Tacit collusion in price-setting duopoly markets: experimental evidence with complements and substitutes.
Anderson, Lisa R. ; Freeborn, Beth A. ; Holt, Charles A. 等
1. Introduction
For decades, economists have studied oligopoly behavior using
laboratory experiments. Much attention has been devoted to identifying
factors that facilitate tacit collusion. We contribute to this
literature by studying the effect of demand structure on the ability of
subjects to collude within the price-setting model. Specifically, we
consider Bertrand substitutes and Bertrand complements. (1) In the case
of substitutes, the model generates upward-sloping reaction functions in
prices. Hence, theory predicts that if one seller moves away from the
Nash solution toward the collusive outcome, the other seller has a
unilateral incentive to respond by raising price toward the collusive
outcome. Alternatively, in the case of complements, the model generates
downward-sloping reaction functions. So a unilateral deviation from the
Nash solution toward the collusive outcome will provide a unilateral
incentive for the other seller to adjust price in the opposite
direction. Based on the slopes of the reaction functions, it is
reasonable to expect that sellers of substitute goods might find it
easier to collude tacitly than do sellers of complement goods (Holt
1995). This argument is not entirely compelling because the incentives
are in terms of myopic best responses to past decisions of the other
seller.
Moreover, the idea is somewhat at odds with economic intuition
because sellers offering competing (substitute) products could
reasonably be expected to engage in aggressive price-slashing behavior.
In contrast, sellers of complementary goods might view the other person
as more of a partner than a rival, thus fostering cooperation.
Two related articles suggest that subjects in experiments find it
easier to tacitly collude when market structure generates upward-sloping
reaction functions. In a recent survey, Suetens and Potters (2007)
compare results from a series of Bertrand and Cournot experiments and
conclude that subjects colluded more when the decision task was choosing
price versus choosing quantity. One possible explanation for this
finding is that reaction functions are upward sloping in Bertrand
(substitutes) games and downward sloping in Cournot games. However, the
different market choice variables (price versus quantity) cannot be
ruled out as an explanation for the observed differences in collusion.
As a follow up to this survey, Potters and Suetens (2008) conducted
experiments with no market framing. They included treatments with
upward-sloping and downward-sloping reaction functions. Consistent with
the survey in their previous work, they conclude that there is more
collusion with upward-sloping reaction functions in their experiment
without market framing.
Many market experiments have focused on identifying other
conditions that are favorable to seller collusion. Engel (2007)
organizes the results from this vast literature in a meta-analysis that
covers 107 articles. These studies span a wide range of experimental
design features, including the number of firms per market, whether or
not subjects have multiple interactions with the same rival, whether or
not subjects face capacity constraints, whether or not firms offer more
than one product, the degree of product differentiation, and the amount
of information subjects receive about rivals' decisions and
earnings. (2) Our design differs from all of the previous studies on
collusion in the sense that we focus on the effect of market structure
(substitutes versus complements) within the price-setting model. Hence,
we eliminate any differences in behavior that might result from choosing
quantity rather than price. Further, we include market framing in our
experiments because tacit collusion is generally a market phenomenon.
Our experimental design is described in detail in the next section,
section 3 presents our results, and section 4 concludes.
2. Experimental Design
We recruited 128 subjects from undergraduate classes at the College
of William and Mary. Subjects participated in a repeated symmetric
duopoly price-setting game in either a complements treatment or a
substitutes treatment. (3) Table 1 summarizes the equilibrium values
derived from the experimental parameters. The complements design is
based on the following demand curve: [Q.sub.1] = 3.60 - 0.5[P.sub.1] -
0.5[P.sub.2], where [Q.sub.1] represents the quantity sold by firm 1,
[P.sub.l] represents the price set by firm 1, and [P.sub.2] represents
the price set by firm 2. The Nash equilibrium price is $2.40 in this
treatment. The substitutes design is based on the following demand
curve: [Q.sub.1] = 3.60 - 2[P.sub.1] + [P.sub.2], and the Nash price is
$1.20. In both designs, there is no marginal cost of production, and
there is a fixed cost of $2.18 per round. With this fixed cost, earnings
from collusion are 50% higher than earnings at the Nash equilibrium.
Subjects earn $0.70 per person at the Nash equilibrium, and they earn
$1.06 per person at the collusive outcome. Another important feature of
this set of parameters is that the difference between the collusive
price and Nash price is the same ($0.60) in both designs. In addition,
the collusive price is the same for both designs and is $1.80. (4)
Figure 1 presents the best-response functions for the two treatments.
Notice that the collusive (joint profit maximizing) price is below the
Nash price in the complements case on the left side, and it is above the
Nash price in the substitutes case, shown on the right.
The Appendix contains instructions for the experiment. Subjects
selected prices, and pairs were able to go at their own pace. Half of
the subjects in each treatment interacted for 10 rounds, and half
interacted for 20 rounds. (5) To avoid end-game effects, subjects were
not told the number of rounds in advance. (6) Subjects were told that
they were matched with the same person for each round. In addition,
subjects were told the equation for demand, and it was common knowledge
that all subjects within a session faced the same demand curve and
costs. Finally, at the end of each round, subjects were told the price
charged by the other seller. Average earnings were $6.42 in the sessions
with 10 rounds and $10.74 in the sessions with 20 rounds. Earnings also
varied considerably based on the treatment, as described below.
