Spite and overbidding in second price
all-pay auctions
A theoretical and experimental investigation
Oliver Kirchkamp∗
Wladislaw Mill†
February 6, 2016
An extensive literature provides evidence for overbidding in auctions. Risk
aversion, joy of winning, anticipated regret etc. do not always offer sufficient
explanations. We show that the second-price all-pay auction presents a very
interesting environment to study spite and overbidding.
In equilibrium we should expect spiteful participants to increase their bids
for small valuations and decrease their bids for high valuations. We measure
spite and bidding behaviour for our participants and find that, indeed, the
more spiteful participants also follow a bidding pattern which is theoretically
in line with more spitefulness.
Keywords: Auction, Overbidding, Spite
1. Introduction
Auctions are an important part of everyday life. Not only do we find explicit auction
institutions in markets. Auctions are also often a good model of non-market interaction.
For example, fights between animals (Riley, 1980; Smith, 1974)1 , competition between
firms (Fudenberg and Tirole, 1986; Ghemawat and Nalebuff, 1985; Oprea et al., 2013),
the voluntary provision of public goods (Bilodeau et al., 2004), legal expenditures in
litigation environments (Baye et al., 2005), the settlement of strikes, fiscal and political
stabilization, the timing of exploratory oil drilling, and many more (see Hrisch and
Kirchkamp, 2010, p. 1) are applications of (all-pay-)auctions.
∗
Department of Economics, Friedrich-Schiller-University Jena, Germany
Department of Economics, Friedrich-Schiller-University Jena, Germany and International MaxPlanck-Research School
1
Smith (1974) uses a war-of-attrition game with common valuations to model fights between animals.
†
1
While risk neutral Bayesian Nash equilibria can be derived for many auction types,
participants in lab-experiments do not always follow these equilibria. Many authors find
overbidding (or over-dissipation): actual bids are higher than equilibrium bids in all-pay
auctions,2 in rent-seeking contests,3 in winner-pay auctions.4 A wide literature discusses
possible adjustments of equilibrium behaviour to explain overbidding.5 However, some
papers indicate that explanations like risk-aversion, joy of winning, anticipated regret
etc. are not in line with all aspects of observed behaviour.6 Some authors suggest spite
as a possible explanation.7
Here we present the second-price all-pay auction as an interesting environment to
study bidding behaviour. We show that in the Bayesian Nash equilibrium of this auction
spiteful subjects should overbid up to a certain (high) threshold. In our experiment we
measure separately spitefullness and bids in auctions. Bidding behaviour confirms the
theoretical prediction: More spite leads to higher bids.
We think that we can make three contributions to the literature:
1. We extend the theoretical model of spiteful behaviour to first-price and secondprice all pay auctions.
2. With the help of experiments we investigate bidding behaviour in the second-price
all-pay auction.
3. Most importantly, we can show that spite is an important determinant of overbidding.
The remainder of the paper is structured the follows: In Section 2 we briefly summarise
the relevant literature. In Section 3 we present the model and the theoretical predictions.
Section 4 will explain the design of the experiment. In Section 5 we show the results of
the experiment. Section 6 concludes.
2. Literature
In this paper we study consider second-price all-pay auctions with sealed-bids and private
information.8 Furthermore, we restrict our attention to the case where the ex-post
2
(Noussair and Silver, 2006; Ernst and Thni, 2013; Goeree et al., 2002; Ong and Chen, 2013; Lugovskyy
et al., 2010).
3
Potters et al. (1998)
4
Morgan et al. (2003); Andreoni et al. (2007); Barut et al. (2002).
5
Filiz-Ozbay and Ozbay (2007, 2010); Cooper and Fang (2008); Andreoni et al. (2007); Cox et al.
(1985, 1988); Fibich et al. (2006); Kagel and Levin (1993); Kirchkamp and Reiss (2008); EngelbrechtWiggans and Katok (2009); Kirchkamp et al. (2008); Armantier and Treich (2009).
6
Kagel and Levin (1993); Kirchkamp et al. (2008); Engelbrecht-Wiggans and Katok (2009); Andreoni
et al. (2007).
7
Morgan et al. (2003); Nishimura et al. (2010); Andreoni et al. (2007); Bartling and Netzer (2014);
Nishimura et al. (2010); Kimbrough and Reiss (2012).
8
Equilibria for all-pay auctions with common values are provided by Hendricks et al. (1988)
andKovenock et al. (1996). Sacco and Schmutzler (2008) provide mixed strategy equilibria for
2
allocation (once all bids are known) is not stochastic.9 In addition, we will consider first
and second price all-pay auctions only.10 We also assume here that the playing subjects
are already specified.11
2.1. Factors other than spite that influence bids
One of our aims is to better understand the influence spitefulness has on bidding behaviour. Besides spitefulness a large number of other factors, internal and external to
the bidders, have already been studied.
Among the external factors, Katok and Kwasnica (2008) show that the speed of the
ticking clock in Dutch auctions affects bids. Cox and James (2012) show that the structure of the presented games have an influence on the bidding behavior of the participants.
Kirchkamp et al. (2009) show that outside options also influence bids.
Among the factors more internal to bidders Nakajima (2011) explains the deviation
of the Dutch auction from the first price auction with the Allais paradox. Gth et al.
(2003), Dittrich et al. (2012), and Ockenfels and Selten (2005) relate bidding behaviour
to learning effects. Kagel et al. (1987) and Hyndman et al. (2012) show that the provision
of information influences overbidding. Chen et al. (2013) provide empirical evidence that
even the menstrual cycle has an influence on the bidder’s decision.
2.2. Literature on overbidding
In many experiments overbidding has been found and explained with different motives,
ranging from risk aversion, over spite, to anticipated regret.
Filiz-Ozbay and Ozbay (2007, 2010) explain overbidding with the emotion of anticipated regret in winner-pay auctions. They assume that players anticipate their regret
after a wrong choice.
Cooper and Fang (2008) are discussing whether joy of winning is a possible explanation
for overbidding. Andreoni et al. (2007) provide evidence against against joy of winning.
Cox et al. (1985, 1988) suggest risk-aversion to explain overbidding. Fibich et al.
(2006) introduce risk averse players in all-pay auctions to explain overbidding. Kagel
and Levin (1993), Kirchkamp et al. (2008), and Engelbrecht-Wiggans and Katok (2009)
argue that risk aversion might by itself not enough to explain overbidding.
common value auctions where the prize is influenced by the own behavior. Feess et al. (2002) show
a pure equilibrium strategy in case of handicapped players. Klose and Kovenock (2015) show equilibria for the case of externalities which depend on the bidders’ identities. Bertoletti (2006) show
equilibria for common value all-pay auctions with reserve price. Dechenaux and Mancini (2008) and
Baye et al. (2005) are model ligation systems with all-pay auctions. The case of affiliated valiations
is studied by Krishna and Morgan (1997).
9
The survey by Dechenaux et al. (2015) also covers rent-seeking games where the ex-post allocation
is stochastic and where also bidders who did not submit the highest bid have a chance to win the
auction.
10
Intermediate situations between the first and second price all-pay auction are studied by Albano
(2001).
11
Bos (2012) considers the situation where the number of bidders is uncertain.
3
Anderson et al. (1998) are introducing bounded rationality as a reason for overbidding
in all-pay auctions. Similarly, Kirchkamp and Reiss (2008) claim that overbidding may
result from bidding heuristics. Armantier and Treich (2009) argue that bidders are
unable to assess winning probabilities correctly.
Andreoni et al. (2007) suggest that spite may be on of the determinants of overbidding.
Also Bartling and Netzer (2014) discuss that “spiteful preferences are an important determinant of overbidding in the second-price auction” (Bartling and Netzer, 2014, p.23).
To the best of our knowledge, only Morgan et al. (2003) and Nishimura et al. (2010)
introduce spite into theoretical bidding behavior. Different from our paper, Nishimura
et al. (2010) and Morgan et al. (2003) study spite in common-valuations-auctions and
winner-pay auctions, respectively. Nevertheless, our theoretical part can be seen as an
extension of Morgan et al. (2003).
2.3. Literature on spite
Besides papers suggesting spite being a determinant of overbidding, there are papers
showing data suggesting that spite is not uncommon in general. D’Almeida et al. (2014)
claim to show that children can behave spiteful in economic games. Chun et al. (2011)
observe that Koreans and Chinese behave less spiteful than Japanese. Fehr et al. (2008)
show experimentally that spiteful behavior is rather wide spread in the least developed
parts of India. Kimbrough and Reiss (2012) observe consistent spiteful behavior in a
second-price winner-pay auction, and Marcus et al. (2014) developed a psychological
test for spitefulness.
In the next section we will present a model that explicitly incorporates spite in the
context of an all-pay auction.
3. Model
In the following we will derive the risk neutral Bayesian Nash equilibrium for spiteful
bidders in the first-price all-pay auction and in the second-price all-pay auction.
3.1. First-price all-pay auction
Two bidders i and j have private valuations for a prize with with values vk , k ∈ {i, j}.
Both submit a bid bk following a monotonic bidding function βk (vk ) with first derivative
and inverse βk−1 (bk ) = vk . Valuations are distributed according to F ,
β ′ (x) ≡ dβ(x)
dx
i.e. v ∼ F (0, v), and f (x) = dFdx(x) . The prize is allocated to the player with the highest
bid. If vi = vj , the prize is distributed randomly. We assume that both bidders have
the same utility function u(x).
Following a standard argument in equilibrium βk (0) = 0. is the utility function and
we assume bidders to be risk neutral.
To integrate spite into the model we assume that the losing participant gains an
additional disutility of α-times the payoff of the winning player. Assume without loss of
4
generality that bj > bi , i.e. bidder j is the winner. Then α · (vj − βj (vj )) reflects bidder
i’s spite.12 The absence of spite is equivalent α = 0. We assume that α ∈ [0, 1), i.e. we
do not consider α < 0 which could represent sympathy or profit sharing.
The payoff of player i is as follows:
ΦI-AP
(βi , vi )
S



u(vi − βi (vi ))
if βi > βj (i wins)
=
if βi = βj (a tie)


u(−βi (vi ) − α(vj − βj (vj ))) if βi < βj (j wins)
u( vi −β2i (vi ) )

