EMPIRICAL RELEVANCE OF AMBIGUITY IN FIRST PRICE
AUCTION MODELS∗
GAURAB ARYAL† AND DONG-HYUK KIM‡
A BSTRACT. We study the identification and estimation of first-price auction models with independent private values under ambiguity about the
valuation distribution. We model the bidders’ preference by the maxmin
expected utility of [Gilboa and Schmeidler, 1989], and provide sufficient
conditions to nonparametrically identify the valuation distribution and the
degree of ambiguity, separately from the risk aversion coefficients (CRRA,
CARA). Moreover, under multiplier preferences of [Hansen and Sargent,
2001], the ambiguity parameter is identified only when bidders are risk
neutral. Analyzing data from an experiment, we find that the subjects,
though informed of the true distribution, are still ambiguity averse.
Keywords: first price auction, identification, Bayesian econometrics, robust inference, ambiguity aversion, risk aversion
JEL classification: C11, C44, D44, E61
1. I NTRODUCTION
We propose a novel approach to inference in first-price auctions with independent private values when there is ambiguity about the true distribution of values (willingness-to-pay), i.e. bidders consider many distributions
as reasonable candidates for the true valuation distribution. We propose
sufficient conditions to nonparametrically identify the valuation distribution, bidders’ attitude toward ambiguity, separately from their attitude toward risk. Econometrics of auction data conventionally assumes that the
Date: January 21, 2013.
∗ The authors thank John Kagel for providing the experimental data used in this paper. The authors also thank Ali Hortaçsu and seminar audience at University of Chicago.
Aryal acknowledges the financial support of CBE internal grant and Kim acknowledges
the financial support from the Business School Research Grant, UTS.
† The Australian National University. e-mail:[email protected]
‡ The University of Technology, Sydney. e-mail: [email protected] .
1
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ARYAL AND KIM
valuation distribution is commonly known by the bidders. In practice, however, this assumption can be unrealistic, especially when bidders’ appraisal
process is complex (e.g., seismic prospecting for mineral auctions) or when
bidders have not yet learned the restructured market system after a shock
(e.g. T-bill auctions after a financial crisis). This paper, for the first time,
dispenses with such an assumption to study bid data.
Ambiguity in probability judgements has been a central issue in economics since [Keynes, 1921; Knight, 1921], culminating to a position of eminence
with [Ellsberg, 1961]. Ambiguity in decision process also arises when model
primitives are partially identified [Manski, 2000; Aryal and Kim, 2012]. Under ambiguity, a decision maker is either unable to pin down the probability
of payoff relevant states or concerned with model misspecification. Allowing for ambiguity in auctions is not only important for robust inference but
also important for (empirical) auction design: the revenue equivalence theorem fails [Lo, 1998], the first price auction is suboptimal [Bose, Ozdenoren,
and Pape, 2006], and the optimal reserve price should decrease with ambiguity [Bodoh-Creed, 2012]. It is, therefore, critical to determine the presence
and extent of bidders’ ambiguity. The main contribution of this paper is to
do just that.
To model ambiguity, we consider the maxmin expected utility (MEU,
henceforth) by [Gilboa and Schmeidler, 1989]. We also consider the multiplier preferences (MP) by [Hansen and Sargent, 2001] briefly as an alternative. Under MEU, every bidder has a unique set of equally reasonable valuation distributions, considers the most pessimistic utility at each bid, and
chooses a bid to maximize this worst utility. We assume that bidders have
an identical set of distributions that contains the true distribution. What is
just assumed is also common knowledge among the bidders. The cardinality of the set represents the degree of ambiguity.
Each bidder exaggerates the probability of competing with high-value
bidders. We provide a way to recover this (pessimistic) belief; see [Manski, 2004] on the importance of measuring expectation (or beliefs). In particular, we propose conditions under which the observed bid is sufficient
AMBIGUITY IN FIRST PRICE AUCTION
3
to identify the underlying valuation distribution and the utility. One difficulty with MEU is that it states the existence of a unique set of distributions without characterizing such a set, but how we model this set affects
inference through the worst distribution in it. For instance, if we use εcontamination to represent the set, the model with ambiguity is equivalent to
the classic expected utility (EU) representation of von Neumann and Morgenstern. To address this issue, we innovate a distortion function that maps
the true probability into a new but pessimistic one, i.e., a curve below the 45
degree line on [0, 1] like a Lorenz curve. Under some regularity conditions,
the distortion function uniquely represents the bidders attitude toward ambiguity. Our identification strategy is to derive this function from the bid,
from which the identification of the valuation distribution and the utility
follows.
If the utility function is nonparametric, the model is not identified even
when bidders participation is exogenous (exclusion restriction).1 The model
primitives are, however, identified under the exclusion restriction if the utility exhibits either constant relative risk aversion (CRRA) or constant absolute risk aversion (CARA), i.e., we separate risk aversion and ambiguity
aversion though both lead to overbidding.2 The discrepancy between the
distortion function and the 45 degree line can be used to test for ambiguity
aversion.
MP, on the other hand, posits that each bidder has an initial estimate of the
valuation distribution, but fears misspecification and considers other distributions at a cost. We set this initial estimate to be the true distribution and
represents the degree of ambiguity by her confidence that this distribution
is indeed the right one. [Strzalecki, 2011], however, shows that there is no
way of disentangling risk aversion from the concern about model misspecification: MP can be represented by the EU paradigm with a more concave
utility function. Hence, the model with risk averse bidders under MP is
1 This discrete exclusion restriction is sufficient to just-identify nonparametric utility
and the valuation distribution without ambiguity; see [Guerre, Perrigne, and Vuong, 2009].
2 This also means that ambiguity erodes market power and might have implications
to markets characterized by ambiguity, such as R&D and innovation that are beyond the
scope of this paper.
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ARYAL AND KIM
equivalent to the conventional auction model with (more) risk averse bidders. The latter is shown to be identified in [Guerre, Perrigne, and Vuong,
2009]. Under the exclusion restriction, the valuation distribution is nonparametrically identified but the utility function is uncovered only up to a
multiplicative constant, which is the confidence parameter. This parameter
is identified only when the bidders are risk neutral.
In this paper, we analyze the widely studied experimental data of [Dyer,
Kagel, and Levin, 1989] under MEU. In the experiment, subjects were told
that their values were independently drawn from a uniform distribution,
and asked to bid for two different auctions, one with three bidders and the
other with six bidders (including themselves). Under CRRA and CARA,
separately, we estimate the utility parameters and the distortion function.
For both cases, the posterior probabilities of the distortion function being
close to the 45 degree line are small, suggesting that the subjects did not
understand what is meant by having independently drawn uniform values
or the way to exploit this information in the bidding game.
Finally, this paper contributes to the literature on structural estimation
of auction models studied by [Paarsch, 1992; Guerre, Perrigne, and Vuong,
2000, 2009; Athey and Haile, 2002; Haile and Tamer, 2003], among others.
We also contribute to the literature that recognizes the importance of robustness in economic modeling summarized by [Hansen and Sargent, 2011].
2. F IRST P RICE A UCTIONS WITH A MBIGUITY AVERSE B IDDERS
An indivisible object is to be allocated to one of n ≥ 2 bidders. Each
bidder i observes her own value vi and offers bi without learning her opponents’ values. The highest bidder wins the auction at the price equal to her
own bid and gets the utility u(vi − bi ), or else u(0) = 0.3 Let u be an increasing and strictly concave utility function. The objective of bidder i with
value vi is to choose bi that solves
max u(vi − bi ) × Pr (win) ≡ max u(vi − bi ) × Pr (bi ≥ b j , j 6= i ).
bi
bi
(1)
We assume that v1 , . . . , vn are independently and identically distributed as
F0 (·|n, Z ), on [v( Z, n), v( Z, n)] with Z ∈ Rd as a vector of observed auction
3 Reserve price is assumed to be equal to zero.
AMBIGUITY IN FIRST PRICE AUCTION
5
covariates. Since Z is observed by both the auctioneer and the econometrician, for notational ease, we suppress Z and use F0 (·|n) to denote the
conditional distribution, given Z.