[FIGURE 1 OMITTED]
[FIGURE 2 OMITTED]
3. Results
Analysis of Price Levels
Figure 2 shows the average price per round for both treatments
separated by 10 and 20 round sessions. In the complements treatment, the
average price starts between the collusive price and the Nash price. The
average price rises and falls over time but generally climbs closer to
the Nash price with repetition. It oscillates around the Nash price
after 13 rounds of play. In the substitutes treatment, the average price
also rises and falls over time but is generally below the Nash
prediction. Notice that prices in the 10 round sessions appear to be
slightly closer to the collusive price than prices in the 20 round
sessions for both complements and substitutes. However, because subjects
did not know the number of rounds they would play, there is no
theoretical reason to believe that behavior would differ across those
sessions. Hence, for much of the analysis that follows, we pool data for
the first 10 rounds of play. Over all rounds, the average price is $1.12
in the substitutes treatment and $2.25 in the complements treatment.
This difference in pricing behavior resulted in average earnings per
subject of $0.74 per round in the complements treatment compared to
$0.44 per round in the substitutes treatment. (7)
Overall, Figure 2 suggests that the average price in the
complements treatment is closer to the collusive price than the average
price in the substitutes treatment. To further investigate the amount of
collusion across the two treatments, we define the "collusive
region" as the range of prices within $0.30 of the collusive price.
(8) Next, we identify matched pairs of subjects who priced in this
region. We focus on pairs of subjects rather than individuals who priced
cooperatively because collusion in a duopoly setting is only relevant
and more likely to be sustained when both players choose the cooperative
outcome. Over all 20 rounds, the percentage of pairs in the collusive
region is 20% for complements and 5% for substitutes. (9)
Because subjects were paired for all rounds of the experiment, we
can also examine how well pairs of subjects were able to sustain
cooperation. When we consider a relatively strict definition of
sustained cooperation as pricing in the cooperative region for at least
70% of the rounds played, 3 of the 32 pairs in the complements treatment
and none of the 32 pairs in the substitutes treatment were able to
sustain cooperation. When we consider a very liberal definition of
sustained cooperation as maintaining prices in the cooperative region
for at least 30% of the rounds played, 7 of the 32 pairs in the
complements treatment and 3 of the 32 pairs in the substitutes treatment
were able to sustain cooperation. (10) For each pair of subjects, we
also calculate the percentage of rounds they priced in the collusive
region. On average, pairs of subjects in the complements treatment
priced in the collusive region in 23% of the rounds played. Pairs of
subjects in the substitutes treatment priced in the collusive region in
only 5% of the rounds played. Thus, subjects in the complements
treatment were significantly more likely to price in this region than
subjects in the substitutes treatment. (11)
Within this collusive region, subjects in the complements treatment
appear to price closer to the collusive price of $1.80 than subjects in
the substitutes treatment. Figure 3 shows average prices by round for
the pairs of subjects who priced in the collusive range. The average
price in the collusive region is $1.78 for complements and $1.64 for
substitutes. For each pair that priced in the collusive region, we
calculate the difference between the average price of the pair and the
collusive price. Averaging over all rounds they were in the collusive
region, pairs of subjects in the substitutes treatment deviated from the
collusive price by $0.15, and pairs of subjects in the complements
treatment deviated by only $0.03. The average price deviations in the
substitutes and complements treatments are statistically different from
one another at the 5% level. (12)
[FIGURE 3 OMITTED]
Turning our attention to competitive behavior, we define the
"Nash region" to be prices within $0.30 of the Nash price.
There are also large differences in the proportion of pairs who priced
in this region, even in early rounds of play. Over 70% of pairs in the
substitutes treatment priced in the Nash region in the first round of
decision making compared to only 22% in the complements treatment. In
every round, more pairs priced in the Nash region for the substitutes
than the complements treatment. Looking at average pricing behavior over
all rounds, 79% of the pairs of subjects in the substitutes treatment
priced in the Nash region, while only 44% of the pairs in the
complements treatment priced in the Nash region. Finally, we also
calculate the percentage of rounds each pair of subjects priced in the
Nash region. On average, pairs of subjects in the complements treatment
priced in the Nash region in 38% of the rounds played, while pairs of
subjects in the substitutes treatment priced in the Nash region in 77%
of the rounds played. Thus, subjects in the substitutes treatment were
significantly more likely to price in this region than subjects in the
complements treatment. (13)
As an additional check on the way in which behavior differs across
the two treatments, we calculate a standard measure of collusiveness for
each duopoly pair: [rho] = ([P.sub.actual] - [P.sub.Nash])]
([P.sub.collude] - [P.sub.Nash]). Note that positive values of p
indicate collusive behavior, zero indicates pricing at the Nash
prediction, and negative values indicate supracompetitive pricing. Using
the average collusiveness measure for each subject pair as the unit of
observation, we obtain 32 observations for both the complements and
substitutes treatments. We find [rho] = 0.24 in the complements
treatment, and [rho] = - 0.13 in the substitutes treatment. These values
are significantly different from each other at the 1% level. (14)
Although subjects do not know the number of rounds, it is possible
that the degree of collusiveness decreases as subjects have the
opportunity to learn. In fact, the average prices in Figure 2 appear to
be approaching the Nash equilibrium toward the end of play. Table 2
presents the average degree of collusiveness over all subject pairs
calculated for all rounds played and for early and late rounds of the
experiment. The final column of Table 2 displays the t-test for equality
of [[rho].sub.substitutes] and [[rho].sub.complements]. Note that there
is significantly more collusive behavior (as indicated by the t-tests)
in the complements treatment than in the substitutes treatment in four
of the five comparisons presented in Table 2. The only comparison
without significant differences between the two treatments is the one in
which we only consider the final five rounds of the 20 round sessions.
