Supply Function Competition, Market Power and the
Generalised Winner’s Curse: A Laboratory Study.
Anna Bayona∗, Jordi Brandts†and Xavier Vives‡§
October 2015
Abstract
We design an experiment to understand whether informational frictions can lead
to market power in the context of supply function competition. Costs are uncertain
and sellers have private information about them. The unique equilibrium predicts that
positively correlated costs lead to steeper supply functions and less competitive outcomes
than with uncorrelated costs. We find evidence of those testable predictions which are
common in both treatments, and we also confirm that behaviour in the uncorrelated
costs treatment is in accordance with the theoretical prediction. Furthermore, our
data shows that differences in behaviour and outcomes between treatments are not
significant. This result can be explained in terms of the generalised winner’s curse.
An analysis of initial choices reveals that only 10% of the subjects in the positively
correlated costs treatment are “strategically sophisticated”. A dynamic analysis shows
that there are significant differences in how subjects learn in the two treatments: Best
response dynamics plays a substantial role in the uncorrelated costs treatment while it
has an insignificant role in the positively correlated costs treatment.
Keywords: Laboratory Experiment; Supply Function; Market Power; Generalised
Winner’s Curse; Bounded Rationality.
JEL Codes: C92, D43, L13.
∗
ESADE Business School.
Institut d’Anàlisis Econòmic (CSIC) and Barcelona GSE.
‡
IESE Business School.
§
We thank José Apesteguía, Maria Bigoni, Colin Camerer, Margaret Meyer, Rosemarie Nagel, Cristina
Lopez-Mayan, Albert Satorra, Arthur Schram and Jack Stecher for useful comments and discussions. The
financial support of the Spanish Ministry of Economics and Competitiveness (Grant: ECO2014-59302-P),
the Generalitat de Catalunya (Grant: 2014 SGR 510), Banco Sabadell and La Caixa Foundation are greatly
acknowledged.
†
1
1
Introduction
We designed a laboratory experiment to understand whether informational frictions can lead
to market power, in a market where bidders compete in supply functions. Each bidder
has incomplete information about his costs and receives a private signal. Costs may be
correlated or uncorrelated among bidders. In this context, our motivation is to investigate in
the laboratory whether bidders learn from prices and whether a higher cost correlation leads
to enhanced market power. This setting is relevant in electricity and financial markets.
The work of Vives (2010, 2011) provides us with a theoretical framework to analyse such
environments using the theoretical concept of supply function equilibria (see, for example,
Klemperer and Meyer (1989)). He finds a unique equilibrium which relates the information
structure to both market behaviour and the competitiveness of the outcomes. With positively correlated costs, the model predicts that the supply function’s slope is steeper and the
intercept is lower than when costs are uncorrelated. The theoretical mechanism which explains these results is as follows. A seller receives a private signal which is informative about
his random costs. A fully rational seller who is strategic must also realise that when costs are
positively correlated, a high price conveys the bad news that costs are high, and therefore he
should compete less aggressively than if costs were uncorrelated in order to protect himself
from adverse selection. This thought process closely resembles the inference that sellers must
make in common value auctions1 . As a consequence, the main theoretical prediction when all
sellers are fully rational is that private information and strategic behaviour lead to a greater
degree of market power, resulting in larger expected prices and profits than when costs are
correlated than when they are uncorrelated. In the positively correlated costs treatment, if
all sellers fail to understand the adverse selection effects of the correlation among costs then
behaviour and outcomes of the two treatments will be indistinguishable. The mechanism
which relates higher cost correlation to enhanced market power is closely related to demand
reduction in multi-unit auctions. The generalised winner’s curse (see Ausubel et al. (2014))
extends the concept of the winner’s curse to multi-unit auctions. Essentially, the generalised
winner’s curse reflects the idea that winning a larger quantity is worse news than winning
a smaller quality, since winning a larger quantity implies that other bidders do not value
the good as much as they might. Therefore, rational bidders refrain from competing too
aggressively.
In our experiment we compare behaviour and outcomes in two treatments: positively
correlated costs and uncorrelated costs. In each treatment, subjects were randomly assigned
1
Our experiment has correlated values with the correlation being less than 1. Furthermore, in relation to
the classical experiments on common value auctions in which a single unit is being auctioned, our setting has
a multi-unit unit divisible good auction.
2
to independent groups of twelve subjects, each consisting of four markets of three sellers.
Within each group, we applied random matching between rounds to keep the one-shot nature
of the theoretical model. The buyer was simulated and subjects were given the role of sellers.
Subjects received a private signal about the uncertain cost and were subsequently asked to
submit a linear supply function. Like in the theoretical model, and in contrast to most of
the experimental literature, we used a normally distributed information structure which may
be a good approximation of the distribution of values in natural occurring environments.
After all decisions had been made, the uniform market price was calculated and each subject
received detailed feedback about his own performance, the market price, and the behaviour
and performance of rivals in the same market. At the end of the experiment, we conducted
a post-experimental questionnaire that asked about participant’s demographic information,
bidding behaviour and understanding of the game. Subjects were given incentives which
consisted of fixed and variable parts. The variable part depended on each participant’s
performance during the game.
Our experimental data conforms to some of the theoretical predictions. First, we confirm
that from the beginning of the experiment average behaviour in the uncorrelated costs treatment is close to the theoretical prediction, and that over time, the average supply function
further tends towards the equilibrium supply function. Second, we find that those testable
predictions that are common in both treatments are observed in the data. In particular, we
observe that the supply function’s intercept is increasing in a bidder’s signal realisation. This
means that subjects understand that a higher signal implies a higher average intercept of the
marginal cost and therefore he should set a higher ask price for the first unit offered, leading
to a higher supply function intercept. We also observe that the inverse supply function slope
is unrelated to the signal received but this depends on the features of the competition and
information environments.
In the positively correlated costs treatment, the average supply function is qualitatively
steeper and has a lower intercept than the average supply function of the uncorrelated costs
treatment. However, differences in behaviour and outcomes between treatments are not
statistically significant. In addition, during the course of our experiment, we do not observe
that the average supply function converges over time towards the equilibrium supply function
of the positively correlated costs treatment, which means that naïve behaviour does not die
out. Furthermore, we cannot reject the hypothesis that the average market price and profits
are the same in both treatments. This means that we reject our original hypothesis that
there is greater market power in the positively correlated costs than in the uncorrelated costs
treatment. In the positively correlated costs treatment, bidders forgo a large percentage
of ex-ante expected profits, a common trait in auctions where bidders ignore the adverse
3
selection effects of correlated value auctions.
Given that average behaviour in the control treatment is broadly in accordance with the
theoretical prediction, the results indicate that subjects in the positively correlated costs
treatment ignore the correlation amongst costs and consequently also ignore its adverse
effects. This is evidence of the generalised winner’s curse, whereby bidders compete too
aggressively in the positively correlated costs treatment.
We provide a further behavioural explanation of the results of our experiment, which is
composed of two parts. The first is based on initial choices before subjects have had a chance
to learn. The second focuses on the adjustment of behaviour over time. The combination of
the two provides an explanation of the observed experimental results. In the uncorrelated
costs treatment, during the first five periods of play we find that the median supply function
is in accordance with the equilibrium and that there is a substantial proportion of subjects
whose choices can be accurately described by the equilibrium prediction. With respect to the
evolution of choices over time, we can explain the adjustment of behaviour in the uncorrelated
costs treatment with a combination of imitation, reinforcement and best response learning.
Analysing initial play in the positively correlated costs treatment we find that there are
very few choices of “strategically sophisticated” subjects (around 10%) and the median choice
is far away from the equilibrium prediction. It is therefore difficult that non-sophisticated
subjects learn from the “strategically sophisticated” subjects since the probability of being
matched with a “strategically sophisticated” subject is very low. The analysis of the dynamics
of choices over time shows that best response learning does not play a role in explaining the
adjustment of behaviour over time in the positively correlated costs treatment.
The paper is organised as follows. Section 2 reviews the related literature. Section 3
explains the model of competition in supply functions with private information. Section 4
lays out the experimental design. Section 5 presents the main experimental results. Section
6 gives explanations for the observed behaviour. Section 7 provides a robustness check and
Section 8 concludes.
2
Related Literature
Our experimental paper studies competition in schedules with an information environment
that encompasses both positively correlated costs and uncorrelated costs. Our information
environment is reminiscent of Goeree and Offerman (2003) since we also use normally distributed values and error terms. However, their information environment compares behaviour
in common and uncorrelated private values in a single unit second-price auction, while ours
compares behaviour in correlated and uncorrelated costs in a supply function uniform price
4
auction.
With respect to the competition environment, some early experiments used bid functions
in auctions with incomplete information (e.g. Selten and Buchta (1994)), but few experiments
have tried to analyse competition in supply functions in the laboratory. Exceptions are the
work of Bolle et al. (2013) which focuses on the testable predictions of the Supply Function
Equilibrium concept, and Brandts et al. (2014) which compare the testable predictions of
alternative models of supply function bidding. Our experiment focuses on a framework where
market power is driven by the small number of firms, increasing marginal costs, and private
information about positively correlated costs. Outside the laboratory, Hortaçsu and Puller
(2008) provide an empirical evaluation of strategic bidding behaviour in multi-unit auctions
using data from the Texas electricity market. They find evidence that large firms bid in
according to the theoretical benchmark, while smaller firms significantly deviated from this
benchmark.
To the best of our knowledge, we have conducted the first laboratory experiment that finds
evidence of the generalised winner’s curse in a multi-unit auction with interdependent values.
