Auctions with a Profit Sharing Contract

Auctions with a Profit Sharing Contract

Vineet Abhishek,  Bruce Hajek1  and Steven R. Williams Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign.
Contact: abhishe1@illinois.edu and b-hajek@illinois.eduDepartment of Economics, University of Illinois at Urbana-Champaign.
Contact: swillia3@illinois.edu
1footnotemark: 1
Working paper. First draft: February 14, 2011. This draft: July 25, 2012.
Abstract

We study the problem of selling a resource through an auction mechanism. The winning buyer in turn develops this resource to generate profit. Two forms of payment are considered: charging the winning buyer a one-time payment, or an initial payment plus a profit sharing contract (PSC). We consider a symmetric interdependent values model with risk averse or risk neutral buyers and a risk neutral seller. For the second price auction and the English auction, we show that the seller’s expected total revenue from the auction where he also takes a fraction of the positive profit is higher than the expected revenue from the auction with only a one-time payment. Moreover, the seller can generate an even higher expected total revenue if, in addition to taking a fraction of the positive profit, he also takes the same fraction of any loss incurred from developing the resource. Moving beyond simple PSCs, we show that the auction with a PSC from a very general class generates higher expected total revenue than the auction with only a one-time payment. Finally, we show that suitable PSCs provide higher expected total revenue than a one-time payment even when the incentives of the winning buyer to develop the resource must be addressed by the seller.

1 Introduction

Auctions are commonly used to sell resources such as licenses to use a particular wireless spectrum bandwidth or oil and gas drilling rights in a tract of land.111See Cramton (1997) and Milgrom (2004) for reviews of the FCC spectrum auctions, and Porter (1995) and Cramton (2009) for reviews of oil rights auctions. The auction of a resource is different from the auction of an item for consumption. A resource in turn can be developed by the winning buyer to generate profit. Once a buyer develops the resource, the profit may become known to both the buyer and the seller through observable elements such as sales data, quality of the resource, market condition, etc. This raises the possibility of the seller selecting ex-ante as part of his auction a contract that specifies how the net profit will be split ex-post between the winning buyer and the seller.

We compare in this paper the following two forms of payment: (i) the seller either charges the winning buyer a one-time payment at the end of the auction stage; or (ii) the seller receives an initial payment from the winning buyer at the end of the auction stage followed by a  profit-sharing contract (henceforth, PSC) in which he additionally receives a prespecified share of the realized profit from the resource. This is motivated by a current example: the FCC spectrum auctions (e.g., auction 73) are of the first type, while the 3G spectrum auctions in India require that a winning buyer pay a spectrum usage charge equal to a percentage of his profit in addition to the spectrum acquisition fees.222See Federal Communications Commission (2007) and Department of Telecommunications (2008) for details. We investigate whether or not there are economic reasons for the seller to prefer auctions with a PSC over auctions with only a one-time payment. The solution to this problem is nontrivial because strategic buyers adjust their bids in the auction stage in response to the payment they are required to make according to the PSC.

We address this problem using the symmetric interdependent values model of Milgrom and Weber (1982). This model includes two extremes as special cases – the independent private value model and the pure common value model – as well as a continuum of interdependent value models between these two extremes. The value of the resource to a buyer is not known to him before it is developed. However, each buyer has some information about the resource, known only to him. This information may be informative to other buyers as it can refine their respective estimates. The value of the resource to the winning buyer is publicly observable once it is developed by him. We work with risk averse or risk neutral buyers and a risk neutral seller; the seller of the resource is usually a large organization (e.g., the government of a country) with large cash reserves and possibly owns multiple such resources.

