Endogenous budget constraints in auctions

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Abstract

In prior literature, bidders' budget constraints have been shown to change revenue and efficiency rankings among auction formats. These results, however, are based on the implicit assumption that the nature of the budget constraint is unaffected by auction rules. I extend the standard symmetric model of auctions for a single good to include principals that optimally constrain their bidder to mitigate an agency problem between the two. I show that the outcomes of the first- and second-price auctions generally agree with those from auction models without budget constraints with the exception that the first-price auction is shown to be more efficient when signals are affiliated.

Introduction

Bidders likely face budget constraints in many real-world auctions, especially in the sale of valuable assets such as wireless spectrum (Bulow et al., 2009, Cramton, 1995, Salant, 1997), and these constraints potentially have important strategic effects on the outcomes of auction models that cannot be captured in standard frameworks.

Current literature on the subject argues that incorporating budget constraints into the standard independent private values model invalidates some well-known results like the revenue equivalence theorem (Myerson, 1981, Riley and Samuelson, 1981). For example, in a model where bidders valuations are private and i.i.d. Che and Gale (1998) show that the first-price auction both raises more revenue and is more efficient than the second-price auction with budget constraints. Further work extends these results to show that the all-pay auction dominates the first-price auction in terms of revenue and efficiency (e.g., Che and Gale, 1996, Maskin, 2000, Pai and Vohra, 2014).

The earlier literature offers various explanations for the underlying cause of the budget constraints, including imperfect capital markets and agency problems (Che and Gale, 1998).2 However, these papers derive their results from models where the budget constraint is treated as an exogenous random variable.3 A potential advantage of this approach is that it allows one to be agnostic about the source of the budget constraints and focus on the strategic effects introduced by the constraints, but it ignores the possibility that the process generating the budget constraints may be affected by a change in auction rules.4

If one tries to describe explicitly an agency problem that generates budget constraints for the bidders, it seems that a description of the auction rules should be included as well. Explicitly including a description of the auction rules in the agency problem would allow the budget to vary according to the rules, an effect that cannot adequately be captured in a model that treats the budget constraints as a primitive. The purpose of this paper is to explore how budget constraints might vary between different auction formats when the mechanism generating the budget constraint is made explicit.5

I develop a model where the bidder's budget constraint is the endogenous result of an agency problem between the bidder and a principal responsible for funding the bidder's bid (Section 3). The details can be summarized by the following situation. An item is auctioned to one of N firms. Within each firm there is a manager interested in purchasing the asset, but the manager must get funding approval from the firm's board of directors.6 The board bases its decision on a noisy signal of the asset's value, expecting the manager to have better information about the asset's value at the auction, perhaps due to his specialized knowledge about the industry the asset will operate in. Although the manager receives equity compensation, an agency problem arises because the board of directors knows that the manager will tend to overpay for the asset relative to its true value to the firm because the manager has an empire-building motive or simply receives some private payoff from managing the asset.7

Results from the model suggest that budget constraints do not invalidate the standard independent private values results from the auction literature when they are treated as endogenous choices. Restricting attention to the case of independent signals between bidders, I find no difference in the expected revenue or efficiency between the first- and second-price auctions (Theorem 1). Although a special case, the independence case corresponds to much of the existing literature on budget constraints.

In the more general case where bidders' information is allowed to be affiliated I can partially characterize the relative performance of the first- and second-price auctions, showing that the first-price auction is more efficient (Theorem 3). The effect on revenue is less clear due to the complicated nature of the model and two counteracting effects. However, when the principal's information about the bidder's signal is precise, the budgets chosen take the form of bids in the Milgrom and Weber (1982) model and as in that model the second-price auction raises more expected revenue (Proposition 2).

Much of the intuition for the results comes from reframing the problem for the principals in terms of one where the principals decide on what types of bidders should be constrained. In this setup, it becomes clear that after conditioning on the expected bids of the different types of bidders, the principal expects each type of bidder to make the same expected payment in the first- and second-price auction when bidder's signals are independent, so she has no incentive to constrain them differently.