Our results are related to the literature of single unit correlated or common value auctions,
which also finds that subjects ignore the adverse selection effects of the correlation among
values. There is ample evidence of the winner’s curse in single-unit auctions (e.g. Kagel
and Levin (1986); Goeree and Offerman (2003), and a large amount of evidence summarised
by Kagel and Levin (2010)). The neglect of the adverse selection effects of the correlation
among values has also been found in laboratory experiments which focus on other strategic
contexts, such as in strategic voting (e.g. Esponda and Vespa (2014)); bilateral negotiations
(e.g. Samuelson and Bazerman (1985)); trade with adverse selection (e.g. Holt and Sherman
(1994)); social learning (e.g. Weizsäcker (2010)) and zero-sum environments (e.g. Carrillo
and Palfrey (2011)). Our results are also related to the emerging literature on correlation
neglect, such as in Eyster and Weizsäcker (2010).
The behavioural explanation of our results has two parts. The first part analyses strategic
thinking during the first few rounds of bidding before subjects have had a chance to learn.
This analysis is inspired by the level-k model of strategic thinking (e.g. Nagel (1995)).
However, our analysis is descriptive and does not fit the level-k formally since it does not
provide a good description of our data. The second part analyses the evolution of choices
over time and is connected to the work of Huck et al. (1999) and Bigoni and Fort (2013),
where they analyse learning in a Cournot setting. Our analysis is more complex since our
subjects set supply functions with incomplete information.
An alternative explanation of the winner’s curse, which is not discussed in our paper,
has been provided by Eyster and Rabin (2005) with the cursed equilibrium concept. In the
5
positively correlated costs treatment, subjects would be fully cursed if they would bid as in
the equilibrium of the uncorrelated costs treatment. However, we do not use this model to
analyse the behaviour of our experiment since it is based on an equilibrium concept which
cannot explain the observed non-equilibrium behaviour of our experiment.
3
Theoretical Background
We use the framework of Vives (2010, 2011) to guide us with the experimental design. There
are a finite number of sellers, n, who compete simultaneously in a uniform price auction.
Each seller submits a supply function. Seller ii’s profit is
λ
πi = pxi − θi xi − x2i ,
(1)
2
where xi are the units sold, θi is a random cost parameter, p is a uniform market price
and λ > 0 represents a parameter which measures the strength of transaction costs. The
market clearing condition allows us to find the uniform market price, p. The random cost
parameter θi is normally distributed with θi ∼ N (θ̄, σθ2 ). The demand is inelastic and equal2
to q .
The information structure is as follows. A seller does not know the value of the cost shock,
θi , before setting the supply schedule and receives a signal with si = θi + i , where the error
term is distributed i ∼ N (0, σ2 ). Sellers’ random costs parameters may be correlated with
corr(θi , θj ) = ρ. When ρ = 1, the model is equivalent to a common costs model; when ρ = 0
to an uncorrelated costs model; when 0 < ρ < 1 to a correlated costs model. Error terms
are uncorrelated with the random cost shocks and among themselves. In the experiment our
treatment variable is the correlation among costs, ρ.
Because of both the quadratic payoff function and the normally distributed information
structure, we focus on linear supply schedules. Linear supply functions are an approximation
of the types of supply functions that bidders may submit in real markets3 . Given the signal
received, a strategy for seller i is to submit a price contingent schedule, X(si , p), which is of
the form
X(si , p) = b − asi + cp.
(2)
The three coefficients (a, b, c) determine the supply function. The interpretation of these
coefficients is as follows: a is a bidder’s response to the private signal; b is the fixed part
of the supply function’s intercept, where the supply function’s intercept, f , is: f = b − asi ;
2
3
Since q > 1 this is a multi-unit auction.
See, for example, Baldick et al. (2004).
6
and c is the supply function’s slope. We shall discuss the experimental numerical results
in terms of the inverse supply function since it reflects more clearly how participants made
their decisions. The inverse supply function can be written as p = b̂ + âsi + ĉX(si , p), whose
; â = ac ;
coefficients can be related to the coefficients of the supply function (2) as: b̂ = −b
c
ĉ = 1c for c 6= 0, where the inverse supply function’s intercept is fˆ = b̂ + âsi .
Vives (2011) finds a unique supply function equilibrium and describes how the equilibrium parameters (a, b, c) depend on the information structure (θ̄, σθ2 , σ2 , ρ) and on the market
structure (n, q, λ). Refer to Appendix A for the formulae that characterise the equilibrium
supply function and outcomes. Figure 1 summarises graphically the comparative statics of
the unique symmetric linear Bayesian Nash supply function equilibrium. The equilibrium
prediction for a fully rational bidder’s behaviour can be summarised as follows. When costs
are positively correlated, a high price conveys the bad news that the bidder’s costs are high.
Therefore, in equilibrium, bidders submit steeper schedules than when costs are uncorrelated
to protect themselves from adverse selection. Additionally, for the same signal realisation,
the equilibrium inverse supply function’s intercept will be lower when costs are positively
correlated than when costs are uncorrelated. As a result, the model also predicts that equilibrium market outcomes will be less competitive when costs are positively correlated than
when costs are uncorrelated: the expected market price and profits are larger with positively
correlated costs than with uncorrelated costs due to adverse selection. Additionally, outcomes in both treatments lie between the Cournot and competitive benchmarks since supply
functions have a positive slope in both treatments.
The thought process that subjects must do in the positively correlated costs treatment
resembles the inference that sellers must do in correlated costs auctions. If subjects do not
understand the adverse selection effects of the correlation among costs then they fall prey
of the generalised winner’s curse (Ausubel et al. (2014))4 and would bid as if costs were
uncorrelated5 . If all subjects were fell prey of the generalised winner’s curse then we would
expect behaviour and market outcomes in both treatments to be indistinguishable.
Summarising this discussion, we derive testable predictions which emerge from the comparative statics of the unique Bayesian Nash equilibrium predictions of the theoretical model
when all sellers are fully rational. For these predictions to hold, we require that a bidder:
(1) forms correct beliefs about the environment; (2) forms correct beliefs about other bidders’ behaviour; (3) given these beliefs, the bidder chooses the action that best satisfies his
preferences. Additionally, the theoretical model presupposes that all bidders are ex-ante
4
The generalised winner’s curse extends the winner’s curse to multi-unit auctions.
Notice that our experimental design does not allow us to distinguish from: (1) Subjects ignore the
correlation among costs or (2) subjects neglect the informational content of prices (as in the cursed equilibrium
concept of Eyster and Rabin (2005)).
5
7
symmetric.
Figure 1: Bayesian Nash Equilibrium supply function predictions for the uncorrelated and
positively correlated costs treatments when all agents are fully rational.
Note: The theoretical predictions for each treatment have been computed for three different signal realisations:
a high signal with value 1,200; a signal equal to the ex-ante value of θi with value 1,000 and a low signal with
value 800.
The following two predictions are common in both treatments and refer to a bidder’s
behaviour:
(A) In each treatment the inverse supply function’s slope is positive and unrelated to a
bidder’s signal realisation.
(B) In each treatment the inverse supply function’s intercept is non-zero and increasing
in a bidder’s signal realisation.
The following two predictions are a result of the comparative statics with respect to the
correlation among costs of the unique Bayesian Nash equilibrium prediction:
(C) The supply function is steeper in the positively correlated costs treatment than in
the uncorrelated costs treatment (and therefore has a higher inverse supply function slope
when costs are correlated than when costs are uncorrelated).
(D) For the same signal realisation, the inverse supply function’s intercept is lower in the
positive correlated costs treatment than in the uncorrelated costs treatment.
Predictions C & D imply that a bidder’s behaviour is different in environments with uncorrelated costs from environments with positively correlated costs. The following prediction
refers to market outcomes and it is a consequence of the previous behavioural predictions:
(E) For the same signal realisation, the expected market price and profits are larger in
the positively correlated costs treatment than in the uncorrelated costs treatment.
8
Prediction E implies that the positively correlated costs treatment will display greater
market power than the uncorrelated costs treatment due to the positive correlation among
costs.
In the context of our multi-unit uniform price auction, if subjects ignore the adverse
selection effects of the correlation among costs, and therefore do not understand that when
costs are positively correlated winning a larger quantity is worse news than winning a smaller
quantity, or in other words, if subjects fall prey of the generalised winner’s curse, then
predictions (C), (D) and (E) may not be satisfied6 .
4
Experimental Design
Sessions were conducted in the LINEEX laboratory of the University of Valencia. The participants were undergraduate students in economics, finance, business, engineering and natural sciences. All sessions were computerised7 . Instructions were read aloud, questions were
answered in private and subjects were not allowed to communicate throughout the sessions.
Instructions explained all the details of the market rules, the distributional assumptions of
the random costs, signals and the correlation among costs. Before starting the experiment,
we tested participants’ understanding. Refer to Appendix B for the instructions of the experiment.
We ran the experiment with 144 undergraduate students8 , half of which participated in
uncorrelated costs treatment and half in the positively correlated costs treatment. Each
treatment had 6 independent groups of 12 members each. Students competed for 2 trial
rounds followed by 25 rounds. In each round, each independent group had 4 markets of 3
sellers in each market. We chose a market size of three since this is the minimum market
size that does not lead to collusion in other similar environments (e.g. Dufwenberg and
Gneezy (2000) in a Bertrand game, and Huck, Normann and Oechssler (2004) in a Cournot
market). In order to keep the spirit of the one-shot nature of the theoretical model, we
applied random matching between rounds to minimise the effects of collusion in repeated
play9 . Thus, the composition of each of the four markets varied each round within a group.
6
Since costs and signals follow a normal distribution we may expect that a subject can fall prey of the
news curse, whereby he ignores the prior information and takes the signal at face value (e.g. Goeree and
Offerman (2003)). In our model, we can show that a bidder that falls prey of the news curse sets the same
supply function slope as that of a bidder that falls prey of the generalised winner’s curse, and therefore, these
two curses are not easily distinguished.
7
Using z-tree (Fischbacher (1999)).
8
Before running the experiment, we conducted two pilots (one in each treatment) in order to pin down
the experimental design.
9
Notice that random matching within a group means that we have less independent observations than if
we had chosen fixed markets. However, we prioritised this in order to keep the spirit of the theoretical model.
9
Table 1 summarises the structure of our experimental design.
Table 1: Experimental design.
