Not Optimal but Reasonable: Revisiting
Overbidding in Sealed Bid Auctions
Sotiris Georganas∗, Dan Levin†,Peter McGee‡
September 1, 2014
Abstract
Bidding ones value in a second-price, private-value auction is a weakly dominant solution (Vickrey, 1961). Repeated experimental studies find more overbidding than underbidding, resulting in overbidding on average. In an experiment, we vary the monetary consequences of losing an auction by sometimes
mitigating realized losses and sometimes magnifying them. The dominant
strategy is unaffected by these manipulations but bidding behavior responds
predictably: the greater the cost to losing an auction, the less overbidding
we observe. These results suggest that subjects fail to discover the dominant
strategy—despite the fact that the dominant strategy only requires rationality from bidders—but respond in a common sense way to out-of-equilibrium
outcomes. We propose a model of minimal irrationality that can explain the
results of our experiment, but can also offer insight into bids above the risk
neutral Nash equilibrium in first-price auctions. Our findings lend themselves
to models in which less than fully rational bidders respond systematically to
out-of-equilibrium incentives, but we find that our model better fits the effects
of our manipulations than most of the existing such models we consider.
Keywords: auctions, dominant strategy, out of equilibrium payoffs
∗
Department of Economics, City University London
Department of Economics, Ohio State University
‡
Department of Economics, National University of Singapore
†
1
1
Introduction
In a sealed-bid second-price auction (SPA) with private valuations, where the highest
bidder wins and pays the second highest bid, bidding one’s value is a weakly dominant
strategy (WDS, Vickrey 1961). This strategy requires only that each bidder behave
rationally and it is unaffected by the number of rivals or their valuations, a bidder’s
risk preferences, or beliefs regarding rationality of rivals. Repeated experimental
studies have found that subjects deviate from the WDS by overbidding much more
than underbidding, resulting in overbidding on average. By contrast, experimental
evidence from the strategically equivalent ascending English auction demonstrates
almost immediate convergence to the dominant strategy (e.g. Kagel, Harstad and
Levin (KHL) 1987; Kagel and Levin (KL) 1993; Kagel 1995).1 While overbidding
relative to the risk-neutral, Nash equilibrium (RNNE) has also been frequently found
in first-price auctions (FPA) (e.g., Kagel 1995 and KL), the “usual suspects,” e.g.,
risk aversion, beliefs about others’ play, biases in perceptions of probabilities, etc.,
that may explain overbidding in FPAs are of no avail in SPAs.
The contrast between SPAs and English auctions suggests that subjects discover
the WDS in the English auction but not in the SPA; why is this the case? The
cognitive process that leads to the discovery of the WDS in an SPA is far from
trivial and an experimental subject may be unable to recognize it without significant
experience or training. In an English auction, on the other hand, a subject needs to
answer a simple question for herself: “Am I ‘in’ or ‘out’ ?” Answering this question
1
Given the differences in the strategy spaces, it is not strictly accurate to say that the two
auction formats have the same dominant strategy.
2
will lead a bidder to drop out at his value.
In this paper we investigate whether experimental subjects who fail to recognize the availability of a WDS in an SPA are still responding to more conventional
and commonly experienced incentives regarding expected gains and losses. In other
words, is it possible that subjects who do not recognize the surprising existence of
a dominant strategy are nonetheless employing strategies cued by more familiar incentives? To test our hypothesis in SPAs, we introduce a parameter that changes
the expected payoff as a function of one’s bid but does not affect the WDS. The
parameter multiplies realized losses by some amount, 0 < β ! 1, where β = 1 is
the standard case.2 Consistent with previous results we find that when 0 < β ≤ 1,
overbidding is pervasive. In contrast, when we change β to 20, overbidding is significantly reduced and underbidding is more prevalent. Overbidding when 0 < β ≤ 1
results in very few (5.8% of auctions) and fairly small losses (median loss of $0.10).
This is a product of our design: the domain of valuations is quite large relative to
the number of bidders, so the second highest bid is almost always below the highest value, even with overbidding. This allows us to rule out a “hot stove” type of
learning whereby losses reduce overbidding. Instead, it appears that the dramatic
reduction in overbidding occurs when β is exogenously and publicly increased and
can be attributed to changes in expected out-of-equilibrium payoffs.
Subjects who do not bid their value are likely still motivated by expected pay2
Several authors have argued that, in English or first-price auctions with private values, the
exact shape of the expected payoff functions matters. This is argued in Harrison (1989) and tested
in, among others, Goeree et al. (2002), Georganas (2011) and Georganas and Nagel (2011). Our
manipulation of β also changes the shape of the expected payoff function for bids that are greater
than an individual’s value.
3
offs in a common sense way. KHL conjectured that subjects are aware that higher
bidding increases the probability of winning the auctions but underestimate the additional cost associated with it (the dominant strategy argument). However, such
simplistic (behavioral) decision processes will still be affected by our manipulations
in a predictable way, in contrast to the dominant strategy predictions, because cost
estimates will be affected by β. Instead of looking for dominant strategies, we suggest
that minimally irrational bidders are guided by a desire to maximize their profits
combined with an inability to fully grasp the intricacies of the auction environment.
We do this by modeling reasonable bidders who recognize (i) a higher bid increases
the probability of winning, and (ii) the bidder may understate negative payoffs to
higher bids. These behaviorally plausible ingredients underpin our simple model
that can explain the effect of the manipulation of β in our experiment. Moreover,
although the purpose of our proposed new model is mainly to explain overbidding in
SPAs, it also provides a new and complementary explanation of overbidding relative
to the RNNE in FPAs. We compare our model of minimally irrational bidders to
several alternative models but find that all but one consistently falls short relative
to minimally irrational bidders.
Our strong findings suggest that incentives outside equilibrium affect behavior
in predictable ways in the laboratory, and probably in the field as well, even when
equilibrium analysis predicts otherwise. Goeree et al. (2002) show a similar result in
a FPAs, but our study provides an additional, and in our view important, insight. In
FPAs, as in many other games where Nash equilibrium is the solution concept, best
responding requires cardinal computations. Since such computations often involve
4
a high degree of complexity and a heavy mental cost, we do not expect that the
outcome in FPAs will exactly reflect the point prediction of Nash equilibrium. It
is much less surprising to find that the subjects’ calculations, possibly involving
heuristics, approximations and simplification rules, will be affected by a change in
the incentives, even if these ought to have no effect on Nash equilibrium. In an
SPA with independent private values, the dominant strategy can be reached with
just “ordinal logic” of dominance, without even a need for common knowledge of
rationality.3 Thus, one would expect the solution norm— bid your value —to have
its best chance for success in this environment. Our study shows that behavior is
still guided by some degree of conscious profit maximization, but subjects’ decision
processes fail to recognize a characteristic that is very seldom present in real life
games: the dominant strategy.
