Appendix: Nash Equilibria
In this section we derive the Nash Equilibria of the generalized assortative matching voluntary contribution game.
We first define the game and derive some useful properties of it. Later, we show which equilibria exist for homogeneous and heterogeneous players for different values of the public good game efficacy.
Notation
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The expected payoff of player i is
$$\begin{aligned} E_{i}\left( \alpha _{i},\alpha _{-i}\right) =\left( w_{i}-s_{i}\right) +\sum _{k=1}^{M}Pr\left( k\mid \alpha _{i},\alpha _{-i}\right) \cdot \left[ Q\cdot \left( S_{-i}^{k}+w_i\alpha _{i}^\gamma \right) \right] \end{aligned}$$
(2)
with \(M=\frac{N}{{\mathcal {S}}}\) the number of groups in the population, \(S_{-i}^{k}\) the sum of the effective contributions in agent’s i group minus his own and \(Pr\left( k\mid \alpha _{i},\alpha _{-i}\right) \) the probability of being ranked kth given \(\alpha _i\) and \(\alpha _{-i}\). We indicate with \({\mathcal {S}}\) the size of the groups.
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We say that players i, j are in class \(C_r\) if \(s_i = s_j = s^r\). We write \( c_{r}\equiv \left| C_{r}\right| =D_{r}\cdot {\mathcal {S}}+\widetilde{c}_{r} \); \(D_{r},\widetilde{c}_{r} \in N\cup \left\{ 0\right\} \).Footnote 10 Note that by definition \(0 \le \widetilde{c}_{r} < {\mathcal {S}}\).
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We call h the highest effective contribution and H the number of players s.t. \(\alpha _i = h\); hence \(H\equiv \left| C_{1}\right| \).
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We indicate with z the number of players playing the strategy \({\bar{\alpha }}\).
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Full heterogeneity means that \(w_i \ne w_j \;\; \forall i\ne j\).
Let us first compute the within group best response.
Lemma 1
The best response within a group is: \(\alpha _i={\bar{\alpha }}\) if \(0<\gamma <1\) and \(\alpha _i = 0\) if \(\gamma \ge 1\).
Proof
The within group payoff is defined as following:
$$\begin{aligned} \phi _{i}\left( \alpha _{i}\mid \alpha _{-i}\right) =w_i\left( 1-\alpha _{i}\right) +Q\sum _{j\in G_{i}}\alpha _{j}^{\gamma }w_j \end{aligned}$$
with \(G_i\) being the group to which agent i belongs and \(\alpha _i \in \left[ 0,1\right] \).
The first order condition reads:
$$\begin{aligned} \frac{\partial \phi _{i}}{\partial \alpha _{i}}=-w_i+Qw_i\gamma \alpha _{j}^{\gamma -1}{\mathop {=}\limits ^{FOC}}0 \Rightarrow \alpha _i = \left( \frac{1}{Q\gamma }\right) ^{\frac{1}{\gamma -1}}\equiv {\bar{\alpha }} \end{aligned}$$
(3)
implying the following payoff for agent i:
$$\begin{aligned} w_i\left( 1-{\bar{\alpha }}\right) + w_iQ{\bar{\alpha }}^\gamma +C \end{aligned}$$
(4)
with C defined as \(\sum _{j\in G_{i},j\ne i}\alpha _{j}^{\gamma }w_j\).
The payoff for the corner strategies instead is \(w_i + C\) for \(\alpha = 0\) and \(w_iQ + C\) for \(\alpha = 1\).
