Abstract
When every individual’s effort imposes negative externalities, self-interested behavior leads to socially excessive effort. To curb these excesses when effort cannot be monitored, competing output-sharing partnerships can form. With the right-sized groups, aggregate effort falls to the socially optimal level. We investigate this theory experimentally and find that while it makes correct qualitative predictions, there are systematic quantitative deviations, always in the direction of the socially optimal investment. Using data on subjects’ conjectures of each other’s behavior we investigate altruism, conformity and extremeness aversion as possible explanations. We show that deviations are consistent with both altruism and conformity (but not extremeness aversion).
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Notes
The formation of groups has been investigated in public goods environments by Page et al. (2005), Ahn et al. (2008, 2009), Brekke et al. (2009), and Charness and Yang (2010). There is a related literature on endogenous institution formation by Kosfeld et al. (2009), Sutter et al. (2010), and Isaac and Norton (2013).
Establishing how subjects partition themselves endogenously is important, since, in the field, participants choose how many groups to form. If players turned out always to choose a suboptimal number of groups or, having chosen the optimal number of groups, turned out to invest differently than when assigned to them exogenously, then our laboratory society would never achieve efficiency even if, as in Schott et al. (2007), subjects make socially optimal choices when the optimal group structure is exogenously mandated.
Recruited subjects included graduate students at University of Michigan and individuals from outside the university. These subjects were randomly distributed across different treatments.
The same formulation also occurs in the innovation-tournament literature (Baye and Hoppe, Theorem 1), the sports-contest literature (Dietl et al. 2008, Eq. 2 with \(\alpha =\gamma =1),\) and the “rent-seeking” literature (Chung 1996, Eq. 2). For other literatures where this model appears, see the excellent book-length survey by Konrad (2009). In each of these literatures, player \(k\) chooses effort/investment \(x_{k}\) and achieves payoff \(\frac{x_{k}}{x_{k}+X_{-k}}v(x_{k}+X_{-k})-cx_{k}\) where \(v(\cdot ),\) the reward function, gives the value of the output or prize. When the payoff function of each player is rewritten as \(x_{k}(P(x_{k}+X_{-k})-c), \) where the strictly decreasing function \(P(x_{k}+X_{-k})=\frac{v(x_{k}+X_{-k})}{x_{k}+X_{-k}},\) the paternity of this ubiquitous model becomes apparent: it is the Cournot model (1838) in disguise (with \(x_{k}\) reinterpreted as effort instead of output). In the Cournot model, of course, the negative aggregate spillover problem results in larger industry output than a monopolist would choose. These literatures all assume that \(v(\cdot )\) is strictly concave but differ in whether this function reaches a maximum (as in the Cournot model) or is strictly increasing (as in Baye and Hoppe 2003); the qualitative results in these literatures are unaffected by this minor difference in assumption. We chose the simpler of the two formulations as easier to explain to subjects. We assumed that \(P(X)\) decreases linearly, which implies that \(v(X)\) is a parabola that passes through the origin, rises to a maximum at \({\hat{X}}\), and then returns to the \(X\) axis again at \(2{\hat{X}};\) for \(X>2{\hat{X}}, v(X)=0.\)
Note that in our experiment, voting is used to select group sizes and not some policy regarding the level of effort/investment. Voting is found to be useful as a tool for policy selection in common pool resources or public goods literature (Walker et al. 2000; Tyran and Feld 2006; Putterman et al. 2010).
The order of the group sizes was changed across sessions. The sequence of group sizes for each cost treatment was as follows: 1236, 6312, 2163, 3621, and 1623 for \(c=1\); 1236, 3261, 6312, 2163, and 6231 for \(c=20\); 1236, 3621, 2163, 6312, and 1623 for \(c=55\); 1236, 6312, 2163, 3261, and 6132 for \(c=100\). Note that in each case the optimal group size (see column 2 of Table 1) was presented at each location at least once.
As will be explained in more detail in Sect. 3, when divided into optimal group sizes, subjects’ total payoff will be close to the socially optimal payoff.
Total investment within each group is uniquely determined in the equilibrium, but not the investment of individual members of a group. Therefore, we focus on the mean investment level. See Heintzelman et al. (2009) for more details.
In the absence of any effort cost (\(c=0\)), it is socially optimal to put everyone in a single group since that would internalize the negative externality; hence, in this extreme case (\(c=0\)), equilibrium investment in Proposition 1 (for \(m=1\)) equals socially optimal investment in Proposition 2. When \(c>0,\) however, such internalization is not optimal; it would generate massive free riding since effort cannot be monitored.
In this experiment, subjects faced a discrete action space. Though the theoretical predictions were generated from a game with continuous actions, the assumption of discrete actions does not change the predictions. More specifically, suppose agents choose a noninteger investment level \(x\) for Project B in the symmetric equilibrium of the continuous investment game. Then, in the discrete version, there is an equilibrium in which every player chooses the integer above \(x\) or below \(x\), or mixes between the two. As a result, both the actions and the payoffs in the discrete and continuous versions are very similar.
