On the optimality of not allocating☆
Introduction
This note illustrates a novel strategic use of the option of not allocating. It has been well known since Myerson (1981) that in order to maximize revenues, the optimal mechanism may require the seller to retain the object. In a setting with externalities, Jehiel et al. (1996) have shown that the seller may be better off not selling at all. In the bargaining literature, it is known that the option of value destruction can be strategically exploited to improve the buyer’s bargaining position; see for instance Dasgupta and Maskin (2007). A common feature of the above papers is that not allocating, or voluntary destroying value, are instruments used by one of the participants in the mechanism to increase his/her own surplus at the expense of that of some other party. Instead, we point out that not allocating can be a tool to increase expected social surplus. This work is part of our research agenda on second best efficiency; see our companion papers Hernando-Veciana and Michelucci, 2011, Hernando-Veciana and Michelucci, 2013. Our approach differs from most of the literature on efficient auctions, which focuses on environments where the first best allocation is feasible; see Maskin (2003) for a review. From a technical point of view, we adapt the ironing techniques introduced by Myerson (1981) to characterize the second best allocation.
Section snippets
The model
One unit of an indivisible good is put up for sale to a set of 2 potential buyers. The seller’s value is assumed to be zero. Let be a vector where corresponds to the realization of an independent random variable with distribution and with a strictly positive density in a bounded support . Buyer privately observes and gains a von Neumann–Morgenstern utility if she gets the good for sale at price , and utility if she does not get the good and pays a
Feasible allocations and first best efficiency
We are interested in the set of allocations that can be implemented. According to the revelation principle, there is no loss of generality when restricting to direct mechanisms. A direct mechanism is a pair of measurable functions , where is an allocation and a payment function. Let an allocation be a measurable function , where , , and where denotes the probability that the good is allocated to when the vector of types is . We say
Second best efficiency and the optimality of not allocating
In our environment the unique symmetric equilibrium of standard auctions (e.g. FPA, SPA, EA) allocates the good to the buyer with highest type, who is the buyer with lowest value. Consequently, standard auctions implement the allocation that induces the lowest expected surplus among the allocations that always allocate the good to one of the buyers. Definition We say that an allocation is second best efficient if it is feasible and it maximizes .
The first best allocation is
Conclusions
We provide a novel rationale for a seller/social planner to credibly commit to retain the object. Interestingly, reserve price and entry fees are not helpful in implementing the most efficient allocation because they are not conditional on the type (or bid) of all the buyers.
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Cited by (2)
Reduced form implementation for environments with value interdependencies
2016, Games and Economic BehaviorInefficient rushes in auctions
2018, Theoretical Economics
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We would like to thank Jacob Goeree and Philippe Jehiel for helpful discussions. Angel Hernando-Veciana also acknowledges the financial support of the Spanish Ministry of Economics and Competitiveness through grant ECO2012-38863.
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CERGE-EI, a joint workplace of Charles University and the Economics Institute of the Academy of Sciences of the Czech Republic, Politickych veznu 7, 111 21 Prague, Czech Republic.