On the reserve price in all-pay auctions with complete information and lobbying games

Abstract

We show that the seller’s optimal reserve price in an all-pay auction with complete information is higher than in a standard auction. We use our results to re-consider some findings of the literature that models lobbying games as all-pay auctions. In particular, we show that the so-called Exclusion Principle appears to rely crucially on the implicit assumption of a “weak” (in terms of bargaining power) seller, and does not hold if she regards bidders’ valuations as iid according to a monotonic hazard rate. Our preliminary results for the case of independent but asymmetric bidders make it even more suspicious.

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MPRA
Munich Personal RePEc Archive
On the reserve price in all-pay auctions
with complete information and lobbying
games
Bertoletti, Paolo
Dipartimento di Economia Politica e Metodi quantitativi,
University of Pavia
January 2006
Online at http://mpra.ub.uni-muenchen.de/1083/
MPRA Paper No. 1083, posted 07. November 2007 / 01:30

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January 2006

On the reserve price in all-pay auctions with complete
information and lobbying games



Paolo Bertoletti§
Dipartimento di economia politica e metodi quantitativi
University of Pavia







Abstract
We show that the seller’s optimal reserve price in an all-pay auction with
complete information is higher than in a standard auction. We use our results to
re-consider some findings of the literature that models lobbying games as all-
pay auctions. In particular, we show that the so-called Exclusion Principle
appears to rely crucially on the implicit assumption of a “weak” (in terms of
bargaining power) seller, and does not hold if she regards bidders’ valuations as
iid according to a monotonic hazard rate. Our preliminary results for the case of
independent but asymmetric bidders make it even more suspicious.
Keywords: all-pay auctions, reserve price, economic theory of lobbying.
JEL Classification: D44, D72.


§ Faculty of Economics, University of Pavia, Via San Felice, 7 - I-27100 - Pavia, ITALY; Email:
paolo.bertoletti@unipv.it; Tel. +39 0382 986202; Fax +39 0382 304226. I thank Guido Ascari and Lorenzo Rampa for
attracting my attention to these topics and for helpful discussions. Useful suggestions were also provided by participants
in seminars at the Universities of Turin, Padova, Pavia and Salerno, and in particular by Marco Pagnozzi. Pietro Rigo
generous first-order advice on order statistics is gratefully acknowledged. Carolina Castagnetti gracefully provided the
relevant MATLAB simulations. I am also very grateful to Domenico Menicucci who pointed out a number of mistakes
in a previous version: also those remaining are mine.

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1. Introduction
Auction models are prototypes of competitive settings, and they are used in several branches of
economic literature. In particular, the so-called (first-price) all-pay auction is used (among others)
by Hillman and Riley (1989), Baye et alii (1993) and Che and Gale (1998) to model the lobbying
process. This type of auction fits the lobbying game well, since a lobbyist's contribution is not
typically returned if his efforts are unsuccessful,1 and indeed this literature has elaborated a number
of interesting results. In particular, Hillman and Riley (1989) prove that, if there is some asymmetry
among bidders/lobbyists, the politically contestable rent is not totally dissipated even in the case of
a large number of potential contenders. In addition, Baye et alii (1993) show that a seller/politician
wishing to maximize political rents may find it in her best interest to exclude certain lobbyists from
her "finalist" short list (so-called "Exclusion Principle"), particularly those lobbyists valuing the
political prize most (in order to rise incentives to spend for the likely losers). Che and Gale (1998)
also show that the imposition of an exogenous cap on individual lobbying contributions may have
the adverse effect of increasing total expenditure (by increasing competition among lobbyists).
The previous results contribute to the economic literature on the lobbying process, and in
addition also to the pure theory of auction. From the former perspective, it is intriguing that some of
them (namely the Exclusion Principle and the possibly adverse effect of a cap on expenditure)
appear not to hold in the alternative class of models so-called à la Tullock (1980): see Fang (2002).
From the latter perspective, it has to be stressed that the quoted literature refers to the case of
complete information,2 which is a somewhat unusual assumption in auction theory. In fact, it is
somewhat unclear which informational assumptions are made: in particular, the role and the
information available to the designer (if any) of the auction are left unexplained.
In this paper we start with placing the all-pay auction model with complete information in the
context of the auction literature, and derive from this comparison some implications for the
economic theory of lobbying (see Boylan, 2000 for a similar approach). In particular, we wish to
argue that the previous results are implicitly but crucially based on the assumption that what would
be called the “reserve price” in auction theory is null. In turn, this raises the questions of why the
"seller" (say a politician) cannot set a positive reserve price, and of what information is ex ante
available to her. The only consistent explanations seem to be that the politician who receives the
lobbies' contributions does not know their preferences, or has very little bargaining power. After
characterizing the equilibrium of the all-pay auction with a reservation price exogenously given, we

1 This feature is also shared by other economic and social games, such as patent races and sports.
2 Hillman and Riley (1989: pp. 29-30) also deal with the case of incomplete information among contenders, and Che
and Gale (1998: p. 648) claim that their result would hold even under incomplete information if there were asymmetry
among bidders.

