ( x, y ), ( x' , y ' )  R  H , x  x' and y  y ' imply
u1 ( x, y )u1 ( x' , y ' )  u1 ( x, y ' )u1 ( x' , y )
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Theorem 1
• ORT implies that u(x,y’)-u(x,y) is a strictly
convex function of u(x,y) whenever y’>y.
• Further normalizing the marginal utilities of
u(x,y) and u(x,y’) to be the same at the “status
quo” level of zero, then
~ with E ( w
~ )  , and a  0,
For random income w
~, y )  u (a, y ) implies
Eu( w
~
~
Eu( w, y ' )  u (a, y ' )  Eu( w, y )  u (a, y )
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Ex Post Efficiency
• Define ex post efficiency to be an outcome that
maximizes the sum of all buyers’ expected utility
surplus.
• By Theorem 1, ex post efficiency is attained if the
risky asset is sold to the buyer who is least risk
averse (or most risk tolerant).
• Ex post efficiency implies an ex post equilibrium
in that no one wishes to trade further once all
information becomes public.
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Theorem 2
• Assume that the seller has the knowledge about
the buyers’ utility and value functional forms as
in a VCG context (although not the buyers’
private types).
• Assume that the buyers do not (necessarily)
know each other’s value function, but they have
the common knowledge about the seller’s
knowledge.
• Then the seller can use a direct mechanism and
implement an efficient asset allocation in ex post
equilibrium.
Direct Mechanism
•
•
•
•
Each bidder reports his type.
The highest type wins the asset.
Random resolution of a tie.
The payment of the winner is t(2) solving:
( 2)
( 2)
~
Eu( x  v (h( 2) , h(1) )  t ; h( 2) )  u(0, h( 2) )
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Theorem 3
• Assume that the seller knows nothing about the buyers
utility and value functional forms.
• Assume that the buyers share common knowledge about
each other’s utility and value functional form as in
Dasgupta and Maskin (2000).
• Then the seller can sell the asset efficiently through an
English (button-) auction with a sufficiently low (nonbinding) reserve price.
• In both Theorems 2 and 3, no assumption is needed as to
the players’ knowledge about the joint distribution of the
buyers’ private types.
English Auction
• The following dropping-out strategies in an
English (button-) auction constitute an ex post
equilibrium:
i
i
~
Eu( x  v (hi ,..., hi , h( n  m ) ,..., h( n ) )  t ; hi )  u (0, hi )
given the history of h( n  m ) ,..., h( n ) .
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Theorem 4
• Consider a mean-preserving spread that
increases the asset’s payoff risk.
• Assume either that the buyers exhibit CARA or
DARA, or that the added noise has a log-concave
density function and the value function is
constant.
• Then all buyers are strictly better off at the
interim stage with higher asset’s risk.
• A fundamental reason for this “increasing risk
effect” is that the potential winner’s expected
utility surplus is a convex function of the pivotal
bidder’s utility.
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• Ex post equilibrium as a solution concept has a strong
theoretical foundation (Bergemann and Morris, 2005).
This study shows that the concept is applicable to a
broader context allowing the players to possess
heterogeneous risk preferences.
• The paper contributes a theoretical framework in which
heterogeneous risk attitudes can be conveniently
studied.
• The results render further support for the use of English
auctions in practice.
• Another insight gained from this study is that the
preference of buyers for higher risk derives from their
heterogeneity in risk preferences, extending that of Eso
and White (2004) to risk preferring, neutral, and nonDARA buyers.
Conclusion
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• For experimental studies about the effect of ex post risk
in auctions, allowing subjects to differ in their risk
preferences will facilitate drawing conclusions as this
personal difference is perhaps one of the most difficult
dimension to control.
• A promising line of future research is to see whether, and
to what extent, the results obtained for the single risky
asset case can be extended to situations in which there
are multiple units of a homogeneous, or heterogeneous,
risky assets for sale.
• New issues may arise as to those related to the notion of
efficiency, the implied income effects, the possible effects
of diversification, and so on.
Conclusion (continued)