(1)
We follow the standard approach and assume that bidder i with valuation vi makes a
bid b. The expected utility of this bidder is given as follows:
E(b, v) =
∫
βj−1 (b)
|0
u(v − b)f (vj )dvj +
{z
}
bidder i wins and obtains the prize
∫
|
v
βj−1 (b)
u(−b − α(vj − βj (vj )))f (vj )dvj
{z
(2)
}
bidder i loses and pays his own bid
and additionally experiences spite
Rearranging the FOC leads to:
(u(v − b) − u(−b − α(βj−1 (b) − βj (βj−1 (b))))f (βj−1 (b))
β ′ (βj−1 (b)) = ∫ β −1 (b)
∫
j
u(v − b)′ f (vj )dvj + βv−1 (b) u(−b − α(vj − βj (vj )))′ f (vj )dvj
0
j
Assuming a symmetric equilibrium bidding function, we obtain the following condition:
((v − b) − (−b − α(v − b)))f (v)
= (1 + α)vf (v) − αf (v)b
(3)
1
Solving the ODE (3) with the initial value b(0) = 0 we obtain the symmetric equilibrium
bidding function bI-AP
:
S
β ′ (v) =
bI-AP
(v)
S
=e
−αF (v)
∫
v
(α + 1)sf (s)e
αF (s)
0
(
)
1 ∫ v αF (s)
α+1
v − αF (v)
ds =
e
ds
α
e
0
(4)
For α = 0, Equation (4) becomes the familiar equilibrium bidding function for all-pay
auctions without spite:
∫ v
bI-AP := bI-AP
=
sf (s) ds
(5)
α=0
0
For uniformly distributed valuations, F (x) = x, we have
(
(v)
bI-AP
S
α+1
e−αv − 1
=
v+
α
α
)
.
(6)
The left part of Figure 1 illustrates in particular the following:
12
Note that spite is similar but not equivalent to the negative aspect of inequality aversion. The latter
would consider relative gain α · (vj − βj (vj ) − βi (v) ).
5
α=2
v=1
0.8
1
0.9
0.6
0.5
0.6
0.8
b
b
0
0.4
0.4
0.7
0.6
−0.5
0.5
0.2
0.2
0.4
0.0
0.0
−1
0.2
0.4
0.6
0.8
0.3
0.2
00.1
1.0
0.2
v
0.4
0.6
0.8
1.0
α
Figure 1: Equilibrium bids in first-price all-pay auctions
Equilibrium bids in first-price all-pay auctions for different valuations v and different levels of spite α
with uniform distributions of valuations (see Equation 6).
Proposition 1. The bidding function in the first-price all-pay auction is increasing in
the bidder’s valuation:
dbI-AP
S
≥0
dv
Proof
(
∫ v
dbI-AP
1
S
= αF (v) −αf (v) (α + 1)sf (s)eαF (s) ds + (α + 1)vf (v)eαF (v)
dv
e
0
)
(α + 1)f (v) ∫ v αF (s)
=
e
ds ≥ 0
eαF (v)
0
The right part of Figure 1 illustrates the following:
Proposition 2. Bids in the first-price all-pay auction are increasing in spite:
dbI-AP
S
≥0
dα
The proof is shown in Appendix A.
In particular bids in the first-price all-pay auction with spite (α > 0) are larger or
− bI-AP ≥ 0.
equal than bids without spite (α = 0), i.e. bI-AP
S
6
Proposition 3. The difference between equilibrium bids with spite and without spite is
increasing in bidder’s valuation:
d(bI-AP
− bI-AP )
S
≥0
dv
The proof is shown in Appendix B.
Proposition 3 is actually quite intuitive. Consider two bidders competing for an object
in the first-price auction. For a low valuation the probability of winning is low, too. In
this situation bidders will suffer from spite almost always. As a result the overall value
of the auction is small and bids will be small, too. If, on the other hand, valuations are
high bidders will want to avoid the disutility from losing by increasing their bids. As a
result we see that when spite increases also bids increase.
3.2. Second-price all-pay auction
The model of the second-price all-pay auction is quite similar to the already mentioned
model of the first-price all-pay auction. Consider again two risk-neutral bidders, i and j
with utility function u(x). Valuations vi and vj are distributed according to F , i.e. v ∼
F (0, v), and f (x) = dFdx(x) . Both bidders submit a bid βk (vk ) with k ∈ {i, j}. In
the second-price all-pay auction both players pay the second highest bid. The prize is
allocated to the bidder with the highest bid.
We assume that for the candidate equilibrium we have βk (0) = 0 and that β ′ (x) = dβ(x)
dx
and βk−1 (bk ) = vk exist.
As before we model spite as a disutility for the losing bidder i of α · (vj − βi (vi )) where
α describes the amount of spite and (vj − βi (vi ) the gain of the winning bidder j. We
again consider α ∈ [0, 1).
The payoff of player i is as follows:
ΦII-AP
(βi , vi )
S


u(vi − βj (vj ))

if βi > βj (i wins)
=
if βi = βj (a tie)


u(−βi (vi ) − α(vj − βi (vi ))) if βi < βj (j wins)
u( vi −β2i (vi ) )