Refer to Appendix C for the screenshots used for running the experiment. In each round,
all subjects received a private signal and were subsequently asked to choose two ask prices:
one for the first unit and one for the second unit offered10 . With these two ask prices,
we constructed a linear supply schedule, which was depicted graphically on each subject’s
screen. The participant could then revise the ask prices several times until the participant
was satisfied with the decision. The buyer was simulated.
Once all supply schedules had been submitted, each bidder received feedback on market
parameters (uniform market price), his own performance (revenues, production costs, transaction costs, units sold and profits), the performance of the other two market participants
(units sold, profits and supply functions), and the values of the random variables drawn (his
own value and value of the other two participants in the same market). Participants could
consult the history of their own performance. Other experiments show that feedback affects
behaviour in the laboratory. For example, Offerman, Potters, and Sonnemans (2002) show
that in a Cournot game, different feedback rules can generate outcomes which go from competitive to collusive. Given the complexity of the experiment, we choose to give maximal
feedback after each round in order to maximise the potential learning of participants. After
each participant had checked his feedback, a new round of the game would start. Note that
in each market and each round, we generated three random unit costs from a multivariate
normal distribution. In each round and for each participant, unit costs and signals were
independent draws from previous and future rounds.
In order to implement the experiment, we specified numerical values for the parameters
of the theoretical model, as shown in Table 2 below. In doing so, we applied three main criteria: (1) The existence of a unique equilibrium; (2) sufficiently differentiated behaviour and
outcomes between the two treatments11 ; and (3) simplification of the participant’s computational requirements. Importantly, notice that the uncorrelated costs treatment had ρ = 0,
10
From each participant’s two dimensional decision, (AskP rice1, AskP rice2), we can infer the slope and intercept of each participant’s inverse supply function, which is of the form p = fˆ+ĉX(si , p). The inverse supply
function slope is ĉ, where ĉ = AskP rice(2) − AskP rice(1) and the intercept is fˆ, where fˆ = AskP rice(1) − ĉ.
11
Refer to Table 11 in Appendix A for the predictions of behaviour and outcomes in each treatment.
10
the control treatment, and the positively correlated12 costs treatment had ρ = 0.6. Refer
to Appendix A for the equilibrium supply function and outcomes given the experimental
parameters of Table 2 and for a statistical description of the distribution of the random costs
and errors that were used in the experiment.
Table 2: Experimental parameters.
We imposed certain market rules, which were inspired by the theoretical model and made
the experiment implementable. First, we asked each seller to offer all the 100 units for sale.
Second, we asked sellers to enter a non-decreasing and linear supply function. Third, ask
prices had to be zero or positive. Fourth, we told bidders that the simulated buyer would
not purchase any unit at a price above the price cap of 3,600. We imposed this price cap in
order to limit the potential gains of sellers in the experimental sessions. This price cap was
not present in the theoretical model but we chose its value high enough so that it did not
distort equilibrium behaviour. The only difference between treatments was the correlation
among costs and consequently the distribution of random costs and signals.
At the end of the experiment, participants answered a questionnaire with personal information and questions about each subjects’ reflections after playing the game. Refer to
Appendix D for the questionnaire. Once the questionnaire was completed, each participant
was paid in private. Each subject received 10 Euros irrespective of his performance during
the experiment. Each player started with 50,000 experimental points. During the experiment, subjects won or lost points. At the end of the experiment, points were exchanged for
Euros at the rate 10,000 experimental points to 1 Euro. With student subjects, the average
payment per subject varied from 10 to 27.8, with an average of 20.8. Session length was
between 2 and 3 hours.
12
In order to have maximally differentiated predictions, we would have liked to set a higher correlation
among unit costs. However, with an inelastic demand, the range for which a unique equilibrium exists is
reduced and ρ = 0.6 was the maximal correlation which satisfied our implementation criteria.
11
5
Experimental Results
First, we present an overview of aggregate bidders’ behaviour and outcomes. Second, we
examine behaviour conditional on the private signal. Third, we examine time trends. Fourth,
we formally evaluate the testable predictions of Section 3 using individual level data across
rounds.
5.1
Aggregate behaviour and outcomes
We first overview bidders’ behaviour and outcomes in each treatment, when decisions and
outcomes are aggregated across all rounds. Figure 2 shows a graphic illustration of the
observed average supply function and each treatment’s corresponding theoretical prediction.
Table 3 displays the average supply function in each treatment, which is characterised by
the supply function slope and intercept. We discuss the results in terms of the inverse
supply function13 , characterised by InterceptPQ and SlopePQ, since it reflects more clearly
how participants made their decisions in the experiment. Outcomes can be summarised
by market price and profit. For each of these variables, the table shows the total number
of observation, the averages in each treatment, standard deviations (s.d.) at the individual
level (or market level for the market price) and the corresponding theoretical predictions. We
employ both parametric (t-tests) and non-parametric tests, with all the p-values reported for
two sided tests unless otherwise specified. In this section, we use the group average as the
unit of analysis (which aggregates individual choices/outcomes within the group and over
time).
In terms of behaviour, both Figure 2 and Table 3 show that in the uncorrelated costs
treatment the average supply function is close to the theoretical prediction: We cannot reject
the hypothesis that the inverse supply function slope is the same as the theoretical prediction
(Wilcoxon signed-rank test, n1 = 6, n2 = 6, p = 0.600; t-test, n1 = 6, n2 = 6, p = 0.914) and,
at the 10% significance level, we also cannot reject hypothesis that the inverse supply function
intercept is the same as the theoretical prediction (Wilcoxon signed-rank test, n1 = 6, n2 = 6,
p = 0.046; t-test, n1 = 6, n2 = 6, p = 0.054). The finding that behaviour in the uncorrelated
costs treatment (control treatment) is on average consistent with the theoretical model is
important since it will allow us to use the control treatment as benchmark for our analysis.
13
A high inverse supply function slope (low c) means that the supply function is steep in the
(Quantity, AskP rice) space.
12
Figure 2: Average experimental and equilibrium supply functions in each treatment.
Note: SF refers to supply function, T0 to the uncorrelated costs treatment and T1 to the positively correlated
costs treatment.
Table 3: Average behaviour and outcomes and their corresponding theoretical predictions in
each treatment.
In the positively correlated costs treatment, there is a large divergence between the average supply function and the theoretical prediction: the average supply function is flatter
(Wilcoxon signed-rank test, n1 = 6, n2 = 6, p = 0.028; t-test, n1 = 6, n2 = 6, p = 0.000) and
has a higher intercept than the Bayesian Nash equilibrium prediction (Wilcoxon signed-rank
test, n1 = 6, n2 = 6, p = 0.028; t-test, n1 = 6, n2 = 6, p = 0.004).
Comparing the average supply function of the two treatments, we notice that as predicted
by the Bayesian Nash Equilibrium, the average inverse supply function of the positively cor13
related costs treatment has a higher slope (7,79 in the positively correlated costs treatment
and 6,05 in the uncorrelated costs treatment) and lower intercept than in the uncorrelated
costs treatment (899,25 in the positively correlated costs treatment and 950,11 in the uncorrelated costs treatment). The difference between the average supply functions in the two
treatments is qualitatively in accordance with the theoretical model. However, differences
between treatments are substantially smaller than predicted. In fact this difference is not
statistically significant in terms of inverse supply function intercept (Mann-Whitney U-test,
n1 = 6, n2 = 6, p = 0.749; t-test with unequal variance, n1 = 6, n2 = 6, p = 0.397) and slope
(Mann-Whitney U-test, n1 = 6, n2 = 6, p = 0.337; t-test with unequal variance, n1 = 6,
n2 = 6, p = 0.240).
We also observe that the heterogeneity of choices is larger in the positively correlated
costs treatment than in the uncorrelated costs treatment. An analysis of behaviour at the
group level reveals that this larger heterogeneity is mainly driven by two groups and three
subjects in these groups. These subjects set an AskPrice for the first unit equal to 0 (or a
very low number) which is independent of the signal received, very steep supply functions
and obtain on average negative or very low profits during the course of the experiment.
We interpret these findings as follows. Since behaviour of both treatments is on average
close to the equilibrium prediction of the uncorrelated costs treatment, we conjecture that
some subjects in the positively correlated costs treatment neglect the correlation among costs
and do not take into account the informational content of the price when bidding. In other
words, average behaviour suggests that the representative subject falls prey to the generalised
winner’s curse.
Turning to average market prices, we note that the difference in average market prices
between treatments is statistically insignificant (Mann-Whitney U-test, n1 = 6, n2 = 6,
p = 0.749; t-test with unequal variance, n1 = 6, n2 = 6, p = 0.420). In both treatments,
the average market prices are lower than their corresponding theoretical predictions: 92.5%
and 72.8% of the theoretical predictions in the uncorrelated and positively correlated costs
treatment, respectively.
Average profits in each treatment are substantially lower than their corresponding theoretical predictions14 : average profits are 45% and 11% of the theoretical predictions in the
uncorrelated and positively correlated costs treatment, respectively. Bidders in the correlated
costs treatment bidders forgo a large percentage of ex-ante expected profit, a common trait
in auctions where bidders ignore the adverse selection effects of the correlation among costs.
14
The theoretical predictions assume that all subjects play the Bayesian Nash Equilibrium in the corresponding treatment. Note that if one subject realises that the opponents are not playing the Bayesian Nash
Equilibrium then it is not optimal for him to play the Bayesian Nash Equilibrium and therefore, ex-ante
expected profits are different. Section 6 discusses this further.
14
The average difference in profits between the two treatments is not statistically significant
(Mann-Whitney U-test, n1 = 6, n2 = 6, p = 0.423; t-test with unequal variance, n1 = 6,
n2 = 6, p = 0.506).
