Assortative Matching with Inequality in Voluntary Contribution Games
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Abstract
Voluntary contribution games are a classic social dilemma in which the individually dominant strategies result in a poor performance of the population. However, the negative zerocontribution predictions from these types of social dilemma situations give way to more positive (near)efficient ones when assortativity, instead of random mixing, governs the matching process in the population. Under assortative matching, agents contribute more than what would otherwise be strategically rational in order to be matched with others doing likewise. An open question has been the robustness of such predictions when heterogeneity in budgets amongst individuals is allowed. Here, we show analytically that the consequences of permitting heterogeneity depend crucially on the exact nature of the underlying publicgood provision efficacy, but generally are rather devastating. Using computational methods, we quantify the loss resulting from heterogeneity visavis the homogeneous case as a function of (i) the publicgood provision efficacy and (ii) the population inequality.
Keywords
Assortative matching Public goods Heterogeneity Inequality Computational economics1 Introduction
Suppose a population of agents faces the collective action (Olson 1965) challenge to provide public goods by means of simultaneous, separate voluntary contributions games (Isaac et al. 1985). In each one, the collective would benefit from high contributions but individuals may have strategic incentives (Nash 1950) to contribute less. Such situations, also known as ‘social dilemmas’, are related to collective management of ‘commonpool resources’ (Ostrom 1990; Schlager and Ostrom 1992) and often result in underprovisioning of the public good (i.e. tragedy of the commons as in Hardin 1968) because of the misalignment of collective interests and strategic incentives.
Generally, grave underprovision of the public good is the unique Nash equilibrium when individual contribution decisions are independent of the matching process. Andreoni (1988)’s model of a linear public goods game with random rematching of groups is the bestknown experimental instantiation of this, and numerous studies have reported corresponding decays in contributions when such games are played in the laboratory (Ledyard 1995; Chaudhuri 2011). Predictions may change dramatically, however, when agents are matched ‘assortatively’ instead, that is, based on their precommitted choice on how much to contribute so that high (low) contributors are matched with other high (low) contributors. Such mechanisms have been coined ‘meritocratic groupbased matching’ (Gunnthorsdottir et al. 2010a), short ‘meritocratic matching’ (Nax et al. 2014).^{1} Under meritocratic matching, new equilibria emerge through assortative matching that are as good as nearefficient (Gunnthorsdottir et al. 2010a; Nax et al. 2014). Indeed, when better (i.e. more efficient) equilibria exist, humans have been shown to consistently play them in controlled laboratory environments (Gunnthorsdottir et al. 2010a, b; Nax et al. 2017; Rabanal and Rabanal 2014).
In this paper, we address the important question of how robust the positive predictions stemming from assortative matching are. To assess this, we generalize the baseline model on two dimensions. On the one hand, we consider a range of publicgoods provision efficacies that nests the standard marginalpercapitarateofreturn (‘mpcr’) model as a special, linear case. On the other hand, we allow heterogeneity in players’ budgets, expressing the ex ante inequality amongst individuals. In other contexts, heterogeneity has been shown to ‘help’ cooperation (Perc 2011). Our work, in particular, builds on one prior attempt at generalizing the standard model in terms of heterogeneity by Gunnthorsdottir et al. (2010b), who consider two levels of budgets in the standard case of mpcrlinear payoffs.
Methodologically, we blend analytical and computational approaches. Our results summarize as follows. We show analytically that the consequences of permitting heterogeneity in terms of provision of the public good depend crucially on the exact nature of the underlying publicgood provision efficacy, but generally are devastating. Indeed, all nearefficient Nash equilibria that exist under homogeneity fall apart when heterogeneity is allowed. Instead, we are either back at the negative allcontributenothing equilibrium or new, previously impossible, complex mixedstrategy Nash equilibria emerge. In the latter case, the expected level of resulting publicgood provision depends crucially on (i) the publicgood provision efficacy and (ii) the population inequality. These mixed equilibria are virtually impossible to characterize and to evaluate analytically for general cases. We therefore use computational methods and quantify the loss resulting from heterogeneity visavis the homogeneous case as a function of parameters regarding (i) and (ii). Thus, our analysis provides novel insights regarding the possible consequences in terms of making wrong predictions when assuming a homogeneous population, which in many realworld cases may be unrealistic.
The rest of this paper is structured as follows. Next, we set up the model including details about our computational algorithm. Section 3 contains the paper’s results. Section 4 concludes. An “Appendix” contains details of the analytical results.
2 The Model
 1.
Actions Each player makes a simultaneous and committed unilateral decision regarding how much of his endowment to contribute. We will indicate with \(\alpha _i \in \left[ 0,1\right] \) the percentage of \(w_i\) contributed by player i. We indicate with \(\alpha = \left\{ \alpha _i \right\} \) the array of strategies obtained in this way and with \(\alpha _{i}\) the strategy vector obtained excluding agent i.
 2.
Matching Players are ranked by their effective contributions \(s_i = w_i \alpha _i\) (from highest to lowest with random tiebreaking). They are then assigned to \(M=\frac{N}{{{\mathcal {S}}}}\) equalsized groups of size \({{\mathcal {S}}}\), such that the \({\mathcal {S}}\) highestranking players are assigned to the first group, the \({\mathcal {S}}\) secondhighest ranking players are assigned to the second, etc.
 3.Outcome Payoffs \(\phi _i\) realize based on the contribution total in each group. Each player receives the amount that he did not contribute plus the sum of all the contributions made by the members of his group multiplied by a factor Q (the marginalpercapitarateofreturn):with \(G_i\) being the group to which agent i belongs.$$\begin{aligned} \phi _{i}\left( \alpha _{i}\mid \alpha _{i}\right) =w_i \left( 1\alpha _{i}\right) +Q\sum _{j\in G_{i}}\alpha _{j}^{\gamma }w_{j} \end{aligned}$$(1)
The parameter \(\gamma \) in the return from the common pool is a measure of the “goodness” of the publicgood provision efficacy. For high values of \(\gamma \) (superlinear payoffs) even a high percentage of contributions has little effect on the common pool, thus making the public good less fruitful. On the other end, for low \(\gamma \) values (sublinear payoffs) even a small contribution allows players to obtain large benefits from cooperation. It is important to notice that the value of \(\gamma \) determines also the maximum efficiency ^{2} of the system ranging from \(w_i +Q\sum _{j\in G_{i}}w_{j}\) for \(\gamma \rightarrow 0 \) to \(w_i\) for \(\gamma \rightarrow \infty \).
2.1 Simulation
 1.

