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Repeated interactions and endogenous contractual incompleteness

Experimental evidence


This paper empirically investigates the interaction between repeated transactions and endogenous contractual incompleteness. We design an indefinitely repeated games experiment between identifiable players. In this experiment, the probability of continuation and the level of shared information vary over the treatments. The level of contractual completeness is decided by participants at each period. Our results show that past interactions are a stronger determinant of the level of investment in contractual completeness than the perspective of future business.

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  1. Let us mention that this result depends on our assumption about the strategy of the players, i.e., trigger strategy. This assumption is generally made in the literature about relational contracting (Baker et al. 1994, 2002; Levin 2002; Halac 2012). Yet, other strategies could lead to different equilibria.

  2. Unforeseen contingencies can appear in \(j\le n\) contractual relationships.

  3. A more general model showing that there is no pooling equilibrium and that each type of seller has no incentive to deviate to misrepresent as another type is provided in Desrieux and Beuve (2015). The intuition is that the type L seller gains nothing to postpone the decision to hold up in future, and the type H seller is always better off cooperating than deviating.

  4. More formally, \(\delta =e^{-r\Delta }\) where r is the rate of time preference and \(\Delta \) is the length of the period. However, if we add a probability \(\mu \) of continuation from one period to the next, then with probability \((1-\mu )\) there is no gain, and with probability \(\mu \), the gain is discounted at rate \(\delta =e^{-r\Delta }\). Then, the expected discounted value of the gain is \(\delta '=\mu \delta =\mu e^{-r\Delta }\). Thus, the situation is the same as if \(\mu =1\) and \(r'=r-\frac{ln(\Delta )}{\mu }\), hence the dual interpretation of the discount rate.

  5. An example of game instructions is provided in Appendix. Results of the control questionnaire: \(m=9.2/10\); \(\sigma \) 1.1. When at least one participant received a mark inferior than 5, we explained the rules of the game a second time.

  6. Earnings in ECUs: \(m=583\), \(\sigma =182\). Earnings in Euros: \(m=14.6\), \(\sigma =4.6\). Duration of the session: between 15 and 30 minutes depending on treatments.

  7. In practice, even if sellers can read the contractual agreement, buyers may have more knowledge about future contingencies that can happen and whether the agreement is more or less complete to deal with all these future contingencies. For instance, a firm can ask a supplier some widgets and may make some efforts to learn about the future contingencies (e.g., changes that could happen in her environment) to guess whether these widgets are likely to be modified or not. The seller (the supplier) may be unaware of the future contingencies the buyer may face and that could force to renegotiate the agreement.

  8. See Cochard et al. (2004) for an interesting study of the “reciprocity hypothesis” in repeated investment game.

  9. It is important to note here that buyers cannot know which sellers are actually available on the market. At the beginning of each round, they can propose to any of the six sellers present in the lab without knowing if they are already engaged in three relationships. Such information would indeed act as a substitute for public information.

  10. Before analyzing our results, we show that there is enough evidence to reject the proposition that samples of observations of our main variables are generated by the same stochastical process. The non-parametric Wilcoxon test reports are presented in Table 6 in the appendix.

  11. The correlation matrix of all the variables used in the estimates is provided in Appendix; Table 10.

  12. i.e., the number of cooperative choices divided by the number of incomplete contract situations in the past and in the relationship with a given buyer.

  13. We use “Reputation” here as defined in Sect. (5.1.2).

  14. This is consistent with our theoretical findings in Sect. 2. Cooperation is expected to be higher under public information. Since buyers adapt their investment regarding to past behavior of sellers, they should observe more cooperation and invest less in contractual completeness. However, the coefficient PublicInfo is only significant at the 10 % level, and we could have expected a stronger impact. This result may come from our restrictive theoretical assumptions (trigger strategy). In practice, other strategies can be chosen by the parties, so that the impact of PublicInfo is slightly weaker than we could expect. In addition, buyers may also get more incentives to invest in contractual completeness under public information, as they observe all sellers’ behavior. They know when they are associated with a non-cooperative seller and adapt their additional investment accordingly.

  15. Similarly, we do not include interaction terms for collinearity reasons.

  16. The measure of completeness adopted by Argyres et al. (2007) and Ryall and Sampson (2009) relies on the level of details (declared by contracting parties or observed though extensive document analysis) put into the contract, while Crocker and Reynolds (1993) describe contractual completeness through the pricing provisions (which defines the leeway of partners to adjust prices through time) adopted by the contracting parties.

  17. Compared to the study of Crocker and Reynolds (1993), the experimental approach allows us to observe the entire story of relationships. History between parties start at the first period and the observed behaviors cannot be related to unobservable past events. Moreover, the empirical works as that of Crocker and Reynolds (1993) focus on relationships between one buyer and one or two sellers, while we can create in lab an environment where buyers and sellers are numerous and identically distributed in the population.


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Conflict of interest

The authors declare that they have no conflict of interest.

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Correspondence to Jean Beuve.




Non-parametric tests

It is crucial, before analyzing the results, to determine whether there is enough evidence to reject the proposition that samples of observations of our main variables (i.e., Cooperation and Additional Investment) are generated by the same stochastical process. As suggested by many studies, this is evaluated by using non-parametric testing methods. The non-parametric Wilcoxon test reports are presented in Table 6 where “Z” is the Wilcoxon score and “P > Z” is the significance level at which the null proposition of no difference in distribution is rejected. Most of the time, tests reject the null proposition of no difference in distribution of our variables between treatments. Particularly, the null proposition is rejected at 1 percent level when we compare short-run versus long-run treatments on the one hand and private versus public information treatments on the other hand.

Table 6 Samples, non-parametric test results (Wilcoxon scores)

Logit analysis of buyers’ additional investment—2

Table 7 Logit analysis of Buyers’ Additional Investment - Quartiles for L.%Cooperation and L.%Reputation

OLS estimations

Table 8 OLS analysis of Sellers’ Cooperation
Table 9 OLS analysis of buyers’ additional investment


Table 10 Correlations

Experimental instructions

The following instructions were handed out to the participants in the LI treatment.


You participate in an economic experiment which takes place in a computer room. We inform you that during this experiment, it is purely forbidden to have talks. If you follow carefully the instructions, you will make gains and you will be paid in cash at the end of the experiment.

Please note that the following guidelines are applicable to all candidates.

The currency used during the experiment is the ECU (Experimental Currency Unit), and all the transactions will only be denominated in ecus. At the end of this session, your gains will be paid in euro according to the following exchange rate: 40 ecus for 1 EURO.

Parts of the experiment

The participants of the experiment are assigned to two different groups:

  • Group A: 6 participants: from \(A_{1}\) to \(A_{6}\).

  • Group B: 6 participants: from \(B_{1}\) to \(B_{6}\).

The participants have to assume the role of their group (A or B) and number (from 1 to 6) until the end of the experiment. This way, you can identify the other participants with who you are going to interact throughout the experiment. You start the experiment with 0 ecus.

The experiment is composed of an undetermined number of periods.

The relationship

In order to improve your decision-making and to optimize your understanding of the decisions made by the other participants, there is some information about the relationships’ functioning.

During this experiment, you are going to make other participants your partners in. These partnerships allow your partner and you to make some profits.

Players A will suggest partnerships to Players B. Players B will have the choice to accept or refuse the partnerships’ suggestions made by Players A.

You will be able to make at the most three different partnerships per periods. It is also possible that according to the other participants’ choices, you will have 0 partnerships sometimes.


When you create a partnership for the first time, both of the participants (A and B) will have to invest 6 ecus. This investment has to be made only once. This way, if you repeat the partnership during the following period, the two participants will not have to invest 6 ecus again. Yet, if two participants who were already in a partnership before decide to get into a partnership again, both of them will have to invest again 6 ecus because they did not repeat the partnership during the following period.

“Situations” and “choices”

When a partnership is created, two types of situations can happen (according to the probabilities given between parentheses):

  • Situation 1 (probability: 50 %): the partnership yields 40 ecus which are automatically spitted up into two equals part. A and B both receive 20 ecus.

  • Situation 2 (probability: 50 %): the partnership yields 40 ecus but in this case, the distribution of the profits depends of the choice of the B participant. B will choose between

    • Choice 1: A receives 20 ecus and B receives 20 ecus.

    • Choice 2: A receives 10 ecus and B receives 30 ecus.

Additional Investment

At the beginning of each period, players A can decide to make an additional investment of 2 ecus in order to change the probabilities assigned to the situations 1 and 2. This additional investment allows players A to obtain the situation 1 with a probability of 75 % and the situation 2 with a probability of 25 % (to compare with the 50–50% probability when there is not additional investment) for the current period and the concerned partnership.


At the end of each period, players A are informed about the choices made by all players B during the past periods. This way, players A can notice the distribution (in percentage) of the choices made by players B between the choice 1 and the 2 during all the past periods.

The sequence of one period

  • Step 1: Players A suggest partnerships to players B. (At most 6 propositions)

  • Step 2: Players B accept or reject partnership offers from players A. (At most 3 accepted partnerships)

  • Step 3: Players A choose at most 3 players B among positive answers.

  • Step 4: Players A choose the level of their investment. (Players B do not know the investment level chose by players A)

  • Step 5: All participants learn the repartition of their partnerships between situations 1 and 2.

  • Step 6: Players B choose surplus sharing in case of situations 2.

  • Step 7: All participants learn their gains for the current period, and then their cumulative gains for the whole experiment.

  • Step 8: Players A learn the choice made by all players B in their partnerships since the beginning of the experiment.

  • Step 9: Players A can decide to stop some of their partnerships (or all of them) or to suggest to players B to pursue some of their partnerships (or all of them).

  • Step 10: Players B accept or reject partnership’s renewals’ offers from players A.

  • Step 11: Players A can suggest partnerships to players B with who they were not associated within the current period.

  • \(\rightarrow \) Go back to Step 2.

Length of the experiment

The experiment entails at least 6 periods, after which the experiment continues period by period with a probability 0.8. In other words, at the end of the sixth period, there are 8 in 10 chances to play an extra seventh period. At the end of this seventh period, there are 8 in 10 chances to play an extra eighth, and so on.


The figure below summarizes gains associated with different situations and actions of players.

figure a

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Beuve, J., Desrieux, C. Repeated interactions and endogenous contractual incompleteness. Theory Decis 80, 125–158 (2016).

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  • Contractual incompleteness
  • Cooperation
  • Repeated games
  • Reputation
  • Experiment