Abstract
The frequency of interaction facilitates collusion by reducing gains from defection. Theory has shown that under imperfect monitoring flexibility may hinder cooperation by inducing punishment after too few noisy signals, making collusion impossible in many environments (Sannikov and Skrzypacz in Am Econ Rev 97:1794–1823, 2007). The interplay of these forces should generate an inverse Ushaped effect of flexibility on collusion. We test for the first time these theoretical predictions—central to antitrust policy—in a laboratory experiment featuring an indefinitely repeated Cournot duopoly, with different degrees of flexibility. Results turn out to depend crucially on whether subjects can communicate with each other at the beginning of a supergame (explicit collusion) or not (tacit collusion). Without communication, the incidence of collusion is low throughout and not significantly related to flexibility; when subjects are allowed to communicate, collusion is more common throughout and significantly more frequent in the treatment with intermediate flexibility than in the treatments with low or high flexibility.
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Notes
 1.
In the extreme case, in which subjects could adjust their actions almost continuously, the median rate of cooperation was as high as 90%. At the other extreme, in which subjects could adjust their actions only once, cooperation rates were close to zero.
 2.
Vives (2001) describes this result as “counterintuitive.” One might speculate, for example, that it is possible to delay punishments until more convincing information becomes available that the other player is really defecting. Such a strategy unravels though since such a delay will strengthen the incentives of the other player to defect.
 3.
 4.
 5.
 6.
A convenient feature is that the Nash and minmax payoff are the same. This makes it easier to derive the best possible collusive outcomes.
 7.
This implies, for example, that there is a probability of 44% that the realized price will deviate by at least 1 unit from the expected price.
 8.
Experiments on risk taking show that high decision flexibility may also generate a more myopic evaluation of outcomes (Gneezy and Potters 1997). Because of the discrete design the number of expected periods (and potential collusion profits) increases slightly in \( \varDelta \). This could also lead flexibility to affect negatively the amount of collusion, although we believe the difference to be negligible.
 9.
See Table D.1 in Online Appendix D for further details on sessions and treatments.
 10.
The shocks were drawn independently across pairs, across periods and across sessions. We bounded the support for the price shocks, so as to prevent prices from going negative. In theory, the probability that a shock realization makes the truncation relevant is: 0.1% if both subjects defect; 0.006% if only one subject defects; 0.0002% if both subjects cooperate. In practice in our experiment the price would have gone negative in 6 out of 11216 cases (0.053%).
 11.
If subjects had played noncooperatively throughout, they would have had zero profits on average and earned only the starting endowment, amounting to 10 Euro. Playing cooperatively throughout would earn them 29.4 Euro in total in expectation. Playing the best cooperative equilibrium in \(\varDelta \) = 2 would generate expected total earnings of 23.8 Euro. This illustrates that the incentives to cooperate are substantial.
 12.
A subject could in theory attain negative cumulative profits. The probability of this happening when subjects always defect is 3%, but it is much lower when subjects cooperate. In 17 out of 1680 matches, it occurred that a subject had a negative total payoff at some point, and one subject out of 240 ended the experiment with a negative balance, thus earning nothing. The opposite possibility that a subject would gain an unreasonably high profit of above, say, 100 Euro, is negligible.
 13.
The length of initial cooperation is zero if at least one player defected in period 1.
 14.
Controlling for the length of the match makes quite a difference. By chance, matches were substantially longer in treatment \( \varDelta =2 \) than in the other two treatments, which suppressed average cooperation rates. At the same time, the length of a match has a positive effect on the length of initial cooperation. If a match ends early, any ongoing cooperation by necessity also ends. A concern one might have is that subjects overestimate the longterm prospects of cooperation in treatment \(\varDelta =\) 2 if they are experiencing relatively long matches in this treatment. To check for such an effect we add the length of the previous match as a regressor to the regressions in Table 5. This regressor is never significant though, and it only marginally affects the coefficients for \(\varDelta = \) 1 and \(\varDelta =\) 3.
 15.
In Table D.3 and Figure D.2 in Online Appendix D we replicate the analysis of subjects’ reaction to observed signals, conditional on their past action. Similar to what we observed in the treatments without communication, subjects tend to switch to defection when they observe low prices. However, they are more forgiving, in the sense that the fraction of subjects who switch back from defection to cooperation tends to be higher than without communication, regardless of the observed signal.
 16.
The reported excerpts are taken from the subjects’ communications, as classified by the coders.
 17.
Online Appendix F contains some sample chats to give an impression of how subjects used the opportunity to communicate.
 18.
A tobit regression pooling data from the three treatments together reveal that the effects of “Agreements” and “Leniency” are significantly smaller in \( \varDelta =1 \) and in \( \varDelta =3 \) than in \( \varDelta =2 \). Regression results are available from the authors, upon request.
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Acknowledgements
We wish to thank the editor, the associate editor and two anonymous reviewers for their helpful comments and suggestions. We also thank Andrzej Skrzypacz for his precious advice, and Charles Angelucci, Jeff Butler, Gabriele Camera, Martin Dufwenberg, Christoph Engel, Ernst Fehr, Tobias Klein, Wieland Mueller, Karl Schlag, Jean Robert Tyran and participants at the European ESA Meeting in Innsbruck, the MBEES Workshop in Maastricht, the Industrial Organization Workshop in Otranto, the CRESSE conference in Rhodes, the EIEFBolognaBocconi IO Workshop, the ESRC London Experimental Workshop (LEW), the Amsterdam Symposium on Behavioral and Experimental Economics (ABEE) and seminars at Humboldt University Berlin, the Stockholm School of Economics, Erasmus University Rotterdam, Tilburg University, University of California Santa Cruz, IFN Stockholm, the MPI for Collective Goods in Bonn, the Technische Universität Berlin, the University of Vienna and the University of AixMarseille for valuable comments.
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Giancarlo Spagnolo gratefully acknowledges research funding from Konkurrensverket (the Swedish Competition Authority).
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Appendix
Appendix
A Theoretical analysis
In this Appendix we outline how the nonmonotonic effect of flexibility (\(\varDelta \)) on the sustainability of collusion is derived. The sustainability of collusion depends on the punishment strategy adopted by the players. Nonetheless, the authors prove that it is possible to compute a robust lower bound by finding the best symmetric equilibrium with Nash reversion as a punishment, and a robust upper bound by finding the best symmetric equilibrium with the minimax payoff of 0 as a punishment. Both bounds are valid for both symmetric and asymmetric equilibria. With our parameters, the upper and the lower bound coincide, as the Nash equilibrium profit coincides with the minimax payoff in our game (both are equal to zero).
We can follow Sannikov and Skrzypacz (2007) to show that in our set up collusion ( \(q_{i}=3\)) can be sustained when \(\varDelta =2\), but not when \(\varDelta =1\) or \(\varDelta =3\). From Abreu et al. (1986) we know that the best stronglysymmetric equilibrium payoff of this game can be achieved by the following strategy profile:

Players start in the collusive state and choose quantities \(q_{C},q_{C} \) (for us it will be \(\left( 3,3\right) )\).

As long as the realized price is in region \(P_{+},\) players remain in the collusive state. If the price is outside this region, they move to the punishment state forever after.

Because in our game minimax has the same payoffs as the static Nash equilibrium, in an optimal equilibrium once the players reach the punishment state they play \(\left( 4,4\right) \) forever.
We now characterize the region \(P_{+}\) and it’s complement \(P_{\_}\). Let \(G\left( Q\right) \) be the probability that the price will be in \(P_{+}\), and V the expected profit of the collusive equilibrium. Each player’s IC constraint is:
which can be rewritten as:
If the IC constraints are satisfied, then the expected profit in this equilibrium is:
which yields:
Note that V is decreasing in \(\delta \) and increasing in \(G(2q_{C})\).
Sannikov and Skrzypacz (2007) show that the optimal \(P_{+}\) region (that maximizes V) corresponds is a tail test. There is a cutoff \(\hat{p}\) such that above \(\hat{p}\) are in \(P_{+}\) and prices below are in \(P_{\_}\).
If a tail test is adopted, then
where \(\phi (\mu ,\sigma ^{2},x)\) is the probability density function of a normal distribution with mean \(\mu \) and variance \(\sigma ^{2}\), evaluated at x. Using the parametrization in our experiment, with \(p(Q)=12q_{1}q_{2}\) and \(\sigma =1.3\), we can rewrite the ICconstraint as a function of \(\hat{p}\) and calculate when it can be satisfied at different levels of \(\varDelta \).
Numerical calculations show that the lefthand side of the ICconstraint (1) is convex, and that when \(\varDelta =2\) it is positive for cutoff prices \(\hat{p}\in [4.758,5.060]\), while it takes negative values for any \(\hat{p} \ge 0\) when \(\varDelta =1\) or \(\varDelta =3\).
This implies that in the infinite horizon Cournot duopoly game with imperfect public monitoring collusion is sustainable in equilibrium when \(\varDelta =2\), while no collusive equilibrium is sustainable when \(\varDelta =1\) or \(\varDelta =3\).
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Bigoni, M., Potters, J. & Spagnolo, G. Frequency of interaction, communication and collusion: an experiment. Econ Theory 68, 827–844 (2019). https://doi.org/10.1007/s0019901811464
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Keywords
 Cartels
 Cournot oligopoly
 Flexibility
 Imperfect monitoring
 Repeated games
JEL Classification
 C73
 C92
 D43
 L13
 L14