In our environment, however, bidders do not know the true distribution,
but they consider many reasonable distributions. There are, therefore, multiple “winning probabilities,” for which the EU paradigm can be inappropriate. Let Pn be a convex set of all strictly increasing continuous distribution
functions defined over [v(n), v(n)] for a given n ∈ N := {n ∈ N : 2 ≤ n < ∞}
such that F0 (·|·) ∈ Pn .4 There is also a need to model each bidder’s beliefs about others’ beliefs about the set of distributions because it will affect
the equilibrium behavior. First, we assume that the number of bidders n
in an auction is common knowledge. Second, we also assume that even
though the bidders do not know the identity of the true distribution, it is
commonly known that the values for all bidders are drawn from the same
distribution and that distribution belongs to the set Pn . This assumption of
symmetry-in-beleifs among the bidders keeps the model tractable because
we do not have to model higher order beliefs explicitly, something that cannot be avoided if bidders had different beliefs about the data generating
process. A justification for this is that bidders have access to a common
training data from which they can learn, e.g. in timber auctions bidders are
taken for a tour to assess the value of the timber. Collectively, we make the
following assumptions:
Assumption 1. It is common knowledge among the bidders that:
(1) There are n ∈ N bidders with an identical utility function u : R+ → R+
with u0 > 0, u00 < 0, and u(0) = 0.
(2) Their values v1 , . . . , vn are independently and identically distributed.
(3) The true valuation distribution F0 (·|n) ∈ Pn with density f 0 (·|n) > 0
is unknown to the bidders, but any information about F0 (·|n) other than
realized values is shared among the bidders.
4 Intuitively this convexity assumption is without loss of generality because all prefer-
ences that are valid with a given set of distributions are also valid for the convex hull of this
set. In other words, the partial order of preference is invariant to any convex combination
of members in the set.
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ARYAL AND KIM
We focus only on a symmetric pure strategy Bayesian Nash equilibrium.
In particular, every bidder conjectures that her opponents use a strictly increasing (pure) bidding strategy, and announces a bid that is a best response to that conjecture and at equilibrium the conjecture turns out to
be true.5 When the distribution F0 (·|n) is common knowledge, there is a
unique symmetric Bayesian Nash equilibrium in pure strategy characterized by a strictly increasing bidding function β n : [v(n), v(n)] → R+ ; see
Theorem 6 [Athey, 2001]. This bidding strategy maps the latent value to the
observed bid. [Guerre, Perrigne, and Vuong, 2000] shows that when bidders
are risk neutral, this map can be inverted to link each bid to a unique value,
thereby identifying F0 (·|n). [Guerre, Perrigne, and Vuong, 2009; Campo,
Guerre, Perrigne, and Vuong, 2011] extend this result to allow for risk averse
bidders. In the remaining of this section, we extend these results to the MEU
and MP representations.
2.1. Maxmin Expected Utility. The seminal article [Gilboa and Schmeidler,
1989] proposes an axiomatic representation of preferences for decision makers (bidders) with multiple priors (valuation distribution) about the state of
nature (opponents’ values). We assume
Assumption 2. The preference ordering of each bidder satisfy
(1) Assumptions A1-A6 in [Gilboa and Schmeidler, 1989].
(2) Monotone Continuity.
Assumption 2 (1) coincides with axioms in EU, except it allows decision
makers to weakly prefer any convex combination of indifferent lotteries to
each individual one instead of restricting the combination to be indifferent–
ambiguity aversion, and uses a weaker version of independence. Let Ω be
the set of the states of nature, ũ(·) the utility function, and A the set of
all feasible actions. [Gilboa and Schmeidler, 1989] shows that a decision
maker’s preference ordering satisfies assumption 2 if and only if there is a
unique set of distributions C over Ω such that she prefers an action a to b
5 In our model, bidders do not have the option of using “ambiguous strategies”, i.e.
they do not have access to a subjective randomizing device (such as Ellsberg urns); see
[Bade, 2011; Riedel and Sass, 2011].
AMBIGUITY IN FIRST PRICE AUCTION
7
whenever
min EP ũ( a) ≥ min EP ũ(b).
P∈C
P∈C
We begin by proposing a way to adapt the set of distributions to represent
the strategic effects of ambiguity. If every bidder had a different set of distributions, we would have to explicitly model the beliefs of other bidders
about that set, and that bidder’s beliefs about what others believe. As mentioned earlier, to gain traction we follow the tradition of [Harsanyi, 1967],
and interpret the auction as a game of imperfect information among bidders, where although all bidders are ambiguous about the true distribution,
it is common knowledge that there is one unique set of distribution that
contains the true distribution. From assumptions 1 and 2, this implies that
every bidder uses the most pessimistic distribution to determine her expected utility and chooses a bid accordingly. The next step is to model a set
of distributions that is sufficiently general to model ambiguity in auctions,
but always has a unique lowest distribution, which first order stochastically dominates all other distributions in the set; see [Gilboa and Marinacci,
2010]. We can then study the bidding behavior with respect to that lowest
distribution.
To motivate our choice of such a set, we begin with an example that illustrates how our choice of a set affects the inference. In particular, we consider
the ε-contaminated model that is widely used in both economics and in robust statistics, and show that the valuation distribution is nonparametrically
identified but the model is observationally equivalent to the first price auction model without ambiguity. [Nishimura and Ozaki, 2006; Kopylov, 2008]
provide an axiomatic justification for using ε-contamination set to model
ambiguity in decision making.6
Example 1. Let ε ∈ (0, 1) be given and consider risk neutral bidders. Under the
ε-contaminated model, the set consists of all (well behaved) distributions that can
be represented as a (1 − ε) mixture of the true distribution F0 (·|n), i.e.
Γ0 = { F : F = (1 − ε) F0 + εR with R ∈ Pn }
6 Many articles adopt ε-contaminated model; see [Huber, 1973; Berger, 1985; Berger
and Berliner, 1986; Nishimura and Ozaki, 2004; Bose, Ozdenoren, and Pape, 2006; Bose
and Daripa, 2009; Cerreia-Vioglio, Maccheroni, Marinacci, and Montrucchio, 2011].
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ARYAL AND KIM
Let F ∗ (·|n) be the most pessimistic distribution and f ∗ (·|n) its density. Then
(
(1 − ε) F0 (v|n), v < v
F ∗ (v|n) = (1 − ε) F0 (v|n) + min R(v|n) =
0
R∈Γ
1,
v = v.
So, an ambiguity averse bidder with valuation v is given by
max
z∈[v(n),v(n)]
[v − β n (z)] [(1 − ε) F0 (z|n)]n−1 =
max
z∈[v(n),v(n)]
[v − β n (z)] F0 (z|n)n−1 .
Therefore, the model is equivalent to the one without ambiguity, and the identification of F0 follows from [Guerre, Perrigne, and Vuong, 2000]. The idea is as follows.
The symmetric and increasing bidding strategy β n (·) is the solution to
β0n (v) =
[v − β n (v)](n − 1) f 0 (v)
.
F0 (v|n)
Let G (·|n) be the distribution of equilibrium bid b := β n (v), i.e., G (·|n) =
1
0
−1
F0 [ β−1 (b)|n] and its density is g(b|n) := f 0 [ β−
n ( b )| n ] /β n [ β n ( b )]. Using the
transformation between the bid and the value, we use the first order condition to
identify v as
G (b|n)
,
( n − 1) g ( b | n )
which identifies F0 . This argument can extend to the case of risk averse bidders with
v = b+
some exclusion restrictions; see [Guerre, Perrigne, and Vuong, 2009].
Lemma 1. If the set of distributions is given by ε-contamination of F0 (·|n) then
the model is equivalent to the first price auction model without ambiguity.
Intuitively, the equivalence transpires because ambiguity, as measured by
ε in this setting, does not affect the relative winning probability for almost
every bidder (except the one with v(n); a measure zero event), and thereby
keeping bidders’ optimal behavior intact. The bid data does not, therefore,
distinguish the ε-contamination model from the model without ambiguity.7
Since the structure of ambiguity is linear (or multiplicative) to the true distribution, the worst distribution F ∗ (·|n) has a positive probability mass at
v̄(n), i.e., ε = Pr [v = v̄(n)], which is because the worst R in the example
above is the Dirac measure on v(n). Moreover, in the limit with ε → 1, the
7 The bidding behavior is not affected in the mechanism of first price auction. Nev-
ertheless, since the optimal mechanism depends on ε as shown by [Bose, Ozdenoren, and
Pape, 2006], it is still important to distinguish the models with and without ambiguity.