With affiliated signals, there is an incentive to relax the constraint (by constraining fewer types of bidders) in the first-price auction relative to the second-price auction leading to the efficiency result. The intuition for this follows from logic similar to that behind the Linkage Principle (Milgrom and Weber, 1982), that the expected payment in the second-price auction is more responsive to the bidder's private information. A constrained bidder in the model bids the amount that would be bid by a lower type. The logic of the Linkage Principle suggests that a bidder in the first-price auction bidding as a lower type has a stronger incentive to increase his reported type because the payment being made is lower (for a bidder making the same report with the same information), so the same increase in the probability of winning from increasing the report is more valuable in the first-price auction. This incentive is passed on to the principal in this model, who consequently relaxes the budget constraint (see Section 6.4 for more detail).

After discussing the related literature in the next section, Section 3 presents the model. Section 4 explains the approach taken toward characterizing equilibria in the model. Section 5 then describes the equilibrium behavior of the bidders. Section 6 provides the main results and is organized according to the different cases being considered. Section 7 concludes. All proofs are in Appendix A.

Section snippets

Related literature

Following early work on the effect of budget constraints on auction outcomes when valuations are known (Che and Gale, 1996), Che and Gale (1998) are the first to compare revenue between auction formats when both budgets and valuations are treated as private information. A further extension of this work is given in Che and Gale (2006) which develops techniques for comparing revenues between auction formats when types are multidimensional and independent. Several papers (Fang and Perreiras, 2002,

Model

The model extends the Milgrom and Weber (1982) (henceforth MW) model of auctions with affiliated values to allow for a budgeting stage prior to a sealed-bid auction for a single asset. To each of N bidders I add a principal responsible for setting their bidder's budget constraint prior to the auction (there are a total of 2N players in the game). Each bidder plays the role of the manager in the firm and each principal plays the role of the board of directors. Both are assumed to be equity

Analytical approach

I characterize symmetric equilibria in which principals select budgets according to an increasing function w(si) and bidders given their own budget w submit bids of the formB(ti,w)=min{b(ti),w}, with b(t) increasing. That there should exist such a choice b(t) independent of w is suggested by the observation that conditional on ti bidder i's inferences about the opposing bid distribution, which depends on (tj,sj) for ji, are unaffected by the value of si.15

Bidders' strategies

Consider the problem of bidder i with signal ti and budget w in a first- or second-price auction, and suppose that all opponents' bids are determined according to b(min{tj,tˆ(sj)}) for some increasing tˆ(s) and increasing b(t).17 Suppose that w is in the range of b, let tˆ=b1(w), and consider the bid b(min{ti,tˆ}). This bid

Equilibrium and results

This section builds on Lemma 1, characterizing the principals' equilibrium strategies in both auction formats. I frame each principal's problem in terms of a choice of tˆ[0,1] which is understood to determine the budget bA(tˆ;tˆ), as in Lemma 1. This setup implicitly restricts the principal's choice of budget to be in the range of bA(;tˆ), but as argued above there are choices of budgets from this set that weakly dominate any choices outside of the set (see Footnote 16). Therefore, an

Conclusion

Given results from the classic symmetric auction models, it has been an important question for auction theory to ask how robust these results are to changes in the environment. Budget constraints are cited as an example of a feature of real-world auctions that would lead to a failure of the revenue equivalence theorem. However, results from models that incorporate budget constraints largely treat them as exogenous. In the model presented here I show that incorporating budget constraints

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    I am grateful to Lawrence Ausubel, Peter Cramton, Daniel Vincent, Emel Filiz-Ozbay, Oleg Baranov, Terence Johnson, Sushant Acharya, Mallesh Pai, Thayer Morrill, anonymous referees, and participants in workshops at North Carolina State University, the Kansas Workshop on Economic Theory, the University of Maryland, Wake Forest University, and the Federal Trade Commission for advice and suggestions. Funding for this research was provided by the Roger and Alicia Betancourt Fellowship in Applied Economics, and a graduate assistantship at the University of Maryland.

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