We piloted an alternative voting mechanism which has a unique subgame- perfect equilibrium. In it, voting for the partnership solution should always occur in the first stage. In this alternative mechanism, after each subject had voted, one of the six subjects was randomly chosen to be “dictator,” and the vote he had cast determined the partnership structure. Since every subject anticipates with positive probability being chosen as the dictator, each subject should vote for his or her most preferred alternative (the partnership solution). However, we were unable to distinguish empirically behavior under the more familiar plurality rule from that under the more contrived dictatorial rule. Hence, we chose to use the more familiar plurality rule in this paper.
The two exceptions are when cost is 20 and group size is six and when cost is 55 and group size is one. In these cases, investments are not significantly different than the predicted levels.
For group sizes greater than 1, complete free riding is not observed as predicted. This is consistent with behavior observed in public goods experiments. It has been documented that subjects do not free ride completely (see Ledyard 1995).
Data are clustered by 20 sessions.
For robustness checks, we have also conducted OLS regressions. None of the qualitative results change with this method. Results are available upon request.
In our experiment, socially optimal level is sometimes higher than the Nash contribution level and sometimes lower. When we control for the location of the socially optimal investment level compared to the Nash level, we then ask whether the order of the treatments and experience of 5 rounds within that treatment matter. We repeated the same regression for these two cases. Our qualitative results on the effect on cost and groupsize are exactly the same as before. In addition, we still do not find any effect of round. However, now we see some effect of the block when socially optimal level is below the Nash equilibrium level. The log odds of investing closer to the Nash level is slightly higher for later blocks. We see a much smaller effect of block when the Nash level is higher than the socially optimal level.
An anonymous referee suggested the possibility that subjects flagged groups of size 6 as “bad” since there is a jump between group sizes 3 and 6 (due to the integer constraint). While this may have some affect on choices, Table 6 do not show any strong biases.
We use conditional logit model (instead of multinomial logit model) since we have alternative-specific regressors. Alternative-specific regressors are those that vary across both cases and alternatives. Case-specific regressors do not vary across alternatives.
We have also run a specification to see whether there is any change in choices over the rounds, but we did not observe any effect.
Since ties are broken randomly, even though there are equal numbers of votes for group sizes 2 and 3 when cost is 20, group size 2 won the voting more frequently than group size 3.
We focus on the partnership solution, since votes are more often for the optimal group size. Therefore, there are not too much data available on the other group sizes. In fact, there are too few data points for many of the nonoptimal group sizes, which makes statistical testing not very meaningful.
We have also used a normalized measure of efficiency. We took the difference between the observed average payoff and the minimum possible payoff and divided this with the difference between the socially optimal payoff and the minimum possible payoff. This alternative measure gives very similar efficiency levels, and all our qualitative results are the same.
The efficiency levels reported in Table 9 as well as voting percentages reported in Table 6 could be different if the experiment did not have subjects experience exogenous group sizes before endogenous group size selection. We conjecture that this sequencing choice, if anything, tends to understate observed efficiency levels and the percentage of votes for partnership solution in the experiment. We base our conjecture on two considerations: in public goods environments (albeit without institutional change), efficiency declines over time. In addition, if we had only run the endogenous choice sessions, subjects would have had no opportunities to be misled by anything that they had experienced earlier in the experiment and would likely have responded only to the induced incentives.
Note that since group size can only be an integer, partnership solution size does not imply that the highest possible level of efficiency will be reached at that group size. It only implies, given selfish individuals, that this predicted group size will bring higher efficiency compared to other group sizes. While for \(cost=1,20,55,\) the data are consistent with this prediction, for \(cost=100\), higher efficiency is achieved for group size of 3. As Table 2 shows, the gap between the Nash equilibrium investment level at the partnership solution and the socially optimal investment level is highest for \(cost=100\). This explains why higher levels of payoffs are reached at \(cost=100\) compared to the predicted payoff of 640 (see Table 8), but not at the other cost levels.
Following the approach of Noussair et al. (1995), we run the following specification:
$$\begin{aligned} Investment_{i}=\alpha \left( \frac{1}{t}\right) +\beta \left( \frac{t-1}{t}\right) +\epsilon _{i}, \end{aligned}$$where \(i\) indicates observations and \(t\) represents the period number. Notice that as \(t\) gets larger, \((\frac{1}{t})\) approaches zero, and \((\frac{t-1}{t})\) approaches 1. Hence, the constant \(\beta \) gives the asymptotic estimate of investment, and the constant \(\alpha \) gives an estimate for the initial investment.
In contrast to these findings, Schott et al. (2007) report no systematic departures from self-interested behavior. Perhaps the difference in our results arises because we had fewer subjects per session (6 subjects instead of 12 subjects) or because we explored a range of cost parameters while they confined themselves to a single cost parameter (outside of our range).