5.2
Behaviour conditional on the private signal
The previous analysis has not conditioned choices on the private signal and therefore left
out an important aspect of behaviour. We first present evidence of aggregate behaviour and
then analyse individual choices. In Section 3 we have seen that the theoretical framework
predicts in both treatments that the inverse supply function slope is independent of the signal
received, while the inverse supply function intercept increases with the signal realisation. We
illustrate it in Figure 3 below. In each treatment, we classify signals into quartiles and analyse
the average inverse supply function slope and intercept for each signal quartile using all the
choices from both treatments.
Figure 3: Experimental inverse supply function intercept and slope in each signal quartile and
the corresponding theoretical prediction in each treatment.
Note: T0 refers to the uncorrelated costs treatment and T1 to the positively correlated costs treatment.
Figure 3 shows that in both treatments the average inverse supply function intercept
15
increases with the signal in each quartile (a regression15 of the the inverse supply function intercept on signal gives a coefficient of 0.856 with p=0.000 in the uncorrelated costs treatment
and a coefficient of 0.926 with p=0.000 in the positively correlated costs treatment), while the
inverse supply function slope remains approximately constant in each signal quartile (in both
treatments a regression of the inverse supply function slope on signal gives an insignificant
coefficient with p=0.703 in the uncorrelated costs treatment and p=0.655 in the positively
correlated costs treatment).
Next, we analyse individual choices of ask prices of the first unit offered in relation
to the equilibrium predictions. Figure 4 displays the theoretical and experimental ratio
AskP rice1/Signal, where AskP rice1 is the ask price of the first unit offered and displays
similar characteristics to the inverse supply function intercept but reflects more clearly how
subjects took their decisions during the experiment16 .
In the uncorrelated costs treatment, the Bayesian Nash Equilibrium predicts that the
ratio of AskP rice1/Signal is between 0.92 and 1.2, with the mean and median equal to 1.
In other words, in the uncorrelated costs treatment the Bayesian Nash Equilibrium predicts
that AskP rice1 is on average equal to the signal received. Experimental choices of the
uncorrelated costs treatment displayed in Figure 4 (bottom left panel) illustrate that the
AskP rice1/Signal ratio has a similar mean to its theoretical counterpart, average ratio is
0.96, with larger standard deviation than predicted due to heterogeneity in individual choices.
In the positively correlated costs treatment, the theoretical prediction sets the ratio
AskP rice1/Signal between 0.58 and 0.73, with the mean and median equal to 0.68, meaning that the equilibrium predicts that the AskP rice1 is lower than the signal received. The
experimental distribution has a mean of 0.91 and median of 0.99, meaning that, for a given
signal, the AskP rice1 is on average larger than predicted by the Bayesian Nash Equilibrium
in this treatment and meaning that subjects in the positively correlated costs treatment are
strongly guided by the signal received when setting AskP rice1 (refer to the bottom right
panel of Figure 4). Consequently, the ratio AskP rice1/Signal is not substantially different
in the positively correlated from the uncorrelated costs treatment. Furthermore, the distribution of experimental AskP rice1/Signal has a larger standard deviation in the positively
correlated costs treatment (s.d. equal to 0.27) than in the uncorrelated costs treatment (s.d.
equal to 0.18). We also notice that there are 6.28% of the choices in the positively correlated
costs treatment and 1.39% of the choices in the uncorrelated costs treatment that set a very
low AskP rice1 (which are less than 100).
15
The regressions of this sub-section have used the group over time as unit of observation. The number of
observations is equal to 150 (6 groups x 25 periods) in each treatment.
16
The relationship between AskP rice1 and InterceptP Q is AskP rice1 = InterceptP Q + slopeP Q.
Remember that in each round each subject had to decide (AskP rice1, AskP rice2).
16
Figure 4: Histogram of the theoretical and experimental ratio of AskPrice1 over Signal received
for individual choices in all periods of the experiment.
Note: For the elaboration of this histogram, we have not plotted the two observations that had an
Experimental AskPrice1/Signal which was greater than 2.
This evidence suggests that bidding behaviour is qualitatively consistent with the most
general testable predictions of the theoretical model which are common in both treatments.
In addition, in accordance with the theoretical model, 97% of the bids are supply functions
with a non-zero intercept. The other 3% of the bids fix a price (as in Bertrand competition).
However, this section also illustrates systematic divergences between behaviour and Bayesian
Nash equilibrium predictions. For example, in the positively correlated costs treatment, for
any signal received, the experimental distribution of the AskP rice1 has a higher mean than
theory predicts and it is not substantially different than in the uncorrelated costs treatment.
Consequently, subjects do not take into account the effects of the correlation among costs
when setting the supply function intercept.
5.3
Time trends
Figure 5 shows the evolution over time of the inverse supply function intercept and slope in
each treatment and the corresponding theoretical predictions. The change in behaviour is
more pronounced in the first 10 periods of play, and more specifically the change in the first
5 periods is particularly marked.
17
Figure 5: Evolution over time of the inverse supply function slope and intercept in each
treatment.
Note: SF refers to supply function, T0 to the uncorrelated costs treatment and T1 to the positively correlated
costs treatment.
We find that the evolution over time is different in each treatment. The panel on the
left of Figure 5 shows that the inverse supply function intercept increases over time in the
uncorrelated costs treatment (a regression of the inverse supply function intercept on period
gives a coefficient of 2.27, with p=0.002) while it does not present any particular time trend in
the positively correlated costs treatment (a regression of the inverse supply function intercept
on period gives an insignificant coefficient with p=0.927). The panel on the right shows that
the average inverse supply function slope becomes smaller over time in the uncorrelated costs
treatment (a linear regression17 of the inverse supply slope on period gives a coefficient of
-0.096 with p=0.000), while there is no significant time trend in the positively correlated costs
treatment (a linear regression of inverse supply function slope on period gives an insignificant
coefficient with p=0.121).
In the uncorrelated costs treatment, both the inverse supply function intercept and slope
tend towards the theoretical prediction as the number of rounds increases. However, in
17
The regressions of this sub-section have used the group over time as the unit of observation. The number
of observations is equal to 150 in each treatment.
18
the positively correlated costs treatment, we do not observe that the difference between the
average supply function and the theoretical prediction becomes smaller over time during the
25 rounds of play.
We now turn to the evolution of market outcomes over time. Table 4 below shows the
evolution of the average market price and profit in blocks of five periods.
Table 4: Evolution over time of average market price and profit in each treatment.
In each treatment, the table shows that there is no time trend in the average market price
(in both treatments a regression of the market price on period gives an insignificant coefficient
with p=0.561 in the uncorrelated costs treatment and p=0.817 in the positively correlated
costs treatment). We observe that profits increase in the uncorrelated costs treatment after
the 10th round but this trend is not observed in the positively correlated costs treatment (in
both treatments a regression of the market price on period gives an insignificant coefficient
with p=0.103 in the uncorrelated costs treatment18 and p=0.556 in the positively correlated
costs treatment).
5.4
Tests of Hypotheses
In this sub-section, we put our analysis on firmer statistical grounds by using individual
choices and outcomes across rounds and formally test the hypotheses A-E presented in Section
3. We use a random effects panel data approach. We aim to formally evaluate whether there
is a difference in market behaviour in the positively correlated costs treatment in relation to
the control treatment. Table 5 reports the panel data random effects regression results of
estimating the equation for the supply function slope and intercept jointly, using a seemingly
unrelated regression (SUR) approach for panel data, which assumes that disturbances across
equations may be correlated and gives us efficient estimates.
18
Despite the fact that we observe a time trend in profits in the uncorrelated costs treatment after the
10th round, this time trend not statistically significant when data is aggregated at the group level. Section
5.4 will show that this time trend is statistically significant when we consider choices at the individual level
over time.
19
Table 5: Seemingly Unrelated Panel Data Random Effects Regression for supply function
choices.
Notes: The unit of observation is the individual choice across time. Dependent variables are the inverse
supply function slope (SlopePQ) and the inverse supply function intercept (InterceptPQ). The independent
variables include the dummy variable D_Treatment, which is 0 in the uncorrelated costs treatment and 1 in
the positively correlated costs treatment; Period which is the number of round; D_Treatment*Period which
is the interaction term between D_Treatment and Period and Signal, which is the signal received. Each
equation was estimated with group dummies.
This estimation has been conducted using the Stata xtsur command Nguyen (2010).
Standard errors (s.e.) are provided in parenthesis. *, **, *** denote the 10%, 5% and the 1% significance
levels, respectively.
In Table 5(1), we find the first set of regressions which estimate jointly the inverse
supply function slope and intercept with the following regressors19 : a treatment dummy
(D_Treatment) which is 0 for the control treatment and 1 for the positively correlated costs
treatment; Group dummies and a constant. The following result summarises our findings:
RESULT 1: We cannot reject the hypothesis that supply functions are the same
in both treatments.
This means that we reject testable predictions C & D, which concern different behaviour
in the two treatments. From the evidence and tests of Section 5.1, we have found that
behaviour in the uncorrelated costs treatment is on average in accordance with the predictions
of Bayesian Nash Equilibrium. In addition, in the positively correlated costs treatment, the
divergence between the average experimental behaviour and the theoretical prediction is large.
19
Table 5 (1) report the regression results of : SlopeP Qit = β0 + β1 DT reatmenti + uit , where and
InterceptP Qit = β0 + β1 DT reatmenti + vit with correlation of the disturbances across equations. See
Baltagi (2008).
20
Therefore, we infer that on average subjects in the positively correlated costs treatment do not
take into account the correlation among costs nor its effects when bidding, and on average
behave as if costs were uncorrelated. We call this phenomenon the generalised winner’s
curse, which is a generalisation of the winner’s curse in a multi-unit auction setting with
interdependent values.