(15)
Analysis of Price Dynamics
Thus far, the analysis has focused on price levels, but pricing
dynamics also reveal differences in tacit collusion across substitutes
and complements. Here we examine how subjects respond to prices set by
their partner in the previous round. Looking at rounds 2 through 10 (or
20), for each subject, we calculate the Nash best response to the
partner's price in the previous round. Next, we compare the price
actually chosen in a round to the best-response price for that round.
For substitutes, if a subject prices higher than the best-response price
in any given round, that price choice can be classified as
"cooperation inducing." For complements, if a subject prices
lower than the best-response price, that price choice can be classified
as cooperation inducing. For each subject, we calculate the percentage
of total prices chosen in rounds 2 through 10 (or 20) that were
cooperation inducing. Figure 4 shows the distribution of
cooperation-inducing prices by treatment. The pair of bars on the far
left side of Figure 4 shows subjects who made cooperation-inducing price
choices in less than 10% of the rounds played. Notice that only one
subject in the complements treatment fell into this category, while 10
subjects in the substitutes treatment were in this category. At the
other extreme, 12 subjects in the complements treatment made
cooperation-inducing moves in every round, while only four subjects in
the substitutes treatment did so. Overall, Figure 4 shows that subjects
in the complements treatment choose cooperation-inducing prices more
often than subjects in the substitutes treatment.
[FIGURE 4 OMITTED]
We also use Nash best-response prices to identify collusive pairs.
Again looking at rounds 2 through 10 (or 20), for each subject, we
calculate the Nash best response to the partner's price in the
previous round. Next, we calculate the deviation from the Nash price by
subtracting the best response price for the actual price chosen in that
round. In the substitutes treatment, a positive value for the deviation
is indicative of cooperation-inducing behavior. In the complements
treatment, a negative value for the deviation is indicative of
cooperation-inducing behavior. For each player, we calculate the average
deviation over all rounds played (not including round 1), and the
resulting number gives a measure of how collusively that person priced.
As a benchmark, in the complements treatment, if a pair of subjects
chooses the joint profit maximizing price each round, they will each
have a deviation of -$0.90 per round. In the substitutes treatment, if a
pair of subjects chooses the joint profit maximizing price each round,
they will each have a deviation of $0.45 per round. (16) By definition,
Nash behavior in both treatments results in a deviation of $0 per round.
For each pair of subjects, we plot their average deviations per round in
the four-quadrant diagrams shown in Figure 5. The joint profit
maximizing pairs of deviations are also plotted and labeled in the
figure. Notice that more than half of the points in the complements
graph fall into the lower left-hand quadrant, where both subjects had
negative average deviations. This indicates that a majority of pairs
chose cooperation-inducing prices in the complements treatment. However,
only two pairs of subjects came close to the joint profit maximizing
point of (-$0.90, -$0.90). In the substitutes treatment, very few pairs
priced in the collusive (upper right-hand) region on the graph. The
majority of pairs were clustered high in the supracompetitive (lower
left) region.
[FIGURE 5 OMITTED]
We next use Nash best-response prices to analyze behavior in the
context of a learning direction theory (see Selten and Buchta [1998] and
Capra et. al [1999] for more details about this model). Using the
individual's best response to the other person's price in the
previous round, we categorize each price change according to whether it
was a movement toward or away from the best response. In both
treatments, the number of movements toward the best response was not
significantly different from the number of movements away from the best
response (p = 0.19 for complements and p = 0.61 for substitutes). (17)
Finally, we analyze the dynamics of individual price-setting
behavior using econometric methods. To identify whether players mimic
the price changes of their partners, we regress the change in each
seller's price in round t on the other seller's change in
price in round t - 1. The lag in the other's price is necessary
because people do not observe others' prices until after the round
has ended. Table 3 displays the results from estimating models of
[DELTA][price.sub.it] = [[beta].sub.0] + [[beta].sub.1]
[DELTA][price.sub.jt-1] + [[epsilon].sub.it]. All models cluster
standard errors at the pair level. We present models that control for
individual heterogeneity using fixed effects (model 1) or random effects
(model 2). We perform a Hausman test, which checks the more efficient
model (random effects) against the less efficient but consistent model
(individual fixed effects). For both substitutes and complements, we
cannot reject the null that the coefficients estimated by the
random-effects estimator are the same as the ones estimated by the
fixed-effects estimator. (18) In all models, the effect of the other
player's change in price is positive, suggesting that a price
change by one player is generally followed by a move in the same
direction by the other player, regardless of treatment. Note that the
coefficient on the other's lagged price is larger in the
substitutes treatment than in the complements treatment. Also, the
coefficient in the substitutes treatment is significantly different from
0 at the 1% level, while the coefficient in the complements treatment is
significant at the 10% level. Potters and Suetens (2008) find similar
results and suggest that reciprocal behavior in the substitutes
treatment may be explained by the slope of the reaction functions.