2
Equilibrium Predictions and Design
We employ standard second-price, sealed-bid auctions for one unit of an indivisible
good. The WDS predicts that players bid their values in equilibrium. We introduce
a factor β by which we multiply eventual negative profits of the bidders. For any β
the equilibrium remains the same because no subject earns negative profits in equilibrium. However, the bidders’ expected payoff functions do change, given that their
opponents follow the equilibrium strategies; in Figure 1 we plot, for different values
of β, the expected payoff function for a bidder whose rivals bid their values. In theory
3
A recent working paper by Levin, Peck and Ivanov (2013) is devoted to separating such failures
of “ordinal logic” (or “insight”) from the failures of and/or biases in Bayesian updating.
5
this change to the expected payoff function should not affect bidding because it only
changes out-of-equlibrium payoffs, but does bidding behavior change nonetheless? If
so, how?
The data come from nine experimental sessions conducted at Ohio State University. Students were recruited via e-mail and sessions took place in the Experimental
Economics Lab. The experiment was programmed and conducted with the software
z-Tree (Fischbacher 2007). In every session, subjects participated in 62 secondprice, sealed-bid auctions—2 trial auctions followed by 60 paying auctions—with
either three or six bidders per auction. Subjects were randomly and anonymously
rematched between auction periods. At the beginning of each auction, subjects
privately observed their own independent private values denominated in an experimental currency unit (ECU), but they did not observe the values of others.
All
values were drawn from a uniform distribution on the interval [0, 100], which was
common knowledge. At the end of each period the bidder who obtained the item
was informed of the price and his profit, while bidders who did not obtain the item
received no information about the price or the bids of others. The instructions can
be found in the appendix.
We multiplied negative profits by a parameter, β, which took on three values
in every session: 1, 0.1, and 20. Beta took on one of these values for periods 119, another for periods 20-39, and the final value for periods 40-60. Subjects knew
the value of β, that it is the same for all bidders, and that they would be made
aware when it changed; subjects did not know when β would change, how many
times it would change, or what its magnitude would be. All subjects were given
6
starting balances of 150 ECUs to cover the possibility of losses. Profits and losses
were added to this balance and the balance was paid at the end of each session. We
ran two sessions with 3 bidder auctions and a β order of 1, 0.1, 20, two sessions with
6 bidder auctions and a β order of 1, 0.1, 20, two sessions with 3 bidder auctions
and a β order of 1, 20, 0.1, and three sessions with 3 bidder auctions and a β order
of 20, 1, 0.1. In sessions with three bidder auctions, the exchange rate was $1=20
ECU, while the exchange rate in sessions with six bidder auctions was $1=14 ECU.
The exchange rates were different in order to equalize the expected payoff between
sessions with different group sizes. In the event that a player went bankrupt, they
were no longer permitted to bid and were paid a participation fee of $8. Due to
the uneven numbers after a bankruptcy, at the beginning of every period after a
bankruptcy two players were randomly assigned to sit out that period. In the six
sessions that started with β = 1, there were two bankruptcies; in the three sessions
that started with β = 20 there were 10 bankruptcies, with all but one occurring in
the first 10 periods. Complete session details can be found in Table 1.
3
3.1
Results
3 bidders with β order 1, 0.1, 20
Average differences between values and bids can be found in Table 2, and figure
2 plots subjects’ bids against their values. The effects of changes in β are visible
and drastic, in contrast to the theoretical prediction of no change at all. Subjects
bid more than their values when β = 1, which is typical of results in the literature
7
(e.g., KHL and KL). When we reduce the punishment for negative outcomes from
β = 1 to β = 0.1 in period 20, there is an immediate effect as the average difference
between bid and value more than doubles from approximately 2.8 to 7.5— an increase
equal to approximately 5% of the support of values. This effect is significant—Mann
Whitney U test comparing bidding in periods 10-19 to bidding in periods 20-29 yields
p < 0.001—and does not diminish while β = 0.1. When β rises to 20 in period 40 and
punishment for negative outcomes is severe, the overbidding largely disappears, with
average overbids falling from 7.5 to 1.1 (Mann Whitney U test comparing periods
30-39 and 40-49 yields p < 0.001). In some periods we even observe underbidding
on average, as can be seen in Figure 3, which plots the mean difference between bids
and values over time. The difference in bidding between β = 1 and β = 20 is also
highly significant (Mann-Whitney U yields p< 0.001).4
Subjects’ bids are not completely homogeneous. Some subjects tend to overbid
more and others tend to overbid less. Still, when β = 0.1 more than 20% of the
subjects’ bids exceeded their values by 10 units or more, as can be seen in Figure
4. In order to investigate the effects of β while allowing for individual heterogeneity,
we estimate the following random effects Tobit model5
Bidij = φ0 + φ1 V alueij + φ2 Dβ=0.1,j + φ3 Dβ=20,j + ωi + ϵij
Bidij is subject i’s bid in auction j, V alueij is his private value, Dβ=0.1,j is a dummy
4
Merlob, Plott, and Zhang (2012) document a similar sort of “overbidding” in the context of
a procurement auction. Their experiments compare different auction designs that in some cases
allows for costless reneging, which has an effect not unlike our manipulation of β.
5
In the experiment, subjects could not bid more than 100, which can be seen in the groups of
bids clustered at 100 in Figure 2. We use a Tobit model to account for the censoring of bids.
8
equal to 1 if β = 0.1 in auction j and 0 otherwise, Dβ=20,j is a dummy equal to 1 if
β = 20 in auction j and 0 otherwise, ωi captures a subject-specific random effect, and
ϵij is an econometric error term with mean zero. The results can be found in column
1 of Table 3. Consistent with the theory, the bidder’s value has a highly significant
impact on his bidding behavior and we cannot reject the null hypothesis that φ1 = 1
(F-test p = 0.341), though overbidding is significant even when β = 1.6 Nonetheless,
the estimated effects for the dummy variables for β values are both significant: when
β = 0.1 bids are more than 5 ECUs higher than when β = 1 (p < 0.001), and when
β = 20 bids are approximately 1.4 ECUs lower than when β = 1 (p = 0.008). These
changes as a result of changes in β amount to 10.6% and 2.6% of the average bid
(53.8 ECUs).