Since \(0<Q<1\) the payoff for \(\alpha = 0\) is always bigger than the one \(\alpha = 1\). Now we need to check when \(\phi _i\left( {\bar{\alpha }}\right) > \phi _i\left( 0\right) \) and hence when
$$\begin{aligned} Q\left( \frac{1}{Q\gamma }\right) ^{\frac{\gamma }{\gamma -1}}>\left( \frac{1}{Q\gamma }\right) ^{\frac{1}{\gamma -1}}. \end{aligned}$$
If \(0< \gamma <1\) we can rewrite the above expression as \(Q\left( Q\gamma \right) ^{\frac{\gamma }{1-\gamma }}>\left( Q\gamma \right) ^{\frac{1}{1-\gamma }}\) and hence as \(\gamma ^\gamma > \gamma \); a condition that is always true for \(0< \gamma <1\). If \(\gamma >1\) we instead obtain the condition \(\gamma ^{-\gamma } > \gamma ^{-1}\), never true for \(\gamma >1\).
If \(\gamma =0\), the FOC trivially results in \(\alpha = 0\).
Hence, the best response for player i if the group placement is independent from \(\alpha _i\) is \(\alpha _i={\bar{\alpha }}\) for \(0<\gamma <1\) and \(\alpha _i = 0\) for \(\gamma >1\). \(\square \)
NE for Homogeneous Players
Here we compute what are the Nash Equilibria in the case of homogeneous players. The proof presented here follows closely the one in Gunnthorsdottir et al. (2010a), changing only to adapt it to the generalized payoff. Note that for homogeneous players we have \(w_i = w \;\; \forall i\).
We first of all note that, for homogeneous players, all players playing the within group Nash Equilibria is a best response.
Lemma 2
\(\alpha _i = {\bar{\alpha }}\) \(\forall i\) is a Nash Equilibrium for every value of \(\gamma \).
Proof
This is obviously an equilibrium. Since the mpcr Q is smaller than 1, there is no profitable deviation in being the only one contributing more than \({\bar{\alpha }}\). A player i deviating to a higher contribution would have the guarantee to be placed in the best group. The best group, however, would be such only because of him, thus making it not profitable to deviate. \(\square \)
In order to prove the existence of the high contribution equilibrium, we first observe that in order for the equilibrium to exist, the following conditions have to hold:
Lemma 3
If there are some strategies \(\alpha _i\) s.t. \(\alpha _i > {\bar{\alpha }}\), then in equilibrium we have to have \(H > {\mathcal {S}}\) and \(\left( H \; mod \; {\mathcal {S}} \right) > 0\). Hence a player contributing more than \({\bar{\alpha }}\) has a non-zero probability to be grouped with a player contributing less than him.
Proof
If the number of the highest contributors were a multiple of the group size, one of those player could unilaterally deviate and reduce his contribution by an amount small enough to remain in the same group and still profit from the deviation. For the same reason, that number of players has to be bigger than \({\mathcal {S}}\). If it were smaller, in fact, high contributors would benefit by deviating to \({\bar{\alpha }}\). \(\square \)
Lemma 4
If there are some strategies \(\alpha _i\) s.t. \(\alpha _i > {\bar{\alpha }}\), then the highest contribution h cannot be smaller than w. I.e. \(\alpha _i = 1 \; \forall i \; \in C_1\).
Proof
We call \(\alpha _h\) the strategy s.t. \(\alpha _h w = h\). From Lemma 3 we know that (one of) the highest contributor(s) i has a non-zero probability to be grouped with some other player contributing less. Hence, if agent i were playing \(\alpha _h < 1\), he could increase his contribution by an arbitrary small amount and be placed for certain in the best group.