Initialization: We assign the initial endowments to each player sampling them from a truncated Gaussian distribution with mean \(W_0\) and standard deviation \(\sigma \). The distribution is truncated at \(w_i=0\) and \(w_i= 2W_0\) so that endowments are always positive and symmetrically distributed around \(W_0\).
 2.

Setting initial strategies: The initial strategy \(\alpha _i \) of each player is set to 0, hence the simulation starts in the fully defective state.^{3}
 3.

Dynamics of play: Each player updates his strategy in the following way:

With probability p he keeps playing the strategy played the round before.^{4}

With probability \(1p\) he myopically best responds to the strategies played by the other agents the round before:
The agent checks what would his rank and consequent payoff be for each of the possible strategies^{5} he can play, given that the other agents keep playing the strategies played the round before. He then chooses the strategy that results in the highest payoff.
 4.

Group matching: Based on the contribution of each player, groups are formed as described above and payoffs are then materialized.
 5.

Update and repeat: The endowment of each player is then reset to his initial one and the algorithm repeats from step 3.
3 Results
The above game exhibits different Nash equilibria depending on the value of \(\gamma \) and on the players having different or the same initial endowments.
As already shown in Gunnthorsdottir et al. (2010a), in the case of linear payoff (\(\gamma = 1\)) and homogeneous players the voluntary contribution game with assortative matching has multiple pure strategy Nash Equilibria: one is non contribution by all players and the others are almost Pareto optimal equilibria in which nearly all players contribute their entire endowment and few (less than the group size) contribute nothing. It is easy to see (see “Appendix”) that this result actually holds for any value of \(\gamma \) bigger than a threshold value \(\bar{\gamma }\), with \(\bar{\gamma } < 1\) and depending on groups size, mpcr and the total number of players. The only difference with the linear payoff case is that for \(\gamma < 1\) the within group Nash Equilibrium is to contribute \({\bar{\alpha }} = \left( \frac{1}{Q \gamma }\right) ^{\frac{1}{\gamma 1}}\). Hence, for sublinear payoffs such that \(\gamma > \bar{\gamma }\), the Nash Equilibria are: all players contribute \({\bar{\alpha }}\) and all players contribute their total endowment except for few players that contribute \({\bar{\alpha }}\). For very small values of the exponent of the public good provision, the numbers of non contributors^{6} in the near efficient equilibrium becomes too high to be sustained. Hence, even though the nearly efficient outcome would be highly rewarding, for \(\gamma \le \bar{\gamma }\) the only existing equilibrium is that all players contribute \({\bar{\alpha }}\).^{7}
A key property of the game, necessary for the nearly efficient equilibria to exist, is that ties in the ranking placement are broken at random. In all these equilibria, in fact, there exists a mixed group where fully contributive players are grouped with defectors, i.e. players contributing the within group Nash Equilibrium. In order for this to be a NE, the fully contributive players must have a sufficiently high probability to be grouped only with other full contributors so that they would not benefit (in expectation) by decreasing their contribution and be placed with certainty in the defectors group.
If each player is endowed with a different initial wealth, however, things change drastically. The heterogeneity of the players implies that there cannot be an equilibrium where more than one player contributes the same positive percentage of his endowment: Since players are ranked based on their effective contributions \(s_i = w_i \alpha _i\), if more than one player were to contribute the same percentage, the one with the highest endowment would have a profitable deviation due to the existence of the mixed group. He could in fact contribute slightly more and be guaranteed to be placed in a better group. If the two players were already contributing everything, the wealthiest player could instead contribute slightly less and still be guaranteed to remain in the same group. For the above reason, the nearly efficient Nash Equilibria in which almost all players contribute everything does not exist for heterogeneous players. Moreover, any unique contribution \(\alpha _i\) such that \({\bar{\alpha }}< \alpha _i < 1\) is also clearly not a Nash Equilibrium due to the fact that it would be possible to contribute less and still be placed in the same group.
Furthermore, for sublinear payoffs (\(\gamma < 1\)) the equilibrium in which all players contribute the within group Nash Equilibrium does not exist. Indeed, for \(\gamma < 1\) we have that \({\bar{\alpha }}>0\) and if all players were to play \({\bar{\alpha }}\), players with a lower endowment would have a profitable deviation by increasing their contributions and being grouped with players with a higher endowment. Hence, for heterogeneous players and sublinear payoffs there exist no pure strategy Nash Equilibria for the game.
For superlinear payoffs (\(\gamma \ge 1\)), however, the within group Nash Equilibrium is to contribute nothing and hence the pure strategy equilibrium in which no player contributes anything continues to exist. Consequently, there are no mixed strategy Nash Equilibria for this game.
In this table we show which equilibria exist for homogeneous and heterogeneous players depending on the value of \(\gamma \)
\({\gamma \le \bar{\gamma }}\)  \({\bar{\gamma }< \gamma <1}\)  \({\gamma \ge 1}\)  