(7)
Again we assume that bidder i with valuation vi makes a bid b. The expected utility of
this bidder is given as follows:
E(b, v) =
∫
|
βj−1 (b)
0
u(v − βj (vj ))f (vj )dvj +
{z
}
bidder i wins and obtains the prize
and pays the loser’s bid
∫
|
v
βj−1 (b)
u(−b − α(vj − b))f (vj )dvj
{z
bidder i loses and pays his own bid
and additionally experiences spite
Rearranging the FOC yieds:
β
′
(βj−1 (b))
(u(v − b) − u(−b − α(βj−1 (b) − b))f (βj−1 (b))
=
∫
(1 − α) βv−1 (b) u(−b − α(vj − b))′ f (vj )dvj
j
7
}
(8)
6
2.0
5
4
1.0
b
b
1.5
α=1
0.75
0.5
0.25
3
2
0
v=1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
−0.5
0.5
1
0
0.0
0.2
0.4
0.6
0.8
1.0
0.2
0.4
v
0.6
0.8
1.0
1.2
α
Figure 2: Equilibrium bids in second-price all-pay auctions
Equilibrium bids in second-price all-pay auctions for different valuations v and different levels of spite
α with uniform distributions of valuations (see Equation 11).
Assuming a symmetric equilibrium we obtain
β ′ (v) =
v + α(v − b))f (v)
v(1 + α)f (v)
α(b))f (v)
=
−
(1 − α)(1 − F (v)
(1 − α)(1 − F (v) (1 − α)(1 − F (v)
(9)
Solving the differential equation (9) with initial value b(0) = 0 gives us the symmetric
equilibrium bidding function bII-AP
:
S
(
)
∫v
α
∫ v
α
1
α+1
(1 − F (s)) α−1 ds
α+1
0
v−
=
(1−F (v)) 1−α
sf (s)(1−F (s)) α−1 ds =
α
1−α
1−α
0
(1 − F (v)) α−1
(10)
For α = 0, Equation 10 becomes the familiar equilibrium bidding function for secondprice all-pay auctions without spite:
bII-AP
(v)
S
∫
II-AP
b
:=
bII-AP
α=0
=
0
v
sf (s)(1 − F (s))−1 ds
For uniformly distributed valuations, F (x) = x, we have
(v) =
bII-AP
S
(
)
)
α
(α + 1) (
(1 − α) (1 − v) 1−α − 1 + vα
α(2α − 1)
(11)
Figure 2 illustrates this case. The left graph in the Figure shows that, as in first-price
all-pay auctions, bids are monotonically increasing in valuations. The right part shows
that, different from first-price all-pay auctions, bids are only increasing in spite for small
8
valuations. If the valuation is sufficiently large, equilibrium bids decrease when spite
increases.
In the Figure we see that for small valuations bids are, as in the first-price case, higher
with spiteful bidders. For large valuations bids rather decrease with spite.
To simplify the notiation we assume in the following that valuations v ∈ [0, 1]. It is
easy to see the following:
Proposition 4. The bidding function in the second-price all-pay auction is increasing
in bidder’s valuation:
dbII-AP
S
≥0
dv
Proof
(
2α−1
1
α
dbII-AP
1+α
α ∫v
S
1−α
α−1
sf (s)(1 − F (s)) α−1
=
(1 − F (v))
f (v) v(1 − F (v))
−
dv
1−α
1−α 0
|
{z
}
(
q≥0
= q v(1 − F (v))
α
α−1
− v(1 − F (v))
≥ 0
α
α−1
∫
v
+
0
(1 − F (s))
α
α−1
)
)
Interestingly, there is no equivalent for Proposition 2 for second-price auctions. Bids are
not necessarily increasing if bidders are more spiteful. On the contrary: As illustrated
in Figure 2, if valuations are sufficiently large, more spite can actually lead to a decrease
in bids.
3.3. Predictions
The predictions we have obtained in Section 3.1 for the first-price all-pay auction are
that spiteful bidders will bid more, in particular when their valuation is high. This is
interesting, but this might also be difficult to disentangle from bidding more due to, say,
risk aversion.
In this sense the predictions we have obtained in Section 3.2 for the second-price all-pay
auction are more specific. Spiteful bidders will bid more, in particular for intermediate
valuations. In our experiment we will, therefore, use the second-price auction as a
workhorse.
We have the following hypotheses:
Hypothesis 1 Increased spite will lead to higher bids
Hypothesis 1 is the main hypothesis of this paper and it is based on the theoretical
prediction, that spiteful agents will overbid compared to non-spiteful agents. We will
show in the result section that, indeed spite leads to higher bids.
9
Hypothesis 2.1 Overbidding will be increasing in valuation for a give level of spite as
long as the valuation is not too large.
Hypothesis 2.2 For very high valuations and a given level of spite we will find underbidding.
Hypothesis 2 are mainly to check for the specific predictions of the model. In the
result section we will find support for hypothesis 2.1 but we will not find any support
for Hypothesis 2.2.
4. Design of the experiment
In the experiment, we first measure spite. We will discuss the different measures of spite
in Section 4.1. Then we observe bids in the second-price all-pay auction which we will
discuss in Section 4.2.
4.1. Measures of Spite
We use three different measures for spite. We will discuss the measure by Marcus et al.
(2014) in Section 4.1.1. The measure by Kimbrough and Reiss (2012) will be discussed
in Section 4.1.2 and our own measure in Section 4.1.3.
4.1.1. Spite according to Marcus et al.
The questionnaire by Marcus et al. (2014) is based on 17 not-payoff relevant questions
at the end of the experiment. Examples of the 17 questions are the following:13
• If I am checking out at a store and I feel like the person in line behind me is rushing
me, then I will sometimes slow down and take extra time to pay.
• I would rather no one get extra credit in a class if it meant that others would
receive more credit than me.
Participants were asked to indicate their agreement on a scale between 1 and 5. The
distribution of spite in this measure is shown in the left part of Figure 3.
4.1.2. Spite according to Kimbrough and Reiss
The second measure is a modification of the design by Kimbrough and Reiss (2012) who
want to observe spiteful behavior in an second-price auction.
First participants were asked to bid in ten auctions with one opponent.14 Then participants were informed about the outcome of the ten auctions, i.e. who had won the
auction and the winner’s bid. Thereafter, participants had the opportunity to increase
13
14
All questions are shown in Appendix D.1.
See Appendix D for details.
10
1.0
0.8
0.6
0.4
1.0 1.5 2.0 2.5 3.0 3.5 4.0
Female
Male
0.0
0.2
0.8
0.4
0.0
0.2
0.6
1.0
0.8
0.4
0.6
Own measure
0.2
0.0
Empirical CDF
Kimbrough-Reiss
1.0
Spite-Score
0
20
40
60
80 100
0
10
20
30
Measure for Spite
Figure 3: Distribution of Measures for Spite
or to keep their own bid. This increase could not be more than the difference between
the winner’s and the loser’s bid. Hence, bidders could not change the allocation, only
the payoff of the winner. Participants who had lost would still lose the auction (and
hence, have a payoff of 0) but they could increase the price the winner had to pay. At the
end participants were asked to indicate their willingness to pay for this bid adaptation.
Participants who had increased their losing bid were considered spiteful. The spitemeasure is a continuous measure between 0% (no adjustment) and 100% (100% if the
loser increases the own bid up to the winner’s bid). A distribution of spite based on this
measure is shown in the middle of Figure 3.
4.1.3. Our own measure for spite
Our own spite measure asks participants to decide six times among 9 possible allocations
similar to the SVO-Slider measure by Murphy et al. (2011); Murphy and Ackerman
(2014). Figure 4 shows all six sets.15 For each set participants had to chose their
preferred allocation.
In all six sets the allocation with the highest payoff for the other player maximises
also the own payoff. Any deviation from this allocation reduces the payoff of the other
player and never increases the own payoff. Such a deviation can, hence, be seen as a
sign of spite. This deviation is costless in sets IA1, RG1 and PS1, it is costly in IA2,
RG2, and PS2.
While one reason for these deviations can be spite, there are other explanations.
Deviations in sets IA1 and IA2 can be a sign of “inequality aversion”. Deviations in sets
RG1 and RG2 can be a sign of “concerns for relative gain”.
15
Details of the allocations are shown in Appendix D.2.
11
100
IA1
Other payoff
90
RG1
IA2
RG2
80
PS1
70
PS2
60
50
50
60
70
80
90
100
Own payoff
Figure 4: The six allocation sets for our own slider measure
For each of the six sets players choose one allocation. For each set we consider the pareto efficient
allocation not spiteful. Less efficient allocations will be considered more spiteful.
12
Notable only 9 participants were behaving extremely spitefully and only one participant was willing to pay for this behavior. A distribution of the combined spite measure
is shown in the right graph in Figure (3).
4.1.4. Other controls
Additionally we controlled for risk-aversion by implementing the task by Holt and Laury
(2002), we controlled for social value orientation and inequality aversion by using the
slider measure by Murphy et al. (2011); Murphy and Ackerman (2014) and we controlled
for rivalry, as this could be linked to spite, by implementing the questionnaire of Back
et al. (2013).
4.2. Design of the auction
After having measured spite16 participants play the second-price all-pay auction. We
explained in great detail (with video-instructions for each behavioral measure) the rules
of the second-price all-pay auction. Participants played the auction for 15 rounds with
stranger matching within matching groups of 6.
Each round participants were asked to indicate their biding function for valuations
between 0 and 100. Participants indicated their bid for valuations of 0, 10, 20,. . . , 90,
100. Bids for intermediate valuations were linearly interpolated. To give more feedback
each pair of bidders plays not only one but ten auctions for a random pair of valuations.
For each of the ten auctions participants were told how much their valuation their own
bid, and their the opponent’s bid was. Participants were told the outcome of the auction
and how much they had won or lost. Figure 5a shows an example of the bidding interface.
Figure (5b) shows an example of the feedback interface.
4.3. Payment of one random task
Participants were paid at the end of the experiment for one random task, i.e. either
one lottery from the risk-measure or one allocation from the SVO-Slider-Measure or
the Spite-Measure or the adaptation of Kimbrough and Reiss (2012) or one of the allpay auctions.17 To ensure the average payoff of each behavioral task to be similar we
adjusted the payoff ratio of ECU (experimental currency unit) to Euros in each task.
Moreover participants were given a higher endowment if the all-pay auction was paid,
as the payoff auction could lead to negative payoff.
16
The implementation of Kimbrough and Reiss (2012) and our all-pay auction were counterbalanced
as both parts are auctions and we want to control for order effects here.
17
In case of the all-pay auction only one of the 10 × 15 auctions about which we gave feedback was
played out.
13
(a) Interface of the bidding stage.
(b) Interface of the feedback stage.
Imputing the bidding function for the possible valuations between 0 and 100. The bidding function
will be drawn after the input of the respective
bids.
Mapping the 10 random valuations and the respective bids on the bidding function. Additionally subjects could see the opponents bid, whether
they won and the amount they won/lost.
Figure 5: Auction Interface
5. Results
We conducted the experiments in June 2015 in the laboratory of the school of economics
of the University of Jena (Germany). We recruited 138 participants in 8 sessions using
the online recruiting platform ORSEE (Greiner, 2004). We implemented the experiment
using z-Tree (Fischbacher, 2007). Instructions were presented as a 25 minutes video
followed by test questions for the all-pay auction and for the auction of the spite-measure
by Kimbrough and Reiss (2012). The entire experiment lasted for about 100 minutes.
Participants earned on average 15.83e, which is slightly above the minimum wage in
Germany. We had 45% male and 55% female participants with a median age of 24.
Participants were on average in their third year of studying and about 12% were students
of business or economics.
5.1. Measures of Spite
Figure 6 shows the joint distribution of the three measures for spite. There is no visible correlation. For the three instruments we find a Cronbach α of 0.0371 (CI=[0.0172,0.0908]). The three instruments seem to measure different things.
Having said that, we find substantial consistency within each scale. For the questionnaire (Marcus et al., 2014) we find a Cronbach α of 0.868 (CI=[0.818,0.933]). For our
own measure we find a Cronbach α of 0.555 (CI=[0.41,0.733]).
Neither the questionnaire nor our own measure seems to be strictly one-dimensional.
For the questionnaire we find that the first element of a principal component analysis explains 33.3% of the variance, (CI=[25.2,40.6]). For our own measure we find
that the first element of a principal component analysis explains 87.2% of the variance,
(CI=[81.7,91.4]).
• The adaptation of Kimbrough and Reiss (2012) could be mainly measuring retal-
14
80
60
40
20
0
1.0 2.0 3.0 4.0
Spite-Score
100
4.0
80
3.5
Spite-Score
Kimbrough-Reiss
Kimbrough-Reiss
100
60
40
3.0
2.5
2.0
20
1.5
0
1.0
0
10 20 30
Own measure
0
10 20 30
Own measure
Figure 6: Joint Distribution of Measures for Spite
iation or punishment, as the amount of lost auctions is a strong predictor of the
amount of spite—this means, that participant are increasing the bid to harm the
opponent to “punish him for winning”.
• The questionnaire is measuring the general motivation towards spiteful behavior.
It is, however, not necessary describing how people will in fact behave.
• Our own measure is potentially in general measuring the attitude towards spite in
an behavioral measure.
As there is, in general, no way to disentangle which of the three spite-measures is “really”
measuring spite we will look at the combined measures.
5.2. Aggregated Behavior
Figure 7 shows the average bidding behavior across all 15 rounds compared to predicted
behavior. The left graph in the Figure shows median bids, the right graph shows median
overbidding, i.e. the median difference between the bid and the bid of a non-spiteful
bidder. Clearly, median bids are not fit spiteful bids too well. Still, spiteful participants
indeed bid more than non-spiteful participants, in particular if their valuations are low.
To check for Hypothesis 1, i.e.. increased spite will lead to higher bids, we will use a
mixed-effects model as spite, risk, social value orientation (SVO) etc. are fixed effects but
the individuals and the matching-group are random effects. We compare four different
models which differ only in the controls C1 , . . . , C4 .
15
b − bII-AP
b
100
Spite=0
Spite=0.3
Spite=0.6
Spite>Median
Spite<Median
50
0
20
40
60
80
100
50
Spite=0
Spite=0.3
Spite=0.6
Spite>Median
Spite<Median
0
0
20
v
40
60
80
100
v
Figure 7: Median bids and overbidding in the auction
The left graph shows median bids. The right graph shows median overbidding.
Bidi,t,j,v
C1
C2
C3
=β0 + β1 Spitei + νi,j + ηj + ϵi,j,k,l + CM
=0
=β2 t + β3 v
=C2 + β4 1Gender=♀ + β5 Riski + β6 rivalryi + β7 SVOi + β8 IAi
C4 =C3 + β9 Spitei × v
(12)
(13)
(14)
(15)
(16)
where νi,j is a random effect for bidder i in group j, ηj is a random effect for group j,
and ϵi,j,k,l is the residual.
Estimation results are shown in Table (1). For all our specifications spite has a positive
and significant influence on bids behavior. This is in line with Hypothesis 1.
Risk and (obviously) valuation also have a significant and positive influence on bids.
Furthermore, female bidders bid more than men. The decrease in bidding over the
rounds could be interpreted as a learning effect of overbidding.
Result 1 Spite has a significant and positive effect on bids.
To investigate Hypothesis 2, increased overbidding in valuation and underbidding for
high valuations, we look at the right part of Figure 7. The Figure shows overbidding,
i.e. the difference between actual bids minus the theoretical bids for a non-spiteful agent.
We can see that observed bids seem to be in line with Hypothesis 2. To formally
investigate Hypothesis 2 we again use a mixed effects model. To model the dependency
of oberbidding on the valuation we use linear splines where knots of the spline function
16
(1)
Spite
Male
Risk
Rivalry
SVO
IA
Period
Spite × Valuation
Valuation
Constant
Observations
Log Likelihood
Akaike Inf. Crit.
Bayesian Inf. Crit.
Notes:
3.51∗
(1.70)
53.68∗∗∗
(3.43)
23,760
−123,402.10
246,816.20
246,864.60
(2)
3.52∗
(1.70)
(3)
4.37∗
−0.40∗∗∗ (0.05)
(2.00)
(6.27)
6.09∗ (2.98)
−1.04 (3.14)
0.40 (0.25)
−1.93 (2.55)
−0.40∗∗∗ (0.05)
0.64∗∗∗ (0.01)
27.42∗∗∗ (3.15)
23,760
−120,479.30
240,972.60
241,029.10
0.64∗∗∗ (0.01)
27.54∗∗∗ (6.67)
23,760
−120,462.00
240,948.10
241,045.00
−19.05∗∗∗
(4)
4.19∗
(2.01)
(6.27)
6.09∗ (2.98)
−1.04 (3.14)
0.40 (0.25)
−1.93 (2.55)
−0.40∗∗∗ (0.05)
0.004 (0.004)
0.64∗∗∗ (0.01)
27.54∗∗∗ (6.67)
23,760
−120,466.20
240,958.40
241,063.40
−19.05∗∗∗
p : ∗ ∗ ∗ < .001 ∗ ∗ < .01∗ < .05
Table 1: Estimation of Equation 12.
Spite is the sum of the three (normalised) spite measures. IA is the sum of the (normalised) inequality
aversion score obtain from the slider measure and the (normalised) score obtained from inequality
allocation of our own spite measure.
are at valuations 25, 50 and 75. We compare three different models:
′
Bidi,t,j,v − bII-AP =β0 + β1 Spitei + νi,j + ηj + ϵi,j,k,l + β3 t + CM
(17)
′
C1 =β4 1v∈[0,25] + β5 1v∈[25,50] + β6 1v∈[50,75] + β7 1v∈[75,100]
(18)
′
′
C2 =C1 + Spitei · (β8 1v∈[0,25] + β9 1v∈[25,50] + β10 1v∈[50,75] + β11 1v∈[75,100] )
(19)
C3′ =C2′ + β12 IAi + β13 1Gender=♀ + β14 Riski + β15 rivalryi + β16 SVOi (20)
where νi,j is a random effect for bidder i in group j, ηj is a random effect for group j,
and ϵi,j,k,l is the residual.
Estimation results are shown in Table (2). Figure 8 shows estimation results for the
fitted spline.
First we can see that generally behavior seems to be in line with theory: Overbidding
first increases and then decreases.
Furthermore, the interaction of spite and valuation seems to be most pronounced for
smaller valuations, i.e. between 25 and 50. Indeed, this is similar to what we have seen
also for equilibrium bids in Figure 2.
As before we find risk to be a significantly positive determinant for overbidding as
gender is a highly significant factor of overbidding.
Note, that neither rivalry, social value orientation nor inequality aversion seem to have
a significant impact.
17
C1′
Male
Risk
Rivalry
SVO
IA
Spite
V ∈ [0, 25]
V ∈ [25, 50]
V ∈ [50, 75]
V ∈ [75, 100]
Period
Spite·V ∈ [0, 25]
Spite·V ∈ [25, 50]
Spite·V ∈ [50, 75]
Spite·V ∈ [75, 100]
Constant
Observations
Log Likelihood
Akaike Inf. Crit.
Bayesian Inf. Crit.
Notes:
3.52∗ (1.70)
13.59∗∗∗ (1.14)
14.24∗∗∗ (0.95)
−16.50∗∗∗ (1.02)
−95.69∗∗∗ (1.03)
−0.40∗∗∗ (0.05)
27.82∗∗∗ (3.20)
23,760
−120,829.30
241,678.60
241,759.30
C2′
C3′
2.77 (1.75)
13.59∗∗∗ (1.14)
14.24∗∗∗ (0.95)
−16.50∗∗∗ (1.02)
−95.69∗∗∗ (1.03)
−0.40∗∗∗ (0.05)
0.66 (0.63)
1.66∗∗∗ (0.52)
0.53 (0.56)
0.60 (0.57)
27.82∗∗∗ (3.20)
23,760
−120,823.10
241,674.30
241,787.40
−19.05∗∗∗ (6.27)
6.09∗ (2.98)
−1.04 (3.14)
0.40 (0.25)
−1.93 (2.55)
3.62 (2.03)
13.59∗∗∗ (1.14)
14.24∗∗∗ (0.95)
−16.50∗∗∗ (1.02)
−95.69∗∗∗ (1.03)
−0.40∗∗∗ (0.05)
0.66 (0.63)
1.66∗∗∗ (0.52)
0.53 (0.56)
0.60 (0.57)
27.94∗∗∗ (6.69)
23,760
−120,805.90
241,649.80
241,803.20
p : ∗ ∗ ∗ < .001 ∗ ∗ < .01∗ < .05
Table 2: Estimation results for Equation 17 (overbidding)
The table shows estimation results for the different controls C1′ , C2′ , and C3′ . Splines have knots at
valuations 25, 50, and 75. Spite is the sum of the three spite measures. IA is the sum of the inequality
aversion score obtain from the slider measure and the score obtained from inequality allocation of our
own spite measure.
18
0 20 40 60 80 100
C10
C20
C30
b − bII-AP
50
0
Spite:
-50
-5
0
5
0 20 40 60 80 100
0 20 40 60 80 100
v
Figure 8: Estimation results for the spline from Equation 17 (overbidding)
The Figure show splines for different controls C1′ , C2′ , and C3′ and for different (normalised) levels of
spite.
Result 2a Spite significantly contributes to overbidding.
Result 2b The interaction of spite and valuations is, in line with theory, only significant
for moderate valuations.
Result 2c Gender and risk-attitude influence overbidding significantly.
6. Discussion and Conclusion
In this paper we want to contribute with the help of theory and with experiments to a
better understanding of bidding behaviour. We propose, in particular, that spite could
be a relevant factor to explain bids. As a workhorse we are using the second-price
all-pay auction. In the theoretical part we have seen that spitefulness should have a
substantial influence on bids in this auction. For the participants in our experiment
we measure spite using a questionnaire, an allocation task and an experimental design
similar to Kimbrough and Reiss (2012). In line with theory more spiteful bidders make
relatively higher bids for small valuations and relatively smaller bids for large valuations
(compared to the non-spite equilibrium) .
Besides this rather robust finding of our experiment, some limitations should be noted.
First, we can explain only a part of the deviation from non-spite equilbria. Other forces,
such as risk-aversion, joy of winning, anticipated regret or bounded rationality, might
play a role, too.
We have also seen that spite seems to be hard to define. The three spite measures we
are using seem to measure different things. It is not so clear how spite is connected to
19
anger, retaliation, inequality aversion or even concerns about justice? Our approach, to
simply sum up the normalised values of each measure is rather pragmatic. It might be
interesting to further disentangle the contribution of each of the three measures to the
joint effect.
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A. Proof of Proposition 2
Proof We want to show the following:
dbI-AP
1
S
= αF (v)
dα
e
∫
v
=
0
(∫
0
v
sf (s)eαF (s) ((α + 1)(F (s) − F (v)) + 1) ds
)
sf (s) eαF (s)−F (v) ((α + 1)(F (s) − F (v)) + 1) ds
|
{z
}
: =Q(α,F (v),F (s))
≥ 0
Let us rewrite F (v) = w and F (s) = z and therefore F (z)−1 = s and F (w)−1 = v are
the case. Hence, Q(α, w, z) := eα(z−w) ((α + 1)(z − w) + 1).
When we now consider the derivative of Q(α, w, z) in α we see that:
d(Q(α, w, z))
= (−z + w) eα (z−w) (−α z + α w − z + w − 2)
dα