AMBIGUITY IN FIRST PRICE AUCTION
9
ε-contamination model becomes unreasonable – F ∗ (·|n) converges to the
Dirac measure on v(n), implying that every bidder believes that there is no
chance to win.
To get around such problems, we restrict ourselves to distributions that
are absolutely continuous with respect to F0 (·|n).8 We begin our model
by formalizing these assumptions (symmetry, non-multiplicative ambiguity and absolute continuity). Let Qn ⊆ Pn be a convex subset of all the
probability distributions such that (a) Qn 3 F0 (·|n) and (b) every element
F (·|n) ∈ Qn is strictly increasing and differentiable. Hence, Qn is convex
and weak∗ compact [Parthasarathy, 1967] and contains distributions that
are absolutely continuous with respect to F0 (·|n); see Theorem 6 [Gilboa
and Marinacci, 2010].9
Since the probability of a bidder winning at a bid b depends on the joint
likelihood of everybody else bidding less than b, which in turn depends the
joint distribution of the values, we have to define such a distribution. In an
auction with n bidders, for every bidder i, define
Γ̃−i = co { F (·|n) × F (·|n) × . . . F (·|n) : F ∈ Qn },
|
{z
}
(n−1)times
as the set of joint probability distributions of all (n − 1) bidders valuation,
where coA denotes the convex hull of the set A. The set Γ̃−i represents
bidder i0 s beliefs about other bidders’ values. By assumption bidders are
symmetric, this set is the same for all bidders, i.e. Γ̃−i = Γ̃− j for all i 6= j
we may drop out the index −i. Moreover, since every member of Γ̃ is a joint
distribution of a random sample of size n − 1, for simplicity, we use Γ to
refer to the unique set of marginal distributions that are associated with the
set of joint distributions Γ̃.
In line with [Gilboa and Schmeidler, 1989], the size of this set determines
the degree of ambiguity. For example, if it is singleton, there is no ambiguity,
and the model becomes EU. Alternatively, if Γ is a closed ball with a diameter ε > 0 around F0 (·|n) for an appropriately chosen metric, the larger ε, the
8 The lowest distribution F ∗ (·|n) in example 1 is not absolutely continuous with respect
to F0 (·|n), a.e.
9 The assumption 2 (2) implies the set Q is weakly compact; see [Chateauneuf, Macn
cheroni, Marinacci, and Tallon, 2005].
10
ARYAL AND KIM
bigger Γ, which means the more ambiguity. We assume that all the bidders
are symmetric in terms of the ambiguity and information structure, i.e,
Assumption 3. The fact that F0 (·|n) ∈ Γ and the set Γ itself are common knowledge among all bidders.
Though every bidder can compute the lowest probability of winning by assumption 3, the econometricians do not know Γ. We are interested in inferring this lowest probability of Γ and therefore the true valuation distribution
from the bid, under the assumptions that bidders choose their bids to maximize their expected utility with respect to the worst distribution. Since Γ
is convex and weakly compact, its lower envelope F ∗ (·|n) belongs to Γ and
unique, and so is the density f ∗ (·|n) almost everywhere.10
Let D [ F0 (v|n)] := F ∗ (v|n) = minF∈Γ F (v|n), ∀v ∈ [v(n), v(n)] that maps
quantile to quantile, i.e.
D ( p) := F
∗
h
F0−1 ( p|n)|n
i
(2)
This function D converts the true probability F0 (·|n) to the most pessimistic
probability F0∗ (·|n), i.e., D ( p) ≤ p, capturing the degree of bidders’ ambiguity aversion; we call it distortion function, i.e., when there is no ambiguity,
i.e., EU, there is no distortion, i.e., D ( p) = p. (An analogy is the Lorenz
curve for income inequality; no inequality = 45 degree line.)
Now, we study the equilibrium bidding function and the identification of
model primitives. We focus on the equilibrium bidding strategy that is a
best response to a bidder when all other bidders adopt a strictly increasing,
symmetric pure strategy bidding function. [Athey, 2001] shows that the
best response itself is a strictly increasing bidding strategy, β n (·), such that
a bidder with a value v behaves as if her value is z that solves
max min u [v − β n (z)] F (z|n)n−1 = max u [v − β n (z)] D [ F0 (z|n)]n−1 .
z ∈R+ F ∈ Γ
z ∈R+
(3)
The first-order condition with respect to z gives
−u0 [v − β n (z)] β0n (z) D [ F0 (z|n)] + u [v − β n (z)] (n − 1) D 0 [ F0 (z|n)] f 0 (z|n) = 0
10 It is a lower envelope in the sense that it is first-order-stochastically-dominated by
every distribution in the set, i.e. F (v|n) ≥ F ∗ (v|n) for all v ∈ [v(n), v(n)] for all F (·|n) ∈ Γ.
AMBIGUITY IN FIRST PRICE AUCTION
11
at z = v. By rearranging terms,
D [ F0 (v|n)]
1
u [v − β n (v)]
= 0
.
u0 [v − β n (v)]
D [ F0 (v|n)] (n − 1) f 0 (v|n)/β0n (v)
Let λ( x ) := u( x )/u0 ( x ) for x ∈ R, then λ0 (·) ≥ 1 and hence it is invertible.
Let H ( p) := D ( p)/D 0 ( p) for p ∈ [0, 1], or alternatively
i
h
−
1
h
i f 0 F0 ( p|n)n
i.
h
H ( p) = F ∗ F0−1 ( p|n)n
f ∗ F0−1 ( p|n)n
(4)
Note that H (·), unlike λ(·), is not necessarily monotone with a slope greater
than 1. Substituting λ(·) and H (·) in the first order condition gives
λ [v − β n (v)] =
H [ F0 (v|n)]
(n − 1) f 0 (v|n)/β0n (v)
(5)
2.1.1. Identification. Let G (·|n) be the distribution of equilibrium bid b :=
β n (v) for v ∼ F0 (·|n), i.e., G (b|n) = F0 [ β−1 (b)|n] and its density is
g(b|n) :=
f 0 [ β−1 (b)|n]
.
1
β0n [ β−
n ( b )]
Let vγ and bγ be the γ-th quantile of the value and the equilibrium bid. Since
γ = F0 (vγ |n) = G [ β n (vγ )|n] = G (bγ |n), for every γ ∈ [0, 1], (5) gives
λ ( v γ − bγ ) =
H (γ)
( n − 1 ) g ( bγ | n )
(6)
for every γ ∈ [0, 1]. Under the i.i.d. assumption, g(·|n) is nonparametrically
identified from the bid data, but the model primitives are not in general
identified without additional assumptions, including the ones on the set Γ;
see [Guerre, Perrigne, and Vuong, 2009] for the case of H (γ) = γ.
In the remaining subsection, we explore sufficient, yet plausible, conditions under which the model primitives are identified. The following proposition establishes a negative result that does not depend on the structure of
the set Γ.
Proposition 1. Under assumptions 1, 2 and 3, the valuation distribution F0 (·|n)
is not identified by the knowledge of bid distribution, i.e., g(·|n).
12
ARYAL AND KIM
Proof. It would be sufficient to consider risk neutral bidders. Then, (6) can
be written as (vγ − bγ )(n − 1) g(bγ |n) = H (γ). Since γ = F0 (vγ |n),
[ F0−1 (γ|n) − bγ ](n − 1) g(bγ |n) = H (γ),
∀γ ∈ [0, 1].
This one equation has two unknowns: F0−1 (γ) and H (γ).
The proposition thus motivates an additional exclusion restriction. Hence,
we introduce the one that has been widely employed in the literature since
[Guerre, Perrigne, and Vuong, 2009];
Assumption 4. Exogenous Participation: for all n ∈ N , F0 (·|n) = F0 (·) and
v(n) = v and v(n) = v.11
Unfortunately, however, when the utility function unspecified, this exclusion restriction is not sufficient to identify the model structure.
Proposition 2. Suppose the econometrician identifies g(·|n1 ) and g(·|n2 ) where
n j ∈ N with n1 6= n2 . Then under assumptions 1–4, the model structure [u(·), F0 (·)]
is not nonparametrically identified.
