All-Pay Auctions with Handicaps

Abstract

Corruption in the public sector erodes tax compliance and leads to higher tax evasion. Moreover, corrupt public officials abuse their public power to extort bribes from the private agents. In both types of interaction with the public sector, the private agents are bound to face uncertainty with respect to their disposable incomes. To analyse effects of this uncertainty, a stochastic dynamic growth model with the public sector is examined. It is shown that deterministic excessive red tape and corruption deteriorate the growth potential through income redistribution and public sector inefficiencies. Most importantly, it is demonstrated that the increase in corruption via higher uncertainty exerts adverse effects on capital accumulation, thus leading to lower growth rates.

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Electronic copy available at: http://ssrn.com/abstract=1625131
Discussion Paper No. 781










ALL-PAY AUCTIONS WITH HANDICAPS


Minoru Kitahara
Ryo Ogawa












June 2010


The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

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Electronic copy available at: http://ssrn.com/abstract=1625131
All-Pay Auctions with Handicaps
Minoru Kitahara∗
Ryo Ogawa†
First Draft: March 2010
This Version: June 16, 2010
Abstract
This paper analyzes an all-pay auction where the winner is de-
termined according to the sum of the bid and a handicap endowed
to all players. The bidding strategy in equilibrium is then explic-
itly derived as a “piecewise affine transformation” of the equilibrium
strategy in an all-pay auction without handicaps. The paper also dis-
cusses the allocation rule implemented in the equilibrium and pro-
vides a comparison of revenue.
Journal of Economic Literature Classification Numbers: D44, D82, D86.
Keywords: asymmetricauctions; all-payauctions; handicapauctions.
∗JSPS Research Fellow, Graduate School of Social Sciences, Tokyo Metropolitan Uni-
versity, 1-1 Minami-Osawa, Hachioji-shi, Tokyo 192-0397, Japan, and Visiting Research
Fellow, Population Research Institute, Nihon University, 12-5 Goban-cho, Chiyoda-ku,
Tokyo 102-8251, Japan (mkitahar@tmu.ac.jp)
†Corresponding author. ISER, Osaka University, 6-1 Mihogaoka, Ibaraki, Osaka 567-
0047, Japan (r-ogawa@iser.osaka-u.ac.jp).
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1Introduction
In many economic environments, agents compete by expending resources
to win prizes. Typical examples include lobbying, bribery, promotion con-
tests, R&D races, economists competing based on the number of publi-
cations, and so on. In the auction theory literature, such situations are
considered using the model of an all-pay auction where different players
spend money or exert effort as “bids” and the player that spends the most
“wins the auction” and obtains the final prize. As the “bid” in these mod-
els is a sunk cost borne by all players, regardless of the ultimate winner,
they are known as “all-pay” auctions.12
In reality, it is often the case that some players have an advantage over
others at the beginning. For instance, the success of lobbying today de-
pends not only on the activity undertaken today but also on the accumu-
lation of past efforts. Likewise, hereditary advantages also play some role
in bribery, while the sponsor of a contest probably undertakes some dis-
criminatory treatment in favor of one group of the players. From the view-
point of auction theory, such (dis)advantages are regarded as “handicaps”
in the all-pay auction; that is, the highest bidder is not always the winner,
as the player whose bid amount less his “handicap” is the highest wins
the auction. The present paper is devoted to the study of an all-pay auc-
tion where the advantages of players are described as handicaps existing
among agents.3
1Amann and Leininger [1] and Krishna and Morgan [11] consider all-pay auctions
with incomplete information while Noussair and Silver [21] conduct experimental work.
The recent theoretical work on all-pay auctions includes Parreiras and Rubinchik [22].
2Ando [2], Che and Gale [4, 5], Kaplan and Wettstein [9], and Moldovanu and Sela [19,
20] apply the all-pay auction model to contests, with recent theoretical work including
Siegel [25].
3In recent work, Mares and Swinkels [13, 14] study the first price handicap auction
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The paper presents two main results in Section 3. The first of these is