= (z − w) eα (z−w) (α + 1) (z − w) +2
|
{z
≤0
}
| {z } |
|
= ≤0
24
≤2
{z
≥−1
{z
≥0
}
}
Also considering the derivative of Q(α, w, z) in w we see that:
(
)
d(Q(α, w, z))
= −1 + (−z + w) α2 + (−2 − z + w) α eα (z−w)
dw











α (z−w) 
= −e
1 + α (α + 1) (z − w) +2


| {z } | {z }


≤2
≥−1


|
{z
}
≥0
= ≤0
Our goal is to show that the following equation holds (which would be Proposition 2)
∫
F (w)−1
0
F (z)−1 Q(α, w, z)dz ≥ 0
Using the derivatives of Q(α, w, z) in w and in α we get:
∫
F (w)−1
0
∫
F (z)−1 Q(α, w, z)dz ≥
F (1)−1
−1
∫
F (w)−1
∫
0
∫
0
F (1)−1
∫
F (1)−1
F (z)−1 Q(1, 1, z)dz
0
F (z)−1 ez−1 ((2)(z − 1) + 1) =
F (z) Q(1, 1, z)dz =
0
F (z)−1 Q(1, w, z)dz ≥
F (1)−1
F (z)−1 ez−1 ((2z − 1)) dz
=
|
0
{z
}
=Q(1,1,z)
Considering this we see that this is positive even for the lowest possible values.
∫
1
0
Q(1, 1, z)dz = 3 · e−1 − 1 ≈ 0.1 ≥ 0
Now we can also show that the function Q(α, w, z) is increasing in z
(
)
d(Q(α, w, z))
= − −1 + (−z + w) α2 + (−2 − z + w) α eα (z−w)
dz