In Table 5(2), we show the second set of regressions, which include additional variables to
the previous regressions in the estimation20 : a variable for the round number (Period ), the
interaction term between the round number and treatment (D_Treatment*Period ), and the
signal received (Signal ). The results presented formally confirm some of the stylised facts that
have already been presented. These results show that supply functions are not significantly
different between treatments once we have controlled for the signal received and the evolution over time. In the equation for the inverse supply function slope we note that: Period
has a significant and negative coefficient, meaning that submitted supply functions become
flatter as the number of rounds increases; D_Treatment*Period has a significant and positive
coefficient, meaning that the decrease of the inverse supply function slope is less pronounced
in the positively correlated costs treatment. We also find that Signal is not statistically significant in determining the inverse supply function slope. Turning to the regression for the
inverse supply function intercept, the results show that: Period has a significant and positive coefficient, meaning that the inverse supply function intercept increases as the number
of rounds increases; D_Treatment*Period with a significant and negative coefficient that is
nearly identical to coefficient of Period, meaning that the inverse supply function intercept
does not vary with Period in the positively correlated costs treatment; and Signal with a
significant and positive coefficient, as predicted by the theoretical model. We can summarise
these findings in the following two results.
RESULT 2: As predicted by the theoretical model, in each treatment the average
inverse supply function’s intercept is increasing in a bidder’s signal realisation,
while the average inverse supply function’s slope is positive and unrelated to a
bidder’s signal realisation.
We interpret this result to mean that we cannot reject testable predictions A & B and
therefore the theoretical model is good at predicting those testable predictions which are
common in both treatments.
20
Table 5 in columns (3) and (4) report the regression results of: SlopeP Qit = β0 + β1 DT reatmenti +
β2 P eriodt + β3 (DT reatment ∗ P eriod)it + β4 Signalit + uit and InterceptP Qit = β0 + β1 DT reatmenti +
β2 P eriodt + β3 (DT reatment ∗ P eriod)it + β4 Signalit + vit , with correlation of the disturbances across equations. See Baltagi (2008).
21
RESULT 3: The evolution of supply functions over time is different in the two
treatments.
Result 3 suggests that the learning process may be different in the two treatments, and this
is explored further in Section 6.2. In the uncorrelated costs treatment, as the number of
rounds increases the inverse supply function tends towards the equilibrium supply function.
In the positively correlated costs treatment, we do not observe that the difference between
the average supply function and the equilibrium prediction becomes smaller over time during
the course of the experiment, suggesting that naïve behaviour does not die out.
In order to evaluate testable prediction E, we use individual market prices and profits
over time as the unit of observation. Table 6 reports the results of random effects panel
regressions, where standard errors have been clustered at the group level.
Table 6: Panel Data Random Effects Regression for Market Outcomes.
Note: The unit of observation for (1) and (2) is the market across time. The unit of observation for (3)
and (4) is the individual across time. Dependent variable for regressions (1) and (2) is the market price.
Dependent variable for regressions (3) and (4) is profits. The independent variables include the dummy
variable D_Treatment, which is 0 in the uncorrelated costs treatment and 1 in the positively correlated costs
treatment; Period which is the number of round; D_Treatment*Period which is the interaction term between
D_Treatment and Period, and Signal, which is the signal received. The standard errors have been clustered
at the group level since we allow intergroup correlation.
This estimation has been conducted using the Stata “xtreg, re” command.
Standard errors (s.e.) are provided in parenthesis. *, **, *** denote the 10%, 5% and the 1% significance
levels, respectively.
22
Regressions (1) and (3) show the results of regressing market price and profit on a constant
and the dummy variable for treatment (D_Treatment)21 , respectively, while (2) and (4)
augment the set of regressors to include the round number (Period ), the interaction term of
treatment and period (D_Treatment*Period ) and the signal received22 (Signal ).
The results show that we cannot reject the hypothesis that market prices are the same in
the two treatments, both when the treatment dummy is used as the only regressor and also
when we include the additional controls. The signal received is the only variable which has a
positive and significant coefficient in determining market price (see column (2) in Table 6).
We find similar results for profits: Treatment is not significant in regression presented in
column (3) of Table 6 and neither in regression presented in column (4) of Table 6, indicating
that there are no statistically significant differences between profits in the two treatments.
The signal received has a negative and significant coefficient in regression of profits, reflecting
the negative correlation between signal (and also unit costs) and profits. We also find that
profits increase slightly over time in the uncorrelated costs treatment, but we do not find any
evolution over time of profits in the positively correlated costs treatment.
These results provide further evidence that we can reject testable prediction E, which
hypothesised that there would be greater market power in the positively correlated than in
the uncorrelated costs treatment due to information based reasons (the positive correlation
among costs). We summarise these results as follows:
RESULT 4: We reject the hypothesis that there is greater market power in the
positively correlated costs than in the uncorrelated costs treatment.
6
A behavioural explanation
This section provides a behavioural explanation of the results described in Section 5. We first
discuss some theoretical considerations. Second, we conduct an analysis of initial behaviour.
Third, we describe how behaviour evolves over time. Taken together, all the parts constitute
an explanation of the results that we observe.
21
In Table 6, regression (1) reports the results of M arketP ricemt = β0 + β1 DT reatmentm + vmt and
regression (3) reports the results of P rof itit = β0 + β1 DT reatmenti + uit
22
In Table 6, regression (2) reports the results of M arketP ricemt = β0 + β1 DT reatmentm + β2 P eriodt +
β3 (DT reatment ∗ P eriod)mt + β4 Signalmt + vmt and regression (4) reports the results of P rof itit = β0 +
β1 DT reatmenti + β2 P eriodt + β3 (DT reatment ∗ P eriod)it + β4 Signalit + uit .
23
6.1
Theoretical considerations
Bidding in the positively correlated costs treatment is cognitively more demanding than in
the uncorrelated costs treatment. In the positively correlated costs treatment, the equilibrium reasoning requires that subjects have common knowledge of the rationality of rivals in
understanding that the market price is informative about the level of costs. Contrastingly,
in the uncorrelated costs treatment, the market price is not informative about the level of
costs. In the positively correlated costs treatment, if a sophisticated subject believes that
rivals are naïve, in the sense that naïve rivals do not take into account the correlation among
costs, then it is individually optimal for him to set a flatter supply function with a lower
intercept than the equilibrium of the positively correlated costs. This result was first noted
by Camerer and Fehr (2006) in the context of games of strategic complementarities with
sophisticated and boundedly rational subjects.
To illustrate these considerations, Figure 6 shows the best response strategy to the average
supply function of the first five periods of bidding in each treatment.
Figure 6: Average supply function during the first 5 periods of bidding, best response supply
function to the former and equilibrium supply function in the uncorrelated costs and positively
correlated costs treatment.
Note: This graphs represents the best response function to the average supply function of the first
5 periods of bidding in each treatment. The best response and equilibrium supply functions have
been calculated for a signal equal to 1.000.
24
The figure shows that in the uncorrelated costs treatment, the average supply function,
the best response supply function and the equilibrium supply function are very close to
each other. However, in the positively correlated costs treatment, the best response supply
function is less steep and has a higher intercept than the equilibrium supply function. In
other words, in the positively correlated costs treatment, a subject that best responds to rivals
who do not take the correlation of costs into account has an incentive to set a flatter supply
function than the equilibrium prediction, thus making behaviour and outcomes between the
two treatments more similar.
6.2
Behaviour in the first five periods of bidding
We first focus on describing subjects’ first few choices during the course of the experiment,
interpreted as the first five periods of bidding. The first five choices are important since
they reveal subjects’ decisions before they have had a chance to learn, inform us of subjects
hypothetical thinking, and influence subsequent choices.
Figure 7 illustrates the empirical cumulative distribution function (hereafter ECDF) for
the inverse supply function in each treatment during the first five rounds of bidding. We
notice that the ECDFS of slopePQ are nearly identical in both treatments for inverse supply
function slopes which are lower than the equilibrium slope of the uncorrelated costs treatment
(slopePQ=6 ). However, for inverse supply function slopes which are larger than 6, then
the ECDFs of both treatments start differing: the ECDF of the positively correlated costs
treatment is slightly shifted to the right, meaning that there is a larger proportion of subjects
that set steeper supply functions than in the uncorrelated costs treatment. This difference
is statistically significant (Kolmogorov-Smirnov test, n1 = 360, n2 = 369, p = 0.023). The
difference between the two ECDFs indicates that some subjects in the positively correlated
costs treatment set a steeper supply function to protect themselves from the adverse selection
of the correlation among costs and we identify these subjects as “strategically sophisticated” 23 .
A simple quantification shows that there would be approximately 10% of subjects that would
be classified as “strategically sophisticated” in the positively correlated costs treatment24 . We
note that the median of the inverse supply function slope is far from the equilibrium of the
positively correlated costs treatment.
Turning to the ECDF of the inverse supply function intercepts, we notice that the ECDFs
are nearly identical in both treatments except for very low inverse supply function intercepts.
23
Notice that subjects that play the equilibrium would be also classified as “strategically sophisticated”.
This classification uses the relative difference in choices between treatments of inverse supply function
slopes which have slopePQ>6. In order to find the approximate average number of subjects, we simply
divided the difference in choices by 5 (since there were 5 periods).
24
25
This is due to the fact that few subjects in the positively correlated costs treatment set
intercepts which are very low and not responsive to the signal received. In fact, the difference
is not statistically significant (Kolmogorov-Smirnov test, n1 = 360, n2 = 369, p = 0.698).
Overall, we conclude that already in the first five periods of bidding, the median inverse
supply function of both treatments is closer to the equilibrium supply function of the uncorrelated costs treatment and far away from the equilibrium supply function of the positively
correlated costs treatment, but the slight difference in the ECDFs of the inverse supply function slopes is due to the presence of few subjects in the positively correlated costs treatment
which are “strategically sophisticated”.
Figure 7: Empirical Cumulative Distribution Function of the inverse supply function slope
(SlopePQ) and intercept (InterceptPQ) of the first five periods of bidding using individual
decisions.
Note: ECDF stands for empirical cumulative distribution function; T0 for the uncorrelated costs
treatment and T1 for the positively correlated costs treatment.
Next, we analyse how many subjects and individual choices are within a pre-defined area
of the equilibrium choice in the corresponding treatment during the first five periods of
26
bidding25 , before subjects have received feedback on own and others’ performance26 . This
analysis will give us an upper bound on the percentage of subjects and choices that behave
as in the equilibrium of the corresponding treatment during the first five periods of bidding.