Our prime focus is on two simple PSCs. First is a profit only sharing contract (henceforth, POSC) where the seller takes a fixed fraction of the positive profit from the winning buyer but does not take any negative profit (loss) from him. Second is a profit and loss sharing contract (henceforth, PLSC) where the seller takes a fraction of both positive and negative profit from the winning buyer. For the second price and the English auctions, we show that:

  1. The seller’s expected total revenue from the auction with a POSC or a PLSC is nondecreasing in the share fraction , and in particular, is higher than the expected revenue from the auction with only a one-time payment (i.e., ). Hence, there are economic reasons to prefer a POSC or a PLSC over only a one-time payment.

  2. For the same share fraction , the auction with a PLSC generates higher expected total revenue than the auction with a POSC. We show that the revenue superiority of the PLSC for a given  over the corresponding POSC primarily stems from the joint effect of positive dependence between the values and the signals that is assumed in this paper, and the nature of PLSC and POSC; the effect of risk aversion is aligned with these factors. This revenue superiority is shown to hold strictly in an example in Section 3.3 with risk neutral buyers.

  3. We leverage the intuition gained from the analysis of the POSC and PLSC to move beyond simple PSCs and show that the auction with a PSC from a very general class generates higher expected total revenue than the auction with only a one-time payment. Moreover, the PLSC with the share fraction is revenue optimal over the general class of PSCs for which the seller’s marginal share is bounded by . The seller can therefore accomplish all of his goals in revenue enhancement using linear sharing contracts.

An obvious question is: “If the value of the resource ultimately becomes known to the seller, then why not simply tax ex-post the entire profit from the resource, subject only to participation constraints on the buyers?” This scenario in fact results by letting the share fraction in the POCS or PLSC approach one.333The case of is mathematically degenerate in our model. Since the seller takes the entire profit ex-post from the winning buyer, a buyer will either decline to participate or place any bid that gives him zero profit. The dependence of a buyer’s bid upon his signal in our analysis of the case of therefore does not apply for . There are several reasons why it is worthwhile to consider a share fraction  less than one and other nontrivial PSCs between the seller and the winning buyer, some of which are enumerated in Section 5. A primary reason is that development of the resource requires effort or expertise from the winning buyer; otherwise, the seller might not need to sell the resource. Section 7 models the relationship between the winning buyer and the seller using a principal-agent model. We show through an example that the revenue maximizing share fraction  in either a POSC or a PLSC can be strictly less than one because of the necessity of providing proper incentives to the winning buyer to develop the resource. However, a one-time payment is never revenue optimal: we show that in our principal-agent model, the auction with a PLSC and a suitably small share fraction generates higher expected total revenue than the auction with only a one-time payment.444As explained in Section 7, it is difficult to analyze more general PSCs or even the POSC in a principal-agent relationship. A numerical example in Section 7 suggests, however, that: (i) as with the PLSC, a POSC with sufficiently small share fraction produces more expected revenue for the seller than a one-time payment; (ii) for fixed , a POSC causes the winning buyer to choose a larger effort than the corresponding PLSC and may thereby produce a larger expected revenue for the seller. The generality of these two observations remains to be explored. The advantages of the PSCs thus carry over to the case in which seller must provide the winning buyer with incentives to develop the resource.

Related work: There are a number of approaches in the literature to increase the seller’s revenue, either by exploiting the informational structure or by making payments contingent on the ex-post realization of the values. The distinguishing aspects of our work are: allowing for risk aversion among buyers; a substantially more general informational environment;555While most of the revenue ranking results in auction theory are through applications of the linkage principle (see Krishna (2002)) and assume various smoothness conditions, our proofs rely entirely on the properties of concave functions and stochastic ordering with no or only mild assumptions of smoothness. consideration of very general transfers ex-post; analysis for the second price and the English auctions; and consideration of the incentives of the winning buyer to develop the resource.