3.2
Additional Treatments
We ran additional treatments to make sure the results are not artefacts of our experimental design. For example, the amount of overbidding found in previous experimental second price auctions often changes with the number of bidders—though
this is not predicted by standard theory. Alternatively, it might also be possible that
subjects are influenced by the order in which they encounter the different values of
β.
Figure 5 plots the mean difference between bids and values over time for all
sessions and Figures 6-8 are scatter plots of bids versus values for all sessions. The
effects of β in these figures do not obviously differ across treatments. The mean
6
We fail to reject the null hypothesis that φ1 = 1 for any treatment.
9
difference between bids and values for these sessions can be found in Table 2 and
we find the same pattern in the mean overbidding across all sessions: it is largest
when β = 0.1 and smallest when β = 20. Non-parametric tests are consistent
with these means, as Mann Whitney U tests analogous to those discussed for the 3
bidder auctions with β order 1, 0.1, 20 yield significant differences in the expected
direction when β changes. Finally, in columns 2-4 of Table 3, we estimate the same
Tobit model as in column 1 for these treatments. Across all treatments, bids are
significantly higher when β = 0.1 than when β = 1 and significantly lower when
β = 20 than when β = 1. There are some differences in the magnitudes of these
effects, but there is no obvious pattern in the effects of various parameters across
sessions. For example, we cannot reject the null hypothesis that there is no difference
between the coefficients estimated for auctions with 3 bidders and a β order 1, 0.1,
20 and with 3 bidders and a β order of 20, 1, 0.1 (χ2 , p = 0.771). Our results are
robust to changes in the experimental design.
3.3
Do subjects learn to stop overbidding?
Our specific design allows for an additional important and interesting observation
regarding the evolution of bidding behavior over time when the β order is 1, 0.1, 20.
There is significant overbidding in the first 40 periods with β = 1 and β = 0.1 and
we see little learning in the direction of value bidding in these periods, but there is a
sharp and drastic reduction (almost elimination) of overbidding in period 40.7 One
7
We perform a test to detect learning in periods 20-40, where β = 0.1, in the following way. We
divide the data in 4 time blocks, with 5 periods in each block. We then run a Kruskal-Wallis H test
for the mean deviations (bid-value) in each time block. The test fails to reject that the 4 samples
10
possible explanation for the decline in overbidding in period 40 is that subjects who
have overbid in earlier periods are chastened by losing money, a sort of “hot stove”
learning. The evidence in Table 5 does not support this explanation for the changes
we see in bidding behavior in period 40. Before β increases to 20, few auctions result
in realized losses. They range from a minimum of 4.3% of auctions for the case with
six bidders when β = 1, to 9.3% with three bidders and β = 0.1. The average loss is
also quite small, ranging from 1.1 ECUs with six bidders and β = 0.1, to 10.8 ECUs
with six bidders and β = 1. Moreover, this small and infrequent negative feedback
for the first two levels of β appears to have no effect on those who experience it.
Of the 31 bidders who lose money when β = 1, 29 also lose money when β = 0.1.
Of the six bidders who lost money when β = 20, 5 lost money at all three levels of
β and the sixth lost money when β = 0.1. Thus, we cannot attribute the sudden
and sharp decline in overbidding we see when β increases to 20 to the tempering
impact of previously suffered losses, but it is consistent with subjects’ “common
sense” cognitive response to the change in incentives.
4
Modeling Minimally Irrational Bidders
The overlooked availability of a WDS must be the starting point of any explanation
to overbidding in SPA. We formalize the intuition behind the conjecture laid out
in KHL by modeling a bidder with “minimal irrationality” who understands that
there are possible gains and losses associated with higher bids but may overstate the
additional probability of winning due to higher bidding and/or the losses associated
are drawn from the same distribution, so we do not detect any time trend.
11
with it.8
Model
Let there be n risk-neutral bidders, each of whom privately observes her value xi ,
i = 1, ..., n. It is common knowledge that the xi ’s are i.i.d draws from a distribution
with a cumulative density function F (t), where F ′ (t) = f (t) > 0 on [0, 1], F (0) = 0,
and F (1) = 1. For our purposes, we assume that the xi ’s are drawn from the
generalized uniform distribution, F (t) = tα! and α
! ≥ 1, where α
! = 1 corresponds to
the uniform distribution used in almost all laboratory SPAs and FPAs.
Assumption 1. Symmetry
Each bidder believes that all other (n−1) bidders use the same, strictly monotonic
bidding function b(t) in an SPA (B(t) in FPA), with an inverse bidding function σ(b)
(Ψ(B) in FPA).
Assumption 2. Upon winning, a bidder’s gross payoff is x.
Assumption 3. Bidders believe that signals are i.i.d from a c.d.f. F (t) = tα with
α≥α
!.
This captures the notion that a bidder of type x > 0 potentially exaggerates
the benefits of bidding higher. He believes that, conditional on having the highest
signal, the probability that a rival’s signal, y ≤ x, is close to x, i.e., y ∈ [x − ε, x] for
0 < ε < x, is higher than the probability given by the priors.
8
“Bidding in excess of x in the second-price auctions would have to be labeled as a clear mistake,
since bidding x is a dominant strategy irrespective of risk attitudes. Bidding in excess of x is likely
based on the illusion that itmproves the probability of winning with no real cost to the bidder as
the second-high-bid price is paid.”(p. 1299, 1-5)
12
Assumption 4. A bidder in an SPA may understate possible losses when he bids
more than his value b > x and wins at price p ∈ (x, b). In this event the bidder
believes that the expected payment is γ(x) b+x
, with 0 < γ(x) ≤ 1, i.e., expected losses
2
are x − γ(x) b+x
.
2
Assumption 5. A bidder never bids above his private value in an FPA, i.e., B(x) ≤
x for all x.
4.1
Second Price Auctions
The maximization problem of a bidder in an SPA in our model is:
α(n−1)
max[σ(b)]α(n−1) {x − [ [σ(x)]
]θ (x) + [ [σ(b)]
[σ(b)]α(n−1) n
b≥0
α(n−1) −[σ(x)]α(n−1)
[σ(b)]α(n−1)
]γ(x) b+x
]},
2
where θn (x) denotes the expected price if a bidder wins at a price below his value.