examination of the entry–exit decisions of players. When there are hand-
icaps in the auction, there is a positive probability that a disadvantaged
playerfindsitoptimaltobidzerowhenhisvalueisverysmall.4Uponcon-
sideration, an advantaged player also finds it beneficial to lower his bid;
that is, the advantaged player has some positive probability of winning
evenwhenhebidszero. TwoexamplesinSection3.1illustratethesestrate-
gies in the equilibrium. In the examples, each player has an “entry point”,
and the player bids zero if his value is lower than this point, whereas he
makes some positive bid otherwise. A more interesting problem is if there
is some case where a player with a heavy handicap totally abandons win-
ning (i.e., he always bids zero regardless of the value). Example 2 in Sec-
tion 3.1 depicts the case where a player with a heavy handicap adopts a
strategy of bidding zero for all possible values (in other words, his entry
point is set to the upper bound of the value).
The second result found is a simple derivation of the equilibrium strat-
egy. Theoretically, it is not obvious if the analysis of asymmetric auctions
is straightforward. Indeed, it is generally understood in the literature that
asymmetries in some auctions lead to numerous complications. For in-
stance, although an equilibrium exists in asymmetric first-price auctions,5
(FPHA) along with other asymmetric settings. Our concept of a handicap is the same
as in their paper; namely, we treat bids and handicaps in terms of summation. We may
also consider another form of “handicap” where the bid of the disadvantaged player is
discounted through multiplication (for the two-player case of multiplicative handicaps,
see Feess et al. [7]). However, in the present paper we restrict our attention to summative
handicaps.
4Amann and Leininger [1] present two numerical examples of two-player all-pay auc-
tions under asymmetric distributions. In their examples, and under some parameter val-
ues, the bidding distribution of one player “has an atom at 0”, i.e., the player exits from
the auction when his or her value is lower than some threshold.
5The existence of pure strategy equilibria in asymmetric auctions (or in games with
incomplete information more generally) is shown using a variety of techniques by
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a closed-form expression of the equilibrium strategy is sometimes unavail-
able, and therefore it is difficult to investigate problems with revenue or
efficiency or the equilibrium allocation.6A study more closely related to
the present analysis is Mares and Swinkels [13, 14] in that they consider a
first-price auction with various handicap formulations. However, the mo-
tivation of their paper is not restricted to deriving the equilibrium strategy
in a closed form, and they argue that the equilibrium strategy is usually
complicated and so remains an open question. Our result then contrasts
sharply with the complications found in many asymmetric auction stud-
ies.
Proposition 1 encompasses the key finding of our paper. This shows
that the equilibrium strategy is written as a “piecewise affine transforma-
tion” of the equilibrium strategy in an all-pay auction without handicaps.
Section 3.2 presents the central idea underlying this result where we argue
that the optimality of the equilibrium strategy in an auction without hand-
icaps directly proves the optimality of the equilibrium strategy in a hand-
icap auction. Section 3.3 then discusses the allocation rule implemented
in the equilibrium. In asymmetric auctions, it is rather obvious that in-
efficient allocation takes place with some positive probability. However,
details of the allocation rule in the equilibrium are not straightforward.
Here, we discuss that the shape of the hyperplane dividing the type space
into each player’s winning is “kinked”.
Following this, Section 4 examines the uniform distribution case and
Athey [3], Jackson and Swinkels [8], Lebrun [12], Maskin and Riley [17], and Reny [24].
6Maskin and Riley [16] examine asymmetric first-price auctions in detail. For some
class of distributions, it is shown that equilibrium strategies in an asymmetric first-price
auction can be explicitly derived (see Plum [23] and Cheng [6]). Marshall et al. [15] ana-
lyze the computation of equilibrium under the asymmetry of distributions in first-price
auctions.
4