There are two sets of papers especially relevant to our work. The first set of papers use a mechanism design approach to maximize revenue for the seller. For the cases of statistically dependent private values, a common value model in which the types of the buyers are independent conditional on the common value, and interdependent values with statistically dependent types, the seller can extract the full surplus from the buyers using one-time payments.666See Crémer and McLean (1988, 1985), McAfee et al. (1989), and McAfee and Reny (1992) for details. The Wilson critique of mechanism design (Wilson (1987)) is useful for distinguishing our approach from the papers on full surplus extraction. While these papers define transfers between the seller and the buyers in terms of the probability distribution of their types, our approach instead uses two common auctions that are defined without reference to the buyers’ beliefs.777Except in the special case of independent private values, however, the two auctions that we consider require that the buyers have common knowledge of their beliefs in formulating their bids. The work on full surplus extraction made honest reporting into a Bayesian-Nash equilibrium; while making an honest report does not require that common knowledge of beliefs among buyers, this common knowledge is required if a buyer is to verify that honest reporting is incentive compatible. Along with a fractional sharing contract, these auctions are the sort of simple and familiar trading procedures that Wilson advocates.

Mezzetti (2007) considers a model in which the value of the item to any risk neutral buyer is a function of his type and the types of the other buyers; a buyer who receives the item realizes its value ex-post. Mezzetti proposes a two-stage reporting procedure in which all buyers first report their types followed by a subsequent report by the winning buyer of the item’s realized value. All losing buyers are compelled to pay a large fine ex-post if the reported value of the winning buyer is inconsistent with the earlier reports by the losing buyers. This insures honest reporting by all buyers and allows full surplus extraction. Practically, however, one must question the willingness of buyers to participate in a mechanism in which they may be required to pay a large fine ex-post depending upon: (i) the reports of the other buyers in the first stage; (ii) the ex-post report of the winning buyer who, while having no incentive to lie, also has no incentive to be honest.

The second set of papers use ex-post transfers to enhance the revenue of the seller. Hansen (1985) constructs examples of an oil tract sale and the sale of a firm to other firms. It is shown in each example that the seller benefits by retaining a share of the profit from the future enterprise. The examples are restricted to the cases of independent private values with risk neutral buyers. Riley (1988) shows in a symmetric interdependent values model that if the value of a resource is observed ex-post, even if imperfectly, then the seller can increase his expected revenue in a first price auction by either requiring a winning buyer to pay a predetermined royalty or by having the buyer establish a royalty rate through bidding. Along with using an auction mechanism different from what we use, Riley’s analysis is restricted to risk-neutral buyers and linear ex-post transfers.

Ex-post information is also used by McAfee and McMillan (1986) and Laffont and Tirole (1987) to analyze procurement auctions in which the winning bidder privately exerts effort to reduce his costs. This raises the principal-agent issue that is mentioned above. These papers demonstrate that the auctioneer can profit from contracting a cost-sharing arrangement with the winning bidder. Both papers, however, are limited to the case of independent private values, which is ill-suited to modeling the sale of resources such as spectrum bandwidth or mineral rights. Moreover, each paper assumes specific utility functions for the bidder.888McAfee and McMillan (1986) assumes that each bidder either has the same constant absolute risk aversion (CARA) function as his utility of money or each bidder is risk neutral, while Laffont and Tirole (1987) assumes risk neutrality. McAfee and McMillan’s cost sharing contract is closely related to the PLSC, though they consider the first price auction. Our analysis of the PLSC in a principal-agent relationship is for the second price auction and the English auction.

DeMarzo et al. (2005) considers the first and the second price auctions in which bids are selected from a completely ordered family of securities whose ultimate values are tied to the resource being auctioned. DeMarzo et al. define a partial ordering based on the notion of steepness and show that the steeper family of securities provides higher expected revenue to the seller. The two stage payment rules that we consider can be viewed as securities and our revenue ranking of different PSCs is consistent with their idea of steepness. Our paper is more general in allowing for risk aversion among buyers and in its substantially less restrictive informational assumptions. The generality of our informational model is significant for more than mathematical reasons: Abhishek et al. (2011) shows that the revenue ranking of DeMarzo et al. (2005) does not necessarily hold if its informational environment is relaxed. Thus, while our results are consistent with theirs, they do not follow from their results. We also complement their work by analyzing the English auction.