Specifically, with upward-sloping reaction functions, players should
adjust prices in the same direction as a change by the other player.
Conversely, in the complements treatment, reciprocal behavior cannot be
explained by the slope of the reaction functions. With downward-sloping
reaction functions, players should adjust prices in the opposite
direction of a change by the other player.
3. Discussion
We compare collusive behavior in Bertrand duopoly experiments with
substitute goods versus complementary goods. We find moderate tacit
collusion with complementary goods but no systematic tacit collusion
with substitute goods. These results, combined with two recent related
studies, suggest that market structure (that is, slope of the reaction
function) is not the only determinant of collusive behavior in these
experiments. Suetens and Potters (2007) compared measures of collusion
in Bertrand and Cournot games from five separate experimental studies.
All of the studies included Bertrand games and Cournot games and only
modeled competition for substitute goods. Overall, Suetens and Potters
(2007) report some evidence of tacit collusion in Bertrand markets, but
no such evidence is seen in Cournot markets. This difference in results
might be explained by the way that the problem was framed (price choice
vs. quantity choice) or by the demand structure (the Bertrand games had
upward-sloping reaction functions, and the Cournot games had
downward-sloping reaction functions).
We can further explore this by comparing our results for Bertrand
complements to the Cournot results that are reviewed in Suetens and
Potters (2007). Mathematically, our Bertrand complements problem is
identical to a Cournot substitutes game in the sense that reaction
functions are downward sloping, and the Nash equilibrium level of the
choice variable is greater than the collusive level. All five of the
Cournot studies report subjects choosing quantities that were higher
than the Cournot Nash levels. In contrast, we find a moderate degree of
collusion with a similar demand structure but with a price choice,
rather than a quantity choice, problem. This suggests that framing might
be an important determinant of tacit collusion. (19)
To further explore the effect of framing, our results can be
compared to the context-free experiments presented in Potters and
Suetens (2008). They studied four experimental treatments that varied in
terms of the slope of the reaction functions and the relative positions
of the Nash prediction and the collusive solution in the strategy space.
Two treatments had upward-sloping reaction functions, and two treatments
had downward-sloping reaction functions. Each of those pairs had a
treatment with the Nash prediction above the collusive solution and a
treatment with the Nash below the collusive outcome.
The subjects in Potters and Suetens (2008) were Dutch college
students. They played in fixed pairs and were told there would be 30
rounds. Further, at the end of each round, they were given all
information about their opponent's choices and earnings. Unlike the
previous work in this area, the experiment was not framed in a market
context. Subjects did not choose prices or quantities; rather, they
picked a number between 0 and 28. The Nash choice was 14 (the midpoint)
across all treatments, and the collusive choice was either 2.5 or 25.5
depending on the treatment. Potters and Suetens (2008) report
subjects' choices were generally between the Nash prediction and
the collusive outcome in all four treatments, regardless of whether the
Nash was above or below the collusive outcome. They calculate the degree
of collusiveness for all four treatments and conclude that the slope of
the reaction function was an important determinant in the degree of
collusion, but behavior did not vary significantly depending on whether
or not the collusive outcome was higher or lower than the Nash
prediction.
The treatment used by Potters and Suetens (2008) with
downward-sloping reaction functions and the collusive choice lower than
the Nash choice is comparable to our complements treatment. They report
[rho] = 0.24 in this case, which is identical to our finding. Potters
and Suetens (2008) find the highest degree of collusion ([rho] = 0.42)
in their treatment with upward-sloping reaction functions and the
collusive choice greater than the Nash choice. We find very different
results in our comparable (substitutes) treatment ([rho] = -0.13). It is
possible that framing the choice problem as a market interaction tends
to depress cooperative behavior when sellers view themselves as
adversaries. Indeed, in a meta-analysis of oligopoly experiments, Engel
(2007) notes that there is more collusion in experiments with a neutral
frame relative to a market frame. This is also consistent with research
from ultimatum game experiments. Hoffman et al. (1994) found that offers
were closer to the Nash prediction when the game was framed as a market
interaction as opposed to a bargaining interaction.
4. Conclusion
In summary, we find no collusion in price-setting markets with
substitute goods and a moderate amount of tacit collusion in
price-setting markets with complements. In both treatments, prices move
closer to the Nash prediction with repetition, and in general, prices
are closer to the Nash than to the collusive price. Our results are in
contrast with some previous studies and suggest that framing the problem
in a market context might affect behavior in important ways. Comparing
our results to Potters and Suetens (2008), we provide a market framework
and find less cooperation in our experiments with substitute goods. An
obvious direction for future research is to extend this study to
consider a quantity-choice problem with substitute and complement goods.