Indeed the expected payoff for player i playing \(\alpha _h\) is at most:
$$\begin{aligned} E_{i}\left( \alpha _{h}\right) <w\left( 1-\alpha _{h}\right) +Q{\mathcal {S}}w\alpha _{h}^{\gamma }\left( 1-\frac{\tilde{c_{1}}}{c_{1}}\right) +Qw\left[ \alpha _{h}^{\gamma }\tilde{c_{1}}+l^{\gamma }\left( {\mathcal {S}}-\tilde{c_{1}}\right) \right] \left( \frac{\tilde{c_{1}}}{c_{1}} \right) \end{aligned}$$
(5)
where l is the strategy played by an agent \(j \in C_2\) and hence \(l < \alpha _h\). \(\square \)
By deviating, player i would surely be placed in the best group, gaining
$$\begin{aligned} E_{i}\left( \alpha _{h}+\varepsilon \right) =w\left( 1-\alpha _{h}-\varepsilon \right) +Q{\mathcal {S}}w\alpha _{h}^{\gamma }+O\left( \varepsilon \right) . \end{aligned}$$
(6)
We have that \(E_{i}\left( \alpha _{h}\right) < E_{i}\left( \alpha _{h}+\varepsilon \right) \) if
$$\begin{aligned} \varepsilon < Q\frac{\tilde{c_{1}}}{c_{1}}\left[ \left( {\mathcal {S}}-\tilde{c_{1}}\right) \left( \alpha _{h}^{\gamma }-l^{\gamma }\right) \right] \end{aligned}$$
that for a small enough \(\varepsilon \) is true. Hence player i would be better off deviating to \(\alpha _h + \varepsilon \). Consequently, \(\alpha _i = 1 \; \forall i \; \in C_1\).
Lemma 5
If there are some strategies \(\alpha _i\) s.t. \(\alpha _i > {\bar{\alpha }}\), then there cannot be any player j contributing \({\bar{\alpha }}< \alpha _j < 1\).
Proof
Let us call j the player with the highest contribution after all the players contributing h; i.e. \(j \in C_2\) and let us call his strategy \(\alpha _l\).
If there are no ties regarding group membership, j could reduce his contribution to \(\alpha _l - \varepsilon \) and remain in the same group.
If there are ties, j could increase his contribution by an arbitrary small amount and be sure to be grouped together with players belonging to \(C_1\). Similarly to Lemma 4, it is possible to prove that for an arbitrary small \(\varepsilon \) we have:
$$\begin{aligned} E\left( \alpha _l \right) < E \left( \alpha _l + \varepsilon \right) \end{aligned}$$
and hence that \(\alpha _j\) cannot be an equilibrium strategy. \(\square \)
Lemma 6
If there are some strategies \(\alpha _i\) s.t. \(\alpha _i > {\bar{\alpha }}\), then the number of players playing the within group best response is smaller than the group size; i.e. \(z<{\mathcal {S}}\).
Proof
From Lemma 3 we know that \(\left( H \; mod \; {\mathcal {S}} \right) > 0\) and hence
\(\left( \left( N-H\right) \; mod \; {\mathcal {S}} \right) > 0\).
If z were bigger than the group size, a player i contributing \({\bar{\alpha }}\) could increase his payoff by an arbitrary small amount and be placed with certainty in the group containing some players contributing h.
Hence there is a profitable deviation and \(z \ge {\mathcal {S}}\) could not be a Nash Equilibrium. \(\square \)
From Lemmata 3–6 follows that if an equilibrium s.t. \(\alpha _i > {\bar{\alpha }}\) for some i exists, then each player plays either \({\bar{\alpha }}\) or 1. Furthermore, the number of players playing the within group best response is smaller than the group size.
From the lemmata above we can derive under which condition the generalized voluntary contribution game has a Nash Equilibrium with high contributions level. Hence the existence of a highly efficient equilibrium depends on the marginal per capita rate of return, the number of players and the size of the groups.
Theorem 1
For values of \(\gamma \) bigger equal than a threshold value \(\bar{\gamma }\left( Q,N,{\mathcal {S}}\right) \), the generalized voluntary contribution game has Nash Equilibria in which \(z<{\mathcal {S}}\) players contribute \({\bar{\alpha }}\) and all the others \(N-z\) players contribute 1. These equilibria are in addition to the equilibrium where all players contribute \({\bar{\alpha }}\).
For \(\gamma < \bar{\gamma }\), the only NE is that all players contribute the within group best response \({\bar{\alpha }}\).