Outline of this paper: The rest of this paper is organized as follows. Section 2 outlines our model, notation, and definitions. Section 3 analyzes the second price auction with a POSC and with a PLSC, obtains an equilibrium strategy of the buyers, and establishes the revenue consequences of these two PSCs. Section 4 extends the results of Section 3 to the English auction. Section 5 provides comments and extensions of our model. Section 6 studies the revenue consequences of general PSCs. Section 7 analyzes auctions with PSCs in a principal-agent relationship. We conclude in Section 8.

2 Model and Notation

Consider buyers competing for a resource that a seller wants to sell. The value of the resource to a buyer is a realization of a random variable , unknown to him. This is the profit to buyer  from developing the resource in the absence of any payments to the seller, but after taking into account the variable costs. The ’s can take negative values, with the interpretation as loss incurred from developing the resource. Buyer privately observes a signal through a realization of a random variable that is correlated with . The joint cumulative distribution function (CDF) of the random variables ’s and ’s is common knowledge.

Let denote a vector of values; denote the random vector by . A vector of signals and the random vector are defined similarly. We use the standard game theoretic notation of . Similar interpretations are used for , , and . Let denote the joint CDF of . It is assumed to have the following symmetry property:

Assumption 1.

The joint CDF of , denoted by , is identical for each and is symmetric in the last components .

Assumption 1 allows for a special dependence between the value of the resource to a buyer and his own signal, while the identities of other buyers are irrelevant to him. The model reduces to the independent private values model if is independent of for all , to the pure common value model if , and includes a continuum of interdependent value models between these two extremes. Because of Assumption 1, the subsequent assumptions and the analysis in the paper is given from buyer ’s viewpoint.

The set of possible values that each random variable can take is assumed to be an interval . Assume that the joint probability density function (pdf) of the random vector , denoted by , exists and is positive for all . Notice that is symmetric in its arguments. The random variables are not required to have a joint pdf. In particular, this allows to take discrete values or to be a deterministic function of .

Let larger numerical values of the signals correspond to more favorable estimates of the value of the resource. Mathematically, we assume the following form of positive dependence between the value of the resource and the buyers’ signals:

Assumption 2.

For any increasing function , ’s, and ’s such that and for all , the conditional expectation is increasing in and , and nondecreasing in and whenever it exists.999Throughout this paper, “increasing” means “strictly increasing” and “decreasing” means “strictly decreasing”. While the results of the paper extend to the case in which the conditional expectation in Assumption 2 is nondecreasing in the ’s and ’s, the analysis is more complicated.

Assumption 2 trivially implies that the random variable conditioned on is increasing in  and nondecreasing in  in the sense of first order stochastic dominance (henceforth, FOSD). Assumption 2 provides a common sufficient condition for the existence of a pure strategy equilibrium in both the second price and the English auctions; it actually can be relaxed into two weaker conditions based on FOSD, one for the analysis of the second price auction (Lemma 1) and one for the English auction (Lemma 6).101010Assumption 2 is weaker than affiliation, which is commonly assumed in the auction literature.

An auction with a PSC has two stages:

  1. The auction stage: The first stage is an auction to decide who should get the resource and to determine the initial payment to the seller. We focus on the second price sealed bid auction (henceforth, just the second price auction) and the English auction.111111There are many variants of the English auction. We work with the one used in Milgrom and Weber (1982). In a second price auction, the buyer with the highest bid wins and pays the amount equal to the second highest bid. An English auction is an ascending price auction with a continuously increasing price. At each price level, a buyer decides whether to drop out or not. The price level and the number of active buyers are publicly known at any time. The auction ends when the second to last buyer drops out and the winner pays the price at which this happens.