The first-order condition (FOC) for the maximization problem is:
2xα(n − 1)[σ(b)]α(n−1)−1 σ ′ (b) − γ(x)α(n − 1)[σ(b)]α(n−1)−1 σ ′ (b)(b + x)
−γ(x)[[σ(b)]α(n−1) − [σ(x)]α(n−1) ] = 0
After simplifying we have:
2xα(n − 1)[σ(b)]α(n−1)−1 [1 − γ(x)]σ ′ (b) − γ(x)α(n − 1)[σ(b)]α(n−1)−1 (b − x)σ ′ (b)
−γ(x)[[σ(b)]α(n−1) − [σ(x)]α(n−1) ] = 0
(1)
13
Proposition 1. Sincere bidding, i.e., b(x) = x for all x, occurs if and only if for all
x, γ(x) = 1.
Proof. Assume that ∀x ∈ [0, 1], b(x) = x. In such a case equation (1) becomes
2xα(n − 1)[σ(b)]α(n−1)−1 (1 − γ(x)) = 0
implying γ(x) = 1, for all x > 0, as σ ′ (b) = 1 with b(x) := x.
Assume that ∀x ∈ [0, 1], γ(x) := 1. In such a case equation (1) becomes
−{[α(n−1)σ(b)]α(n−1)−1 (b−x)σ ′ (b)+[[σ(b)]α(n−1) −[σ(x)]α(n−1) ]} ! 0 as b(x) " x.
When α = α
! and γ(x) = 1, our model reduces to the standard model and we
arrive at the sincere bidding solution. This does not require that the bidder recognizes
the availability of the WDS. In fact, sincere bidding only requires γ(x) = 1 but allows
α>α
! .9
We are able to make a more general statement about the solution to the bidder’s
maximization problem.
Remark. There is a linear solution to the maximization problem with b(x; α, γ, n) =
δ(α, γ, n)x.
Consider the FOC in equation (1). Recognizing that at the solution σ(b) = x
and σ ′ (b) =
9
1
,
b′ (x)
and simplifying, we can rewrite (1) as
When b(x) > x, the value of α will, however, affect the extent one bids above his value.
14
b
σ(x) α(n−1)
2α(n − 1)[1 − γ(x)] − γ(x)α(n − 1)( − 1) − γ(x)b′ (x)[1 − (
)
] = 0. (2)
x
x
Assume that ∀x ∈ [0, 1], γ(x) = γ < 1 and b(x) = δx, with δ ≥ 1. Equation (2)
can be written as:
1 α(n−1)
2α(n − 1)[1 − γ] − γα(n − 1)(δ − 1) − γδ[1 − ( )
]=0
δ
When δ = 1 then (2) becomes 2α(n − 1)[1 − γ] > 0 when γ < 1, which is the case
for overbidding.
By inspecting (2), it is clear that the LHS is strictly declining in δ, so that there
is a unique δ ∗ that solves
2α(n − 1)[1 − γ] − γα(n − 1)(δ ∗ − 1) − γδ ∗ [1 − (
1 α(n−1)
)
]=0
δ∗
(3)
Having characterized the general solution to the maximization problem, we would
like to be able to say more about how this solution changes with the parameters of the
model. Before we can make any statements about the model’s predictions, however,
it is necessary to make a statement about the relationship between γ and β. Applying
the theory to our experimental treatments, when a bidder wins with b > x and loses,
his expected payment is β[x − γβ b+x
]. A standard SPA corresponds to our treatment
2
in which β = 1, such that the expected losses are 1 × [x − γ1 b+x
]. This allows us to
2
define γβ as a function of γ in a standard auction, i.e., γ1 , which we do by setting
15
[x − γ1 b+x
] = β[x − γβ b+x
], and solving for γβ . When we do this, we see that
2
2
γβ =
β − 1 2x
γ1
+
β b+x
β
(4)
We can substitute b(x) = δx into equation (4) to obtain
γβ =
1 2(β − 1)
[
+ γ1 ]
β δ+1
(5)
We plug (5) into equation (3) for γβ and solve numerically; the results can be
found in Table 5. Assuming α = 1.1 and γ1 = 0.95, the model’s parameters are
quite reasonable, with δ ∗ increasing monotonically in β and with plausible levels of
overbidding, ranging from bidding 9.7% above one’s value when β = 0.1 to 0.5% of
one’s value when β = 20.10
4.2
First Price Auctions
Minimally irrationality of this sort can also provide a reasonable explanation for
overbidding relative to the RNNE in FPAs. The maximization problem of a bidder
in an FPA in our model is:
max[σ(B)]α(n−1) [x − B]
B≥0
(6)
Denote the solution to (3) by B(x; α) and the risk-neutral Nash equilibrium by
B(x; α
!). It is well known that the solution to (3) is given by,
It is also possible to show that δ ∗ is increasing in α and n. These results are available from the
authors.
10
16
B(x; α) = x −
"x
0
tα(n−1) dt
xα(n−1)
=x−
x
α(n−1)+1
=
α(n−1)
x
α(n−1)+1
Proposition 2. B(x; α) ≥ B(x; α
!) as α ≥ α
!.
Proof. The proof is immediate.
Thus, minimally irrational bidders will bid above the RNNE but below their
value, which is consistent with a large body of experimental results.
5
Alternative models
One important test of our model of minimally irrational bidders is how well it fits
the data relative to existing models. We looked at six different models: symmetric
Nash equilibrium (SNE) with normally and logistically distributed errors, SNE with
normal and logistic errors allowing for joy-of-winning (JOW), Quantal Response
Equilibria (QRE, McKelvey and Palfrey (1995)), and QRE allowing for JOW.11 The
details of the fit exercise can be found in Tables 9 and 10 in the appendix, but we
find that the fit all but two of the models is poor relative to minimally irrational
bidders.
We focus on models that take all payoffs—not just those on the equilibrium
path—into account because these seem like good candidates to explain our results.
11
Another possible model is a level-k model of the sort used to explain auction results in Crawford
and Iriberri (2007). This model predicts neither overbidding nor reaction to different values of β,
even allowing symmetric errors. In such a model, players of level one or higher are characterized
by different beliefs, but they all best respond to these beliefs. Since bidding one’s value is a weakly
dominant strategy in a SPA and does not depend on a player’s beliefs (or risk attitudes), players
of all levels are predicted to bid their values, so we do not consider it.