In the positively correlated costs treatment, there are at most 5.6% of the subjects that
set at least one supply function which is broadly in accordance with the equilibrium of
this treatment. During the first five periods, each of these subjects sets on average 1 out
of 5 supply functions which are within the pre-specified area containing the corresponding
equilibrium. Notice that this upper bound is very low and therefore we conclude that there
are very few subjects that behave as in the equilibrium of the positively correlated costs
treatment during the first five rounds of play.
The situation is dramatically different in the uncorrelated costs treatment since there are
78% of the subjects that set an inverse supply function within the area of success of this
treatment. During the first five periods of bidding, each subject sets on average 2.5 out of
5 supply functions which are close to the equilibrium of the uncorrelated costs treatment.
Furthermore, we may ask whether the equilibrium of the uncorrelated costs treatment can
give us a good description of the behaviour in the positively correlated costs treatment. There
are at most 65% of the subjects in the positively correlated costs treatment whose behaviour
can be described by the area of success of the uncorrelated costs treatment, and each of these
subjects sets on average 2 out of 5 supply functions which are close to the equilibrium of the
uncorrelated costs treatment. Our analysis highlights the following result.
RESULT 5 (Initial choices):
i) There is a significant difference the distribution of supply slopes between treatments which is driven by 10% of the subjects which are “strategically sophisticated”
in the positively correlated costs treatment.
ii) In the first five periods of bidding, there are at most 5.6% of the subjects
in the positively correlated costs treatment that set a supply function which is
approximately in accordance with the equilibrium of this treatment, while this
figure rises to 78% in the uncorrelated costs treatment.
25
Given the difficulties of calculating the analytical equilibrium in each treatment we pre-specify an area
surrounding the equilibrium to give us a measure of predictive success, in the spirit of Selten (1991). The
area of success for the positively correlated costs treatment comprises inverse supply function slopes in the
range [25,29] whose intercepts lie in the range [350,950]. For the uncorrelated costs treatment the successful
area comprises inverse supply function slopes in the range [4,8] whose intercepts lie in the range [650,1250].
26
Notice that once subjects have realised that others in the same treatment do not behave as in the
equilibrium of the corresponding treatment, it is individually optimal for them to deviate from the equilibrium,
and therefore, the analysis needs to be done differently after the first few rounds of bidding.
27
6.3
Evolution of choices over time
Our goal is to describe and understand the drivers of the evolution of choices over time in
the two treatments. Our experimental design provided subjects with very complete feedback
after each round, which included each subjects’ own choice, own profits and the choices and
profits of the other two market participants. Subjects may be influenced by the choices
of rivals in the same group since there was random matching within subjects of the same
group. In this high-information environment and with twenty-five rounds we aimed to foster
learning about the equilibrium. A subject’s use of a particular learning model depends on:
(i) the information available and (ii) subjects’ cognitive abilities. Given the complexity of the
experiment, we considered only those learning models in which subjects had the necessary
information to learn and those which were not cognitively too demanding.
We consider that the evolution of behaviour over time can be described by three main
categories of learning models (for an excellent review see Camerer (2003))27 : experiential
learning, in which subjects learn from their own experience; imitation based learning, in
which after the first period subjects choose a strategy which has been previously chosen
by other players in the previous period and can be payoff-dependent or independent; and
belief learning, which are essentially adaptive since subjects update their beliefs based on the
history of play.
Our empirical strategy follows the approach of Huck et al. (1999) and Bigoni and Fort
(2013), whereby we consider a few representative models of each of the three categories
described above. Furthermore, in this sub-section, we consider that the supply function slope
is the relevant strategic variable which best summarises how a subject adjusts his behaviour
over time. We do not analyse the dynamics of the inverse supply function intercept since
we consider that subjects cannot learn about it because of lack of information: we did
not provide feedback on the signals received by the rivals. As a result, the inverse supply
function intercept may be affected by confounding factors, such as how subjects respond to
the private signal. In addition, inverse supply function intercepts are practically identical in
the two treatments and do not evolve over time in the positively correlated costs treatment.
In terms of experiential learning, we focus on “reinforcement learning”, whereby a player
increase the probability of playing a strategy (i.e. the strategy is reinforced) according to the
previous profits. The propensity of a strategy is the cumulative sum of the previous profits
obtained with such strategy, while the relative propensity of a strategy is the propensity of a
strategy divided by the aggregate propensities of all strategies in a given period. Our imple27
There are very few theoretical results on the convergence properties of learning models in the context
of supply function competition with private information. For some general considerations, refer to the
introduction of Chapter 7 of Vives (2008).
28
mentation of the “reinforcement learning” assumes that a player will set a supply function
slope with the highest relative propensity.
We then consider two models based on learning by imitation: (1) “imitation of the average”, where subjects imitate the rivals’ average strategy28 of the previous period; (2) “imitation of the best”, where subjects imitate the strategy of the subject with highest profits in
the previous period, and the most successful player imitates himself29 .
As a representative of belief learning models, we consider “best response dynamics”, whereby
subjects best respond to the rivals’ previous period strategies30 and whose features have been
described in Figure 6.
We estimate the following equation31 :
E
IA
IB
B
4slit = β0 +βE (slit−1
−slit−1 )+βIA (slit−1
−slit−1 )+βIB (slit−1
−slit−1 )+βB (slit−1
−slit−1 ), (3)
where 4slit = slit − slit−1 is the first difference of subject i inverse supply function slope
E
in round t in relation to the slope in round t − 1, slit−1
is the inverse supply function slope
which corresponds to the highest relative propensity based on player i’s last period profit
IA
is the average inverse supply function slope in round t−1 by
(“reinforcement learning”); slit−1
IB
the other two rivals (“imitation of the average”); slit−1
is the average inverse supply function
slope in round t − 1 by the subject with highest profits in the previous period (“imitation of
B
is subject i’s best response inverse supply function slope to the rivals’
the best”); and slit−1
strategy in round t − 1 (“best response dynamics”). We also estimate an augmented version
of equation (3) which includes the four interaction terms with D_Treatment, one for each
learning model.
If one learning model provided a complete explanation of how subjects adjust the supply
function slope over time then the coefficient corresponding to this learning model would be
equal to 1 and the other coefficients would be equal to 0. If various learning models provide an
explanation of how subjects adjust the supply function slope over time then the coefficients
would be positive and significantly different from zero. Then, each learning model with a
28
In each round, the composition of each market changes since we have applied a random matching procedure to determine the players of each market. We take this into account in the dynamic analysis.
29
We have not considered learning models such as imitate the exemplary firm, such as in Offerman et al.
(2002), since it would require subjects to calculate the joint profit maximisation if the three firms produced
a given level of output, and this is not trivial in our supply function with private information environment.
30
We do not consider “fictitious play” since it is cognitively more demanding that “best response dynamics”.
Subjects must remember the average strategies of all previous rounds of play, while “best response dynamics”
only requires subjects to remember the strategies of the other two players in the previous round.
31
In relation to Huck et al. (1999) we have added the reinforcement learning term. In relation to Bigoni and
Fort (2013) we have replaced the ’trial and error’ experiential learning model by the reinforcement learning
model since it has a wider acceptance in the literature.
29
significant coefficient would individually provide a partial explanation of how subjects adjust
their slopes over time.
Table 7 presents the estimation results using a random effects panel data approach with
standard errors clustered at the group level. The first column of Table 7 reports the results of the augmented version of equation (3), which includes the interaction terms with
D_Treatment, and uses data from both treatments. The second column shows the results
of estimating equation (3) for the uncorrelated costs treatment and the third column for the
positively correlated costs treatment.
Table 7: Estimation of the learning equations.
Note: Standard error are provided in parenthesis and are clustered at the group level. *, **, *** denote the
10%, 5% and the 1% significance levels, respectively.
The results show that in both treatments there is not a unique learning model that
explains the adjustment of behaviour over time. In the uncorrelated costs treatment, all
learning models play a substantial role and, contrary to what is expected, some coefficients
are negative. This is due to the multi-collinearity of the regressors32 . The results show
32
In the uncorrelated costs treatment, all regressors of (3) become more similar over time since over the
course of the experiment there is some degree of convergence.
30
that each coefficient is positive and between 0.32 (corresponding to “imitation of the best”)
and 0.42 (corresponding to “best response dynamics”). To summarise, we notice that in the
uncorrelated costs treatment all learning models partially contribute to explain the evolution
of choices over time.
Learning is different in the positively correlated cost treatment since results show that
the best response model does not contribute to the explanation of how subjects learn. We
notice that the best response strategy requires that subjects take into account the correlation
among costs. If subjects ignore it then it is not surprising that best response dynamics does
not contribute to explain the evolution of choices over time in the positively correlated costs
treatment. The other three learning models (reinforcement and the two imitation) jointly
provide a partial description of how subjects adjust their slope over time.
The first column of Table 7 allows us to understand whether the adjustment of behaviour
can be explained by different learning models in the two treatments and therefore contribute
to answer whether subjects learn differently in the two treatments. We notice that we cannot
reject hypothesis that the learning model “imitation of the best” has the same coefficient in
both treatments, while we reject the hypothesis that all the other learning models have the
same adjustment coefficients in the two treatments. In particular, in the positively correlated
costs treatment, “reinforcement learning” has a lower coefficient, “imitation of the average”
has a larger coefficient, and “best response dynamics” has a lower coefficient than in the
uncorrelated costs treatment. The following result summarises these findings.
RESULT 6: There are significant differences between how subjects learn in the
two treatments. The “best response dynamics” learning model plays a substantial
role in the uncorrelated costs treatment while it is insignificant in the positively
correlated costs treatment.
