Sharing rules in Bertrand duopolies with increasing returns

Last registered on April 06, 2022


Trial Information

General Information

Sharing rules in Bertrand duopolies with increasing returns
Initial registration date
April 05, 2022

Initial registration date is when the trial was registered.

It corresponds to when the registration was submitted to the Registry to be reviewed for publication.

First published
April 06, 2022, 1:54 PM EDT

First published corresponds to when the trial was first made public on the Registry after being reviewed.



Primary Investigator

University of Potsdam

Other Primary Investigator(s)

Additional Trial Information

In development
Start date
End date
Secondary IDs
Prior work
This trial does not extend or rely on any prior RCTs.
Despite its empirical relevance, increasing returns to scale are understudied in experimental Bertrand markets. In our setting, returns to scale increase because of avoidable fixed costs. The equilibrium in this game depends on the market's sharing rule (which describes what happens in the case of equal prices). We use Bertrand duopolies to experimentally compare the symmetric sharing rule (where both of the two players serve half of the market demand) with the winner-takes-all sharing rule (where a fair randomization device decides who serves the entire market). We expect that market prices under the winner-takes-all sharing rule are higher because it provides a collusion mechanism that does the symmetric sharing rule does not. This preregistration describes the experimental design, the possible collusion mechanism, the hypotheses, the number of planned observations, and a pipeline of exploratory analyses.
External Link(s)

Registration Citation

Orland, Andreas. 2022. "Sharing rules in Bertrand duopolies with increasing returns." AEA RCT Registry. April 06.
Experimental Details


We conduct a lab experiment. In our between-subject experimental design, we vary the sharing rule for an otherwise identical Bertrand duopoly with increasing returns to scale. We label our treatments SYM (for the symmetric sharing rule) and WTA (for the winner-takes-all sharing rule).
Intervention Start Date
Intervention End Date

Primary Outcomes

Primary Outcomes (end points)
Subjects' decisions and market outcomes:
- set prices (40 per subject, 80 per cluster/duopoly),
- market prices (40 per cluster/duopoly).
Primary Outcomes (explanation)
While our theoretical prediction for pricing behavior is that average set prices in SYM are (slightly) higher than in WTA, we expect that market prices are higher in WTA than in SYM (this forms our testable Hypothesis 1, as listed below). We base this expectation on a collusion mechanism that is more plausible in WTA than in SYM. (Footnote: An alternative strategy that can be applied in both treatments is alternation. Alternation means that one firm sets a price of 8.75 or 9.00 while another firm (purposefully) sets a lower price, and that subjects take turns between periods and thus share the monopoly profit equally. We will consider this strategy in our exploratory analysis.) Sharing the market comes at a severe efficiency loss in SYM, compared to WTA. (Footnote: Assume that the two firms tacitly coordinate on a price. In our experiment, the expected profit in WTA is 0.32 euros greater than in SYM for all prices. Our line of reasoning is the following: (i) When, all other things equal, coordination is more profitable in WTA, we expect to observe more coordination in this treatment. (ii) When coordination is more profitable for higher prices, we should observe more coordination on these prices.) This opens the following channel for collusion in WTA: Coordination on one price is more profitable regardless of the price and both firms simultaneously set the same price that ensures the maximum profit, either 8.75 or 9.00, and let the random draw decide about the receiver of the entailed profit (Hypotheses 2a and b).

In the following, we list our hypotheses and the related statistical tests:

Hypothesis 1: Market prices in WTA are higher than in SYM.

Test: We calculate the average market price across all periods for each pair. Then we use the Wilcoxon-Mann-Whitney test to compare average market prices between the two treatments.

Hypothesis 2a: Subjects in WTA more often set equal prices than subjects in SYM.

Test: We calculate the number of periods where both firms set the same price. We then use the Wilcoxon-Mann-Whitney test to compare these numbers between the two treatments.

Hypothesis 2b: Subjects in WTA more often set prices of 8.75 or 9.00 than subjects in SYM.

Test: We calculate the number of periods where both firms set a price of 8.75 or 9.00. We then use the Wilcoxon-Mann-Whitney test to compare these numbers between the two treatments.

Secondary Outcomes

Secondary Outcomes (end points)
Subject-specific covariates:
- gender,
- final math grade in school (a variable between 1 and 5),
- study a field with economics courses (dummy-variable),
- experience with game theory (dummy-variable),
- risk aversion (incentivized, a variable between 0 and 10),
- individual decision times per period (in seconds).
Secondary Outcomes (explanation)
Robustness checks & further explanatory questions:
- We test Hypothesis 1 using set prices (instead of market prices) and regressions.
- Are the distributions of set prices different between the treatments? We plot histograms of the price distributions in both treatments and use a clustered Chi-squared test to examine this question.
- Is the dispersion of set prices in SYM larger than in WTA (as predicted by the supported price range)?
- In SYM, duopoly profits in the price range between 2.75 and 5.00 are negative; in WTA the duopoly profits in this range are either positive or zero (depending on the random draw). Thus, set prices in WTA might be higher than in SYM because subjects avoid setting prices in the price range between 2.75 and 5.00 in SYM (but not in WTA). We examine this by comparing the shares of set prices in this range by treatment.
- Do subjects in our experiment use alternation to collude? Do they use alternation more often in SYM than in WTA (where they have the strategy at hand explained in Primary Outcomes)?
- Can we observe a convergence of market outcomes over time?
- Can past periods' prices and outcomes explain current periods' behavior (using an impulse-response model)?
- Can the answers to the questionnaire items explain observed behavior?