Proof. We begin by stating (without a proof) the rationalizability lemma that
is adapted from [Guerre, Perrigne, and Vuong, 2009]’s Lemma 1.
j
j
j
Lemma 2. Let G j (·|n j ) be the joint distribution of (b1 , b2 , . . . , bn j ), conditional on
n j for j = 1, 2. There exists an IPV auction model with risk aversion and maxmin
expected utility [u(·), F0 (·)] that rationalizes both G1 (·|n1 ) and G2 (·|n2 ) if and
only if the following conditions hold:
j
j
nj
j
(1) G j (b1 , . . . , bn j |n j ) = ∏i=1 Gj (bi |n j ), where Gj (·|n j ) is the bid distribution form auction with n j bidders.
(2) ∃λ : R+ → R+ and ∃ H : [0, 1] → R+ such that λ(0) = 0, H (0) = 0
and λ0 (·) ≥ 1 such that ξ 0 (·) > 0 on [b, b] where
ξ (b, u, G, n, H ):
j
H ( G (b |n ))
(a) ξ (b, u, Gj , n j ) = b + λ−1 (n −1j) g (b jj|n ) , j = 1, 2.
j
j
j
h
i
H (γ)
H (γ)
(b) ∀γ ∈ [0, 1], bγ1 + λ−1 (n −1) g(b1 |n ) = bγ2 + λ−1 (n −1) g(b2 |n ) .
1
γ
1
2
γ
2
11This also means that the set P is the same for all n ∈ N and because Γ will also be
n
the same, so will F ∗ (·).
AMBIGUITY IN FIRST PRICE AUCTION
13
Following [Guerre, Perrigne, and Vuong, 2009] we can identify λ−1 (·). Let
[ F (·), λ(·), H (γ) := γ] and [ F̃ (·), λ(·), H̃ (γ) := γι ], with ι ∈ (0, 1) be two
model structures. Let F̃ (·) be the distribution of ṽ defined as follows: For
j
every quantile γ ∈ (0, 1] compute v(γ) = F −1 (γ) and determine bγ =
β[vγ , F (·), n j , H ] and
"
ṽγ =
j
bγ
+λ
−1
γι
#
.
j
( n j − 1 ) g j ( bγ | n j )
Then, it is clear that the two model structures satisfy Lemma 2 and hence
are observationally equivalent.
Informally, for the model under EU, [Guerre, Perrigne, and Vuong, 2009]
shows that the model structure [u(·), F0 (·)] is just-identified by the knowledge of g(·|n1 ) and g(·|n2 ) with n1 6= n2 . But, the model under MEU has
one more element to identify, which is D. Hence, for the same information,
the model under MEU is under-identified. In view of this result, we restrict
ourselves to parametric families of utility functions, CRRA and CARA, that
are most widely used in the empirical literature, nesting risk neutral bidders
as a special case;
Assumption 5. (CRRA) The utility function is CRRA, i.e.,
u( x ) =
x 1− θ
with θ ∈ [0, 1)
1−θ
Assumption 6. (CARA) The utility function is CARA, i.e.,
(
[1 − exp(− xθ )] /θ if θ > 0
u( x ) =
x
if θ = 0
Under assumptions 5 and 6, λ(·) becomes

x

under CRRA

 1− θ ,
1−exp(− xθ )
, under CARA,
λ( x ) =
θ exp(− xθ )


 x,
under risk neutrality.
As propositions 1 and 2 argue, the model is not identified without the exclusion restriction. This is true even with the parametrized utility functions.
It would be, therefore, useful to understand the source of nonidentification
14
ARYAL AND KIM
without the exclusion restriction. Consider the CRRA utility. Since the utility function is strictly increasing and concave, choosing a bid to maximize
MEU is equivalent to choosing a bid to maximize the certainty equivalence
corresponding to a worse distribution, i.e.
arg max
z
12
n −1
[v − β(v)]1−θ
D [ F0 (z|n)]n−1 = arg max[v − β(z)] D [ F0 (z|n)] 1−θ ,
z
1−θ
which means a risk averse bidder with CRRA utility with parameter θ would
bid the same as a risk neutral bidder with more pessimistic distortion, D (·)
1
versus D (·) 1−θ . Hence, as far as the bid data is concerned, there is a substitutability between risk aversion and the “pessimism”. This is because both
lead to over-bidding in the same way and hence cannot be disentangled by
the data. Consider now a bidder with the highest value, v. She should not
distort her winning probability, as she knows she is the highest. Hence, if
she overbids, it must be because of risk aversion, not because of ambiguity.
To make this formal, we introduce the following regularity assumption that
the inverse of the Mill’s ratio corresponding to F ∗ (·) is equal to the inverse
of the Mill’s ratio corresponding to F0 (·) at the highest valuation v, i.e.
Assumption 7. No distortion at the top:
f ∗ (v)
F ∗ (v)
=
f 0 (v)
.
F0 (v)
That is, the highest bidder uses the true probability when she forms her optimal bid, as she knows she wins for sure. Once the risk aversion coefficient
is identified by the highest bidder’s bidding behavior, the exclusion restriction is sufficient to identify the bidder’s attitude toward ambiguity and also
the valuation distribution.
We now formally establish the identification of the model premitives. We
begin with the CRRA specification and consider the risk neutral case as a
corollary of CRRA. We then conclude this subsection by the CARA specification.
Proposition 3. Suppose that bidders’ utility function u is CRRA, i.e, assumption
5. Under assumptions 1, 2, 3, 4 and 7, the valuation distribution F0 (·), the utility
function u, and the distortion function D are identified by g(·|n1 ) and g(·|n2 )
where n1 , n2 ∈ N with n1 < n2 .
12And given our notation of the worst distribution, the certainty equivalence
c.e(v, z, D, F0 , θ ) solves c.e(v, v, D, F0 )θ = [v − β(z)]θ [ D ( F0 (v| N ))]n−1 .
AMBIGUITY IN FIRST PRICE AUCTION
Proof. Under assumption 5, λ( x ) =
x
1− θ
15
⇔ λ−1 (y) = (1 − θ )y. From (6), we
get
v−b = λ
−1
H [ G (b|n)]
( n − 1) g ( b | n )
= (1 − θ )
H [ G (b|n)]
( n − 1) g ( b | n )
j
For each quantile γ ∈ [0, 1], let vγ ∈ [v, v] such that F0 (vγ ) = γ, and bγ :=
j
β n j (vγ ). Then, since G (bγ |n j ) = G [ β n j (vγ )|n j ] = F0 (vγ ) = γ, for each γ ∈
[0, 1], we have
j
v γ = bγ +
(1 − θ ) H ( γ )
j
( n j − 1 ) g ( bγ | n j )
.
(7)
where j ∈ {1, 2}. Equating the quantiles for v under two auctions, we get
# −1
"
1
1
,
(1 − θ ) H (γ) = (bγ2 − bγ1 )
−
(n1 − 1) g(bγ1 |n1 ) (n2 − 1) g(bγ2 |n2 )
which when evaluated at γ = 1 identifies θ since H (1) = 1. Once θ is
identified, identification of H (γ) immediate, which identifies
Z 1
1
dt .
D (γ) = exp −
γ H (t)
and F0 (·) by (7), as well. Moreover, F ∗ (v) = D [ F0 (v)].
As mentioned above, of the highest bidder’s bidding behavior identifies
θcrra . After partialling out the effect of risk aversion, any deviation from the
standard model explains bidders’ attitude toward ambiguity, identifying D,
from which the identification of F0 follows. An immediate corollary is the
identification with risk neutral bidders, which is the case of θ = 0.
Corollary 2. Suppose that bidders are risk neutral. Under assumptions 1, 2, 3 and
4, the valuation distribution F0 (·) is identified by the knowledge of bid distributions
g(·|n1 ) and g(·|n2 ) where n1 , n2 ∈ N with n1 < n2 .
We now establish the identification of the auction model under CARA.
Proposition 4. Suppose that bidders’ utility function u is CARA, i.e, assumption
6. Under assumptions 1, 2, 3, 4, and 7, the valuation distribution F0 (·), the utility
function u, and the distortion function D are identified by g(·|n1 ) and g(·|n2 )
where n1 , n2 ∈ N with n1 < n2 .