PSL  MS  PSH  PSL  MS  PSH  PSL  MS  PSH  
Homog.  ✓  ✗  ✗  ✓  ✗  ✓  ✓  ✗  ✓ 
Heterog.  ✗  ✓  ✗  ✗  ✓  ✗  ✓  ✗  ✗ 
To obtain the mixed strategy equilibrium of the game we resort to computational methods. We simulate agents playing the game for different payoffs and different width of the initial wealth distribution.
We are mainly interested in a comparison between the (unique) equilibria in the case of heterogeneous players and the equilibria reached in the homogeneous case. In particular, we are interested in how much efficiency (see footnote 2) is lost due to the heterogeneity of players. Indeed, even though non contribution by all is a Nash Equilibrium, the quasi Pareto optimal equilibrium is payoff dominant^{8} and hence is the one to which we refer (when it exists). Furthermore, experimental results (Gunnthorsdottir et al. 2010a) have also shown that the nearly efficient equilibria are the one reached by the population.
We observe, as predicted, that for superlinear payoffs the only possible equilibrium is non contribution by all and thus that the loss of efficiency is total. For intermediate sublinear payoff the system achieves different efficiency level, from quite low ones to levels closer to the homogeneous case. The quantitative value of efficiency that the mixed equilibrium achieves depends on the value of the mpcr and the width of the distribution of initial wealth as well as from other parameters. For values of the exponent of the public good provision lower than \(\bar{\gamma }\), we initially observe an increase in efficiency when endowments are heterogeneous. This is due to the fact that for values of \(\gamma \) slightly below \(\bar{\gamma }\), the only equilibrium in the homogeneous case is to contribute \({\bar{\alpha }}\) but it would still be more efficient if more players contributed a higher percentage of their endowment. Finally, for \(\gamma \rightarrow 0\), the heterogeneous system approaches the same efficiency of the homogeneous, on account of the benefits of cooperation being obtainable for an arbitrary small contribution.
Interestingly, one can observe that the efficiency loss doesn’t seem to change much for different widths of the endowment distribution (see online material).
For the width of the distribution equal to 0 we observe, as expected, the Nash Equilibria in case of homogeneous endowments.^{9}
For different values of the marginal per capita rate of return we observe from the simulations (see online material for examples) that the higher the mpcr, the wider is the area with partial efficiency losses in the picture and the smaller is the gain in efficiency around \(\bar{\gamma }\).
Hence we can conclude that for a wide range of payoffs, the more realistic assumption of heterogeneous players leads to a disruptive loss in efficiency when compared to the homogeneous case. For a very limited range of \(\gamma \), however, heterogeneity seems to result in a small increase in efficiency.
4 Summary of Results
In summary, mechanisms based on assortative matching promise large efficiency gains when the interaction is such that it is safe to assume that the population consists only of equals. In the presence of heterogeneity, however, whether and how much assortative matching is likely to gain the population (relative to random matching) depends crucially on the provision efficacy of the public good and on the precise degree of heterogeneity. This implies that guarantees of more equal playing fields in these environments may be as important as implementation of assortative matching.