6.4
Post-Experimental Questionnaire Analysis
The main result of the experiment is that most subjects in the positively correlated treatment
behave as if they do not understand that the market price is informative about the level of
costs. The analysis of the answers of the post-experimental questionnaire provide further
evidence of this interpretation. We asked: (1) “Do you think that a high market price
generally means good/mixed/bad news about the level of your costs?”; (2) “Explain your
answer”. In questions (1) and (2), we aimed to check whether after playing 25 rounds subjects
understood that the price is informative of the level of costs in the positively correlated costs
treatment, and whether there were differences between the two treatments.
Analysis of the results show that there are no substantial differences between the two
31
treatments in the answer to Question 1. In the uncorrelated costs treatment, 29% answered
good, 47% answered mixed and 24% answered bad, while in the positively correlated costs
treatment, 32% answered good, 39% answered mixed and 29% answered bad. The analysis of
the question where subjects explained their answer were more enlightening. Table 8 classifies
the answers of the post-experimental questionnaire into types and provides an example of
a typical answer of the category. Table 8 reports an analysis of the classification of the
post-experimental questionnaire answers into types in each treatment.
Table 8: Classification of post-questionnaire answers.
Note: The answers correspond to the explanation part of: (1) “Do you think that a high market price
generally means good/mixed/bad news about the level of your costs?”; (2) “Explain your answer”.
The answers classified as “Market price is informative about costs” reflect the logic of
the equilibrium reasoning of the positively correlated costs treatment, while the answers
classified as “Market price is not informative about costs” reflects the equilibrium reasoning
of the uncorrelated costs treatment. The answers classified as “High costs imply a high
market price” is somewhat ambiguous since it could reflect either: (1) understanding of the
equilibrium reasoning of the positively correlated costs treatment, or (2) an understanding of
the common prediction in both treatments that a high signal implies high costs and therefore
subjects set a higher intercept. Therefore the we are not going to focus on the interpretation
of this category of answers nor on those classified as “Other factors”.
Table 9: Answers of post-experimental questionnaire (Question 2) in each group of each
treatment.
32
The classification shows that there are approximately twice the percentage of subjects in
the category “Market price is informative about costs” in the positively correlated treatment
compared to the uncorrelated treatment (17% vs. 8%), reflecting that a few more subjects
in the positively correlated costs treatment understand that the market price is informative
about the level of costs. This is consistent with the interpretation provided earlier that there
are few “strategically sophisticated” subjects in the positively correlated costs treatment. In
addition, we note that the answer “Market price is not informative about costs” is the most
prevalent answer in both treatments with no significant differences between how subjects
answered this question in the uncorrelated and positively correlated costs treatments. This
is consistent with our interpretation that: (1) there is a large percentage of subjects that
understand the logic of the equilibrium of the uncorrelated costs treatment and (2) there is a
large percentage of subjects in the positively correlated costs treatment that fail to understand
the adverse selection effects of the correlation among costs in the positively correlated costs
treatment and bid as if costs were uncorrelated (i.e. fall prey of the generalised winner’s
curse).
7
Robustness check: Bayesian Updating
It could be argued that subjects fail to conduct Bayesian updating and this is the reason why
subjects in the positively correlated costs treatment fail to understand that the market price
is informative about the level of costs. In order to further explore this possibility and aid
subjects in their decision making process, we conducted an additional session with two groups
in the positively correlated costs treatment. These sessions had the same experimental design
features as in the baseline treatment except for: (1) In addition to the signal received, we
gave subjects the expected value of their own costs and rivals’ costs conditional on the signal
received (subject i received a signal si and was also given E(θi | si ) and E(θj | si ))33 ; (2) We
explicitly asked subjects to think about what rivals would do and according to their beliefs we
provided a simulation tool where they could provisionally enter a decision and visualise the
provisional market price. They could then revise their decision; (3) The experiment lasted34
for 15 rounds instead of 25.
33
The instructions of these additional treatments are available upon request. In order to explain the
conditional expectations, we told subjects that in each round an expert would give them the expected value
of their unit cost and the unit cost of their rivals.
34
We shortened the number of rounds since these modifications made the experiment longer and since
Section 5.3 has shown that subjects do not adjust much their behaviour in the last few rounds of bidding.
33
Table 10: Robustness Session: Behaviour and outcomes in the positively correlated costs
treatment.
Table 10 presents the summary statistics of outcomes and choices of these two groups in
the positively correlated values treatment. We notice that the average supply function and
outcomes of the robustness session is very similar to the average supply function and outcomes
presented in Table 3 for the positively correlated costs treatment. In other words, aiding
subjects in their decision making process does not seem to affect behaviour and outcomes
of the positively correlated values treatment in a significant way. This seems to suggest
that our results in the positively correlated costs treatment are not due to a bias related
to simple Bayesian updating. Our interpretation that subjects in the positively correlated
costs treatment do not understand that the market price is related to the level of their costs
and therefore should bid less aggressively in order to protect themselves from this adverse
selection is robust.
8
Concluding Remarks
The experiment presented in this paper has analysed bidding behaviour in the laboratory,
in a market where bidders compete in supply functions, have incomplete information about
their costs and receive a private signal. We have used the fully rational unique Bayesian
Nash equilibrium prediction and comparative statics as the benchmark for evaluating our
original hypothesis of whether bidders learn from prices and a higher cost correlation leads to
enhanced market power. Our experiment uses a between subject design with two treatments:
uncorrelated costs and positively correlated costs. The uncorrelated costs treatment serves as
the control treatment. In implementing the experiment, we have chosen numerical values for
the parameters of the model so that the theoretical predictions for behaviour and outcomes
were sufficiently differentiated between the two treatments.
The results show that the theoretical model is useful in organising the observed bidding
34
behaviour in relation to the testable predictions which are common in both treatments.
We also find that behaviour in the control treatment is on average in accordance with the
theoretical model. As a result, the uncorrelated costs treatment provides us with a benchmark
from which to compare behaviour in the positively correlated costs treatment. An analysis of
time trends shows that choices in the uncorrelated costs treatment evolve over time towards
the equilibrium prediction. However, in the positively correlated costs treatment, subjects
submit flatter supply functions with higher intercepts than the equilibrium prediction and
we do not find that behaviour converges towards the equilibrium value during the rounds of
play of our experiment. In addition, differences in bidding behaviour and outcomes between
treatments are less differentiated than the theoretical model predicts and are not statistically
significant.
We interpret this evidence to indicate that subjects in our experiment ignore the adverse
selection effects of the correlation among costs. This finding is consistent with the broad experimental literature of the winner’s curse in common value single unit auctions (e.g. Kagel
and Levin (2010)), the winner’s curse in other strategic settings (e.g. Carrillo and Palfrey
(2011)), and the literature of correlation neglect (e.g. Eyster and Weizsäcker (2010)). Our
experiment is the first to find evidence of the winner’s curse in an environment with interdependent values and multiple units being auctioned, a phenomenon named as the generalised
winner’s curse by Ausubel et al. (2014).
The paper provides a behavioural explanation of the results of the experiment, which
comprises two parts. The first part analyses initial play and the second part the evolution of
behaviour over time. The combination of these two allows us to give provide an explanation
of the observed behaviour. The analysis of initial play shows that the median supply function is consistent with the equilibrium prediction in the uncorrelated costs treatment, and
that a substantial proportion of choices are in accordance with the equilibrium prediction.
In the positively correlated costs treatment, initial behaviour differs substantially from the
equilibrium and there are 10% of choices of subjects that are “strategically sophisticated”,
who understand the adverse selection effects of the correlation among costs. As a result, it
is difficult for unsophisticated subjects to learn from (few) sophisticated rivals. The initial
play influences the evolution of choices over time, which is different in the two treatments.
We find that a combination of reinforcement, imitation and best response learning explains
the behaviour of the uncorrelated costs treatment. However, in the positively correlated
costs treatment, best response learning is not a good descriptor of the evolution of average
behaviour, while imitation and reinforcement can partially explain behaviour. These facts
can explain why in the positively correlated costs treatment behaviour and outcomes do not
evolve towards the equilibrium prediction in the positively correlated costs treatment.
35
The mitigation of market power has been the concern of regulators in electricity markets35 . When costs are positively correlated, our experiment shows that competition in
supply functions with boundedly rational subjects and (few) sophisticated subjects leads to
more competitive market outcomes than predicted by the Bayesian Nash equilibrium concept.
Future work could explore how professional subjects behave in our environment in order to
assess the external validity and consistency of our results with more experienced subjects.
35
See, for example, Hortaçsu and Puller (2008).
36
Appendix A. Theoretical Equilibrium in our Experimental
Design
This section uses results from Vives (2010, 2011). When demand is inelastic and equal q and
2
−1
if n−1
< ρ < 1, σσ2 < ∞ and λ > 0, the model described in Section 3 has a unique linear
θ
Supply Function Equilibrium (SFE) given by
X(si , p) = b − asi + cp,
(4)
where
a=
1
b=
1+M
(1 − ρ)σθ2
(d + λ)−1 ,
(1 − ρ)σθ2 + σ2
(5)
qM
σ2 θ̄(d + λ)−1
−
,
n
(1 + (n − 1)ρ)σθ2 + σε2
!
(6)
and
c=
n−2−M
,
λ(n − 1)(1 + M )
(7)
2
ρnσ
where M = (1−ρ)((1+(n−1)ρ)σ
2 +σ 2 ) represents an index of adverse selection and d =
ε
θ
the slope of the inverse residual demand.
The ex-ante expected market price is equal to
1
(n−1)c
is
(d + λ)q
,
(8)
n
and expected profits of seller i at the SFE given the predicted values with full information
are equal to
E(p) = p̄ = θ̄ +
(1 − ρ)2 (n − 1)σθ4
1
λ q2
,
E[πi | t] = E[π̃(t; d)] = (d + )
+
2
2 n2 n(σε2 + σθ (1 − ρ)) (d + λ)2
!
(9)
where t=(E [θ1 | s] , E [θ2 | s] , ..., E [θn | s]).
In our experiment, given the experimental parameters of Table 2, the numerical equilibrium supply function and expected outcomes in the two treatments can be summarised in
the table below.