Experimental Design

Experimental Design
We base the market environment on the following parametrization of the Bertrand competition. Two symmetric firms, i={1, 2}, play the stage game repeatedly. In each period, they compete by simultaneously setting prices (in 25-cent increments) in the range from 0 to 9. Each firm i has the same cost function C(q_{i}) = [(0 if q_{i} = 0) and (25 + q_{i} if q_{i} > 0)] (thus, the cost function has avoidable fixed costs, and decreasing average costs for positive output). The demand function is given by D(p) = 17 - p.

If firm i sets the lower price, it serves the whole market and makes a profit of pi_{i}(p_{i}, p_{-i})=p_{i} D(p_{i}) - C(D(p_{i})) while the other firm, -i, receives nothing: pi_{-i}(p_{i}, p_{-i})=0. If the two firms set the same price, the resulting duopoly profits differ between the two treatments. In SYM, both firms share the market demand equally if they set the same price. In WTA, both firms have an equal chance of receiving either nothing or serving the entire market (this is determined by a random draw).

At the beginning of the experiment, each subject is randomly and anonymously matched with another subject and interacts with this subject for the entire experiment. The experiment lasts 40 periods, and this and all other design features of their treatment are known to the subjects.

We frame the experiment as price competition, but instead of informing subjects about all details of the underlying model, we explain the experimental game using payoff tables (which was also done in the experiments of some other papers). Our payoff tables show the profit consequences (in euros) for all allowed prices and depending on the decision of the other firm. These tables are part of the experimental instructions and can also be used by the subjects when taking their decisions. Subjects are told that negative numbers stand for losses. To cover these possible losses, subjects start the experiment with an initial endowment of 1.50 euros. After reading the instructions on-screen, subjects have to correctly answer a set of control questions to proceed (and this is known to all subjects, too).

When entering their decision on the computer, subjects will be shown a slider on-screen. Subjects choose a price using the slider and are shown, below the slider, the profit consequences before submitting the chosen price (and this information is updated in real-time when sliding to a different price). After confirming their decision, subjects receive feedback: their price, the other firm's price, and their period profit and total profit so far. At the end of the experiment, we sum up all periods' profits (and losses) and the initial endowment. If subjects incur overall losses at the end of the experiment, their payoff is capped at zero euros.

After the main part of the experiment, we conduct a questionnaire. We ask for the subjects' gender (male/female/non-binary), elicit their cognitive abilities by asking for their final math grade in school (on a range from 1 to 5, best to worst grades in the German system), ask if they study a field with compulsory economics courses, and ask if subjects have experience with game theory (again, to introduce a game theory-dummy). We also conduct a simple, monetarily incentivized task to elicit the subjects' risk aversion: each subject is endowed with one euro and decides about the amount (in 10-cent increments) to invest in a risky asset---with 50% probability the investment is lost, with 50% probability the investment is paid out 2.5-fold.

We invite subjects using a pre-existing database (based on ORSEE). These subjects are exclusively students from various fields of study from the University of Potsdam and other nearby universities. Every subject takes part in one session only. Upon arrival at the laboratory, all subjects are seated at computer workstations with privacy walls. Communication between subjects is prohibited. After the experiment, we pay all subjects in cash and in private. On top of the payoff of the incentivized part of the experiment, subjects receive a show-up fee of five euros (and this show-up fee is not offset against potential losses from the main part of the experiment).
Experimental Design Details
Randomization Method
Students will be recruited via a preexisting database, where only a random sample of all registered students will be invited by e-mail (until a session is filled). A coin flip before each session decides about which of the two treatments is conducted (until the planned number of observations per treatment is reached). In each session, subjects are randomly matched (by a computer) into duopolies/pairs of firms and then stay paired during the entire experimental session.
Randomization Unit
Group level randomization into experimental treatments. Individual randomization into duopolies/pairs of firms. (Interactions will only occur between firms of the same duopoly, so each duopoly is a cluster.)
Was the treatment clustered?

Experiment Characteristics

Sample size: planned number of clusters
50 duopolies/pairs of firms
Sample size: planned number of observations
4000 set prices (=2 firms * 50 duopolies * 40 periods)
Sample size (or number of clusters) by treatment arms
25 duopolies/pairs of firms in each of the two experimental treatments
Minimum detectable effect size for main outcomes (accounting for sample design and clustering)

Institutional Review Boards (IRBs)

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