17
A Nash equilibrium that allows subjects’ errors depend on the expected utility of
each action in a systematic way, such as the logistic error structure in Crawford and
Iriberri (2007), could potentially explain our experimental manipulation of β. Adding
logit errors to the Nash model performs worse than adding normally distributed error
in most cases, despite the fact that the latter fails completely to predict the change
in bidding when we shift β.12 Similarly, QRE posits that players choose an action
with a probability proportional to its expected payoff. QRE improves on the Nash
model with logistic errors, but is still out-performed by the Nash model with normal
errors.
An alternative is a “joy of winning” that can be incorporated by adding an extra
fixed utility, Ui , to the payoff of subject i, conditional on winning the auction.13 It
is easy to show that with such modification a new dominant solution emerges, with
bi (vi ) =
Ui
+vi .14
β
This implication helps as it predicts that players who enjoy winning
will overbid with respect to the Nash equilibrium and the amount of overbidding will
depend inversely on β. The JOW parameter j is found to be positive and yields a
significantly higher likelihood in every case. Nonetheless, SPNE with normal errors
12
The same would be true of any symmetric error specification. The logit errors do so poorly
because a high frequency of underbidding is predicted for intermediate private values with logistic
errors. However this prediction is not confirmed by the data.
13
Note that JOW looks superficially similar to spiteful behavior, analyzed for example in Andreoni
et al. (2007). The difference is that a subject experiences JOW only in the case where she wins,
while a subject exhibits spite even if she does not win but in cases where she raises the price for
other bidders. Thus, JOW yields constant overbidding in the baseline β = 1 case, while spite yields
falling overbidding. Low value players who win only with a small probability bid higher to raise
prices, while high value players overbid much less, since there is a higher probability that they will
be the actual winners. A look at the scatter plots reveals that the constant overbidding predicted
by joy of winning is much more plausible than the behavior predicted if spiteful motives are in
place.
14
Details are provided in the appendix.
18
still provides the best fit among JOW models.
Having established that the SPNE with normal errors fits our data best out of
all the models we consider, in Table 6 we compare how well the SPNE models with
normal errors fare against our model in SPAs, English auctions, and FPAs.15 We find
that our model fares better than the SPNE with just normal errors in all auctions.
On the other hand, the SPNE with normal errors and JOW outperforms our model
in every auction, albeit only slightly. The SPNE with normal errors and JOW may
have slight advantages by the measure of Table 6, but these advantages do not reveal
the full story. In Table 7 we break out the fit by the number of bidders and values
of β. In this case, the SPNE still fares better in only six of the nine cases. In Table
8, we compare the the predicted mean overbidding by minimally irrational bidders
and the SPNE with JOW to the observed mean overbidding. In 4 out of the 7 cases,
minimally irrational bidders are closer to the predicted overbidding that are SPNE
bidders who experience JOW.
6
Conclusions
Implementation by a dominant strategy is most appealing but it is not often available.
We examine a second-price, sealed-bid auction with private values— a mechanism
with incomplete information where bidding one’s value is a WDS. We found, as
others before us have, that many subjects in our experiments deviate significantly
from the WDS. We also find, however, that manipulations which do not to affect
15
The English auction data come from Georganas (2011), and the FPA data come from Dyer et
al. (1989).
19
the WDS but do affect expected payoffs out of equilibrium have predictable and
sizable effects on bidding behavior. One treatment that mitigates the penalty for
losing money increases observed deviation from the dominant strategy, doubling the
baseline percentage of overbidding to almost 40%, while another treatment that
sharply increases the penalty for losing money shrinks the fraction of overbids to
just 5% of all bids.
Our finding of such large and theoretically unpredicted responses to our treatments can inform mechanism designers, theorists and practitioners, who rely only
on equilibrium analysis: even in cases with a dominant strategy, the nature of incentives outside equilibrium must also be considered. Such an approach may help
design more stable and efficient mechanisms inducing behavior that is closer to the
desired norm.
We propose a model of minimal irrationality that considers bidders who fail to
recognize the availability of a dominant strategy. Bidders in this model understand
that raising their bid increases the probability of winning but may either overstate
the increase in the likelihood of winning and/or fail to appreciate the costs associated
with increasing their bids. Our data provide support for bidders of this sort, who are
reasonable if not bidding optimally. This model has the appealing feature that it can
also provide a complementary explanation for the large body of experimental firstprice auction results. In contrast, we fit several existing models designed to explain
overbidding using our data. We find that most of these models perform poorly
even when they consider out-of-equilibrium payoffs that would be affected by our
experimental manipulation, and none of the models outperforms ours consistently.
20
The availability of a “dominant” action that is best irrespective of the other
features of the decision is rare in games with incomplete information and in strategic
situations outside the lab. The behavior of a bidder in a second-price auction who
fails to recognize or discover such available strategy is still likely to be guided by
rules that are useful in a wide range of situations, such as cost-benefit analysis.
In accordance with such lessons learned in more familiar settings, as we vary the
magnitude of the penalty for losses, a natural reaction is to hedge and bid lower
when the penalty is relatively larger and to be more aggressive when the penalty is
relatively lower. Such behavior may not be optimal in a second-price auction, yet it
is sensible when viewed through the lens of its applicability in richer environments.
21
Figure 1: Expected payoff functions in a second price auction with 3 bidders, for the three
different values of β. There are 5 curves in every panel which represent expected utility,
depending on one’s bid for a private value v = 0, 25, 50, 75 and 100
22
Figure 2: Scatter plot of values versus bids in periods 5-19 (left panel), 20-39 (middle) and
40-60 (right panel) in an auction with 3 bidders
23
Figure 3: Mean difference between bids and private values over time in auctions with 3
bidders and β order 1, 0.1, 20.
24
Figure 4: Proportion of subjects bidding more than their private value plus X in a second
price sealed bid auction with three bidders and β order 1, 0.1, 20.