37
Table 11: Numerical equilibrium supply function & outcomes given the experimental parameters.
The next table summarises the statistics of the distribution of sellers’ costs and errors in
the signals which have been drawn from the normal distribution as described in Section 3.
Table 12: Statistical Distribution of Random Costs and Errors used in the Experiment.
Note: Subscripts 1, 2 and 3 correspond to Seller 1, Seller 2 and Seller 3, respectively.
Appendix B. Instructions of the Experiment.
These instructions are for the treatment with positively correlated costs and have been translated from Spanish (except from figures).
38
INSTRUCTIONS
You are about to participate in an economic experiment. Your profit depends on your
decisions and on the decisions of other participants. Read the instructions carefully. You can
click on the links at the bottom of each page to move forward or backward. Before starting
the experiment, we will give a summary of the instructions and there will be two trial rounds.
THE EXPERIMENT
You will earn 10 Euros for participating in the experiment regardless of your performance in
the game. You will gain (lose) points during the experiment. At the end of the experiment,
points are exchanged for euros. 10,000 points are equivalent to 1 Euro. Each player will
start with an initial capital of 50,000 points. Gains (losses) that you accumulate during the
experiment will be added (subtracted) to the initial capital. Players who have accumulated
losses at the end of the experiment will receive 10 Euros for participating. Players with gains
will receive their gains converted to Euros plus the 10 Euro participation fee.
The experiment will last 25 rounds. In the experiment you will participate in a market.
You will be a seller of a fictitious good. Each market will have 3 sellers. Market participants
will change randomly from round to round. At any given time, no one knows who he is
matched with. We guarantee anonymity. The buying decisions will be made by the computer
and not a participant of the experiment. In each round and market, the computer will buy
exactly 100 units of the good.
YOUR PROFIT
In each round, your profit is calculated as shown in the figure below:
Your profit is equal to the income you receive from selling units minus total costs (consisting of production and transaction costs).
39
Some details to keep in mind: You only pay the total costs of the units that you sell. If you
sell zero units in a round, your profit will also be zero in this round. You can make losses when
your income is less than the total costs (production and transaction). The cumulative profit
is the sum of the profit (loss) on each round. Losses will be deducted from the accumulated
profit. Throughout the experiment, a window in the upper left corner of your screen will
show the current round and accumulated profit.
YOUR DECISION
In each round, you have to decide the minimum price that you are willing to sell each unit
for. We call these Ask Prices.
THE MARKET PRICE
Once the three sellers in a given market have entered and confirmed their decisions, the
computer calculates the market price as follows.
1. In each market, the computer observes the 300 Ask Prices introduced by the sellers of
your market.
2. The computer ranks the 300 Ask Prices from the lowest to the highest.
3. The computer starts buying the cheapest unit at the lowest Ask Price, then it buys
the next unit, etc. until it has purchased exactly 100 units. At this time the computer stops.
4. The Ask Price of the 100th unit purchased by the computer is the market price (the
price of the last unit purchased by the computer).
The market price is the same for all units sold in a market. In other words, a seller
receives a payment, which is equal to the market price for each unit he sells. If more than
one unit is offered at the market price, the computer calculates the difference:
Units Remaining= 100- Units that are offered at prices below the market price.
The Units Remaining are then split proportionally among the sellers that have offered
them at an Ask Price equal to the market price.
UNITS SOLD
In each round and market, the three sellers offer a total of 300 units. The computer purchases
the 100 cheapest units. Each seller sells those units that are offered at lower Ask Prices than
the market price. Note that those units that are offered at higher Ask Prices than the market
price are not sold. Those units offered at an Ask Price which is equal to the market price
will be divided proportionally among the sellers that have offered them.
40
MARKET RULES
In each round and market, the computer buys exactly 100 units of the good at a price not
exceeding 3600. In order to simplify the task of entering all Ask Prices in each round, we
request that you to enter:
• Ask Price for Unit 1
• Ask Price for Unit 2
Ask Prices can be different for different units. To find Ask Prices for the other units, we will
join the Ask Price for Unit 1 and the Ask Price for Unit 2 by a straight line. In this way,
we find the Ask Prices for all the 100 units. In the experiment, you will be able to see this
graphically and try different values until you are satisfied with your decision.
We apply the following five market rules.
1. You must offer all the 100 units for sale.
2. Your Ask Price for one unit must always be greater than or equal to the Ask Price of
the previous unit. Therefore, the Ask Price for the second unit cannot be less than the Ask
Price for the first unit. You can only enter integers for your decisions.
3. Both Ask Prices must be zero or positive.
4. The buyer will not purchase any unit at a price above the price cap of 3600.
5. The Ask Price for some units may be lower your unit cost, since unit costs are unknown
at the time when you decide the Ask Prices. You may have losses.
EXAMPLE
This example is illustrative and irrelevant to the experiment itself. We give the example on
paper. Here you can see how the computer determines the market price and units sold by
each seller in a market.
UNIT COST
In each round the unit cost is random and unknown to you at the time of the decision. The
unit cost is independent of previous and future round. Your unit cost is different from the
unit cost of other participants. However, your unit cost is related to the unit costs of the
other market participants. Below we explain how unit costs are related and we give a figure
and explanation of the possible values of unit costs and their associated frequencies. This
figure is the same for all sellers and all round.
41
The horizontal displays the unit cost while the vertical axis shows the frequency with
which each unit cost occurs (probability). This frequency is indicated by the length of the
corresponding bar.
In the figure you can see that the most frequent unit cost is 1000. We obtain 1000 as unit
cost with a frequency of 0.35%. In general terms, we would obtain a unit cost of 1000 in 35
of 1000 cases.
In 50 % of the cases (50 of 100 cases), the unit cost will be between 933 and 1067.
In 75 % of the cases (75 of 100 cases), the unit cost will be between 885 and 1,115 .
In 95 % of the cases (95 of 100 cases), the unit cost will be between 804 and 1,196 .
There is a very small chance that the unit cost is less than 700. This can occur in 1 of
1,000 cases approximately. Similarly, there is a very small chance that the unit cost is greater
than 1,300. This occurs can occur in 1 of 1,000 cases, approximately.
For participants with knowledge of statistics: The unit cost is normally distributed with
mean 1,000 and standard deviation 100.
INFORMATION ABOUT YOUR UNIT COST (YOUR SIGNAL)
In each round, each participant receives information on his unit costs. This information is
not fully precise. The signal that you receive is equal to:
Signal = U nitCost + Error
The error is independent of your unit cost, it is also independent from the unit costs of
other participants and it is independent from past and future errors. The following figure
describes the possible values of the error term and an indication of how likely each error is
likely to occur. This graph is the same for all sellers and rounds.
42
On the horizontal axis you can observe the possible values of the error terms. On the
vertical axis, you can observe the frequency with which each error occurs (probability). This
frequency is indicated by the length of the corresponding bar.
In the figure you can see that the most common error is 0. The frequency of error 0 is
0.66 %. In general terms, this means that in approximately 66 of 10,000 cases you would get
an error equal to 0.
In 50 % of the cases (50 of 100 cases), the error term is between -40 and 40.
In 75 % of the cases (75 of 100 cases), the error is between -69 and 69.
In 95 % of the cases (95 of 100 cases), the error is between -118 and 118.
There is a very small chance that the error is less than -200. This occurs in 4 out of
10,000 cases. Similarly, there is a very small probability that the error is greater than 200.
This occurs in 4 out of 10,000 cases.
For participants with knowledge of statistics: The error has a normal distribution with
mean 0 and standard deviation 60.
HOW YOUR COST IS RELATED TO THE COSTS OF THE OTHER SELLERS
The unit cost is different for each seller and your unit cost is related to the unit cost of the
other sellers in your market. The association between your unit cost and unit cost of another
seller in your market follows the trend:
• The higher your unit cost, the higher will be the unit cost of the other sellers.
• The lower your unit cost, the lower the unit cost of the other sellers.
The strength of the association between your unit cost and unit cost of another seller is
measured on a 0 to 1 scale. The strength of the association between your unit cost and unit
cost of the other seller is +0.6 .
43
Graphically we can see the relationship between your unit cost (horizontal axis) and the
unit cost of another seller (vertical axis) for some strengths of association. The figure that
has a red frame corresponds to an intensity of association of +0.6.
For participants with knowledge of statistics: The correlation between your unit cost and
unit cost of any other player is +0.6 .
END OF ROUND FEEDBACK
At the end of each round, we will give you information about:
• Your profit (loss) and its components (Revenue-Cost of Production - Cost of transaction)
• Market price
• Your units sold
• Other market participants feedback: decisions; profits and unit costs.
You can also check your historical performance in a window in the upper right corner of your
screen. During the experiment the computer performs mathematical operations to calculate
the market price, units sold, Ask Prices for intermediate units, etc. For these calculation we
use all available decimals. However, we show all the variables rounded to whole numbers,
except from the market price.
THE END
This brings us to the end of the instructions. You can take your time to re-read the instructions by pressing the BACK button. When you understand the instructions you can
44
indicate it to us by pressing the OK button at the bottom of the screen. Next you have
to answer a questionnaire about the instructions, unit cost distributions and signals. When
all participants have taken the questionnaire and indicated OK, we will start the practice
rounds. Your profits or losses of the practice rounds will not be added or subtracted to your
earnings during the experiment.
Appendix C. Screenshots of the Experiment
Screen 1: Signal Screen
45
Screen 2: Decision Screen
Screen 3: Feedback about a seller’s own performance
46
Screen 4: Feedback about market performance and other sellers’ in
the same market
Appendix D. End of experiment questionnaire
After the rounds were completed, we asked for 3 demographic questions: Age, gender and
degree studying.
We then asked the following additional questions regarding understanding of the game.
1. Do you think that a high market price generally means GOOD/MIXED/BAD news about
the level of your costs?
2. Explain your answer.
3. Do you think that the other sellers have answered the same as you to the previous question?
4. Explain your answer.
47
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