Proof
For \(N-z\) players contributing 1 to be a NE, we have to show that no full contributor has a profitable deviation to contribute \({\bar{\alpha }}\) and that no player contributing \({\bar{\alpha }}\) has an incentive to play 1. We write
$$\begin{aligned} E_{1}\left( 1\right)= & {} w\frac{{\mathcal {S}}-z}{N-z}Q\left[ \left( {\mathcal {S}}-z\right) +z{\bar{\alpha }}^{\gamma }\right] +\frac{N-{\mathcal {S}}}{N-z}Q{\mathcal {S}}w \end{aligned}$$
(7)
$$\begin{aligned} E_{1}\left( {\bar{\alpha }}\right)= & {} w\left( 1-{\bar{\alpha }}\right) +wQ\left[ {\mathcal {S}}-z-1+\left( z+1\right) {\bar{\alpha }}^{\gamma }\right] \end{aligned}$$
(8)
$$\begin{aligned} E_{{\bar{\alpha }}}\left( {\bar{\alpha }}\right)= & {} w\left( 1-{\bar{\alpha }}\right) +wQ\left[ {\mathcal {S}}-z+z{\bar{\alpha }}^{\gamma }\right] \end{aligned}$$
(9)
$$\begin{aligned} E_{{\bar{\alpha }}}\left( 1\right)= & {} w\frac{{\mathcal {S}}-z+1}{N-z+1}Q\left[ \left( {\mathcal {S}}-z+1\right) +\left( z-1\right) {\bar{\alpha }}^{\gamma }\right] +\frac{N-{\mathcal {S}}}{N-z+1}Q{\mathcal {S}}w \end{aligned}$$
(10)
where we indicate with \(E_{1}\left( \alpha \right) \) the payoff of a high contributor and with \(E_{{\bar{\alpha }}}\left( \alpha \right) \) the payoff of a low contributor.
For the above to be a NE we have to have that (7) > (8) and (9) > (10).
The first condition is equivalent to:
$$\begin{aligned} z \ge \frac{\left( 1-{\bar{\alpha }}\right) N-QN\left( 1-{\bar{\alpha }}^{\gamma }\right) }{Q\left( 1-{\bar{\alpha }}^{\gamma }\right) \left[ N-1-{\mathcal {S}}\right] +1-{\bar{\alpha }}} \end{aligned}$$
(11)
The second condition leads to:
$$\begin{aligned} z \le 1 + \frac{\left( 1-{\bar{\alpha }}\right) N-QN\left( 1-{\bar{\alpha }}^{\gamma }\right) }{Q\left( 1-{\bar{\alpha }}^{\gamma }\right) \left[ N-1-{\mathcal {S}}\right] +1-{\bar{\alpha }}} \end{aligned}$$
(12)
However, it is important to remember that z needs to be smaller than the size of the groups.
Hence, we have that for values of the exponent \(\gamma \) such that (11) is at most \({\mathcal {S}}-1\), the generalized voluntary contribution game has nearly efficient Nash Equilibria. For values of \(\gamma \) s.t. (11) is bigger than \({\mathcal {S}}-1\) the only equilibrium is the one where all players play the within group best response. We call \(\bar{\gamma }\) the value of \(\gamma \) such that eq. (11)\(={\mathcal {S}}-1\). \(\square \)
Hence we obtain the existence of a nearly-efficient high equilibrium depends on the marginal per capita rate of return, the number of players and the size of the groups.
NE for Heterogeneous Players
Here we show that the near efficient Nash Equilibrium cannot exist for heterogeneous players. Furthermore, we derive under which condition the generalized voluntary contribution game has a pure strategy NE.
In the following we prove the lemmata necessary to derive the equilibrium.
Lemma 7
In case of full heterogeneityFootnote 11 the assumptions of Lemma 3 cannot be satisfied.
Proof
Let’s assume that they are and show that this cannot be a NE. We call k the player with the lowest \(w_i\) belonging to \(C_1\).