25
Figure 5: Mean difference between bids and private values over time in all treatments
26
Figure 6: Scatter plot of values versus bids in all treatments when β = 1
27
Figure 7: Scatter plot of values versus bids in all treatments when β = 0.1
28
Figure 8: Scatter plot of values versus bids in all treatments when β = 20
29
Table 1: Summary of sessions
Sessions
2
2
2
3
Bidders
3
6
3
3
Ex. Rate
14
20
14
14
Endowment
150
150
150
150
Periods
60
60
60
60
Subjects
24
42
33
51
β order
1, 0.1, 20
1, 0.1, 20
1, 20, 0.1
20, 1, 0.1
Subject-Auctions
1446
2603
1964
1783
Table 2: Mean difference between bid and value by treatment
β=1
2.79
(6.75)
4.07
(10.26)
4.04
(10.64)
2.37
(7.04)
β = 0.1
7.46
(13.67)
7.54
(12.96)
4.62
(9.94)
5.64
(10.61)
β = 20
1.07
(5.61)
0.43
(4.60)
2.32
(5.72)
1.13
(9.08)
Bidders
3
β order
1, 0.1, 20
6
1, 0.1, 20
3
1, 20, 0.1
3
20, 1, 0.1
Note: Standard deviations in parentheses.
30
Table 3: Estimated coefficients from a random effects Tobit model of the effect of β on
bids
Dependent Variable: Bid
(1)
Private Value
β = 0.1
β = 20
Constant
Observations
(2)
(3)
(4)
0.993***
1.016***
1.003***
1.001***
(0.008)
(0.006)
(0.007)
(0.007)
5.276***
4.492***
1.172**
3.548***
(0.527)
(0.450)
(0.489)
(0.551)
-1.400***
-3.579***
-1.513***
-3.034***
(0.530)
(0.483)
(0.534)
(0.520)
3.244***
3.236***
4.082***
4.878***
(1.314)
(0.876)
(1.024)
(1.052)
1446
2603
1964
1783
* p<0.10, ** p<0.05, *** p<0.01
Note: Standard errors are reported in parentheses. Column 1 is β order 1, 0.1, 20 with 3 bidders, column 2 is β
order 1, 0.1, 20 with 6 bidders, column 3 is β order 1, 20, 0.1 with 3 bidders, and column 4 is β order 20, 1, 0.1
with 3 bidders. Bids are censored at 100.
31
Table 4: Subjects experiencing losses with β order 1, 0.1, 20
Three Bidders
Fraction of Auctions Resulting in Losses
Mean Loss (in ECUs)
Median Loss (in ECUs)
Maximum Loss (in ECUs)
Number of Subjects Who Experience a Loss
β=1
5.4%
9.9
6.0
43.0
11
β = 0.1
9.3%
1.5
1.0
8.0
14
β = 20
0.6%
206.7
140.0
400.0
2
Six Bidders
Fraction of Auctions Resulting in Losses
Mean Loss (in ECUs)
Median Loss (in ECUs)
Maximum Loss (in ECUs)
Number of Subjects Who Experience a Loss
β=1
4.3%
10.8
6.5
88.0
20
β = 0.1
7.1%
1.6
1.1
8.8
27
β = 20
0.4%
65.0
30.0
180.0
4
Table 5: Numerical estimates of parameters for minimally irrational bidders in auctions
with 3 bidders
β
0.10
0.20
0.50
1
2
10
20
δ
1.0965
1.0889
1.0715
1.053
1.0352
1.0093
1.0048
γβ
0.9141
0.9202
0.9345
0.9500
0.9664
0.9909
0.9952
Note: The numerical solution was found using fzero in Matlab, assuming α = 1.1 and γ1 = 0.95. The highlighted
values of β correspond to the values used in the experiment.
32
Table 6: Comparing model fits across auction formats
SPA
English
FPA
13415
26212
37574
α = 1.82, σ = 10.18, γ = 0.95
α = 1.82, σ = 10.46, γ = 0.95
α = 1.78, σ = 8.54, γ = 0.95
13687
26480
38036
σ = 10.99
σ = 10.87
σ = 10.87
13323
26107
37459
σ = 9.92, j = 0.77
σ = 10.30, j = 0.78
σ = 8.45, j = 0.79
Minimally Irrational
SPNE with normal errors
SPNE with normal errors and JOW
Note: The SPA models are estimated including data from β order 1, 0.1, 20 for both the 3 and 6 bidder cases. The
English auction data come Georganas (2011) and the group size is 4. The FPA data come from Dyer et al. (1989)
with 3 and 6 bidder auctions.
Table 7: Estimated Log Likelihoods
Minimally Irrational Bidders
SPNE with normal errors and JOW
β=1
1151
1151
SPA, N = 3
β = 0.1
β = 20
2169
1518
2093
1516
β=1
2263
2295
SPA, N = 6
β = 0.1 β = 20
3609
2831
3547
2825
English
β=1
13002
12995
FPA
β=1
β=1
5515
5517
5546
5491
Note: The first three columns correspond to the 3 bidder auctions for the SPA, while the next three columns
correspond to the 6 bidder auctions. The English auction data come Georganas (2011) with 4 bidders per auction.
The FPA data come from Dyer et al. (1989). The first column of FPA results come from 3 bidder auctions and the
second column come from 6 bidder auctions.
33
Table 8: Comparing Predicted Mean Overbidding to Observed Mean Overbidding
Minimally Irrational
SPNE with normal errors and JOW
Observed
β=1
2.89
0.78
3.26
SPA, N = 3
β = 0.1
β = 20
5.12
0.26
7.84
0.04
7.46
1.07
β=1
3.13
0.78
4.48
SPA, N = 6
β = 0.1
β = 20
5.40
0.27
7.84
0.04
7.54
0.43
English
β=1
0
0.78
1.17
Note: The first three columns correspond to the 3 bidder auctions for the SPA, while the next three columns
correspond to the 6 bidder auctions. The English auction data come Georganas (2011) with 4 bidders per auction.
The FPA data come from Dyer et al. (1989). The first column of FPA results come from 3 bidder auctions and the
second column come from 6 bidder auctions.
References
[1] Andreoni, James; Yeon-Koo Che & Jinwoo Kim (2007) ”Asymmetric Information about Rivals’ Types in Standard Auctions: An Experiment,” Games and
Economic Behavior, vol. 59(2), 240-259.
[2] Crawford, Vincent P. & Nagore Iriberri (2007) ”Level-k Auctions: Can a NonEquilibrium Model of Strategic Thinking Explain the Winner’s Curse and Overbidding in Private-Value Auctions?,” Econometrica, vol. 75(6), 1721-1770.
