1. By Audrey Hu and Liang Zou University of Amsterdam
Efficient Auctions of Risky Asset to Heterogeneous Risk Averse (Preferring) Buyers
By Audrey Hu and Liang Zou
University of Amsterdam

2. 4/12/2018
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?

3. The VCG Mechanism Ex post (or robust) equilibrium
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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.

4. Existing Studies of Risk Averse Buyers
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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).

5. 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
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6. 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:
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7. 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
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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.

10. 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:
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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 (non- binding) 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.

12. English Auction
The following dropping-out strategies in an English (button-) auction constitute an ex post equilibrium:
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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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Conclusion
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 non- DARA buyers.

15. Conclusion (continued)
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Conclusion (continued)
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.