Appendix 1: Instructions for Complements Treatment
(Reprinted with permission from vecon.econ.virginia.edu/admin.php.)
Page 1
Rounds and Matchings: The experiment sets up markets that are open
for a number of rounds. Note: You will be matched with the same person
in all rounds.
Interdependence: The decisions that you and the other person make
will determine your earnings.
Price Decisions: Both you and the other person are sellers in the
same market, and you will begin by choosing a price. You cannot see the
other's price while choosing yours, and vice versa.
Sales Quantity: A lower price will tend to increase your sales
quantity, and a higher price charged by the other seller will tend to
lower your sales quantity. This is because consumers use your product
together with the other's product, so an increase in their price
will reduce your sales.
Page 2
Price and Sales Quantity: Your price decision must be between (and
including) $1.50 and $3.00; use a decimal point to separate dollars from
cents.
Production Cost: Your cost is $0.00 for each unit that you sell.
However, you must pay a fixed cost of $2.18 for a license to operate,
regardless of your sales quantity. So your total cost is $2.18,
regardless of how many or few units you produce.
Consumer Demand: The quantity that consumers purchase depends on
all prices. Your sales quantity will be determined by your price (P) and
by the other seller's price (A): Sales Quantity = 3.60 - 0.50*P -
0.50*A. Negative quantities are not allowed, so your sales quantity will
be 0 if the formula yields a negative quantity.
Sales Revenue: Your sales revenue is calculated by multiplying your
production quantity and the price. Since your sales are affected by the
other's price, you will not know your sales revenue until market
results are available at the end of the period.
Page 3
Earnings: Your profit or earnings for a round is the difference
between your sales revenue and your production cost. If Q is the
quantity you sell, then total revenue is (Q*price), total cost is $0.00
+ fixed cost of 2.18, so earnings = Q*(price) - $2.18.
Cumulative Earnings: The program will keep track of your total
(cumulative) earnings. Positive earnings in a round will be added, and
negative earnings will be subtracted. Working Capital: Each of you will
be given an initial amount of money, $0.00, so that gains will be added
to this amount, and losses will be subtracted from it. This initial
working capital will show up in your cumulative earnings at the start of
round 1, and it will be the same for everyone. There will he no
subsequent augmentation of this amount.
Page 4
In the following examples, please use the mouse button to select
the best answer. Remember, your sales quantity = 3.60 - 0.50*Price -
0.50*(Other Price).
Question 1: Suppose that both sellers choose equal prices and that
the total sales for both sellers combined is Q units, then each seller
has a sales quantity of:
(a) 2Q
(b) Q/2.
Question 2: A higher price will increase both the price-cost margin
and the chance of having a positive sales quantity.(True/False)
(a) True.
(b) False.
Page 5
Question 1: Suppose that both sellers choose equal prices and that
the total sales for both sellers combined is Q units, then each seller
has a sales quantity of:
(a) 2Q
(b) Q/2
Your answer, Ca), is Correct. The sales quantity formula divides
sales equally when prices are equal.
Question 2: A higher price will increase both the price-cost margin
and the chance of having a positive sales quantity.(True/False)
(a) True.
(b) False.
Your answer, (b), is Correct. The chances of making sales go down
as price is increased.
Page 6
Matchings: Please remember that you will be matched with the same
person in all rounds.
Earnings: All people will begin a round by choosing a number or
"price" between and including $1.50 and $3.00. Remember, your
sales quantity = 3.60 - 0.50*Price + - 0.50*(Other Price). Your total
cost is $0.00 times your sales quantity, plus your fixed cost $2.18, and
your total sales revenue is the price times your sales quantity. Your
earnings are your total revenue minus your total cost. Positive earnings
are added to your cumulative earnings, and losses are subtracted.
Rounds: There will be a number of rounds, and you are matched with
the same person in all rounds.
Appendix 2: Instructions for Substitutes Treatment
(copied from veconlab.econ.virginia.edu/admin.htm)
Page 1
Rounds and Matchings: The experiment sets up markets that are open
for a number of rounds. Note: You will be matched with the same person
in all rounds.
Interdependence: The decisions that you and the other person make
will determine your earnings.
Price Decisions: Both you and the other person are sellers in the
same market, and you will begin by choosing a price. You cannot see the
other's price while choosing yours, and vice versa.
Sales Quantity: A lower price will tend to increase your sales
quantity, and a higher price charged by the other seller will tend to
raise your sales quantity. This is because consumers view the products
as similar, so an increase in their price will increase your sales.
Page 2
Price and Sales Quantity: Your price decision must be between (and
including) $0.60 and $2.10; use a decimal point to separate dollars from
cents. An increase in the other seller's price will tend to raise
the number of units you sell.
Production Cost: Your cost is $0.00 for each unit that you sell.
However, you must pay a fixed cost of $2.18 for a license to operate,
regardless of your sales quantity. So your total cost is $2.18,
regardless of how many or few units you produce.