[3] Dyer, Douglas; John Kagel & Dan Levin (1989) ”Resolving Uncertainty About
Numbers of Bidders in Independent Private Value Auctions: An Experimental
Analysis,” RAND Journal of Economics, vol. 2, 268-279.
[4] Fishbacher, Urs (2007) ” z-Tree: Zurich Toolbox for Ready-made Economic Experiments, Experimental Economics”, Experimental Economics, vol. 10(2), 171178.
[5] Georganas, Sotiris (2011) ”English Auctions with Resale: An Experimental
Study”, Games and Economic Behavior, vol. 73(1), 147-166.
34
[6] Georganas, Sotiris & Rosemarie Nagel (2011) ”English Auctions with Toeholds:
An Experimental Study,” International Journal of Industrial Organization, vol.
29(1), 34-45.
[7] Goeree, Jacob K.; Charles A. Holt & Thomas R. Palfrey (2002). ”Quantal Response Equilibrium and Overbidding in Private-Value Auctions,” Journal of Economic Theory, vol. 104(1), 247-272.
[8] Harrison, Glenn (1989) ”Theory and Misbehavior of First-Price Auctions”, American Economic Review, vol. 79(4), 749-762
[9] Kagel, John H. (1995) Auctions: A Survey of Experimental Results In: Kagel,
John H. and Alvin Roth (eds.) The Handbook of Experimental Economics,
Princeton University Press, Princeton NJ
[10] Kagel, John H & Dan Levin (1993) ”Independent Private Value Auctions: Bidder Behaviour in First-, Second- and Third-Price Auctions with Varying Numbers
of Bidders,” Economic Journal, vol. 103(419), 868-79, July.
[11] Kagel, John H. ; Ronald M. Harstad & Dan Levin (1987) ”Information Impact
and Allocation Rules in Auctions with Affiliated Private Values: A Laboratory
Study”, Econometrica, vol. 55(6), 1275-1304
[12] Levin, Dan; James Peck & Asen Ivanov (2013) ”Separating Missing Insight
from Bayesian Updating: An Experimental Investigation,” Ohio State University
Working Paper
35
[13] McKelvey, Richard & Thomas R. Palfrey (1995) ”Quantal Response Equilibria
in Normal Form Games,” Games and Economic Behavior, Elsevier, vol. 10(1),
6-38.
[14] Merlob, Brian ; Charles Plott & Yuanjun Zhang (2012) ”The CMS Auction:
Experimental Studies of a Median-Bid Procurement Auction with Non-Binding
Bids”, The Quarterly Journal of Economics, vol. 127(2), 793-828.
[15] Vickrey, William (1961) ”Counterspeculation and Competitive Sealed Tenders,”
Journal of Finance. Vol. 16(1), 8-37.
A
A.1
Appendix
Proofs
Lemma 3. In a second price auction where negative payoffs are multiplied times β
and bidders have a heterogeneous joy of winning Ui , the dominant strategy equilibrium
is bi (vi ) =
Ui
β
+ vi .
Proof. Expected profits in the auctions are Πi = prob{bi > max(b−i )}(vi + Ui −
E[max(b−i )|bi > max(b−i )]), and subjects choose a bi (vi ) to maximize Πi . Consider
bidding b0 < bi (vi ) =
Ui
β
+ vi , when it matters i.e. you win with bi (vi ) but lose with
b0 . It also means that the price the winner pays is p ∈ [b0 , bi (vi )]. If p > vi , winning
earns Ui − β(p − vi ) ≥ Ui − β(( Uβi + vi ) − vi ) = 0. When p ≤ vi , strict positive payoffs
are assured. With a similar step we show that bidding bi (vi ), weakly dominates
bidding b0 > bi (vi ).
36
A.2
Examining the consistency of joy of winning
The fit of the SNE and the behavioral models is improved when we allow for joy of
winning. But is joy of winning a consistent explanation across all different values of
β that we have used in the experiments? As we have seen, joy of winning gives a
clear prediction for every β. A person who understands the dominant strategy but
has joy of winning j, will overbid by exactly j when β = 1 but will overbid by 10j
when β = 0.1. On the other hand she has to overbid by only j/20 when β = 20. In
following Figure 9 we examine if individual players’ behavior is consistent with this
model.16
Figure 9: Consistency of the joy of winning hypothesis for each individual across the three
values of β.
Every dot is a single observation, plotting the average overbidding of a player for
two different values of β. In the left panel for example, we have mean overbidding
16
The bidding results of the 3 bidder and 6 bidder case are pooled, as the number of bidders
makes no difference for the overbidding predicted by the joy of winning model.
37
when β = 1 in the x axis and mean overbidding when β = 0.1 in the y axis. According
to the theory all observations should lie on a straight line through the origin with
slope βx /βy . What we see is that many observations lie on or close to the line.
However there are still some observations, especially in the second and third panel
that lie far away from this line. This indicates that although heterogeneous joy of
winning could at first sight partly explain overbidding in second price auctions, it is
not a fully consistent explanation.
A.3
Goodness of fit tables
Table 9: Model fit with β order 1, 0.1, 20. SNE denotes the symmetric Nash equilibrium.
-LL is the negative log likelihood, BIC the Bayesian Information Criterion. Note a level k
model is the same is the SNE.
SNE (norm errors)
SNE (log. errors)
QRE
SNE (norm. errors)
SNE (log. errors)
QRE
-LL
BIC
σ
-LL
BIC
λ
-LL
BIC
λ
-LL
BIC
σ
-LL
BIC
λ
-LL
BIC
λ
3 bidders
N = 336
β=1
1158.4
2317.9
7.60
1162.3
2330.4
1.58
1161
2327.8
1.56
6 bidders
N = 588
β=1
2260.5
4527.3
11.3
2464.1
4934.6
0.45
2417.7
4841.7
0.41
N = 480
β = 0.1
1998.5
3998.1
15.56
1947.9
3902
0.63
1960.5
3927.2
0.68
N = 840
β = 0.1
3461.5
6929.7
14.98
3603.3
7213.3
0.61
3614.6
7236
0.57
N = 462
β = 20
1460.4
2921.9
5.71
1777.9
3561.9
0.28
1747.8
3501.7
0.28
N = 882
β = 20
2601.7
5210.3
4.6
2812.5
5631.1
4.69
2714.8
5436.4
4.82
38
N = 1278
pooled (common prec.)