  2. The profit sharing stage: The second stage is a PSC. We use the term preliminary profit to denote the net obtained after subtracting the auction stage payment made by the winning buyer from the value of the resource. The seller takes a fixed fraction of the preliminary profit (or loss) from the winning buyer.121212For simplicity, we only consider at this point single period profit realization with no discounting. Profit realization over multiple time periods with discounting is discussed in Section 5. We assume that the preliminary profit is observed by both the seller and the winning buyer.

    A POSC and a PLSC are each specified by a share fraction , known to the buyers in the auction stage. If the winning buyer makes a payment  at the end of the auction stage and the value of the resource is revealed to be , then the payment he makes in the second stage in the POSC is (here, ), while the payment in the PLSC is . Notice that corresponds to having no profit sharing stage, i.e., the winning buyer makes only a one-time payment at the end of the auction stage.

The buyers are assumed to be risk averse or risk neutral. Each buyer has the same von Neumann-Morgenstern utility of money, denoted by , which is concave (possibly linear, as in the case of risk neutrality), increasing, and normalized so that . Henceforth, we use the term weakly risk averse to refer to risk averse or risk neutral behavior. The utility function  is over the total profit from the two stages. The seller is assumed to be risk neutral.

In what follows, we assume that all expectations and conditional expectations of interest exist and are finite. Moreover, conditioned on any signal vector , the expected utility of a buyer from developing the resource without any payments is assumed to be positive; i.e., . Thus, the buyers who are competing for the resource expect to make a positive profit from developing it and therefore willing to participate.

Finally, define the random variables to be the largest, second largest, , smallest among . Let and denote a realization of by . Henceforth, in any further usage, , , and are always in the support of random variables , , and , respectively, for and ; and are always in the support of the random vectors and , respectively.

3 The Second Price Auction with a Profit Sharing Contract

This section characterizes the equilibrium bidding strategies in the second price auction with a POSC and a PLSC, and then evaluates the revenue consequences for the seller. We look for a symmetric equilibrium.

The following lemma is a consequence of Assumption 2 and is used extensively in this section.

Lemma 1.

is increasing in and nondecreasing in for any increasing function for which the expectation exists.

Proof.

From the definition of , if and only if at least one of the ’s, , is equal to . Since exists and is positive everywhere, conditioned on , the probability that two or more ’s are equal to is zero. Hence, the event can be thought of as the union of disjoint events, one for each , , such that . By Assumption 1, conditioned on , each of these events occur with equal probability. Therefore,

The result then immediately follows from Assumption 2. ∎

3.1 Profit only sharing contract (POSC)

In a POSC, the winning buyer pays a fraction of any preliminary positive profit to the seller. However, the seller does not bear any share of a loss. We start by defining a function that will be used to characterize the bidding strategies of the buyers:

(1)

Since is decreasing and continuous in , there is a unique that makes the expectation in (1) equal to zero. The function is therefore well-defined. It can be interpreted as follows. If buyer makes a payment in the auction stage, his payment in the POSC stage will be and his total profit will be . Thus, is the payment in the auction stage at which the overall expected utility of buyer  is zero, conditioned on his signal being , and the highest signal of other buyers being .

The next lemma characterizes some important properties of .

Lemma 2.

The function is increasing in , nondecreasing in , decreasing in , and positive for small values of . Moreover, for all , , and ,

(2)

and the inequality is strict everywhere unless and is linear.

Proof.

Since is increasing in and is increasing, Lemma 1 implies that is increasing in and nondecreasing in . It immediately follows from (1) that is increasing in and nondecreasing in . Again, since is decreasing in for and nonincreasing in for , (1) implies that is decreasing in .

We assumed in Section 2 that , and so

It follows from (1) that . It is easy to see that is continuous with respect to . Hence, is positive for small values of .