Consumer Demand: The quantity that consumers purchase depends on
all prices, with more of the sales going to the seller with the lowest
(best available) price in the market. Your sales quantity will be
determined by your price (P) and by the other seller's price (A):
Sales Quantity = 3.60 - 2.00*P + 1.00*A.
Negative quantities are not allowed, so your sales quantity will be
0 if the formula yields a negative quantity.
Sales Revenue: Your sales revenue is calculated by multiplying your
production quantity and the price. Since your sales are affected by the
other's price, you will not know your sales revenue until market
results are available at the end of the period.
Page 3
Earnings: Your profit or earnings for a round is the difference
between your sales revenue and your production cost. If Q is the
quantity you sell, then total revenue is (Q*price), total cost is $0.00
+ fixed cost of 2.18, so earnings = Q*(price) - $2.18.
Cumulative Earnings: The program will keep track of your total
(cumulative) earnings. Positive earnings in a round will be added, and
negative earnings will be subtracted.
Working Capital: Each of you will be given an initial amount of
money of $0.00, so that gains will be added to this amount, and losses
will be subtracted from it. This initial working capital will show up in
your cumulative earnings at the start of round 1, and it will be the
same for everyone. There will be no subsequent augmentation of this
amount.
Page 4
In the following examples, please use the mouse button to select
the best answer. Remember, your sales quantity = 3.60 - 2.00*Price +
1.00*(Other Price).
Question 1: Suppose that both sellers choose equal prices and that
the total sales for both sellers combined is Q units, then each seller
has a sales quantity of:
(a) 2Q
(b) Q/2.
Question 2: A higher price will increase both the price-cost margin
and the chance of having a positive sales quantity.(True/False)
(a) True.
(b) False.
Page 5
Question 1: Suppose that both sellers choose equal prices and that
the total sales for both sellers combined is Q units, then each seller
has a sales quantity of:
(a) 2Q
(b) Q/2
Your answer, (b), is Correct. The sales quantity formula divides
sales equally when prices are equal.
Question 2: A higher price will increase both the price-cost margin
and the chance of having a positive sales quantity.(True/False)
(a) True.
(b) False.
Your answer, (b), is Correct. The chances of making sales go down
as price is increased.
Page 6
Matchings: Please remember that you will be matched with the same
person in all rounds.
Price Choice: All people will begin a round by choosing a number or
"price" between and including $0.60 and $2.10.
Demand: Remember, your sales quantity = 3.60 - 2.00*Price +
1.00*(Other Price).
Cost: Your total cost is $0.00 times your sales quantity, plus your
fixed cost $2.18.
Earnings: Your earnings are your total revenue (price multiplied by
sales quantity) minus your total cost. Positive earnings are added to
your cumulative earnings, and losses are subtracted.
References
Altavilla, Carlo, Luigi Luini, and Patrizia Sbriglia. 2006. Social
learning in market games. Journal of Economic Behavior and Organization
61:632-52.
Capra, C. Monica, Jacob K. Goeree, Rosario Gomez, and Charles A.
Holt. 1999. Anomalous behavior in a traveler's dilemma. American
Economic Review 89(3):678-90.
Davis, Douglas. 2008. Do strategic substitutes make better markets?
A comparison of Bertrand and Cournot markets. Unpublished Paper,
Virginia Commonwealth University.
Dolbear, F. Trenery, Lester B. Lave, G. Bowman, A. Lieberman,
Edward C. Prescott, R. Rueter, and Roger Sherman. 1968. Collusion in
oligopoly: An experiment on the effect of numbers and information. The
Quarterly Journal of Economics 82(2):240-59.
Dufwenberg, Martin, and Uri Gneezy. 2000. Price competition and
market concentration: An experimental study. International Journal of
Industrial Organization 18:7-22.
Engel, Christoph. 2007. How much collusion: A meta-analysis of
oligopoly experiments. Journal of Competition Law and Economics
3(4):491-549.
Feinberg, Robert M., and Roger Sherman. 1988. Mutual forbearance
under experimental conditions. Southern Economic Journal 54(4):985-93.
Garcia-Gallego, Aurora. 1998. Oligopoly experimentation of learning
with simulated markets. Journal of Economic Behavior and Organization
25:333-55.
Garcia-Gallego, Aurora, and Nikolaos Georgantzls. 2001.
Multiproduct activity in an experimental differentiated oligopoly.
International Journal of Industrial Organization 19:493-518.
Hoffman, Elizabeth, Kevin McCabe, Keith Shachat, and Vernon Smith.
1994. Preferences, property rights, and anonymity in bargaining games.
Games and Economic Behavior 7(3):346-80.
Holt, Charles A. 1995. Industrial organization: A survey of
laboratory research. In Handbook of experimental economies, edited by
John Kagel and A1 Roth. Princeton, NJ: Princeton University Press, pp.
349-443.
Holt, Charles A., and Susan K. Laury. 2008. Theoretical
explanations of treatment effects in voluntary contributions
experiments. In Handbook of experimental economics results, Volume 1,
edited by Charles Plott and Vernon Smith. New York: Elsevier Press, pp.