16
ARYAL AND KIM
Proof. As in the previous proof, for each quantile γ ∈ [0, 1], let vγ ∈ [v, v]
j
such that F0 (vγ ) = γ, and bγ := β n j (vγ ). Since
λ( x ) =
1 − exp(− xθ )
1
= [exp( xθ ) − 1],
θ exp(− xθ )
θ
its inverse is
λ−1 (y) = log(1 + θy)/θ.
From (6), for every γ ∈ (0, 1], and j ∈ {1, 2} we have
"
#
1
H (γ)θ
j
vγ = bγ + log 1 +
.
j
θ
( n j − 1 ) g ( bγ | n j )
(8)
Since vγ is the same for both j = 1, 2, we can equate the two equations to get
"
#
"
#
H
(
γ
)
θ
H
(
γ
)
θ
(bγ2 − bγ1 )θ = log 1 +
(9)
− log 1 +
(n2 − 1) g(bγ2 |n2 )
(n1 − 1) g(bγ1 |n1 )
Clearly, θ = 0, i.e. linear utility, solves this equation. For identification we
want to show that there is another θ 6= 0 that also solves the equation. Let
j
R j := [(n j − 1) g(b1 |n j )]−1 . The left hand side of (9) as a function of θ is linear
in θ, starts at the origin and is strictly increasing with a constant slope of
(b12 − b11 ). Let m(θ ) be the right hand side of (9), then because H (1) = 1 (from
assumption 7) m0 (θ ) = [ R1− R2 ]/[(1 + θR2 )(1 + θR1 )] > 0, limθ →0 m(θ ) =
1
< ∞, and m00 (θ ) < 0. That is, m(θ ) is strictly
0, limθ →∞ m(θ ) = log R
R2
increasing and strictly concave, that starts at the origin and converges to a
finite constant from below. Thus, if m0 (0) is greater than the slope of the
LHS of (9), which is (b12 − b11 ) then there is a unique θ > 0 that solves (9).
0
From Lemma 2 condition 2, we know λ0 (·) ≥ 1 and so λ−1 (·) ∈ (0, 1).
Then aggressive bidding in auction with n2 bidders imply the rent under
n1 auction is greater than under n2 auction and hence v1 − b11 > v1 − b12 ⇔
λ−1 ( R1 ) > λ−1 ( R2 ) ⇔ R1 > R2 . Therefore,
b12 − b11 = λ−1 ( R1 ) − λ−1 ( R2 ) < R1 − R2 = m0 (0),
as desired. Once θ is identified, we can identify H (γ) from (9) as
exp((bγ2 − bγ1 )θ ) − 1
H (γ) =
θ
1
(n1 −1) g(bγ1 |n1 )
−
exp((bγ2 −bγ1 )θ )
(n2 −1) g(bγ2 |n2 )
.
AMBIGUITY IN FIRST PRICE AUCTION
17
Once θ and H (γ) is identified, we identify
Z 1
1
dt .
D (γ) = exp −
γ H (t)
and F0 (·) by (8), as well. Moreover, F ∗ (v) = D [ F0 (v)].
The intuition of the identification under CARA is identical to that under
CRRA, though the nonlinearity of λ complicates the proof. We now end
this subsection by an example where the set of valuation distributions is
determined by a total variation norm.
Example 3. For a fixed but unknown ε ∈ [0, 1], let
Γ = { F ∈ P : sup | F0 (t) − F (t)| ≤ ε}.
t
Let v1 solve F0 (v1 ) = ε, then
(
D ( F0 (v)) = F ∗ (v) =
0,
v ≤ v1
F0 (v) − ε, v ≥ v1
Then, it is straightforward to note that any bidder with type less than v1 will bid
zero and for the rest, it will solve
β0n (v) =
[v − β n (v)](n − 1) f 0 (v)
F0 (v) − ε
from which we can recover
1 G (b|n) − ε
,
n − 1 g(b|n)
if v ≥ v1 . The model is not identified but if we equate the quintiles of valuations
v = b+
across two auctions with n1 < n2 , (auction j = 1, 2, respectively) we get
bγ1 +
1
γ−ε
1
γ−ε
= bγ2 +
,
1
n 1 − 1 g1 ( b γ )
n2 − 1 g2 (bγ2 ))
leading to
e = γ−
(bγ2 − bγ1 )(n1 − 1)(n2 − 1) g1 (bγ1 ) g2 (bγ2 )
.
(n2 − 1) g2 (bγ2 ) − (n1 − 1) g1 (bγ1 )
18
ARYAL AND KIM
So, we can first, estimate ε and then recover pseudo-values and identify only the
truncated valuation distribution. This suggests that, unless we see a positive mass
of bidders bidding zero, we can safely rule out that the total variation model.
2.2. Multiplier Preferences. [Hansen and Sargent, 2001] considers a situation where decision makers, i.e., bidders, have an initial estimate of the true
distribution, but are worried about a misspecification of it, and consider
other distributions that are not too far away in terms of relative entropy.13
Within our environment, the initial estimate will be the true distribution
F0 (·|n) and any departure (in terms of the bidding behavior) from it will be
attributed to ambiguity.
Assumption 8. The preference order of each bidder satisfies assumptions A1-A6,
A8 and P2 in [Strzalecki, 2011].
If there are at least three disjoint non-null events, then assumption 8 is necessary and sufficient for each bidder’s preferences to have the multiplier
preference representation, so that a bidder with value v solves
n
o
n −1
max min u[v − β n (z)] F (z|n)
+ αK ( F k F0 ) .
z∈[v,v] F ∈Pn
(10)
Here α ∈ (0, ∞] captures the bidders’ confidence on their initial estimate
F0 (·|n), and can be thought of as the degree of ambiguity. For instance, if
α = ∞, the minimization is solved by F0 (·|n) with the interpretation that the
bidders are certain that F0 (·|n) is the true distribution. It is also known that
(10) is equivalent to
max ωα {u[v − β n (z)]} F0 (z)n−1
z∈[v,v]
(11)
where ωα is given by
x
ωα ( x ) := 1 − exp −
α
(12)
where α > 0 in assumption 6; see appendix A and also [Strzalecki, 2011].
Since (12) is a concave function, the MP representation for ambiguity averse
bidders is equivalent to the EU representation for more risk averse bidders
13 Let K ( F k F ) is the Kullback-Leibler divergence (or the relative entropy) of F with
0
respect to the true distribution F0 .
AMBIGUITY IN FIRST PRICE AUCTION
19
without ambiguity, i.e., U := ωα ◦ u. Intuitively, therefore, the bidders attitude toward risk and that for ambiguity cannot not be separately identified;
Proposition 5. Under assumptions 1, 2, 3, 4 and 8, the distribution function F0
is nonparametrically identified but the utility function u is identified up to a multiplicative constant α by the knowledge of bid distributions g(·|n1 ) and g(·|n2 )
where n1 , n2 ∈ N with n1 < n2 . Moreover, if bidders are risk neutral, α is identified by g(·|n1 ) and g(·|n2 ).
Proof. By [Guerre, Perrigne, and Vuong, 2009], U and F0 are nonparametrically identified by g(·|n1 ) and g(·|n2 ). From (12), we have u( x ) = −α log[1 −
U ( x )]. Hence, if bidders are risk neutral, i.e., u( x ) = x, the MP parameter α
is identified, i.e., α = − x/ log[1 − U ( x )].
As proposition 5 indicates, if bidders are risk neutral, the ambiguity parameter under MP is given by
α=−
x
.
log[1 − U ( x )]
The fact that RHS is a constant provides a testable restriction. When bidders are risk averse, however, the utility function is not identified as MP is
equivalent to EU with more risk averse bidders.
Though MP is observationally equivalent to EU, it is important to consider the bidders’ attitude toward ambiguity, i.e., α, separately from that
toward risk. Failure to identify α would force an investigator to choose the
parameter arbitrarily.14 However, the structure of ambiguity may affect the
optimal auction design just like under MEU [Bose, Ozdenoren, and Pape,
2006; Bodoh-Creed, 2012].
3. E STIMATION U SING E XPERIMENTAL D ATA
In this section, we explain the experimental data used by [Dyer, Kagel,
and Levin, 1989], propose the Bayesian estimation approach, present the
estimation results corresponding to MEU, and test for bidders’ ambiguity
aversion.
