Ambiguity aversion in the all‐pay auction and war of attrition

• Published date: 01 December 2018
• Date: 01 December 2018
• DOI: http://doi.org/10.1111/jpet.12345

Received: 26 January 2018

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Accepted: 23 October 2018

DOI: 10.1111/jpet.12345

ORIGINAL ARTICLE

Ambiguity aversion in the all‐pay auction and war

of attrition

Steven Stong

Department of Economics, University of Iowa,

Iowa City, Iowa

Correspondence

Steven Stong, Department of Economics,

University of Iowa, W312 John Pappajohn

Business Building, Iowa City 52242, IA.

Email: steven-stong@uiowa.edu

Ambiguity aversion is introduced to a class of commonly

applied games including the war of attrition and all‐pay

auction. In contrast to subjective expected utility, the all‐pay

auction is shown to generate less expected expenditure than

the first‐price auction. The war of attrition generates less

expected expenditure than the all‐pay auction and second‐

price auction. In the all‐pay auction, increasing ambiguity

causes low types to bid lower and high types to bid higher. In

the war of attrition, ambiguity can decrease the bids for all

types.

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INTRODUCTION

This paper explores the impact of ambiguity aversion on behavior and aggregate expenditures in the all‐pay

auction (APA; Nalebuff & Stiglitz, 1983) and war of attrition (WOA; Maynard Smith, 1974). Ambiguity can

informally be described as uncertainty for which the probabilities of some events are unknown to the

decision maker. Ellsberg (1961) and subsequent authors show that when there is ambiguity, individuals

frequently make decisions that cannot be explained by subjective expected utility (SEU). The presence of

ambiguity in applications and the importance of ambiguity for behavior motivate the study of ambiguity

aversion in the APA and WOA.

The APA and WOA are contest games in which players compete for a prize by expending resources. Incomplete

information arises because players do not know the value of winning a prize or cost of expending resources for

other players. In many applications, the distribution of values and costs of other players is difficult to learn because

players compete infrequently or the environment changes in novel ways. Situations modeled by a WOA include

firms competing to determine industry standards (Bulow & Klemperer, 1999), firms exiting a crowded market

(Fudenberg & Tirole, 1986), labor strikes (Kennan & Wilson, 1989), provision of a public good (Bilodeau & Slivinski,

1996; Bliss & Nalebuff, 1984), and some online auctions (Platt, Price, & Tappen, 2013). The first three examples

usually arise when there is a change in an industry; the more novel or infrequent the change is, the more ambiguity

would be present. The last two examples will tend to involve ambiguity when players meet infrequently.

Applications of the APA where ambiguity is present may include political competition (Baye, Kovenock, & de Vries,

1993), litigation (Baye, Kovenock, & de Vries, 2005), R&D contests (Dasgupta & Stiglitz, 1980), and competition for

college admissions (Hickman, 2012).

J Public Econ Theory. 2018;20:822–839.wileyonlinelibrary.com/journal/jpet822

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© 2018 Wiley Periodicals, Inc.

Lobbying (Baye et al., 1993) is one application of contests in which ambiguity is likely to be present. Ambiguity is

particularly salient in disputes over state pension systems. In these lobbying contests, players compete to

determine levels of state and individual contributions, levels of benefits, and the retirement age, subject to funding

future obligations. Even estimates of the expected obligations are in great doubt with Healey, Hess, and Nicholson

(2012) citing studies that give the amount of unfunded liabilities in 2012 as being in a range from $730 billion to

$4.4 trillion for U.S. states combined. These authors point out that uncertainty surrounding pensions includes the

uncertainty of shifting demographics and forecasts of future rates of return. The competitors trying to influence

policymakers must also deal with uncertainty about the mood of voters and the degree to which changes to the

pension system are legally allowed. Such uncertainty about the strength and level of motivation of competitors may

be difficult to reduce to a single probability distribution. So lobbies may adopt strategies that are sensitive to the

ambiguity of the situation as modeled in this paper.

The main results derive the impact of ambiguity aversion on behavior and on the aggregate expenditures or

revenue. Relative to the equilibrium with SEU, ambiguity‐averse players with low values bid less in the all‐pay

auction, and players with high values bid more. In the WOA ambiguity aversion may decrease bids for all types. The

expected expenditure is higher in the first‐price auction than in the all‐pay auction, and the WOA generates the

lowest expenditure. Under an additional condition, the WOA generates less expenditure than the second‐price

auction. Next, I will relate this model to previous work on incomplete information games with ambiguity aversion.

The results are also compared to the contest literature with SEU. Ambiguity aversion is also shown to be consistent

with much of the experimental work.

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Related literature

This paper fits into a large literature, beginning with Ellsberg (1961), that demonstrates that ambiguity aversion can

explain observed behavior that is not explained by standard models with SEU preferences. In the finance and

macroeconomics literature, ambiguity aversion has been shown to solve many puzzles regarding asset prices, such

as the equity premium puzzle.

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Mukerji (1998) has shown that ambiguity aversion can explain the incompleteness

of contracts in situations where costly contracting alone gives counter factual predictions. Kagel and Levin (1985)

suggest that ambiguity aversion may be part of the explanation for overbidding in first‐price auctions.

MEU has been increasingly applied to a variety of games and mechanism design environments. In his seminal

work, Lo (1998) applied the MEU model to the first‐price auction where he derived the unique increasing

equilibrium. Following Lo (1998), I model ambiguity using the MEU model of Gilboa and Schmeidler (1989). Players

are assumed to have ambiguous information about the other players' values for a prize. Players do not know the

correct distribution but believe that it is one of a set of distributions over the other players' values. A player's

interim utility is the lowest expected utility generated by any distribution in the set. Given the other players'

strategies, each player chooses an action that maximizes the minimum expected utility. Thus, an ambiguity‐averse

player chooses an action that is robust to the worst case distribution.

With a particular form of ambiguity, Lo (1998) showed that the first‐price auction generates more revenue than

the second‐price auction. Bodoh‐Creed (2012) showed that more generally no revenue ranking exists between the

two auctions. In contrast to this negative result, the ranking that I provide for the APA and first‐price auction holds

under general conditions. The ranking between the APA and WOA and between the WOA and second‐price

auction hold under conditions that can be interpreted economically.

The revenue results are also related to the literature on mechanism design with ambiguity aversion.

Bodoh‐Creed (2012), Bose and Daripa (2009), and Bose, Ozdenoren, and Pape (2006) investigated the

revenue‐maximizing mechanism in abstract mechanism design problems where agents may be ambiguity‐averse.

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See Guidolin and Rinaldi (2013) for a survey. Also see Hansen and Sargent (2008).

STONG

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