846-55.
Huck, Steffen, Hans-Theo Normann, and Joerg Oechssler. 2000. Does
information about competitors' actions increase or decrease
competition in experimental oligopoly markets? International Journal of
Industrial Organization 18:39-57.
Potters, Jan, and Sigrid Suetens. 2008. Cooperation in experimental
games of strategic complements and substitutes. Review of Economic
Studies 76(3):1125-47.
Selten, Reinhard, and Joachim Buchta. 1998. Experimental sealed bid
first price auctions with directly observed bid functions. In Games and
human behavior, edited by D. Budescu, I. Erev, and R. Zwick.
Philadelphia, PA: Lawrence Erlbaum Associates, pp. 79-104.
Sherman, Roger. 1971. An experiment on the persistence of price
collusion. Southern Economic Journal 37(4):489-95.
Suetens, Sigrid, and Jan Potters. 2007. Bertrand colludes more than
Cournot. Experimental Economics 10:71-7.
Lisa R. Anderson, * Beth A. Freeborn, ([dagger]) and Charles A.
Holt ([double dagger])
* Department of Economics, College of William and Mary,
Williamsburg, VA 23187, USA; E-mail lisa.anderson@wm.edu; corresponding
author.
([dagger]) Department of Economics, College of William and Mary,
Williamsburg, VA 23187, USA; E-mail bafree@wm.edu.
([double dagger]) Department of Economics, University of Virginia,
Charlottesville, VA 22904, USA; E-mail holt@virginia.edu.
Financial support from the National Science Foundation (SBR
0094800) is gratefully acknowledged. We thank Sara St. Hillaire for
research assistance and three anonymous reviewers for comments.
Received July 2008; accepted January 2009.
(1) Note that we refer to goods from a consumption perspective
rather than a production perspective. Specifically, we use the term
"substitute goods" to refer to goods with a positive cross
price elasticity of demand. Some studies in this area define the
relationship between goods based on producers' reaction functions,
which can be affected by cost considerations as well as demand effects.
For example, the term "strategic substitutes" refers to
downward-sloping reaction functions, and the term "strategic
complements" refers to upward-sloping reaction functions.
(2) See, for example, Dufwenberg and Gneezy (2000) on the effect of
number of firms, Feinberg and Sherman (1988) on the effect of repeated
matchings, Sherman (1971) on the effect of capacity costs,
Garcia-Gallego and Georgantzis (2001) on multiproduct firms,
Garcia-Gallego (1998) on product differentiation, and Huck, Normann, and
Oechssler (2000) and Altavilla, Luini, and Sbriglia (2006) on the effect
of information about other's choices and earnings. A recent article
by Davis (2008) studies the effect of product differentiation on pricing
behavior and reports that Bertrand markets converge to the Nash
prediction faster than Cournot markets. Note that this finding is
somewhat inconsistent with the results reported in Potters and Suetens
(2008). Further, a high degree of product substitutability speeds the
convergence to Nash prices in Bertrand markets.
(3) Experiments were conducted using the Veconlab website developed
by Charles Holt at the University of Virginia.
(4) Suetens and Potters (2007) report a "Friedman Index"
to measure the sustainability of collusion. It is calculated as the
collusive profit minus the Nash profit (the potential gain from
colluding) divided by the maximum profit from a unilateral defection
minus the collusive profit (the potential gain from defecting on a
collusive agreement). The five studies they review have indexes ranging
from 0.32 to 1.00. The parameters in our experiment generate a Friedman
Index of 0.88 in both treatments, which is higher than three of the five
studies reviewed in Suetens and Potters (2007).
(5) This design feature was motivated by results we observed in
early sessions of the experiment. We had subjects play only 10 rounds of
the experiment, and we noticed a general upward trend in prices over all
10 rounds in the substitutes treatment and the complements treatment. We
added the 20 round sessions to determine whether or not that trend would
persist past 10 rounds.
(6) In an early experimental study of collusion in Bertrand
markets, Dolbear et al. (1968) report that subjects altered behavior in
late rounds of their pilot experiments. More recently, end-game effects
have been observed in public goods experiments. Holt and Laury (2008)
survey articles using the voluntary contributions mechanism and report
that when the number of rounds is known, cooperation declines in late
rounds of the experiment.
(7) Average per round earnings are higher in the complements
treatment than in the substitutes treatment at the 1% level (t-stat =
4.80, p = 0.0000). To test for significant differences in earnings
across the treatments, we calculated the average earnings per round for
each subject, which resulted in a total of 128 observations.
(8) We chose $0.30 as the boundary of the collusive region because
it is the midpoint between the collusive price and the Nash price.
(9) In the substitutes treatment, there are six rounds with no
pairs pricing in the collusive region (rounds 1 and 2 and 16-19). There
is at least one pair that prices in the collusive region for all rounds
of the complements treatment.
(10) There is no evidence that the number of rounds played affects
the ability of subjects to maintain collusion. The pairs of collusive
subjects were distributed roughly equally across the 10 and 20 round
sessions.
