Abstract
This paper studies the effect of social relations on convergence to the efficient equilibrium in 2×2 coordination games from an experimental perspective. We employ a 2×2 factorial design in which we explore two different games with asymmetric payoffs and two matching protocols: “friends” versus “strangers”. In the first game, payoffs by the worse-off player are the same in the two equilibria, whereas in the second game, this player will receive lower payoffs in the efficient equilibrium. Surprisingly, the results show that “strangers” coordinate more frequently in the efficient equilibrium than “friends” in both games. Network measures such as in-degree, out-degree and betweenness are all positively correlated with playing the strategy which leads to the efficient outcome but clustering is not. In addition, ‘envy’ explains no convergence to the efficient outcome.
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Notes
This term that has been used in the social science literature at least since Bogardus (1928).
In this paper, social efficiency is measured as total earnings.
The concept of ‘envy’ we consider throughout the paper is based on Fehr and Schmidt (1999). In their framework they consider that subjects’ utility may decrease as the amount by which opponents’ payoffs exceeds subjects’ own payoffs increases.
Frank (1985) suggests that people are more apt to make comparisons with people who seem less distant.
Note that, in Spain, individuals always have two surnames instead of only one as is common in many other countries.
As the goal of eliciting the social network was to analyze a coordination game played by pairs of friends and pairs of strangers, we prefer to underestimate the social network but assuring that the links reported are true.
The reason for not having a voluntary recruitment was that we wanted to elicit a real social network, so we had to run the experiment in an already existing network (in our case, the classroom network).
See for example Schmidt et al. (2003).
As a consequence, the only Nash equilibrium in mixed strategies for both games is ((1/2,1/2),(1/2,1/2)), that is, playing each strategy at random with an equal probability, independently of the player’s role.
We decided not to change the partners in each round because we were interested in analyzing the convergence to a Nash equilibrium. If matching were random, subjects could not use feedback (opponent’s strategy) to coordinate in a particular equilibrium.
Note that subjects were paid after the coordination game just for the coordination game, for the BCJP game subjects were paid after subjects played it.
In this way we assure in most cases that subjects did not know the identity of her partner in the game. We cannot assure that the groups of friends, at the end of the experiment shared all profits obtained and they divided equally among them. Nevertheless, it seems that this was not a problem given that our main result is that pairs of strangers coordinate more frequently in the efficient equilibrium than pairs of friends.
Another problem that arises when conducting the experiment is that all of the pairs of friends must show up for the same experimental session. Therefore, if one group of friends was comprised of four friends and a member did not show up (we dropped the corresponding partner from the game), the remaining members (2) could visually infer exactly who their partner was in the game.
We designed this “envy’ Game’ with the aim of obtaining a general measure of ‘envy’. Hence, we decided to run the game only for pairs of strangers (not for friends). The main reason is that we were interested in analyzing how a general level of ‘envy’ could affect subjects’ behavior if they were playing in a strategic environment with a friend (or with a stranger). We think that no framing at all was the best option to obtain a general measure of ‘envy’.
See Electronic supplementary material for the instructions of the ‘envy’ game.
This cost is considered just in monetary terms and is not related to time. That is, the cost is the money that the dictator is losing when she takes the envious decision.
A similar work by Charness and Grosskopf (2001) studies “difference aversion” and its influence on self-reported happiness with binary dictatorial decisions.
All measures in the questionnaire were not economically incentivized.
See Electronic supplementary material for the complete instructions for Treatment 1 (Game 1 and friends matching protocol). Instructions for the other treatments are available upon request.
As all participants played the Stage 3 in a very short period of time, we have no reason to think that different ‘envy’ levels are due to implementation factors.
Following Ehrblatt et al. (2007), we establish the threshold in order for a pair to choose the Nash equilibrium in at least 3 consecutive periods.
Table 1 contains observations from all the pairs who participated in Stage 2, including those that didn’t converge either to the efficient equilibrium or to the inefficient one.
Alternate is defined as players alternating between the 2 equilibria from a certain period until the end of the game, being that first period of alternation higher or equal than 20.
Another alternative (included in the Miscoordination cathegory) would be to consider that subjects play according to the unique Mixed-Strategy Nash Equilibrium (MSNE): ((1/2,1/2), (1/2,1/2)). Recall that this is the same for both games. For each pair who coordinated in one of the three cathegories (U, L; D, R; Alternate) we have conducted a test to analyze differences with the MSNE and the least significant one is z=1.715, p=0.043 (one-tailed). For pairs included in the category of Miscoord., the behavior of 36 out of 46 is compatible (not statistically differences found) with the fact of playing the MSNE.