Finally, (1) and an application of Jensen’s inequality gives

where the first inequality is strict if is strictly concave and the last inequality is strict if . ∎

The results of Lemma 2 can be interpreted as follows. Larger values of and imply that is likely to take larger values. Hence, the maximum payment that buyer is willing to make in the auction stage, assuming he also knows , is increasing in and nondecreasing in . A larger value of corresponds to the seller taking a larger fraction of any preliminary positive profit from the winning buyer. Buyer  compensates for this in the auction stage by lowering the maximum amount he is willing to pay; i.e., is decreasing in . Finally, inequality (2) states that the maximum amount that buyer is willing to pay for the resource given and is no more than the expected value of the resource given and , with equality holding only in the case of risk neutrality and no profit sharing in the second stage.

The strategy of a buyer is a mapping from his signal to his bid. Let buyer use strategy  for . The next lemma characterizes an equilibrium bidding strategy for the second price auction with a POSC. The construction here is similar to the derivation in Milgrom and Weber (1982).

Lemma 3.

Let the strategies be identical and defined by for all . Then the strategy vector is a symmetric Bayes-Nash equilibrium (BNE) of the second price auction with the POSC determined by . This is the unique symmetric equilibrium with an increasing and differentiable strategy.

Proof.

Assume that each buyer uses the strategy . We will show that the best response for buyer is to also use the strategy .

Given , let buyer bid . Buyer wins if .131313Since the probability of a tie is zero, we ignore the issue of tie breaking without any effect on the analysis. From Lemma 2, , where . Thus, the expected utility of buyer is:

From (1) and Lemma 2, is positive for and negative for . Thus, the expected utility is uniquely maximized by setting .

Finally, consider an arbitrary symmetric equilibrium strategy vector such that is increasing and differentiable in . We show that uniquely solves buyer ’s first order condition for maximizing his expected utility given the use of by all other buyers. Consequently, is the unique symmetric equilibrium. Define a function as:

(3)

Denote the conditional pdf of given by . Given , let buyer bid . The expected utility of buyer is:

The first order condition for an optimal bid gives

(4)

The second line holds because is positive for all in the support of , and is increasing in because is increasing in . The third line is from (3) and the last line is from (1). The proof is completed by noticing that is an optimal bid because the strategy vector constitutes a symmetric BNE. ∎

It is clear from (1) that may be negative for larger values of . This corresponds to the case where buyer with signal finds the expected net gain from developing the resource after accounting for the part of the positive profit to be paid to the seller, too small to compensate for the potential loss associated with developing the resource. In such a case, buyer will not participate in the auction unless the seller pays him upon winning the auction.141414We thus require that the auction be interim individually rational. The seller, however, expects to gain from the profit sharing stage. He may thus allow the buyers to submit negative bids and make negative payments upon winning, as long as the seller’s expected revenue is positive. In a second price auction with negative bids allowed, if the second highest bid is negative, by charging the winning buyer an amount equal to the second highest bid, the seller effectively pays the winning buyer in the auction stage. We show in Proposition 1 that with our model assumptions, the seller’s expected total revenue is always positive under this scheme. Hence, we eliminate the issue of buyers dropping out by allowing the buyers to submit negative bids.

By symmetry, the seller’s expected revenues from the auction stage and from the POSC stage are the same as the expected payments made by buyer in the auction stage and in the POSC stage, respectively, conditioned on him winning the auction. In the symmetric equilibrium given by Lemma 3, buyer wins if his signal is highest among all the buyers. Thus, the seller’s expected revenue from the auction stage is , his expected revenue from the POSC stage is , and his expected total revenue from both stages, denoted by , is:

(5)

Taking corresponds to no POSC stage, i.e., the second price auction with only a one-time payment.

Proposition 1 below summarizes the revenue consequences of the second price auction with a POSC. The proof is in Appendix A.

Proposition 1.

The following statements hold for the second price auction with a POSC and weakly risk averse buyers:

  1. The seller’s expected revenue from the auction stage (possibly negative) is decreasing in the share fraction , while the expected revenue from the POSC stage is positive and increasing in the share fraction .