=









α (z−w) 
e
1 + α (α + 1) (z − w) +2


| {z } | {z }


≤2
≥−1


{z
}
|
≥0
= ≥0
Now as we know that F (z)−1 is an increasing function and as we know moreover that
∫
∫
d(Q(α,w,z))
≥ 0 and also 01 Q(α, w, z)dz ≥ 01 Q(1, 1, z)dz ≥ 0 we conclude the following:
dz
∫
0
F (w)−1
F (z)−1 Q(α, w, z)dz ≥
∫
F (1)−1
0
25
F (z)−1 ez−1 ((2)(z − 1) + 1) ≥ 0
B. Proof of Proposition 3
Proof To prove that the deviation is increasing in valuation in the first-price auction
we can use the result of 1
)
(
∫ v
d(bI-AP
− bI-AP
1
S
rational )
= αF (v) −αf (v) (α + 1)sf (s)eαF (s) ds + (α + 1)vf (v)eαF (v) − vf (v)
dv
e
0

(α + 1)
= f (v)  αF (v)
e
Now we need to show that
(
↔
↔
↔
(
(α+1)
v
eαF (v)
(α+1)
v
eαF (v)
v
0
(
(α + 1)
e| {z } ds − v  ≥ f (v)
v−v
eαF (v)
≥1
)
αF (s)
)
− v is positive.
−v
)
−1
−F (v)
−F (v)e−F (v)
(
↔ W −F (v)e−F (v)
↔

∫
W (−F (v)e−F (v) )
−F (v)
≥0
)
≥ −(α + 1)e−αF (v)
≥ −F (v)(α + 1)e−αF (v)
≥ −F (v)(α + 1)e−F (v)(α+1)
≥ −F (v)(α + 1)
−1 ≤α
Whereas W(.) represents the Lambert W-function.
W (−F (v)e−F (v) )
− 1 is zero. Therefore, the deviation is increasing in
According to Maple
−F (v)
valuation in the first-price auction for α ≥ 0.
C. Revenue
We have seen that the introduction of spite could explain overbidding in all-pay auctions.
In addition, it would be interesting to see some results on revenue ranking in case of
spiteful bidders.
In this paper we are only looking at two players and therefore the revenue of the seller
is just two times the ex-ante expected payment, which is the bid multiplied with the
probability to pay the bid:
Expected payment: m(v) = Bid[v] · Prob[Paying Bid]
Ex-ante expected payment: E[m(v)] := Expected revenue for one player= 12 ·
Expected revenue for the seller
Now we can look at the revenues of the all-pay auctions with spite:
Proposition 5. The revenue in the first-price all-pay auctions with two players is given
by:
∫ 1 −αF (s)
(e
− e−α )
)
=
2
RS1AP = 2E(m1AP
(α + 1)sf (s)eαF (s) ds
(21)
S
α
0
26
The revenue in the second-price all-pay auctions with two players is given by:
∫
RS2AP
=
2E(m2AP
)
S
1
=2
2
0
α+1
sf (s)(1 − F (s))ds
2−α
(22)
Proof
∫
RS1AP
=
2E(m1AP
)
S
=
∫
=
1
2
0
1
2
bI-AP
f (v)dv
S
f (v)e−αF (v)
∫
∫
1
0
z}|{
=
∫
1
2
∫
1
2
0
∫
RS2AP
=
2E(m2AP
)
S
1
= 2
0
∫
1
= 2
0
∫
1
∫
α+1
1−α
∫
1
s
1
= 2
0
∫
1
(
0
= 2
1
2
0
∫
0
v
1
e−αF (v) (α + 1)sf (s)eαF (s) ds
−α s
(e−αF (s) − e−α )
(α + 1)sf (s)eαF (s) ds
α
1
α
sf (s)(1 − F (s)) α−1 ds(1 − F (v)) 1−α (2(1 − F (v)))f (v)dv
1
(1 − F (v)) 1−α 2f (v)dv
(1 − F (v))
= 2
∫
f (v)e−αF (v) dv(α + 1)sf (s)eαF (s) ds
bII-AP
(2(1 − F (v)))f (v)dv
S
= 2
0
1
s
0
=
∫
2
if α̸=0
(α + 1)sf (s)eαF (s) dsdv
0
0
=
v
2−α
1−α
1
α+1
sf (s)(1 − F (s)) α−1 ds
1−α
1
1 − α 1 α + 1
2
sf (s)(1 − F (s)) α−1 ds
α−2 s 1−α
2−α
(1 − F (s)) 1−α 2
)
1
1−α α+1
sf (s)(1 − F (s)) α−1 ds
2−α 1−α
α+1
sf (s)(1 − F (s))ds
2−α
Using the revenue equivalence principle18 , which says that the expected revenue is the
same through all standard auctions (meaning the winner obtains the object for sure) with
risk-neutral and rational bidders, we get the first interesting result concerning revenue:
18
For a simple explanation see:Krishna (2002) p. 28.
27
Proposition 6. The expected revenue of the first-price and the second-price all-pay
auction in case of spite-experiencing bidders is higher than the expected revenue in case
of rational bidders.
E(m1AP
) ≥ E(mrational )
S
2AP
E(mS ) ≥ E(mrational )
(23)
(24)
Proof We first compare the revenues of spiteful bidders in first-price all-pay auction
and standard-rational revenue
RS1AP − Rrational =
− mrational ) = 2
2E(m1AP
S
∫ 1
α+1∫ 1
sf (s)(1 − F (s))ds
sf (s)(1 − e−α eαF (s) )ds − 2
α
0
0
∫
1
= 2
sf (s)(
α + 1 α + 1 α(F (s)−1)
−
e
− 1 + F (s))ds
α
α
sf (s)(
1 α + 1 α(F (s)−1)
−
e
+ F (s))ds
α
α
0
∫
1
= 2
0
It is quite straightforward that this difference is positive as we know that bI-AP
≥ bI-AP .
S
To see this:
→
→
≥ bI-AP
≥ f (v) · bI-AP
bI-AP
S
f (v) · bI-AP
S
∫1
0
→
→
f (v) ·
bI-AP
S
E(m1AP
)
S
2E(m1AP
)
S
1AP
RS
≥
∫
0
1
f (v) · bI-AP
≥ E(mrational )
≥ 2E(mrational )
≥ Rrational
We now compare the revenues of spiteful bidders in second-price all-pay auction and
standard-rational revenue
∫ 1
α+1
sf (s)(1 − F (s))ds
sf (s)(1 − F (s))ds − 2
2−α
0
0
(
)
∫ 1
α+1
= 2
sf (s)(1 − F (s)) 2
− 1 ds
2−α
0
E(m2AP
− mrational ) = 2
S
∫
1
2
As we are interested just whether the difference is positive we check only: 2 α+1
−1 ≥
2−α
2AP
0 ↔ 2α + 2 ≥ 2 − α ↔ 3α ≥ 0. Therefore E(mS − mrational ) is positive for all α ≥ 0.
∀α ≥ 0
⇒ RS2AP ≥ Rrational
28
We have seen that the revenues are higher if bidders are experiencing spite relative to
the rational case. Therefore, it seems plausible that the revenue equivalence principle
does not hold anymore when bidders experience spite and thus we expect a revenue
ranking on standard-auction formats for spiteful bidders. For that reason we are going
to look at the expected revenues of the four formats (first and second winner-pay and
all-pay auctions respectively) and we will try to identify an ordering of those revenues.
Let us first look at the revenues of the all-pay auctions.
Proposition 7. The expected revenue of the second-price all-pay auction in case of
spite-experiencing bidders is higher than the expected revenue of the first-price all-pay
auction with spiteful bidders.
E(m1AP
) ≤ E(m2AP
)
S
S
(25)
Proof
∫
∫
1 α+1
(e−αF (s) − e−α )
(α + 1)sf (s)eαF (s) ds −
2
sf (s)(1 − F (s))ds
α
2
−
α
0
0
(
)
∫ 1
1 1eαF (s)
2
= (α + 1)
sf (s)
−
−
(1 − F (s)) ds
α
αeα
2−α
0
) =
E(mS1AP − m2AP
S
1
∫
1
= (α + 1)
0
∫
1
= (α + 1)
0
∫
= (α + 1)
0
1
)
(
(2)eα − 3αeα − 2eαF (s) + eαF (s) α + 2αF (s)eα
sf (s)
ds
(2 − α)(α)eα
(
)
(α − 2)eαF (s) + 2(1 + (F (s) − 32 )α)eα
sf (s)
ds
(2 − α)(α)eα
)
(
(α − 2)eα(F (s)−1) + 2(1 + (F (s) − 32 )α)
ds
sf (s)
(2 − α)(α)
|
{z
}
: =n(F (s),α)
If we could show that n(F (s), α) ≤ 0 it would be obvious that the difference is negative.
Therefore, we show that this is negative by proofing that it is not positive.
n(F (s), α)
↔
(α−2)
2
→
(α−2) −1+ 1 α
2
e
2
↔
W( (α−2)
)
2
→
F (s)
(α − 2)eα(F (s)−1) + 2(1 + (F (s) − 32 )α)
>0→
>0
(2 − α)(α)
3
> −e−α(F (s)−1) (1 + (F (s) − )α)
2
3
3
> −e(1+(F (s)− 2 )α) (1 + (F (s) − )α)
2
3
> −(1 + (F (s) − )α)
2
(α−2)
3
α − 1 − W( 2 )
> 2
α
Whereas W(.) represents the Lambert W-function.
3
α−1−W(
As we know due to numeric programs that 2
α
can conclude the difference is negative, as F (s) ≯ 1.
29
(α−2)
)
2
is one except a null set we
Consequently, we know that a seller could make more profit with the second-price than
with the first-price all-pay auction if participants would bid according to our model.
To get a ranking we are going to look now at the expected revenues of the winner-pay
vs. all-pay auction:
Proposition 8. The expected revenue of the first-price all-pay auction in case of spiteexperiencing bidders is higher than the expected revenue of the first-price winner-pay
auction with spiteful bidders.
) ≥ E(m1S )
E(m1AP
S
(26)
The expected revenue of the second-price all-pay auction in case of spite-experiencing
bidders is higher than the expected revenue of the first-price winner-pay auction with
spiteful bidders.
E(m2AP
) ≥ E(m1S )
(27)
S
Proof
RS1
−
RS1AP
=
2E(m1S
−
m1AP
)
S
∫ 1 −αF (s)
(e
− e−α )
1+α∫ 1
α
= 2
(sf (s)F (s) − 2
(α + 1)sf (s)eαF (s) ds
1−α 0
α
0
)
(
∫ 1
F (s)α − F (s) 1 − eα(F (s)−1)
−
= 2(1 + α)
sf (s)
ds
1−α
α
0
|
{z
}
:=g(F (s),α)
Let us now set g(F (s), α) : = g(k, α):
dg(k, α)
=
dk
dg(k,α)
dk
dk
αkα
k
−1
+ eα(k−1)
1−α
(
= α eα(k−1) − k α−2
)
Now we want to prove that the second derivative of g(k, α) is negative. Hence, we assume
the opposite.
(
α eα(k−1) − k α−2
)
→
eα(k−1) k 2−α
→ eα(k−1) k > eα(k−1) k 2−α
→
eα(k) kα
→
kα
→
k
30
>0
>1
>1
> eα α
> W(eα α)
W(eα α)
>
= 1a.s.
α
Whereas W(.) represents the Lambert W-function.
Thus, we know that max(k) = 1 and hence k ≯ 1 and hence we know that