14“...policy recommendations based on such a model would depend on a somewhat
arbitrary choice of the representation. Different representations of the same preferences
could lead to different welfare assessments and policy choices, but such choices would not
be based on observable data.”–[Strzalecki, 2011].
20
ARYAL AND KIM
3.1. Data. The subjects were MBA students at the University of Houston.
There were three experimental runs with six different subjects participating
in each run, for a total of 18 subjects. In these experiments, bidders were
assigned independently and identically distributed (i.i.d.) values v drawn
from the uniform distribution on [0, 30]. In the event that they won, subjects were paid their value minus their bid. Each subject participated in 28
auctions over the course of two hours. The number of bidders was determined at random in the experiment. With probability one-half, there were
n1 := 3 bidders and with probability one-half, n2 := 6 bidders. Subjects
submitted two contingent bids and one noncontingent bid. After the bids
were submitted, a coin was tossed to determine whether the contingent or
noncontingent bids would be used in determining the winner. A second
coin toss determined whether n = n1 or n = n2 . If the contingent treatment
was selected, the first n1 contingent bid was used if n = n1 , and the n2 contingent bid was used if n = n2 . Otherwise, the noncontingent bid was used
so that the bid could not be conditioned on n. After each auction, bids and
corresponding private values were publicly posted on a blackboard.
We only use the contingent bids; see Figure 1.15 Panels (a)–(c) show the
histograms of contingent bids for n1 , n2 , and true values. The vertical dashed
lines on panels (a) and (c) are the upper bound of the value. Panels (d) and
(e) are the scatter diagrams of values against bids along with the 45 degree
line, and panel (f) does similarly between bids n1 and n2 along with the 45
degree line and the marks for the upper bound of values. Total number of
values (thus, the number of bids for each n) is 705.
Some features of the data suggest that the subjects deviate from the usual
bidding strategy. First, four subjects choose to obtain nonpositive utility:
among them, two bid zero while their values are strictly positive (say, 0.1
and 0.5) and two other bid higher than or equal to their own values for both
n1 and n2 , e.g., Figure 1 shows that a subject bid 32 for both n’s, whereas his
or her value was 29.02. Second, 33.5 % of the subjects bid higher for auctions
with n1 than the auctions with n2 , i.e., b1 ≥ b2 , whereas it must be that b1 <
b2 because n1 < n2 . Third, subjects tend to overbid. In particular, 87.1% of
15
For detailed analysis on the data readers are suggested to consult [Bajari and
Hortaçsu, 2005].
AMBIGUITY IN FIRST PRICE AUCTION
21
F IGURE 1. Experimental Data
(a) Contingent Bids (n=3)
80
(b) Contingent Bids (n=6)
80
60
60
40
40
20
20
(c) Values
50
40
30
20
0
0
20
(d) Values vs. Bids (n=3)
20
10
0
0
20
0
10
20
Values
30
0
20
(e) Values vs. Bids (n=6) (f) Bids (n=3) vs. Bids (n=6)
30
Bids (n=6)
Bids (n=3)
30
0
30
Bids (n=3)
0
10
20
10
0
0
10
20
Values
30
20
10
0
0
20
Bids (n=6)
F IGURE 2. Panels (a)–(c) show the histograms of contingent bids for
n1 = 3, n2 = 6, and true values (drawn from U [0, 30]) along with the
upper bound of the value (30) by a dashed vertical line. Panels (d) and (e)
are the scatter diagrams of values against bids along with the 45 degree
line, and panel (f) does similarly between bids n1 and n2 along with the 45
degree line and the marks for the upper bound of values.
the subjects bid higher, in the auctions with n1 bidders, than the equilibrium
bid with no risk aversion under EU, and 61.4% did in the auctions with n2
bidders.
3.2. Estimation and Model Selection Methods.
Bid Density Specification. We use b j to denote the equilibrium bid of an auction with n j bidders with j ∈ {1, 2}. We begin by specifying the joint bid
distribution of (b1 , b2 ) using a flexible parametric model.16
16 We estimate joint density and not just two marginals to account for any dependence
between the bids in two different auctions as they use the same valuation distribution.
22
ARYAL AND KIM
First of all, notice that the upper bounds of the bid data, i.e., (b̄1 , b̄2 ), are
parameters to be estimated. In this case, the associated statistical model is
irregular and standard asymptotic distribution theory does not apply. Here
is a widely used example;
iid
Example 4. Consider X1 , . . . , Xn ∼ Uniform[0, θ ] where θ is the unknown parameter. Then, the maximum likelihood estimator (MLE) for θ is given by θ̂ ML =
max { X1 , . . . , Xn } , and its asymptotic distribution is a shifted exponential distribution, where the true parameter value does not belong to the interior of the support
of the asymptotic distribution of the MLE. The MLE is therefore inefficient; see sections 9.4 and 9.5 of [van der Vaart, 1998].
For such irregular cases, [Hirano and Porter, 2003] shows that the maximum
likelihood estimator (MLE) is generally inefficient, but the Bayes estimator
is efficient. Furthermore, the asymptotic distribution of the two-step estimator proposed in [Guerre, Perrigne, and Vuong, 2000] is yet unknown, possibly because both steps are nonparametric with estimated (pseudo) values
entering the second step. For this reason, we employ a Bayesian approach.
In addition, since the CRRA and CARA parameters are identified by the
bid densities at the upper bounds b̄ j ; see propositions 3 and 4, we need to
employ a statistical model that is flexible, yet behaves well near the boundaries. [Leblanc, 2012] shows that Bernstein polynomial density (BPD) estimators have optimal mean integrated squared error properties and behave
well in terms of bias. In particular, the BPD estimators have uniform bias
over the entire support, which means that the bias at the boundary is not
larger that the bias for the interior. For this reason, we employ the BPD in a
Bayesian framework. We specify the joint bid density as follows.
1 k k
b1
b2
g(b1 , b2 |k, Ψk ) =
ψi,l φi,k
φl,k
∑
∑
b̄1 b̄2 i=1 l =1
b̄1
b̄2
(13)
where φl,k ( x ) is the density function of the beta distribution with parameters
l and k − l + 1, ψi,l ∈ ∆k2 −1 , the k2 − 1 dimensional unit simplex, i.e., ψi,l ≥ 0
for all i, l and ∑ik=1 ∑kl=1 ψi,l = 1, and Ψk is the k × k matrix whose (i, l )th
element is ψi,l .
The parameters such as θcrra and θcara and the function D (and H as well)
are estimated by the marginal densities of b1 and b2 . Hence, in principle, one
AMBIGUITY IN FIRST PRICE AUCTION
23
can obtain the posterior distributions of the marginal densities, separately
for each n, and construct the estimates for (θcrra , θcara , D ). We, however,
estimate the joint density of (b1 , b2 ) and deduce, from it, each marginal density because every subject was asked to bid in two different auctions for the
same value, i.e., b1,i and b2,i are correlated through vi . If we estimate the
marginal densities assuming the auctions with n1 and n2 are independent,
there would be a loss of efficiency.
The marginal density and the CDF of b1 is given by
!
k
1 k
b1
g1 (b1 |k, Ψk ) =
ψi,l φi,k
∑
∑
b̄1 i=1 j=1
b̄1
!
k
1 k
b1
G1 (b1 |k, Ψk ) =
ψi,l Φi,k
∑
∑
b̄1 i=1 l =1
b̄1
(14)
(15)
where Φi,k is the CDF of the beta distribution with parameters i and k − i + 1.
We can define g2 (·) and G2 (·), analogously. As k increases, the BPD in (13)
forms a dense subset of the set of all absolutely continuous distributions
with support [0, b̄1 ] × [0, b̄2 ]. [Petrone, 1999a,b] develop a nonparametric
Bayesian method of density estimation for univariate densities by modeling
k as a random parameter defined on the entire natural numbers. Unfortunately, a multivariate extension of [Petrone, 1999a] has not yet been developed. Instead, we consider a set of models with k ∈ {3, 4, 5} and choose one
using a formal Bayesian method of model selection.