  2. The seller’s expected total revenue from the two stages is positive and is nondecreasing in the share fraction ; i.e., for any ,

    (6)

    In particular, the expected total revenue from the two stages is higher than the expected revenue from the second price auction with only a one-time payment (i.e., ).

Inequality (6) holds even for values of close to one. At these values, the second highest bid can be negative and the seller might have to pay the winning buyer in the auction stage. However, the seller extracts a large expected revenue from the POSC stage and his expected total revenue is still positive.

3.2 Profit and loss sharing contract (PLSC)

Next, we consider the case where the seller takes a fraction of both any preliminary positive profit as well as any preliminary loss from the winning buyer. As in Section 3.1, we define a function that will be used to characterize the bidding strategies of the buyers:

(7)

Since is decreasing and continuous in , there is a unique that makes the expectation in (7) equal to zero. The function is therefore well-defined. It can be interpreted as follows. If buyer makes a payment  in the auction stage, his payment in the PLSC stage will be , and his total profit will be . Thus, is the payment in the auction stage at which the overall expected utility of buyer is zero, conditioned on his signal being , and the highest signal of other buyers being .

The next lemma characterizes some important properties of .

Lemma 4.

The function is increasing in , nondecreasing in , nondecreasing in (increasing if is strictly concave), and positive everywhere. Moreover, for all , , and ,

(8)

where is defined by (1). The left inequality above is strict everywhere except for , and the right inequality is strict if is strictly concave.

Proof.

From Lemma 1, is increasing in and nondecreasing in . Then (7) implies that is increasing in and nondecreasing in . Next, let and  be such that . Since is concave,

(9)

where the inequality is strict if is strictly concave. From (7) and (3.2),

is a decreasing function of . Then from (7), we must have . This shows that is nondecreasing in (increasing in if is strictly concave).

From (1) and (7), . Since is decreasing in while is nondecreasing in , we get . Hence, is positive everywhere. Moreover, , where the equality holds only at . Finally, Jensen’s inequality and (7) imply

which establishes the second inequality (strict inequality if is strictly concave). ∎

Recall from Lemma 2 that the payment that makes buyer indifferent to winning in a POSC given and is nonincreasing in the share fraction ; buyer can only be willing to pay less as the seller takes a larger share in the POSC stage. The corresponding payment in the PLSC, however, can only remain constant or increase as the share fraction increases. This is because a larger not only reduces his preliminary positive profits but also shields him from preliminary losses. If buyers are strictly risk averse, then a larger allows buyer to make a strictly larger payment, reflecting the additional benefit that he receives from transferring risk to the seller.

As in Section 3.1, let denote the strategy of buyer , for each . The next lemma characterizes an equilibrium bidding strategy for the second price auction with a PLSC.

Lemma 5.

Let the strategy of a buyer be . Then the strategy vector is a symmetric BNE of the second price auction with the PLSC determined by . This equilibrium is the unique symmetric equilibrium with an increasing and differentiable strategy.

Proof.

The proof is similar to that of Lemma 3 and is obtained by using the function instead of the function . ∎

The equilibrium bid in a PLSC is always positive. Unlike a POSC, the issue of buyers either dropping out or the seller allowing the buyers to submit negative bids does not arise here.

Again, by symmetry, the seller’s expected revenues from the auction stage and from the PLSC stage are the same as the expected payments made by buyer in the auction stage and in the PLSC stage, respectively, conditioned on him winning the auction. In the symmetric equilibrium given by Lemma 5, buyer wins if his signal is highest among all the buyers. Thus, the seller’s expected revenue from the auction stage is , his expected revenue from the PLSC stage is , and his expected total revenue from both stages, denoted by , is:

(10)

Again, corresponds to the second price auction with only a one-time payment, and so .

Proposition 2 below summarizes the revenue consequences of the second price auction with a PLSC. The proof is in Appendix B.

Proposition 2.