(11) The unit of observation for this t-test is the percentage of
rounds a pair of subjects priced in the collusive region (n = 64, t-star
= 3.56, p = 0.0000).
(12) The unit of observation for this t-test is the average
deviation from the collusive price made by a pair of subjects during the
rounds they priced in the collusive region (n = 64, t-stat = 2.20, p =
0.0186).
(13) The unit of observation for this t-test is the percentage of
rounds a pair of subjects priced in the Nash region (n = 64, t-stat =
5.91, p = 0.0000).
(14) The unit of observation for this t-test is the average p for a
pair of subjects (n = 64, t-star = 4.51, p = 0.0000).
(15) Table 2 also reveals that prices were more competitive in the
20 round sessions than in the 10 round sessions (as indicated by the
lower values of [rho]). For example, averaging over all rounds played in
the substitutes treatment, [rho] = - 0.100 for pairs who played 10
rounds, and [rho] = -0.161 for subjects who played 20 rounds. Since
subjects did not know how many rounds they would play, there is no
theoretical explanation for why subjects would behave differently in
these sessions.
(16) These joint profit maximizing deviations are derived from the
best-response functions as follows. In the substitutes treatment, the
best response to the joint profit maximizing price is [P.sub.A] = 0.90 +
0.25[P.sub.B] = 0.90 + 0.25 (1.80) = $1.35. The deviation between the
joint profit maximizing price and the best response is $1.80 - $1.35 =
$0.45. Similarly, in the complements treatment, the best response to the
joint profit maximizing price is [P.sub.A] = 3.60 - 0.50[P.sub.B] = 3.60
-- 0.50 (1.80) = $2.70. The deviation between the joint profit
maximizing price and the best response is $1.80 - $2.70 = -$0.90.
(17) A more complete version of these results is available from the
authors upon request.
(18) The [chi square] values for the Hausman test are 0.16 for
substitutes and 0.00 for complements.
(19) Alternatively, it might appear that there is more collusion in
the complements treatment because it is harder for subjects to determine
the Nash best response with downward-sloping reaction functions. The
results reported in Potters and Suetens (2008) contradict this
explanation; they report significantly less collusion with
downward-sloping reaction function than with upward-sloping function.
The results of the learning direction analysis described in the results
section also contradict this explanation. If it is more difficult for
subjects to figure out the Nash best response with downward-sloping
reaction functions, in early rounds of play, subjects in the complements
treatment should be less likely to make price changes in the direction
of the Nash best response than subjects in the substitutes treatment.
The learning-direction analysis shows that in round 2, more subjects
responded to their partner's round 1 price by moving in the
direction of the Nash price in the complements treatment (42%) than in
the substitutes treatment (39%). If we take an average over rounds 2
through 5, we find that the number of subjects moving in the direction
of the Nash price is identical across treatments at 52%. Thus, there is
no evidence in early rounds of play that subjects in the substitutes
treatment were more aware of the Nash best response than subjects in the
complements treatment. Over all rounds, subjects in the complements
treatment moved toward the Nash best response 57% of the time compared
to 50% for subjects in the substitutes treatment.
Table 1. Equilibrium Values
Complements Treatment
Price Quantity Profit Per Round
Nash $2.40 1.2 $0.70
Joint profit
maximizing $1.80 1.8 $1.06
Substitutes Treatment
Price Quantity Profit Per Round
Nash $1.20 2.4 $0.70
Joint profit
maximizing $1.80 1.8 $1.06
Table 2. Pair-Level Average Degree of Collusiveness
by Rounds of Play
Substitutes Complements
Mean Standard Mean Standard
Rounds of Play [rho] Deviation [rho] Deviation
Panel A: 10 Round Sessions
All 10 rounds -0.100 0.066 0.370 0.081
Last 5 rounds -0.005 0.068 0.311 0.094
Panel B: 20 Round Sessions
All 20 rounds -0.161 0.081 0.116 0.094
First 10 rounds -0.183 0.085 0.212 0.115
Last 5 rounds -0.121 0.086 0.011 0.125
t-test for
Rounds of Play Equality
Panel A: 10 Round Sessions
All 10 rounds -4.532 ***
Last 5 rounds -2.730 ***
Panel B: 20 Round Sessions
All 20 rounds -2.238 **
First 10 rounds -2.765 ***
Last 5 rounds -0.868
There are 16 pairs of subjects within each of the four groups
represented in the table (10 round complements, 10 round
substitutes, 20 round complements, and 20 round substitutes).
* p < 0.10.
** p < 0.05.
*** p < 0.01.
Table 3. Regression of Change in Individual's Price
Model 1 Model 2
Panel A: Complement goods
Lagged change in other 0.0866 * (0.048) 0.0870 * (0.048)
seller's price
Control for individual Fixed effects Random effects
Panel B: Substitute goods
Lagged change in other 0.1447 *** (0.040) 0.1484 *** (0.039)
seller's price
Control for individual Fixed effects Random effects
Standard errors are in parentheses. Because the independent
variable is lagged change in other seller's price, we cannot
use the first two rounds of prices. This results in 832
observations for each regression.
* p < 0.10.
** p < 0.05.
*** p < 0.01.