4861.4
9723.9
10.86
4984.7
9976.6
0.52
4959.8
9926.8
0.54
N = 2310
pooled (common λ)
8825.1
17658
11.06
9212
18431.8
0.81
9084
18176
0.83
N = 1278
pooled (β-dependent prec.)
4617.3
9241.7
4888.1
9783.4
4869.3
9745.8
N = 2310
pooled (β-dependent λ)
8323.7
16654.6
8879.9
17767.0
8747.1
17501.4
39
(1278 observations)
SNE+joy (norm)
4729.1
9472.5
σ = 9.66, j = 0.77
(2310 observations)
8593.2
17201
σ = 10, j = 0.78
3 bidders
SNE (norm)
4861.4
9723.9
10.86
6 bidders
8825.1
17658
11.06
-LL
BIC
parameters
-LL
BIC
parameters
9212
18432
λ = 0.81
SNE (logit)
4984.7
9976.6
λ = 0.52
9057.9
18131
λ = 1.6, j = 2.2
SNE (logit+joy)
4944.9
9904.1
λ = 0.68, j = 2.4
9084
18176
λ = 0.83
QRE
4959.8
9926.8
λ = 0.54
8850.9
17717
λ = 1.08, j = 6.3
QRE+joy
4838
9690.3
λ = 0.71, j = 5.8
8626
17273
α = 1.8172, γ = 0.9465, σ = 10.1841
Minimally Irrational
4788.5
9598.4
α = 1.8172, γ = 0.9465, σ = 10.1841
Table 10: Model fit in all treatments with β order 1, 0.1, 20. SNE denotes the symmetric Nash equilibrium. -LL is
the negative log likelihood, BIC the Bayesian Information Criterion. Note a level k model is the same is the SNE.
Auction experiment
This is an experiment in the economics of market decision making. The National Science
Foundation and other research organizations have provided funds for conducting this research.
The instructions are simple, and if you follow them carefully and make good decisions you may
earn a CONSIDERABLE AMOUNT OF MONEY which will be PAID TO YOU IN CASH at the
end of the experiment.
1. In this experiment we will create a number of markets in which you will act as buyers of a
fictitious commodity. There will be 6 bidders in each market. Bidders will be randomly
broken up into groups in every period so that the other bidders in your group change from
period to period. There will be 62 trading periods, with the first two being only trial periods
with no actual payoffs. A single unit of the commodity will be auctioned off in each market in
every trading period.
Your task is to a bid for the commodity along with the three other buyers in your market. In
each trading period you will be assigned a private VALUE for the item. This indicates the value
to you of holding the item at the end of a period.
Values
All values in the experiment will be in terms of an experimental currency, the drachma. In each
period bidders will be given a new private value for the good that is being sold. This value is the
amount of drachmas the experimenter will give you if you wind up getting the item at the end of
the auction period. At the start of each auction these values are drawn randomly from a uniform
distribution over the interval (0 , 100). The value you have in one auction has no impact on the
value you will have in the next auction and the value you have has no impact on the value of the
other bidders. No bidder will ever have a value greater than 100 and no bidder will have a value
lower than 0.
The Auction
There will be a second-price sealed bid auction. This means, the bidder with the highest bid in
your market obtains the item and pays the second-place bid. If you are not the highest bidder,
you do not obtain the item. This establishes, for the high bidder, an interim profit of
(PRIVATE VALUE) – (SECOND HIGHEST BID) = INTERIM PROFIT
If the interim profit is positive, then the final profit is equal to the interim profit. That is
(PRIVATE VALUE) – (SECOND HIGHEST BID) = INTERIM PROFIT > 0
INTERIM PROFIT = FINAL PROFIT
If profit is negative, the final profit is equal to a number X times the interim profit. That is
(PRIVATE VALUE) – (SECOND HIGHEST BID) = INTERIM PROFIT < 0
X × INTERIM PROFIT = FINAL PROFIT
For the bidders who do not obtain the item, the profit is zero. The value of X will change during
the experiment but will always be the same for all bidders. Before you begin bidding, you will be
made aware of the value of X and you will be made aware of the new value of X whenever it
changes.
This is all little abstract so let’s give some examples:
Suppose that you have a private value of 68 and you bid 60 and X = 1. The second-highest
bidder submits a bid of 53. You submit the highest bid and pay 53 drachmas, making your
interim profit 15 (68 – 53 = 15). Because your interim profit is positive, your final profit is equal
to your interim profit, 15.
Suppose you have a private value of 48 and you bid 55 and X = 8. The second-highest bidder
submits a bid of 52. You submit the highest bid and pay 52 drachmas. Your interim profit is -4
(48 – 52 = -4). Because your profit is negative, your final profit will be eight times your interim
profit, or -32 (8 x -4 = -32).
Note that in this auction the items for sale have no intrinsic value. You would do us a favor if you
tried to maximize your own earnings, as our objective is to study how real, cool-headed people
are likely to behave outside the lab.
Total payoffs
In the beginning of today’s session you will receive an endowment of 150 drachmas which
includes your show up fee. Every period your payoff from that period is added to this initial
endowment. At the end of the today’s session you will receive your cumulative payoffs – your
earnings from each auction plus your initial endowment. The conversion rate from drachmas
dollars is $1 = 14 drachmas. Note that if at any period your cumulative auction profits are such
that you have exhausted your initial endowment, you will not be allowed to continue playing.
Please, answer these questions, they will help us to know if you understood the instructions
well. When you finish, call the coordinator and he will check the answers.
Suppose your private value is 18. What is a possible value for the other bidders?
a)
b)
c)
d)
any values from 0 to 99
any values from 0 to 100
any values from 0 to 99 except 18
any values from 0 to 100 except 18
Your private value is 60 and you bid 50 in the auction. The highest bid submitted is 50 and X =
1. The second highest bid is 40. What is your final payoff?
_____________________________________________
Your private value is 46 and you bid 50 in the auction. The second highest bid is 72 and X = 8.
What is your final payoff?
_____________________________________________
Your private value is 53 and you bid 60 in the auction. The highest bid submitted is 60 and X=1.
The second highest bid is 57. What is your final payoff?
Your private value is 83 and you bid 88 in the auction. The highest bid submitted is 88 and X=8.
The second highest bid is 87. What is your final payoff?
_____________________________________________
Your private value is 66 and you bid 90 in the auction. The highest bid submitted is 90 and
X=0.1. The second highest bid is 87. What is your final payoff?
_____________________________________________