We have two possibilities: \(\left( a \right) s_k = w_k\) and \(\left( b \right) s_k < w_k\)
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\(\left( a \right) \):
Let’s take a player j s.t. \(j \in C_1 \) and \(j \ne k\). When playing \(s_j = s^1 \), he has an expected payoff of at most (because in the mixed group there could also be players of classes lower than 2):
$$\begin{aligned} E\left( j\right) \le w_{j}-s^{1}+Qs^{1}{\mathcal {S}}\cdot \left( 1-\frac{\widetilde{c}_{1}}{c_{1}}\right) +Q\left[ s^{1}\widetilde{c}_{1}+s^{2}\left( {\mathcal {S}}-\widetilde{c}_{1}\right) \right] \cdot \frac{\widetilde{c}_{1}}{c_{1}} \end{aligned}$$
(13)
where \(\frac{\widetilde{c}_{1}}{c_{1}}\) is the probability that agent j ends up in the group where not all agents belong to \(C_1\).
But agent j could play \(\alpha _\varepsilon \) s.t. \(s_j = s^1 + \varepsilon \), being guaranteed to end up in the first group and thus having an expected payoff of
$$\begin{aligned} E_{\varepsilon }\left( j\right) =w_{j}-s^{1}-\varepsilon +Q\left( {\mathcal {S}}-1\right) s^{1}+Q\left( s^{1}+\varepsilon \right) \end{aligned}$$
But if \( \varepsilon <{\mathcal {S}}s^{1}\left( {\mathcal {S}}-\widetilde{c}_{1}\right) \frac{\widetilde{c}_{1}}{c_{1}}\left( 1-\frac{s_{2}}{s_{1}}\right) \) we have that \(E_{\varepsilon }\left( j\right) > E\left( j\right) \).Footnote 12 Hence there is a profitable deviation for agent j; so there can’t be a NE.
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\(\left( b \right) \):
The same as \(\left( a \right) \), except that now even for k it is profitable to deviate to \(s_k = s^1 + \varepsilon \).
Hence for heterogeneous players, it is impossible to maintain the conditions under which nearly efficient equilibrium was possible. \(\square \)
Lemma 8
In case of full heterogeneity of endowments \(w_i\), with all \(w_i\) having the same order of magnitude, \(\alpha _i = {\bar{\alpha }} \; \forall i\) is not a Nash Equilibrium.
Proof
If all players were playing \({\bar{\alpha }}\), they would be ranked based on their endowments. Hence, a player i with the biggest endowment \(w_i\) smaller than the biggest \({\mathcal {S}}\) endowments would have a profitable deviation by playing \(\alpha _i = {\bar{\alpha }} + \varepsilon \) and be assigned to the best group.
If the endowments are such that \(w_{i+1} > {\bar{\alpha }}w_i\; \forall i,\)
Footnote 13 then there are no profitable deviations and \(\alpha _i = {\bar{\alpha }} \; \forall i\) is a Nash Equilibrium. \(\square \)
From the above lemmas we can derive the following theorem:
Theorem 2
In case of full heterogeneity of endowments \(w_i\), with all \(w_i\) having the same order of magnitude, for \(\gamma \ge 1\), the generalized voluntary contribution game has as only equilibrium non-contribution by all. For \(0< \gamma < 1\), the game has no pure strategy NE and hence it has a mixed strategy Nash Equilibrium.
Proof
From Lemma 7 we know that the nearly efficient NE cannot exist for heterogeneous players. Furthermore, we can prove as in Lemma 5 that there can be no pure strategies such that \({\bar{\alpha }}< \alpha _i < 1\) for any player i.
From Lemma 1 we know that for \(\gamma \ge 1\) the within group best response is to play \(\alpha _i = 0 \; \forall i\). This can be an equilibrium and hence for values of \(\gamma \) bigger than 1 there is a unique pure strategy NE for the generalized voluntary contribution game.
For \(\gamma < 1\), however, the within group best response is to play \(\alpha _i = {\bar{\alpha }}\). But Lemma 8 shows that this cannot be an equilibrium of the game (if the values of the endowment don’t differ too much). Hence for \(\gamma < 1\) there are no pure strategy NE and thus there has to exist a mixed strategy Nash Equilibrium. \(\square \)