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show that the seller would prefer a strictly positive reserve price, which also increases the overall
efficiency of the auction outcome (even if it might decrease the efficacy of the lobbying process
through higher rent dissipation). This casts some doubts on the “adverse” results concerning the
effects of caps on individual spending and of lobbyist exclusion, because they might disappear once
the possibility that the reserve price is optimally set is taken into account. We then argue that more
robust results (if any) should be based on the explicit assumption that the seller faces incomplete
information while setting her reserve price, and investigate such a case by comparing the all-pay
auction with other trading mechanisms. Our preliminary findings in this setting make even more
suspicious the case for the Exclusion Principle.
The paper is organized as follows: section 2 compares the (first-price) all-pay auction with
complete information with the so-called "standard auctions" (see e.g. Klemperer, 2004: section
1.1.2). Section 3 characterizes the equilibrium of the former under a positive reserve price
exogenously given. Section 4 considers the way an optimal reserve price should be set by a seller
facing incomplete information when using an all-pay auction rather than a “standard” auction or a
take-it-or-leave-it offer (a “trading mechanism” sub-optimal for the seller if there is more than a
single buyer). Section 5 discusses the value for the seller of additional competition on the bidders’
side and why our approach casts some doubt on the Exclusion Principle and on the result obtained
by Che and Gale (1998) concerning the effect of caps on the lobbies' expenditure. Section 6 briefly
concludes. Technical proofs are presented in the Appendixes.

2. Reserve price, auction theory and lobbying games
Consider the following setting: n risk-neutral3 agents (the “buyers”) bid for a prize (there is no
resale possibility). Bidder i's (private) valuation of the prize is vi (i = 1, …, n), and we order bidders
in such a way that v1 > v2 >…> vn-1 > vn > 0.4 The rules of the auction can include a reserve
(minimum) price pr ≥ 0, i.e., a price below which the prize is not assigned. In an important
contribution, Milgrom (1987) puts auction theory in the more general context of bargaining theory,
and argues that (winner-pay) standard auctions (namely, the oral ascending or descending-bid and
the first and second-price sealed-bid auctions) under complete information (i.e., assuming that every

3 Agents’ risk neutrality is assumed throughout the paper: however, this assumption makes no difference for the case of
the second-price sealed-bid auction, as it is well known.
4 The possibility of ties in the valuations is ignored here. This can be justified by assuming that the vi are ex-ante
continuously independently distributed, and so that case has a priori a zero probability. Moreover, for standard auctions
the only relevant implication of ties is that v1 = v2 implies full rent dissipation, independently from the level of the
reserve price (if not larger than v1) and from the number of contenders (on the contrary, in a all-pay auction ties may
imply the existence of multiple Nash equilibria which are not necessarily revenue equivalent: see Baye et alii, 1996 and
footnote 14 below).

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details of the setting is common knowledge to all the participants, including the seller) lead to
efficient outcomes. In particular, Milgrom (1987: Proposition 1, p. 7) argues that, under complete
information, the set of perfect5 equilibrium outcomes of the standard auctions consists of the Core
outcomes of the corresponding exchange game (together with the no-trade outcome if pr = v1), as
the minimum price ranges from the "seller" evaluation of the good to be sold to the highest "buyer"
evaluation.6 Indeed, while there are many Nash equilibria in those auction games, the only
"sensible" ones (i.e., such that no agent ever uses weakly dominated strategies) seem to be those
such that the prize is allocated to agent 1 for her bid of r = Max{pr,v2} (notice that a reserve price v0
< pr < v2, where v0 ≥ 0 is the seller's valuation of the prize, would have no effect on the auction
outcome).
In particular, let us indicate with bi the bid of agent i. As it is well known, in the case of a
second-price sealed-bid auction bi = vi, i = 1, …, n, does constitute a (weakly) dominant strategy
equilibrium in which the prize goes to bidder 1 at a price r (as soon as pr ≤ v1). In the case of a first-
price sealed-bid auction, the typical equilibrium has b1 = r, and the strategies of the agents j = 3, …,
n can be specified arbitrarily, provided that bj ∈ [pr,vj) with probability 1 (if pr > vj, agent j bids
arbitrary on [0,pr)). The cumulative distribution function of b2, F2(b2), is continuous7 on a support
(weakly) included in [0,v2] and first-order stochastically dominates F(b) = (v1 - v2)/(v1 - b). Notice
that in both these auctions (and in the correspondent oral ones)8 the equilibrium payoffs are U1 = v1
- r and Uj = 0, j = 2, …, n, while the total payment to the seller is r. Milgrom (1987: p. 8) then
observes that, from the perspective of (cooperative) game theory, the seller's ability to set any
particular reserve price and stick to it measures her bargaining power.
What about an (first-price) all-pay auction version of the previous setting? In such a case,
bidder i receives the prize if bi > Max{bj≠i} and in that case his payoff is vi - bi, whereas his payoff
is - bi if he loses (ties are broken randomly). Assuming pr = 0, Hillman and Riley (1989), and Baye
et alii (1993) and (1996) show that in the unique Nash equilibrium agent 1 uses the uniform
distribution F1(b1) = b1/v2 on the support [0,v2], while agent 2 uses F2(b2) = 1 - v2/v1 + b2/v1 on the
same support (note that this amounts to say that agent 2 randomises between b2 = 0 and the uniform
distribution on [0,v2] with probabilities respectively 1 - v2/v1 and v2/v1). Agents j = 3, …, n bid bj = 0

5 To avoid technical problems, Milgrom (1987: pp. 5-8) actually works with discrete bid spaces.
6 Milgrom (1987: p. 3) also argues that, if the prize can be later resold, the outcome is stable and especially favourable
to sellers in a relatively poor bargaining position with respect to the potential buyers.
7 For pr < v2, agent 2 cannot play b2 ∉ [pr,v2) with a positive probability, since those strategies are (weakly) dominated
for her: thus, she should bid between pr and v2 with probability 1. If pr > v2, then also F2(·) is arbitrary on a support ⊆
[0,pr).
8 As it is well known, the descending ("Dutch") auction is strategically equivalent to the first-price sealed-bid auction,
while in an ascending ("English") auction, similarly to a second-price sealed-bid auction, is clearly a dominant strategy
for each bidder i to stay in the bidding until the price reaches her value vi (see e.g. Klemperer, 2004: section 1.1.4).