⇒
⇒
dg(k,α)
dk
dk
≤ 0.

dg(k,α)
dk 
dk
≤ 0 → min(
dg(k, α)
dg(k, α)
)=
|k=1 = 0
dk
dk
dg(k, α)
≥ 0 → max(g(k, α)) = g(k, α)|k=1 = 0
dk
g(k, α) ≤ 0
Therefore :
∫
1
2(1 + α)
0
(
)
F (s)α − F (s) 1 − eα(F (s)−1)
sf (s)
ds ≤ 0
−
1−α
α
{z
|
}
=g(s,α)≤0
⇒ E(m1S − m1AP
)≤0⇒
S
RS1AP ≥ RS1
The second part of the proposition is just straightforward, as we know due to (10) that
the revenue of the second-price all-pay is higher than the revenue of the first-price all-pay
auction we can easily conclude that
E(m2AP
) ≥ E(m1S ) ⇒
S
RS2AP ≥ RS1
So far we have the following ranking of expected revenues:
Proposition 9. The expected revenues of the standard games (without second-price and
third-price winner-pay) with two spiteful bidders can be ranked generally in the following
manner:
E(m(v)2AP
) ≥ E(m(v)1AP
) ≥ E(m(v)1S ) ≥ E(m(v)rational )
(28)
S
S
Summarizing our results we have seen that in theory spiteful bidders would bid more
than rational bidders. Now, we have seen that this behavior would lead to more revenue
for the seller. Also, we have seen that the revenue equivalence principle is not applicable
with spiteful bidders, and the result is that all-pay auctions are yielding higher revenue
than the first-price winner-pay auction. Moreover, we have seen that the second-price
all-pay auctions yield higher revenue than the first-price all-pay auction. Now, for the
ranking to be complete we have to look at the second-price winner-pay auction.
For the ranking of the second-price winner-pay auction in relation to the first-price
all-pay auction we obtain the following result:
Proposition 10. The expected revenue of the second-price winner-pay auction in case
of spite-experiencing bidders is higher than the expected revenue of the first-price all-pay
auction with spiteful bidders.
E(m1AP
) ≤ E(m2S )
S
31
(29)
Proof
E(mS1AP
−
∫
m2S )
(e−αF (s) − e−α )
(α + 1)sf (s)eαF (s) ds −
α
0
1
1+α∫ 1
(sf (s)(1 − F (s) − (1 − F (s)) α ))ds
1−α 0


1
∫ 1
α(F (s)−1)
α
1 − F (s) − (1 − F (s)) 
1−e
= (1 + α)
sf (s) 
−
ds
α
1−α
0
1
=
≤ (1 + α)

∫
0
∫

= (1 + α)
0
0
(1 + α)
1−α
(1 + α)
=
1−α

1
1
0
(
)
1
sf (s) (1 − F (s)) α − α(1 − F (s)) ds
(∫
1
0
1
α
sf (s)(1 − F (s)) ds − α
∫
1
0
sf (s)(1 − F (s))ds
)
[
∫ 1
1
1
(1 + α)
−α
1
+1 −α
α
=
s(1 − F (s))
|0 − (1 − F (s)) α +1
ds
1−α
1+α
1+α
0
(
−α s(1 −
(1 + α)
=
1−α
[∫
−1 1
F (s))2
|0
2
1
0
(1 − F (s))
−
1
+1
α

∫
0
1
21
(1 − F (s))
2
)]
ds
∫ 1
α
1
ds − (1 − F (s))2 ds
1+α
2
0
]

(1 + α) ∫ 1 2α(1 − F (s)) α +1 − (1 + α)(1 − F (s))2 ds 
=
1−α
(1 + α)2
0
=
=
(1)
2(1 − α)
1
[∫
0
1
1
2α(1 − F (s)) α +1 − (1 + α)(1 − F (s))2 ds

]

∫
(
) 
α−1
1
(1) 
 1

 (1 − F (s)) α +1 2α − (1 + α)(1 − F (s)) α ds ≤ 0
2(1 − α)  0
|
{z
} 
:=M (F (s),α)≤0
(

1 − F (s) − α + αF (s) − 1 + F (s) + (1 − F (s)) α 
sf (s) 
ds
1−α
= (1 + α)
=
1

1
∫

1 − F (s) − (1 − F (s)) α 
sf (s) 1 − F (s) −
ds
1−α
1
∫
1
1 − 1 − α(F (s) − 1) 1 − F (s) − (1 − F (s)) α 
sf (s) 
−
ds
α
1−α
1
To see that 2α − (1 + α)(1 − F (s))
α−1
α
)
≤ 0 we just derive it on F (s) and hence we
32
get:
−1 α − 1
dM (F (s), α)
= − (1 + α) (1 − F (s)) α
(−1)
dF (s)
α
1 − α2
1
= −
α } (1 − F (s)) α1
| {z
|
≥0
≤ 0
{z
≥0
}
Thus, the function is decreasing in F (s). If we look at the maximum we see that it is
still negative:
M (0, α) = 2α − 1 + α = α − 1 ≤ 0
Therefore, we can conclude that:
) ≤ E(m2S )
E(m1AP
S
Unfortunately we cannot replicate the same, or even a similar, property onto the
second-price all-pay auction as can be seen in the following graph:
) and E(mIIS ) with uniform distributions of valuations.
Figure 9: Difference of E(mII-AP
S
33
Figure 10: Revenue Ranking with uniform distributions of valuations
We can see in the Figure 10 that the gap in revenue between the second-price winnerpay and all-pay auction is first negative (the winner-pay auctions yields higher revenue
first) and then turns positive (the all-pay auctions yields higher revenue to the seller).
Therefore, the revenues do not have a clear ordering as can be seen in the Figure 9.
We can extend Proposition 9, but only in uniform distribution and not in general:
Proposition 11. The expected revenues of the standard games with two spiteful bidders
and uniform distributions can be ranked as follows:
for
α ∈ [0, 0.125] and F (v) = v
E(m(v)2S ) ≥ E(m(v)2AP
) ≥ E(m(v)1AP
) ≥ E(m(v)1S ) ≥ E(m(v)rational )
S
S
for
α ∈ [0.125, 1) and F (v) = v
E(m(v)2AP
) ≥ E(m(v)2S ) ≥ E(m(v)1AP
) ≥ E(m(v)1S ) ≥ E(m(v)rational )
S
S
Proof This proof is basically simple analysis but nevertheless for reasons of completeness the reasoning is presented:
E(m(v)2AP
)
S
−
E(m(v)2S )
∫
=
(1 + α)

0
1
sf (s)(1 − F (s))