Prior Specification. We maintain the prior beliefs consistent across the models under consideration. Suppose our prior has the mean of the random distribution G0 with density g0 and our confidence about the beliefs is quantified by a numeric value α0 > 0. If a nonparametric model is employed, then
this prior can be represented by a Dirichlet Process prior with mean parameter G0 and the accuracy parameter α0 ; see [Ferguson, 1973]. For a fixed k,
however, the prior beliefs should be framed by the model with k2 components. In this case, the prior of Ψk is reduced to a Dirichlet distribution with
density
p̃(Ψk ) ∝
k
k
α αi,l −1
∏ ∏ ψi,l0
i =1 l =1
(16)
24
ARYAL AND KIM
where
Z
Z
g0 (b1 , b2 |k, Ψk )db2 db1 ;
[ i−k 1 , ki ] [ l −k 1 , kl ]
again see [Ferguson, 1973]. The mean and the variance of the (i, l ) compoα (1− α )
nent of Ψk under (16) are αi,l and i,lα0 +1 i,l , respectively. We construct the
prior employing (16) by imposing shape restrictions arising from economic
theory, which is to be shown shortly.
Consider first the model with the CARA utility function. Define
αi,l :=
R j (b j |k, Ψk ) := [(n j − 1) g j (b j |k, Ψk )]−1
for j ∈ {1, 2}. Let θcara (Ψk ) be the solution that solves
1 + θR1 (b̄1 |k, Ψk )
(b̄2 − b̄1 )θ = log
1 + θR2 (b̄2 |k, Ψk )
(17)
and
Hcara (γ|k, Ψk ) :=
exp[(bγ2 − bγ1 )θcara (Ψk )] − 1
θcara (Ψk ) R1 (bγ1 |k, Ψk ) − exp[(bγ2 − bγ1 )θcara (Ψk )] R2 (bγ2 |k, Ψk )
Z 1
1
Dcara (γ|k, Ψk ) := exp −
dt
γ Hcara ( t | k, Ψk )
for all γ ∈ [0, 1]. We then construct the prior
pcara (Ψk ) ∝ p̃(Ψk ) · IR+ [θcara (Ψ)]
×IR+ inf Hcara (γ|Ψ) · IR+
inf [γ − Dcara (γ|Ψ)]
γ∈[0,1]
γ∈[0,1]
so that the posterior chooses a Ψk that satisfy the shape restrictions as indicated in the characteristic functions I A ( a) taking 1 if a ∈ A, otherwise,
zero.
Similarly, for the CRRA, let
b̄2 − b̄1
θcrra (Ψk ) := 1 −
R1 (b̄1 |k, Ψk ) − R2 (b̄2 |k, Ψk )
#
"
bγ2 − bγ1
1
Hcara (γ|k, Ψk ) :=
1 − θcrra (Ψk )
R1 (bγ1 |k, Ψk ) − R2 (bγ2 |k, Ψk )
Z 1
1
Dcrra (γ|k, Ψk ) := exp −
dt
γ Hcrra ( t | k, Ψk )
AMBIGUITY IN FIRST PRICE AUCTION
25
for all γ ∈ [0, 1]. Then, we employ the prior
pcrra (Ψk ) ∝ p̃(Ψk ) · I[0,1] [θcrra (Ψ)]
×IR+ inf Hcrra (γ|Ψ) · IR+
inf [γ − Dcrra (γ|Ψ)]
γ∈[0,1]
γ∈[0,1]
Model Selection. The Gaussian Metropolis-Hastings algorithm draws
T
l
Ψ1,k
, . . . , ΨlS,k ∼ pk,l (Ψlk |z T ) ∝ pk,l (Ψlk ) ∏ g(b1,t , b2,t |k, Ψlk ).
t =1
for every (k, l ) ∈ M := {3, 4, 5} × {cara, crra}, i.e., we consider six models.17 Then, in order to choose a model that fits the data the best, we employ
the method of the Bayesian model selection. Let π (·) denote a probability
mass function defined on the model space M that reflects our prior beliefs
for each model in M. Notice that here the model index m := (k, l ) can be
regarded as a parameter in a larger model and we can obtain the posterior
probability π (m|z T ) for each model using Bayes rule. The posterior odd
ratio of m relative to m0 is given by
π (m|z T )
π (m)
M(z T |m)
=
×
0
0
π (m |z T )
π (m ) M (z T |m0 )
(18)
where the marginal likelihood of the model m = (k, l ) is given by
M (z T |k, l ) :=
Z
∏ g(b1,t , b2,t |k, Ψk ) pk,l (Ψk )dΨk .
t
The ratio of any two (marginal) likelihoods is known as the Bayes factor, i.e.,
posterior odd ratio = prior odd ratio × Bayes factor.
We consistently estimate the marginal likelihood using the importance sampling with the draws from the posterior, i.e., it is known that
(
"
#)−1
1 S
1
a.s
→ M(z T |k, l )
T
l
S s∑
=1 ∏t=1 g ( b1,t , b2,t | k, Ψk,s )
(19)
17 We will consider more flexible models in the later version. Nevertheless, the spec-
ifications considered here, say, the ones with k = 3, 4, 5 are fairly flexible because they
approximate the joint density by 9, 16, and 25 basis functions, respectively.
26
ARYAL AND KIM
TABLE 1. Bayes Factors
Utility
CRRA
CARA
k
3
4
5
3
4
5
Bayes Factor 1.000 0.427 0.603 1.487 0.869 0.543
This table shows the Bayes factors of each model relative to the model
(k = 3, CRRA). The models with k = 3 under CARA is best supported by
the data.
The estimator (19) is often unstable, but it works well for our exercise; see
[Kass and Raftery, 1995; Han and Carlin, 2001] for alternative methods of
estimating the marginal likelihoods.
Table 1 shows the Bayes factors of each model relative to the model (k =
3, CRRA). The models with k = 3 under CRRA is best supported by the
data. However, since the Bayes factors are neither close to zero nor very
large, a researcher may choose any of them depending on his own prior beliefs, π (m). We shall, therefore, report the analysis results for all the models
below .
3.3. Results. Table 2 shows the posterior means (the posterior standard deviations in parentheses) of the parameters (θcrra , θcara ) and the upper boundaries (b̄ N =3 , b̄ N =6 ) for each model m ∈ M. The table shows that the point estimates for the parameters are all very similar across the models with small
posterior standard deviations.
Figures 3 and 4 show the posterior predictive density estimates under
CARA and CRRA, respectively. Each row corresponds to the number of
components k ∈ {3, 4, 5}. The first column shows the histogram of bids
for n1 bidders along with the posterior predictive marginal density of the
bid and a 90 % credible band around it. The second column does similarly
for the auctions with n2 bidders, and the last column shows the posterior
predictive joint density of (b1 , b2 ) by its level curves along with scattered
data points. Note that in this figure all the bids are normalized to be between
[0, 1] by the factor of the estimate of the bid upper bounds.
3.4. Bayesian Hypothesis Testing for Ambiguity Aversion. Figure 5 plots
the point-wise posterior mean of the distortion functions (solid line) and the
AMBIGUITY IN FIRST PRICE AUCTION
27
TABLE 2. Point Estimates of θcrra , θcara , b̄n=3 , and b̄n=6
Utility Function
k
θcrra , θcara
b̄ N =3
b̄ N =6
3
0.871
(0.011)
26.029
(0.030)
27.034
(0.032)
CRRA
4
0.888
(0.013)
26.031
(0.030)
27.032
(0.031)
5
0.889
(0.042)
26.039
(0.035)
27.047
(0.045)
3
0.760
(0.047)
26.030
(0.030)
27.034
(0.032)
CARA
4
0.780
(0.056)
26.033
(0.034)
27.041
(0.040)
5
0.781
(0.083)
26.042
(0.043)
27.049
(0.048)
This table shows the posterior mean (the posterior standard deviation) of
the parameters of risk aversion and the upper boundaries of the bid distributions.
2.5 and 97.5 point-wise percentiles (dashed line) along with the 45 degree
line. The rows correspond to CRRA and CARA and the columns correspond
to the number of components k ∈ {3, 4, 5}, respectively. We use the L2 norm k · k2 to measure the discrepancy of D (·) from the 45-degree line in
each case. We are interested in testing the hypothesis that D (·) is equal to
the 45 degree line. Given the estimation errors, we do not, however, expect
the null to be exactly true. Instead, we want to check if the true D (·) can
be reasonably approximated by the 45 degree line. Hence, we perform a
Bayesian hypothesis testing against the null hypothesis that k D (γ) − γk2 <
ε with a small ε > 0. Note that c0 := kγ − 0k2 = 0.5774, i.e., c0 is the L2
distance of the 45-degree line from the horizontal axe. We use ε q := qc0
with q ∈ {0.01, 0.02, 0.05}.