The following statements hold for the second price auction with a PLSC and weakly risk averse buyers:

  1. The seller’s expected revenue from the auction stage is positive and is nondecreasing in the share fraction . The expected revenue from the PLSC stage is also positive.

  2. The seller’s expected total revenue from the two stages is positive and is increasing in the share fraction ; i.e., for any ,

    (11)

    In particular, the expected total revenue from the two stages is higher than the expected revenue from the second price auction with only a one-time payment (i.e., ).

A natural question now is: How does the second price auction with a PLSC compare with the second price auction with a POSC in terms of the expected total revenue they generate for the same share fraction ? The two procedures are identical at . For , since , the seller’s expected revenue from the auction stage is higher in the case of a PLSC than a POSC. However, . Thus, the seller’s expected revenue from the profit sharing stage is higher in the case of a POSC than a PLSC. As a result, it is not immediately clear how the expected total revenue from the two stages in the PLSC compares with the expected total revenue from the POSC. The next proposition resolves this issue. It shows that in the PLSC, the gain from the higher bidding in the auction stage outweighs the loss in revenue in the PLSC stage. The proof is in Appendix C.

Proposition 3.

The second price auction with a PLSC and weakly risk averse buyers generates higher expected total revenue than the second price auction with a POSC and weakly risk averse buyers; i.e., for any ,

(12)

3.3 Revenue ranking for risk neutral buyers

The seller is risk neutral in our model while the buyers are weakly risk averse. The return to the winning buyer is less risky in the PLSC determined by than in the corresponding POSC. In the case of strictly risk averse buyers, the PLSC thus creates more potential gains from risk sharing than the POSC. This suggests that the ranking (12) reflects a successful effort by the seller to garner a portion of these gains in the form of higher revenue by using a PLSC. We explain below, however, why (12) holds even in the case of risk neutral buyers as a consequence of the form of positive dependence among signals and return assumed in Section 2, and the nature of PLSC and POSC. This is followed by an example in which (12) holds strictly in the case of risk neutral buyers. The effect of risk sharing on revenue is thus aligned with but not the primary cause of this ranking.

Fix a share fraction and consider , i.e., buyer wins in both the POSC and the PLSC. Let and be the ex-post total revenue to the seller from the POSC and the PLSC respectively, if the value of the resource is equal to :

(13)
(14)

Buyer in each case pays in the auction stage the bid of the buyer who observed . This is equal to in the POSC and in the PLSC. In our symmetric model with risk neutral buyer (i.e., ), and are the bids that would make buyer indifferent to winning conditioned on in the POSC and the PLSC, respectively:

(15)
(16)

Equations (15) and (16) together imply that the difference satisfies:

(17)

The seller would thus expect to receive the same revenue from the PLSC as in the POSC if the highest and the second highest signals are the same, i.e., . The event in which he sells to buyer , however, is defined by . The difference is a line with slope for and constant for ; it is therefore nondecreasing for all and increasing for . It can then be shown using Lemma 1 the left side of (17) is nondecreasing in buyer ’s signal ; i.e.,

for . The PLSC with share fraction therefore generates a higher expected revenue than the corresponding POSC.

Example 1.

Consider two risk neutral buyers. The signals and are independent and uniformly distributed in . Given the signals of the buyers, the value of the resource to buyer and buyer are Bernoulli random variables  and respectively, with and . Clearly, Assumptions 1 and 2 are satisfied. Also, .

For the second price auction with the POSC determined by , the function is obtained as follows:

Thus, buyer bids if his signal is and buyer bids if his signal is . The seller’s expected revenue from the auction stage is:

Here, we have used the fact that the pdf of is for . The seller’s expected revenue from the POSC stage is:

Since , the above expression can be further simplified to

With some further (tedious) algebra, one can get closed form expressions for the seller’s expected revenue from the auction stage and the seller’s revenue from the POSC stage.

For the second price auction with the PLSC determined by , the function is obtained as follows:

As a consequence, each buyer bids his signal. The seller’s expected revenue from the auction stage is: