Abstract
Cooperation is a pervasive social phenomenon, but more often than not economic theories have little to say about its causes and consequences. In this paper, we explore the hypothesis that cooperative behaviour might be motivated by pure self-interest when the “social” payoff in a game is increasing. We report the results of a series of experiments on the Centipede game. The experiments are organised in two subsequent steps. Subjects first participate in a 2-period trust game, randomly matched with unknown partners. We apply the strategy method in order to elicit their social preferences. On the basis of their pre-game behaviour, individuals are divided into three main social groups: selfish individuals, pure altruists and reciprocators. At the second step of the experiment, subjects play a repeated 6-move Centipede game with an increasing final payoff. Each subject plays twice, in a low and in a high-stake Centipede game, and he/she is informed about his/her co-player social preferences. We provide statistical evidence to identify the origin of cooperation within homogeneous and heterogeneous social groups. We construct a Poisson regression model to assess the determinants of the duration of conditional cooperation in the Centipede.
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Notes
Diverging opinions about the violation of backward induction in the Centipede have been put forward in the literature. Aumann (1995, 1996, 1998) has argued that rational choice theory dictates the player’s maximisation of the expected payoff at every information set, so that the Nash equilibrium unquestionably applies. Instead, Binmore (1988, 1996) endorsed the view that the players consider Bayesian updating at each node they could reach, so that, contrary to orthodox rational choice theory, out-of-equilibrium moves are legitimate. In other words, the infringement of backward induction by the first player’s across move at the first node allows the other player to assign a positive probability to the event that the result of his playing across could result in a pay-off higher than the one he would get by choosing down.
The structure of the trust game we adopted in our experiments is known as the Generalised Trust Game. Respondents sent tokens back not to the sender with whom they were coupled, but to some other anonymous sender present in the same session (see Barr et al. 2005).
As it will be explained in “Results of the experiments”, we identify three social types: selfish individuals, reciprocators and altruists.
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Different approaches to the consideration of the role played by the information in strategic interactions, namely to put each player in a condition to rationalise his opponent’s expected behaviour, have been proposed in the literature. In Gachter and Thoni (2005), players were informed on the social types they were playing with, in repeated public good games. The information on the reciprocating strategies was elicited by a pre-play participation of the subjects in a one-shot public good game. Subjects were then ranked according to their performance in the pre-play game, and such information was disclosed to the other players when participating in the experiments. In our opinion, this methodology provides only very limited information on social preferences. Moreover, it does not address the problems of understanding and possible impact of the method itself on the individuals’ preference functions.
It may be argued that the respondents’ table, composed of ten observations, was quite a complex piece of information to understand in a limited amount of time. However, in some experimental designs, such as oligopoly experiments played as normal form games, complex information on decision variables is provided (see Holt 1995).
For expositional purposes, we report the final node as node 7 for player A and node 8 for player B.
In the low-stake games, the proportion of A players choosing to exit the game at the seventh step drops from 56% in Session 0 to a minimum of 9% (Session 2). By the same token, the proportion of B player exiting the game at the final node varies between 46% in Session 0 and a minimum of 10% in Session 2.
In three cases out of four (Sessions 1–3), the proportion of A (B) players exiting the game at the final node drops from 17% (20%) to 0; whilst in Session 4 the proportion of A (B) exiting the game at the final node is equal to 8% (12%).
The proportion of B players exiting at node 2 in Session 1–4 is higher than the same proportion in Session 0 in four cases out of eight.
Such distribution corresponds to the equal split of the total number of tokens between recipients and senders, in the eight cases (out of ten) in which such equal split can be autonomously implemented by the recipient.
In a similar fashion, Barr et al. (2005) apply the strategy method in the GTG to elicit the individuals’ social preferences. In their paper, the authors focus on several hypotheses on how respondents’ behaviour may be classified in the GTG. The main hypothesis tested is that the latent reciprocating strategies adopted by the respondents are rays, i.e. “they are linear in the amount sent by senders and pass through the (nothing sent, nothing returned) origin” (Barr et al. 2005, p. 16).
According to this hypothesis, social behaviour may be classified as selfish, reciprocating and altruistic, according to the estimated slope of the individuals’ reciprocating strategies’ profiles. As an example, selfish strategies are linear strategies with zero slope. Reciprocating strategies are monotonically increasing strategies with the value of the slope, (β, varying in the interval: 0 < β ≤ 3, where 3 is the value of the multiplying factor, α. We do not adopt this methodology and prefer to follow simple criteria as the convergence of the individual preference profile to a theoretical type.
Overall, 48 individuals in our sample belong to the first category, 8 to the second and 24 to the third one.
There are two aspects that need to be underlined in order to assess the relevance of our results. First, both samples do not contain homogeneous pairs of the type β (the reciprocator players). Second, because of the larger number of selfish players in our experiments, the sample of homogeneous pairs is composed of a higher proportion of selfish players than altruists. Since both aspects may partly explain the high level of cooperation we observe in the heterogeneous groups, we will analyse the evidence of groups and individuals conjointly, in order to evaluate the relative incentive to cooperate of each social type, both when they face a similar type of player and when they face a different type.
In fact, the cumulative frequency of the “E” strategy in steps 5–7 is equal to 58% in the homogenous group and 71% in the heterogeneous ones. By the same token, the frequency of the choice of “E” at step 1–2 is equal to 20% for the former group and 10% for the latter one.
Table 7 actually shows that the highest level of cooperation was found in heterogeneous pairs of selfish and reciprocating players. However, the low number of observations we have in this case suggests caution in the interpretation of the evidence.
The estimation model is: \( Y_i = \frac{{\mu ^Y {\text{e}}^{ - \mu } }} {{Y!}} + u_i \) with μ replaced by equation 2. This model is non-linear in parameters and then requires the “quasi-maximum likelihood” estimation. The estimation of the mean (μ i) for the ith pair is as follows: \( \hat{\mu }_{i} = e^{{\hat{\beta }_{1} + \hat{\beta }_{2} S_{{2k}} + \hat{\beta }_{3} R_{{3k}} + \hat{\beta }_{4} S_{{4j}} + \hat{\beta }_{5} R_{{5j}} }} \)
The computed values of the base of the natural logarithm e, which are needed for the estimation of the μ values, and these latter values, are available on request.
These values are available on request.
Our finding detaches from what has been found in the PG games literature. In an experiment conducted by Gachter and Thoni (2005), subjects were ranked with respect to their contribution in a one-shot PG game and then sorted into groups of individuals with similar ranks. Cooperation in the “alike” groups of like-minded people was found to be significantly higher than in random group composition.
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Acknowledgments
We thank Emanuele Ciriolo, Alessandro Innocenti and Niall O’Higgins for their helpful comments on a previous version of the paper. We also thank Riccardo Ricciuti and Francesco LoMagistro for contributing to the instructions and for technical assistance in the experimental sessions. Financial support from the research programme of the University of Siena is acknowledged. The usual disclaimers apply.
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Farina, F., Sbriglia, P. Conditional cooperation in a sequential move game. Int. Rev. Econ. 55, 149–165 (2008). https://doi.org/10.1007/s12232-007-0037-y
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DOI: https://doi.org/10.1007/s12232-007-0037-y