In the Bayesian framework, hypothesis testings are straightforward; if
the posterior probability of the null hypothesis is less than 1/2, it is rejected, otherwise, accepted. Table 3 summarizes the posterior probability
of the null hypothesis for each (k, q) ∈ {3, 4, 5} × {0.01, 0.02, 0.05} under
the CRRA. All the probabilities are much smaller than 0.5. For the CARA
utility, the probabilities are essentially zero for all pairs of (k, q). As a conclusion, bidders are not only risk averse and but also ambiguity averse with
respect to the valuation distribution. This suggests that the subjects did not
understand what it means by having uniformly distributed values or they
did not know how to use this information.
28
ARYAL AND KIM
1
0
0
0.5
bN=3
50
0
1
bN=6
Density
50
0
0.5
bN=6
0.5
0
1
0
0.5
bN=3
1
0
0.5
bN=3
1
0
0.5
bN=3
1
1
0
0
0.5
bN=3
50
0
1
bN=6
Density
50
0
0.5
bN=6
0.5
0
1
1
0
0
0.5
bN=3
1
50
0
bN=6
50
Density
Density, k=5
Density, k=4
Density, k=3
F IGURE 3. Posterior Predictive Densities under CARA
0
0.5
bN=6
1
0.5
0
This Figure shows some summary of posterior distribution of bid density
estimates: each row represents a distinct model with the number of components k and the columns are the marginal densities for bn=3 and bn=6 ,
and the joint density with the sample (histogram or dots). The utility is
assumed to be CARA.
4. C ONCLUSION
It has been a conventional practice to assume that the valuation distribution is common knowledge among the bidders in the empirical auction
literature. Since most auction theory has been developed based on such assumptions, it is natural to follow it for data analysis too. Moreover, by such
hypotheses, the underlying valuation distribution is identified.
This assumption may, however, be unrealistic. The valuation distribution summarizes the system of market demand and the structure of bidders’ information. For example, when the subprime credit crisis hit the
AMBIGUITY IN FIRST PRICE AUCTION
29
F IGURE 4. Posterior Predictive Densities under CRRA
0
0.5
bN=3
50
0
1
bN=6
Density
50
0
Pred. Density of g(⋅|N=6)
0
0.5
bN=6
Predictive Joint Density
1
0.5
0
1
0
0.5
bN=3
1
0
0.5
bN=3
1
0
0.5
bN=3
1
50
0
0
0.5
bN=3
50
0
1
bN=6
Density
1
0
0.5
bN=6
0.5
0
1
1
50
0
0
0.5
bN=3
1
50
0
bN=6
Density
Density, k=5
Density, k=4
Density, k=3
Pred. Density of g(⋅|N=3)
0
0.5
bN=6
1
0.5
0
This Figure shows some summary of posterior distribution of bid density
estimates: each row represents a distinct model with the number of components k and the columns are the marginal densities for bn=3 and bn=6 ,
and the joint density with the sample (histogram or dots). The utility is
assumed to be CRRA.
Euro money market in 2007, many financial market participants might be
concerned about the ambiguity about the market system of repo auctions
(short-term loan interbank markets) of the European Central Bank. If the
model does not approximate the real world, its prediction and policy recommendations may be unreliable. It is, therefore, important to assess the
suitability of the model assumption to a certain context and to propose an
alternative, if not appropriate.
In order to formally consider the bidders’ attitude toward ambiguity of
the valuation distribution, we model the bidders’ preference by the MEU
30
ARYAL AND KIM
Distortion, CRRA
F IGURE 5. Posterior Predictive Distortion Functions
1
1
1
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
Distortion, CARA
0
0
0.5
1
0
0
0.5
1
0
1
1
1
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0
0
0.5
quantile, k=3
1
0
0
0.5
quantile, k=4
1
0
0
0.5
1
0
0.5
quantile, k=5
1
This Figure shows the posterior distributions of the distortion function D
by its pointwise mean (solid) along with its 90% credible band (dashed).
Each panel shows the 45-degree line. The upper block represents the case
of the CRRA utility, and the lower the CARA.
representation of [Gilboa and Schmeidler, 1989] and the MP representation [Hansen and Sargent, 2001; Strzalecki, 2011], separately. For the former, we provide sufficient conditions under which the valuation distribution and bidders’ attitude toward ambiguity are nonparametrically identified, yet separately from the parameter of risk aversion (CARA, CRRA). For
the latter, we show that the model is observationally identical to the model
with risk averse bidders who is certain about the valuation distribution and
that the valuation distribution is nonparametrically identified but the utility
function is identified only up to a multiplicative constant. We then analyze
a bid sample generated from an experiment within the framework of MEU,
AMBIGUITY IN FIRST PRICE AUCTION
31
TABLE 3. Posterior Probabilities % of k D (γ) − γk2 < ε p : CRRA
q
0.01 0.02
0.05
k = 3 1.058 4.517 23.527
k = 4 0.213 1.527 12.712
k = 5 0.027 0.499 6.445
This table shows the posterior probability of the distortion function being
similar to the 45 degree line for each different tolerance q and model specifications k.
and find that the subjects, though informed of the true distribution, are still
ambiguity averse.
In this paper, we open up a new gate for a research in (empirical) auction
design, which has not yet been investigated. Though we explore here the
simplest setting as a first step, it can be extended to asymmetric bidders and
multi-unit auctions. Also, ambiguity will also affect bidding behavior in a
dynamic auction because ambiguity averse decision makers respond differently to new information than do the expected utility maximizer [Strzalecki,
forthcoming]. So, extending our framework to a dynamic auctions or auctions with affiliated values or pure common values would be important to
understand the effect of ambiguity on efficiency and revenue. For instance,
ambiguity could affect the ”Linkage Principle” and the revenue rankings.
All of these are important and interesting future research questions.
A PPENDIX A. M ULTIPLIER P REFERENCES
In this section, we show that (10) is equivalent to (11). The purpose of
this section is to convince readers about the equivalence relation between
(10) and (11). It exploits a result that is already known in the literature of
decision theory; see [Strzalecki, 2011]. From (10), we have
Z
Z
u
dF + K ( F k F0 )
min
udF + αK ( F k F0 )
= α min
α
F ∈Γ
F ∈Γ
Z
Z
u
dF dF1
= α min
dF + log
·
dF
α
dF1 dF0
F ∈Γ
Z
Z
u
dF1
dF .
= α min
dF + K ( F k F1 ) − log
α
dF0
F ∈Γ
32
ARYAL AND KIM
The second equality holds because for all F ∈ Γ we have F F0 and F0 F.
Let F1 be the candidate equilibrium such that
e−u/α
dF1
:= R −u/α
.
dF0
e
dF0
Using this definition of F1 , the RHS in the last equality become
Z
Z
Z
u
u
u
α min
dF + K ( F k F1 ) −
dF − log exp −
dF0
α
α
α
F ∈Γ
Z
Z
u
exp −
= α min K ( F k F1 ) − log
dF0 dF
α
F ∈Γ
Z
u
= α min K ( F k F1 ) − log
exp −
.
dF0
α
F ∈Γ
We can then see that the second term is independent of F and the minimum
is achieved when K ( F k F1 ) = 0, which happens if and only if F = F1 . Therefore, F1 defined above solves the minimization problem as desired and the
minimum is equal to
Z
u
−α log
exp −
dF0 ,
α
and in auction the utility is u(v − b) if the bidder wins the auction and
u(0) = 0, otherwise. Therefore, (10) can be written as
h
i
u(v − β n (z))
n −1
n −1
F0 (z)
+ exp(0) 1 − F0 (z)
max −α log exp −
α
z∈[v,v]
h
i
u(v − β n (z))
n −1
n −1
= min exp −
F0 (z)
+ 1 − F0 (z)
α
z∈[v,v]
u(v − β n (z))
= max 1 − exp −
F0 (z)n−1
α
z∈[v,v]
which equivalent to (11) with the transformation in (12).
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