2
1
(1 − F (s)) α −1 

−
+
ds
2−α 1−α
(1 − α)
=F (v)=v −1/6
34
1
a (8 a − 1)
(2 a + 1) (−2 + a)
This is again positive when when α ∈ [ 18 , 1).
All other orderings were already proofed. Consequently the proof is complete.
35
D. Spite Measure
D.1. Spite Measure -Questionnaire
The questions of the questionnaire according to Marcus et al. (2014) included the following questions (translated into English):
• I would be willing to take a punch if
it meant that someone I did not like
would receive two punches.
• I would be willing to pay more for
some goods and services if other people I did not like had to pay even
more.
• If I was one of the last students in a
classroom taking an exam and I noticed that the instructor looked impatient, I would be sure to take my time
finishing the exam just to irritate him
or her.
• If my neighbor complained about the
appearance of my front yard, I would
be tempted to make it look worse just
to annoy him or her.
• It might be worth risking my reputation in order to spread gossip about
someone I did not like.
• If I am going to my car in a crowded
parking lot and it appears that another driver wants my parking space,
then I will make sure to take my time
pulling out of the parking space.
• I hope that elected officials are successful in their efforts to improve my
community even if I opposed their
election. (reverse scored)
• If my neighbor complained that I was
playing my music too loud, then I
might turn up the music even louder
just to irritate him or her, even if
meant I could get fined.
36
• I would be happy receiving extra
credit in a class even if other students
received more points than me. (reverse scored)
• Part of me enjoys seeing the people
I do not like fail even if their failure
hurts me in some way.
• If I am checking out at a store and I
feel like the person in line behind me
is rushing me, then I will sometimes
slow down and take extra time to pay.
• It is sometimes worth a little suffering on my part to see others receive
the punishment they deserve.
• I would take on extra work at my job
if it meant that one of my co-workers
who I did not like would also have to
do extra work.
• If I had the opportunity, then I would
gladly pay a small sum of money to
see a classmate who I do not like fail
his or her final exam.
• There have been times when I was
willing to suffer some small harm so
that I could punish someone else who
deserved it.
• I would rather no one get extra credit
in a class if it meant that others
would receive more credit than me.
• If I opposed the election of an official,
then I would be glad to see him or her
fail even if their failure hurt my community.
D.2. Spite Measure—Own Measure
Our own spite measure is measuring spite similar to the social value orientation measures
of Murphy et al. (2011); Murphy and Ackerman (2014). The slider measure of Murphy
et al. (2011); Murphy and Ackerman (2014) are presenting the participants with 6 (or
15 if inequality aversion is also measured) allocational sets. For each set the participants
need to decide between 9 allocations determining the own payoff and the payoff of the
“opponent”. Similarly our spite measure asks participants to decide six times between 9
possible allocation. An overview of the six allocations can be seen in Figure 4.
The table below provides all the details. The leftmost allocation is always the nonspiteful allocation and the rightmost allocation is always the maximally spiteful allocation. In the experiment each set was shown on a separate screen and two sets were
presented in reverse order.
You
IA1
Other
You
IA2
Other
You
RG1
Other
You
RG2
Other
You
PS1
Other
You
PS2
Other
70
◦
100
70
◦
100
100
◦
100
100
◦
100
100
◦
85
100
◦
85
70
◦
98
68
◦
96
100
◦
98
98
◦
96
100
◦
83
98
◦
81
70
◦
96
65
◦
92
100
◦
96
96
◦
92
100
◦
81
96
◦
76
70
◦
94
62
◦
89
100
◦
94
94
◦
89
100
◦
79
94
◦
72
70
◦
92
60
◦
85
100
◦
92
92
◦
85
100
◦
78
92
◦
68
70
◦
91
58
◦
81
100
◦
91
91
◦
81
100
◦
76
91
◦
63
70
◦
89
55
◦
78
100
◦
89
89
◦
78
100
◦
74
89
◦
59
70
◦
87
52
◦
74
100
◦
87
87
◦
74
100
◦
72
87
◦
54
70
◦
85
50
◦
70
100
◦
85
85
◦
70
100
◦
70
85
◦
50
Each of the six tasks are supposed to measure one feature of spite. The sets IA1
and IA2 are measuring spite when it is behaviorally in line with inequality aversion. A
decision maker with positive concerns for social efficiency would choose the allocation
with the highest payoff for the other player since this choice also maximises the own
payoff. A spiteful person but also an inequality averse person would choose possibly a
different allocation. In IA1 being spiteful has no cost. Decision makers get 70 ECU
for sure and can basically reduce the payoff of the opponent. In IA2 spitefulness has a
cost. In both IA1 and IA2 it may be that the motivation of the decision maker of not
maximising the payoff of the other player could be to either harm the other (spite) or
to decrease the overall inequality.
RG1 and RG2 are measuring spite when spite is behaviorally in line with relative
gain. Again, a decision maker with positive concerns for social efficiency would choose
37
Submeasures
IA
IA-WP
RG
RG-WP
PS
PS-WP
∑
No Spite in %
Spite in %
Average Spite
83.00
93.00
93.00
99.00
93.00
99.00
17.00
7.00
7.00
1.00
7.00
1.00
1.53
0.07
0.26
0.01
0.30
0.01
80.00
20.00
2.09
Table 3: The allocation of choices considered (non-)spiteful in the six allocational tasks
of our own spite measure.
(a) Interface of the auction spite measure.
(b) Interface of the feedback of each auction.
Imputing the bidding function for the possible valuations between 500 and 1000. The bidding function will be drawn after the input of the respective
bids.
Mapping the 10 random valuations and the respective bids on the bidding function. Additionally subjects could see the opponents bid (if the
opponent won) and whether they won or lost.
Figure 11: Auction Interface
the allocation with the highest payoff for the other player since this choice also maximises
the own payoff. In RG1 a person who deviates from this choice is considered spiteful as
this person decreases the payoff of the opponent. However, this behaviour would also be
in line with the behavior of an agent who wants to have relative better payoff compared
to the opponent. RG2 is a variant of RG1 where the spiteful choice is costly.
In PS1 and PS2 the efficient outcome implies already a positive relative standing of
the decision maker who can only decrease the payoff of the other player. Here we find
it less plausible to attribute a deviating choice as driven by relative standing. Instead,
we take the last two sets as extreme spite. PS2 is a variant of PS1 where the spiteful
choice is costly.
The allocation of the overall spite in this measure can be seen in Figure ??. The
decisions of the individual set can be seen in Table ??.
38
Figure 12: Interface of the bid adaptation.
To reduce the demand effect participants were allowed to increase their losing but also the winning bid.
The screen was split in two parts so that they made the decisions on the winning part on the right part
of the screen and the losing decisions on the left.
D.3. Spite Measure -Auction Adaptation
The most complicated part of this experiment was actually the adaptation of Kimbrough
and Reiss (2012). In the original paper by Kimbrough and Reiss (2012) participants were
playing in groups of 3, 16 rounds of an auction (a second-price winner-pay auction which
is the design of f.e. eBay). Participants were supposed to bid for an object for which
they were provided a personal induced value v ∼ U [500, 1000]. After this participants
were doing some effort task. Afterwards participants were informed about whether they
won or lost the auction and thereafter they were allowed to increase their bid. However,
they also had a possibility to buy the object, they were competing for, at a random price
p ∼ U [300, 500] if they lost.
We had a similar design to Kimbrough and Reiss (2012) but we changed some aspects
of their design. We excluded the fixed price and we also excluded the real effort task.
Moreover, we played this part only for one round. Additionally, we measured the participants willingness to pay for the bid adaptation. Therefore, the measure consisted
basically of four stages:
Stage 1 Participants (I{1,2} ) submitting initial bids (BI{1,2} ) according to the possible
valuation. This part was accessed by a strategy-like-approach were we presented
the participants with the possible valuation between 500 and 1000 in steps of 50
and asked them to indicate their respective bid (see Figure 11a)
Stage 2 10 random valuations with v ∼ U [500, 1000] were drawn for each subject and
the respective bids from the bidding function of stage 1 were mapped. Each valuation and bid of each pair represent an auction. Participants were now, informed
39
about the highest bid and bidder in each of the 10 auctions (see Figure 11b).
Stage 3 Participants were asked to submit final bids (BF{1,2} ) where BF{1,2} ≥ BI{1,2} .
However, the outcome of the auction could not be changed with the final bids.
Hence, the initial bids determined who won which auction, but the final bids
determined how much the winner need to pay. The participants saw the interface
as can be seen in Figure 12.19
Stage 4 Participants were randomly matched with a new partner and were asked to
state how much they are willing to pay for the final bid to be implemented. The
willingness to pay of each pair was compared and the final bids of the person who
stated a higher willingness to pay were implemented. However, those participants
whose final bid was implemented needed to pay the willingness to pay of their
partner in the stage 4. Hence,the willingness to pay was design-wise also a second
price auction which is supposed to be reveling the true willingness to pay as the
dominant strategy is to pay the true valuation.
E. Linear Bidding
As we have seen, that average bidding behavior can be best described by a linear function
we want to estimate the optimal linear bidding behavior for spiteful agents.
Let us assume that we have two players i and j how are competing for a object which
each player values with vk ∈ [0, 1] with k ∈ {i, j}. The valuation is supposed to be
drawn from a distribution with density function F (v) and distribution function f (v).
We assume also that both are bidding for the object with the bidding function βk (vk )
with k ∈ {i, j}. Additionally u(x) is representing the utility function which is equal for
both players. We also assume that both players have the same spitefulness α ∈ [0, 1)
with α = 0 being the rational player. The utility of the second price all-pay auction is
given by the following equation:
ΦII-AP
(βi , vi )
S



u(vi − βj (vj ))
if i wins, i.e. βi > βj
=
if the bid ties, i.e. βi = βj


u(−βi (vi ) − α(vj − βi (vi ))) if j wins, i.e. βi < βj
u( vi −β2i (vi ) )

(30)
Now, we want to make the following assumptions:
• Subjects are not risk averse etc u(x) = x
• We assume the uniform distribution F (v) = v.
19
An indicator, that there may be a demand effect existing (or participants did not fully understood
this part) is that 41% of the participants increased their bid also in the winning case. It may also
be, that participants wanted to ensure that they won or they experienced joy of winning—but in
any case behavior can be driven by these motivations only if the task is not completely understood.
40
• Both subjects are bidding linear:
βi (vi ) = mi · vi + ti
βj (vj ) = mj · vj + tj
• We postulate the following restrictions on the bidding function:
0 ≥ βk (vk ) ≤ 1.5.
This implies:
t k + mk · v ≥ 0
t k + mk · v ≤ b
t k + mk · v ≤ b
t k + mk · v ≥ 0
↔
↔
↔
↔
tk ≥ 0
tk ≤ 1.5
mk ≤ 1.5 − tk
mk ≥ −tk
As we want to find the bidding function which maximizes the utility independent of
the actual valuation for the object, we maximize the ex-ante payoff. Which is given by:
41
42
bidder i loses and pays his own bid
and additionally experiences spite

and pays the loser’s bid
mj
)
 m +t −t ( mi ·vi +ti −tj
∫1
∫ j mji i ∫

mj


(vi − mj · vj − tj )dvj + mi ·vi +ti −tj (−(mi · vi + ti ) − α(vj − (mi · vi + ti )))dvj dvi +
tj −ti

0

mj

mi


∫ tj −ti (∫
)

1

m

(−(mi · vi + ti ) − α(vj − (mi · vi + ti )))dvj dvi +
 0 i

0

(∫
)

1
1
∫ m

(v
−
m
·
v
−
t
)dv
dvi

j +tj −ti
i
j
j
j
j

0

mi










( m ·v +t −t
)

mj +tj −ti

i i
i
j

∫
∫
∫

mj
1
m
i


(vi − mj · vj − tj )dvj + mi ·vi +ti −tj (−(mi · vi + ti ) − α(vj − (mi · vi + ti )))dvj dvi +
0
0


mj


(∫
)