In all tests conducted in this section we consider 1 observation per pair in each game and matching protocol, in this way we assure that all observations are independent.
To test the robustness of the threshold imposed in the equilibrium convergence definition, we have run all the tests in this section decreasing the convergence threshold to 19 (7 consecutive rounds in which both members of the pair play the same strategy seems to provide enough evidence of convergence to equilibrium). Although the percentages in Table 1 change slightly, the significance levels of the Mann-Whitney tests remain constant.
Further analysis on the speed of convergence to the efficient equilibrium shows that 40% of pairs of friends and strangers converge in the first five rounds in Game 1, while the percentage of convergence in the last five rounds is close to zero. This means that the convergence to the costless efficient equilibrium for both friends and strangers basically occurs in the first rounds of the game.
A further analysis of the data shows that no pairs of friends converge to the efficient equilibrium in the last ten rounds.
Recall that in the ‘envy’ game was only played by 90% of the total subject pool, we lost 10% of our observations to conduct this econometric analysis.
As we expected to observe a higher percentage of ones (efficient strategy) than zeros, we have chosen the probit instead of the logit model.
Although in the context of our experiment it makes more sense to use a bivariate model, after running the regression we observed that the correlation coefficient between error (usually denoted as ρ) is very close to 0. Therefore, the estimated coefficients of explanatory variables are very similar if we consider two univariate probits.
The following variables are dummies: Game j× Friend =1 when the observation is from game j and Friend matching protocol, j=1, 2, Game j× Stranger =1 when the observation is from game j and Stranger matching protocol, j=1, 2. We dropped Game 1 × Stranger to avoid perfect multicollinearity.
The terms and definitions are taken from the book of M.O. Jackson (2008).
We computed these network measures using pajek software.
In this way, we do not have to lose any information when computing a new value for each explanatory variable that captures both the row and the column player effect (for instance, the mean of both variables).
Note that there are only 3 different specifications of the econometric model. However as the model is bivariate, there are 2 equations for each regression. This is why there are 6 columns in Table 2.
We found that explanatory variables ‘envy’ and clustering_column, in-degree_row and in-degree_column, betweenness_row(column) and out-degree_row(column), betweenness_row and clustering_row, in-degree_row and clustering_row are correlated two by two. This is concluded from the following analysis. For pairwise partial correlation coefficients higher than 0.4, we consider a regression in which all independent variables are significant, including only one of the variables which may be correlated (for instance clustering_row). Then we run a new regression adding the variable (in-degree_row) which is possibly correlated with the first one. If the original variable (clustering_row) turns out not to be significant in the new regression, we deduce that these two variables are correlated.
Note that this result does not contradict the previous result of friendship decreasing efficiency. While the first result refers to the density and cohesiveness of your social network, the second one refers to the fact of playing with a friend or not.
Many of the sociodemographic variables are correlated such as: {age, course, hours of work}, {number of rooms, people living at home}, {educational level of the family head, work of the family head}. Given that most of them are not significant and we have a considerable number of explanatory variables, we have decided not to include them in Table 3.
The p-value (and the associated statistic) reported here is the minimum of the three regressions considered in Table 3.
All the tests performed throughout this section are one-tailed.
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Acknowledgements
We are very grateful to Gary Charness for his useful comments at the earliest stages of this project. We would like to thank Antoni Bosch, Pablo Brañas-Garza, Catherine Eckel, Maria Paz Espinosa, Rosie Nagel, Giovanni Ponti and Pedro Rey-Biel for all their helpful comments and suggestions. We would also like to thank Laura Crespo, Teresa García, Elena Martinez, Juan Mora, Ana Moro and Carlos Sánchez for their irreplaceable help in the econometrics. Financial support from the Generalitat Valenciana GV 06/275, the Spanish Ministry SEJ2007-62081/ECON and the Junta de Andalucia SEJ-2547 is gratefully acknowledged.
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Cobo-Reyes, R., Jiménez, N. The dark side of friendship: ‘envy’. Exp Econ 15, 547–570 (2012). https://doi.org/10.1007/s10683-012-9313-0
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DOI: https://doi.org/10.1007/s10683-012-9313-0