In the dictator game, a selfish dictator would contribute nothing to the other player. In the linear public goods game, selfish players would make no contributions to the public account. Positive contributions in either game are interpreted as altruism but, as Ledyard (1995) pointed out, all errors in such games must lie on the side of the Nash prediction interpreted as altruism. In contrast, aggregate investment motivated by altruism in our experiment is predicted to be larger than aggregate investment motivated by self-interest for some sets of parameters (the number of groups and the cost of investing) but smaller than self-interested investment for other sets of parameters. This makes it all the more remarkable that every departure from Nash equilibrium that we observe is in the direction of socially efficient aggregate investment.
Velez et al. (2009) pioneered the approach of using conjecture data to test alternative theories in an environment with negative externalities by simply asking subjects about their beliefs. Instead, we do not directly ask subjects’ conjectures (and, therefore, we do not draw subjects’ attention toward their conjectures). In addition, while subjects do not receive direct monetary payments for their conjectures, they do have significant incentives to enter their correct conjectures since they can learn their payoffs through Situation Analyzer without suffering any computational errors in calculating the consequences themselves.
Subjects were not compelled to use the Situation Analyzer but did so in 2,160 of the 3,000 investment decisions. Subjects received no financial reward (beyond the payoffs in the game) from entering conjectures which turned out to be accurate.
As a further check, we repeated our analysis directly with investment data (without using any conjecture data) and found similar qualitative results. These results are available from the authors upon request.
Note that the Situation Analyzer does not provide information about the payoff consequences for other subjects. In an environment where such information is available, one would expect even stronger evidence of altruism.
Our results do not mean that there is no subject with a preference to avoid extreme decisions. In fact, if we only have ownpay and extreme as explanatory variables, then extreme has a positive and highly significant coefficient. Instead, what we learn is that extremeness aversion does not have any additional explanatory power when we also consider altruism and conformity.
As long as the deviation between the actual investment and the predicted best response is less than one, we assumed the subject best responds.
Using conjecture data, Velez et al. (2009) estimate an average individual best response function and show that conformity is the dominant preference. It is encouraging to see that we also find a strong preference for conformity even though our data derive from a very different economic and experimental environment and we utilize very different estimation strategy.
Our Situation Analyzer restricted subjects from entering conjectures larger than 30 since the 5 other subjects each had 6 tokens to invest. It turns out that, for group sizes 2 and 3, we should have further restricted their conjectures. For example, for a group size of 3, the other 2 members of the group could not feasibly invest more than 12 tokens in Project B and the 3 nonmembers could not feasibly invest more than 18 tokens. In conducting our statistical analysis of conjecture data, we eliminated conjectures that were “out of range” since we regard them as reflecting a misunderstanding (or possibly a subject experimenting with the Situation Analyzer). Our results are unchanged whether or not these conjectures are included.
Future research should address the stability of the partnership mechanism. By stability, we mean migrations of subjects from one existing group to another or from an existing group to a newly formed group. Heintzelman et al. (2009) predict that no such migrations should occur. However, they also predict that migrations to newly formed solo partnerships would occur unless there was a direct cost to such migrations or there was a benefit to team production that would be lost by going solo. Their predictions, however, are based on self-interested behavior. As we have found in our voting treatment, altruism reduces the incentive of subjects to go solo. In addition, preplay communication may affect cooperation levels (Charness 2000; Charness and Dufwenberg 2006; Brandts and Cooper 2007; Chaudhuri et al. 2009; Brandts et al. 2012). For a preliminary study of the effects of such communication on the partnership solution, see Buckley et al. (2009, 2010).
This is based on the investment data from the exogenous groups since there are very few data points for statistical testing for endogenous groups at the nonoptimal group sizes.
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Acknowledgments
An earlier version of this paper was circulated under the title “Size Matters (in Output-Sharing Groups): Voting to End the Tragedy of the Commons.” This research was completed while Salant was a visitor at the Department of Economics at NYU and benefited from the generous suggestions of Guillaume Fréchette and Andrew Schotter. We thank Abigail Brown, Gary Charness, Yan Chen, Rachel Croson, Silvana Krasteva, Erin Krupka, Yusufcan Masatlioglu, Daisuke Nakajima, Yesim Orhun, Helen Popper, Tanya Rosenblat, Mel Stephens, Jason Winfree, and Homa Zarghamee for valuable comments on previous drafts. We also thank the seminar participants at Columbia, Johns Hopkins, Michigan State University, New York University, University of Michigan, Paris Environmental and Energy Economics Seminar (PEEES), Santa Clara University, UCSB, and Yale, and audiences at the Erb Institute, Resources for the Future, the Montreal Natural Resources and Environmental Economics Workshop, the 2009 and 2010 ESA Meetings, the 2010 SEA Meetings, and the 2010 APET Meetings. Experiments were run at the Institute of Social Research at University of Michigan. Subject payments were financed by a grant from the Erb Institute at University of Michigan.
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Cherry, J., Salant, S. & Uler, N. Experimental departures from self-interest when competing partnerships share output. Exp Econ 18, 89–115 (2015). https://doi.org/10.1007/s10683-014-9413-0
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DOI: https://doi.org/10.1007/s10683-014-9413-0