∫ 1
1
mj +tj −ti
(vi − mj · vj − tj )dvj dvi
0
Φi (bi )
mi









( m ·v +t −t
)



∫1
∫1
∫ i imj i j



(vi − mj · vj − tj )dvj + mi ·vi +ti −tj (−(mi · vi + ti ) − α(vj − (mi · vi + ti )))dvj dvi +
 tjm−ti
0

mj
i


 tj −ti (
)

∫
∫

1

 0 mi
(−(m
·
v
+
t
)
−
α(v
−
(m
·
v
+
t
)))dv

i
i
i
j
i
i
i
j dvi
0











)
( m ·v +t −t



∫1
∫ 1 ∫ i imj i j



(vi − mj · vj − tj )dvj + mi ·vi +ti −tj (−(mi · vi + ti ) − α(vj − (mi · vi + ti )))dvj dvi
 0
0
This can be rephrased in the following way:
∫
and
and
and
tj −ti
mi
tj −ti
mi
when 0 ≤
when 0 >
and
tj −ti
mi
tj −ti
mi
when 0 >
when 0 ≤
mj +tj −ti
mi
mj +tj −ti
mi
mj +tj −ti
mi
mj +tj −ti
mi




mi ·vi +ti −tj


∫
∫
1  min(max(
,0),1)
1

mj

E(b) =
(vi − mj · vj − tj )dvj +
(−(mi · vi + ti ) − α(vj − (mi · vi + ti )))dvj 
mi ·vi +ti −tj

 dvi
0  0
min(max(
,0),1)

mj
{z
}
|
{z
}
|


bidder i wins and obtains the prize

>1
>1
≤1
≤1
The general approach will be to find an optimal mi and an optimal ti which maximize
the utility.
Opt
As we are only interested in symmetric equilibrium we will search for a mi Sym which
Opt
i (bi )
fulfills: δΦδm
|mi =mj ,ti =tj = 0. And for a ti Sym which fulfills: δΦδti (bi i ) |mi =mj ,ti =tj = 0.
i
Opt
Opt
ti Sym will depend on α and mi . The same goes for mi Sym , which is depend also
Opt
Opt
on α and ti . Therefore we will find the optimal bidding function bi Sym with mi Sym
Opt
and ti Sym . However, we will see that the optimal bidding function is not within the
defined area of ti .
First we generally derive the ex-ante utility for mi
(
)
(
)2
(
)
(
)(
))
 (
−tj +ti
mi +2 mj 2 + −6 ti +6 tj mj +6 −tj +ti
(−α−1)mj 2 + (−3 α−3)tj +3 ti +2 α mj +3 (α+1)tj +(α−1)ti −α


1/6

mi 3







(
)2 (
((
)
)
(
) )


(2 α−1)ti +(α+1)tj −2 α mi +2 ti −2 tj mj −α −tj +ti mi
mj +tj −ti
(−2+(α+1)mi )mj 2 +


−1/6

mj 2 mi 3





δΦi (bi )
(
)
(
((
)
)
(
)) 3
(−4 α+2)mj +2 α mi 4 + (3 α−3)mj 2 +
−6 ti +3 tj α+3 ti−3 tj +2 mj +3 α −tj +ti
mi
δmi 

1/6
2m 3

m
j
i

(((
)
)
(
))(
)2
(
)3


 +2 ti +1/2 tj α−1/2 ti +1/2 tj mj −1/2 α −tj +ti −tj +ti mi +2 mj −tj +ti


1/6

mj 2 mi 3






((
)
)
(
)



1/6 (3 α−3)mj 2 + −4 mi −6 ti +3 tj α+2 mi +3 ti −3 tj +2 mj +2 α mi+3/2 ti −3/2 tj
mj 2
when 0 ≤
tj −ti
mi
and
when 0 >
tj −ti
mi
and
when 0 ≤
tj −ti
mi
and
when 0 >
tj −ti
mi
and
To get the optimal mi we derive the utility with respect to mi and assume symmetry
= 0 and solve for mi will give us the optimal mi . The symmetric
optimal mi is described by the following function:
δΦi (bi )
|
δmi mj =mi ,tj =ti


mi = − 3αti−2α−2


α+1
















mi = − 3αti−2α−2

α+1




δΦi (bi )
|mj =mi ,tj =ti

δmi






mi = − 3αti−2α−2


α+1














m = − 3αti−2α−2
i
α+1
43
when 0 ≤
tj −ti
mi
and
mj +tj −ti
mi
≤1
when 0 >
tj −ti
mi
and
mj +tj −ti
mi
≤1
when 0 ≤
tj −ti
mi
and
mj +tj −ti
mi
>1
when 0 >
tj −ti
mi
and
mj +tj −ti
mi
>1
m
m
m
m
Now we generally derive the ex-ante utility for ti
)
 (
(−2 α+2)ti −mj −2 tj +α mi +mj −2 ti +2 tj


1/2

mi 2






(
)(
((
)
)
(
) )



mj +tj −ti
(2 α−1)ti +tj −α mi −tj +ti mj −α −tj +ti mi
(mi −1)mj 2 +



mj 2 mi 2
−1/2




δΦi (bi )
)
)
(
)) 2
(
)
(
((
mi
(−2 α+1)mj +α mi 3 + (2 α−2)mj 2 +
−4 ti +2 tj α+2 ti−2 tj +1 mj +2 α −tj +ti
δti


1/2 (
mj 2 mi 2

)((
)
(
))
(
)2


 −2 −tj +ti −1/2 ti +1/2 tj +α ti mj −1/2 α −tj +ti mi −mj −tj +ti

1/2


mj 2 mi 2






((
)
)
(
)


2

1/2 (2 α−2)mj + −2 mi −4 ti +2 tj α+mi +2 ti −2 tj +1 mj +α mi +2 ti −2 tj
mj 2
when 0 ≤
tj −ti
mi
and
mj +tj −ti
mi
≤1
when 0 >
tj −ti
mi
and
mj +tj −ti
mi
≤1
when 0 ≤
tj −ti
mi
and
mj +tj −ti
mi
>1
when 0 >
tj −ti
mi
and
mj +tj −ti
mi
>1
To get the optimal ti we derive the utility with respect to ti and assume symmetry:
= 0 and solve for ti will give us the optimal ti . The symmetric optimal
ti is described by the following function:
δΦi (bi )
|mj =mi ,tj =ti
δti


ti =


















ti =





α−mi+1
2α
when 0 ≤
tj −ti
mi
and
mj +tj −ti
mi
≤1
α−mi+1
2α
when 0 >
tj −ti
mi
and
mj +tj −ti
mi
≤1
α−mi+1
2α
when 0 ≤
tj −ti
mi
and
mj +tj −ti
mi
>1
α−mi+1
2α
when 0 >
tj −ti
mi
and
mj +tj −ti
mi
>1
δΦi (bi )
|mj =mi ,tj =ti

δti






ti =
















t =
i
OptSym
This results in the following optimal solutions for ti (α, mi


ti =



















ti =



δΦi (bi )
|mj =mi ,tj =ti 
δti






ti =
















t =
i
):
α2 −1
2α2 −α
when 0 ≤
tj −ti
mi
and
mj +tj −ti
mi
≤1
α2 −1
2α2 −α
when 0 >
tj −ti
mi
and
mj +tj −ti
mi
≤1
α2 −1
2α2 −α
when 0 ≤
tj −ti
mi
and
mj +tj −ti
mi
>1
α2 −1
2α2 −α
when 0 >
tj −ti
mi
and
mj +tj −ti
mi
>1
44
OptSym
Using the optimal ti (α, mi
OptSym
) we obtain mi (α, ti



mi =
















mi =





α+1
2α−1
when 0 ≤
tj −ti
mi
and
mj +tj −ti
mi
≤1
α+1
2α−1
when 0 >
tj −ti
mi
and
mj +tj −ti
mi
≤1
α+1
2α−1
when 0 ≤
tj −ti
mi
and
mj +tj −ti
mi
>1
α+1
2α−1
when 0 >
tj −ti
mi
and
mj +tj −ti
mi
>1
δΦi (bi )
|mj =mi ,tj =ti

δmi






mi =
















m =
i
):
Therefore the optimal b sofar seems to be:
OptSym
bi (vi )
α+1
α2 − 1
=
· vi + 2
2α − 1
2α − α
To know whether we have a local maximum at the bi (vi )OptSym we need to check
whether the hesse matrix is positive definit.

Hesse = 
1/3 mi −1 + 1/3 mαi 2 − 2/3 mαi − mαi + 1/2 mi −1 + 1/2 mαi 2
− mαi
+ 1/2 mi
−1
+ 1/2
−2
α
mi 2
α
mi
+ mi
−1
+
α
mi 2


Looking at the eigenvalues of the hesse matrix we get:

√
 1/6


13)(2 α mi −α−mi )
mi 2
√
(4+ 13)(2 α mi −α−mi )
mi 2
(−4+
−1/6


 1/6

=



√
2
13)(2 α2 +α−1)
(−4+
−1/6
(1+α)4
√
(4+
13)(2 α2 +α−1)
2





(1+α)4
Obviously the eigenvalues are all negative hence, the Hesse-Matrix is negative definit.
Therefore, the optimal solution is a local maxima.
So we see that the symmetric equilibrium is maximizing the payoff.
However, for all α ≤ 0.5 ti is above 1.5 which is the upper bound and for α > 0.5 ti
is negative. Therefore, the optimal ti cannot be fulfilled.
Furthermore, the bidding function bi (vi )OptSym is above the upper bidding bound (1.5)
for α < 0.5. However, the bidding function for α > 0.5 is at least partly in the allowed
area, which could describe a stepwise bidding function in these parts.
PROBLEMS: Using numerical optimization we find that mi = ti = 0 is optimal for
all alphas if both bidder behave the same. However, if player j sticks to this function it
is best for player i to change the function. Therefore, we do not have a equilibrium.
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