Footnotes
 1.
 2.Definition: Efficiency is defined as the gain from the game relative to the initial endowment:$$\begin{aligned} { Efficiency}=\frac{\sum _{i=1}^{N}\phi _{i}\sum _{i=1}^{N}w_{i}}{\sum _{i=1}^{N}w_{i}}. \end{aligned}$$
 3.
Different initial starting conditions have been explored and they have been observed to have no effect on the final outcome of the simulation.
 4.
Inertia was added to ensure the convergence to the purestrategy Nash equilibrium (if existing). Without inertia, the bestresponse dynamics could oscillate around such an equilibrium.
 5.
The interval \(\left[ 0,1 \right] \) is discretized for computational purposes.
 6.
i.e. people contributing only \({\bar{\alpha }}\).
 7.
All players contributing \({\bar{\alpha }}\) is always an equilibrium for homogeneous players. The proof proceeds like in the linear case: it does not make sense to be the only player contributing more than that because this would only make the player’s groupmates better off at the player’s expense. Contributing less than \({\bar{\alpha }}\) is never beneficial, not even with random rematching of groups, because of the nonlinearity of the payoff function. See the “Appendix” for more details.
 8.
 9.
Note that there is no continuous convergence from the heterogeneous to the homogeneous case. This is a direct consequence of the fact that, as discussed before, as soon as heterogeneous endowments are introduced, there cannot be an equilibrium where more than one player contributes the same positive percentage of his endowment. This makes the existence of a mixed group impossible. See Lemma 7 in the “Appendix”.
 10.
Or alternatively, \(D_{r}\) and \(\widetilde{c}_{r}\) are defined as the unique nonnegative integers such that \(\left C_{r}\right =D_{r}\cdot {\mathcal {S}}+\widetilde{c}_{r}\).
 11.
Actually it is already valid for “enough” heterogeneity.
 12.Because:$$\begin{aligned} A= & {} Q\left[ s^{1}{\mathcal {S}}+\varepsilon \right] \varepsilon> Qs^{1}{\mathcal {S}}\cdot \left( 1\frac{\widetilde{c}_{1}}{c_{1}}\right) +Q\left[ s^{1}\widetilde{c}_{1}+s^{2}\left( {\mathcal {S}}\widetilde{c}_{1}\right) \right] \cdot \frac{\widetilde{c}_{1}}{c_{1}} \\ A> & {} Qs^{1}{\mathcal {S}} + \varepsilon> Qs^{1}{\mathcal {S}}\cdot \left( 1\frac{\widetilde{c}_{1}}{c_{1}}\right) +Qs^{1}{\mathcal {S}}\cdot \frac{\widetilde{c}_{1}}{c_{1}}+Q\left[ s^{1}\left( \widetilde{c}_{1}{\mathcal {S}}\right) +s^{2}\left( {\mathcal {S}}\widetilde{c}_{1}\right) \right] \cdot \frac{\widetilde{c}_{1}}{c_{1}} \\& \varepsilon > Qs^{1}\left( \widetilde{c}_{1}{\mathcal {S}}\right) \frac{\widetilde{c}_{1}}{c_{1}}\left[ \frac{s^{2}}{s^{1}}1\right] \end{aligned}$$
 13.
With the endowments ranked from the highest to the lowest.
Notes
Acknowledgements
We thank the associate editor, the anonymous reviewers, Michael Mäs, Ryan Murphy, Stefano Balietti, and the colleagues at the COSS chair at ETH Zurich for helpful discussions and comments and for helping improving the manuscript. All remaining errors are ours. Financial support from the European Commission through the ERC Advanced Investigator Grant Momentum (Grant 324247) is gratefully acknowledged.
Supplementary material
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