Enhancing voluntary contributions in a public goods economy via a minimum individual contribution level

Abstract

We propose and theoretically analyze a measure to encourage greater voluntary contributions to public goods. Our measure is a simple intervention that restricts individuals’ strategy sets by imposing a minimum individual contribution level while still allowing for full free riding for those who do not want to contribute. We show that for a well-chosen value of the minimum individual contribution level, this measure does not incentivize any additional free riding while strictly increasing the total contributions relative to the situation without the minimum contribution level. Our measure is appealing because it is nonintrusive and in line with the principle of “freedom of choice.” It is easily implementable for many different public goods settings where more intrusive measures are less accepted. This approach has been implemented in practice in some applications, such as charities.

Appendix A Proofs

1.1 Proof of the results of Sect. 3.1

Game $\Gamma (N)$ is a potential game with a concave potential function $\Phi$ (defined as in (3)), where the strategy sets are closed intervals of the real line. It follows that a strategy profile is a Nash equilibrium if and only if it maximizes the potential function. As the potential function $\Phi$ is strictly concave on the convex set on which it is defined, it has a unique global maximum $\varvec{\lambda }^*\in [0,1]^n$ that coincides with the unique Nash equilibrium of $\Gamma (N)$. Such an equilibrium is strict. In particular, the equilibrium $\varvec{\lambda }^*$ is such that for each $i\in N$, $\lambda _i^*$ satisfies the following KKT conditions

$$\begin{aligned} \left\{ \begin{aligned}&h^\prime (G^*(N))-p_i^\prime (\lambda _i^*)+\psi _i^*-\phi _i^* = 0 \\&\psi _i^* \lambda _i^*=0\ \ \phi _i^*(\lambda _i^*-1)=0,\ \ \psi _i^*,\phi _i^*\ge 0, \end{aligned} \right. \end{aligned}$$

(A1)

where $G^*(N)$ denotes the total contribution level at equilibrium. We may observe by (A1) that, owing to the assumption that $h^\prime (0)> p_i^\prime (0)$ for at least one $i\in N$, this equilibrium is $\varvec{\lambda }^* \ne (0,\ldots ,0)$.

As the term $h^\prime (G^*(N))$ is the same for each $i\in N$, it immediately follows that, if the $p_i$s are such that $p_1^\prime (\lambda ) \le ... \le p_n^\prime (\lambda )$, for each $\lambda \in [0,1]$, then $\lambda ^*_n \le \ldots \le \lambda ^*_1$.

1.2 Proof of the results at the beginning of Sect. 3.2

Given the game $\Gamma (N)$, we suppose that an additional $(n+1)$-th individual enters the game.

First, to show that the i-th individual’s contribution at equilibrium for each $i\in N$ is such that $\lambda _i^*(N\cup \{n+1\}) \le \lambda _i^*(N)$, we suppose by contradiction that there exists $i\in N$ such that $0 \le \lambda _i^*(N) < \lambda _i^*(N\cup \{n+1\}) \le 1$. Because of the strict convexity of $p_i$,

$$\begin{aligned} p_i^\prime (\lambda _i^*(N)) < p_i^\prime (\lambda _i^*(N\cup \{n+1\})), \end{aligned}$$

and, from the KKT conditions in (A1), and as $\lambda _i^*(N)<1$ and $\lambda _i^*(N\cup \{n+1\})>0$,

$$\begin{aligned} h^\prime (G^*(N)) \le p_i^\prime (\lambda _i^*(N)) < p_i^\prime (\lambda _i^*(N\cup \{n+1\})) \le h^\prime (G^*(N\cup \{n+1\})). \end{aligned}$$

From the first and from the last term and because of the concavity of h, it follows that

$$\begin{aligned}&G^*(N) > G^*(N\cup \{n+1\}). \end{aligned}$$

(A2)

If the total contribution without individual $n+1$ is strictly greater than that after her entrance, this implies that there exists at least one individual $j\in N$, which verifies the opposite inequality compared to i, i.e., such that $1 \ge \lambda _j^*(N) > \lambda _j^*(N\cup \{n+1\}) \ge 0$. Following the same reasoning as before, we conclude that $G^*(N) < G^*(N\cup \{n+1\})$, which contradicts (A2).

Second, we show that the total level of contribution is such that $G^*(N\cup \{n+1\}) \ge G^*(N)$. We know that $\lambda _i^*(N\cup \{n+1\}) \le \lambda _i^*(N)$ for each $i\in N$. In particular, if the equality holds for each $i\in N$, the thesis follows trivially, as $G^*(N\cup \{n+1\}) = G^*(N)+ \lambda _{n+1}^*(N\cup \{n+1\})$, with $\lambda _{n+1}^*(N\cup \{n+1\})\ge 0$. In contrast, if there exists $i\in N$ such that $\lambda _i^*(N\cup \{n+1\}) < \lambda _i^*(N)$, following the same reasoning as before, then we can conclude that the inequality holds strictly.

Finally, we show that individual utility and social welfare are $U_i(\varvec{\lambda }^*(N\cup \{n+1\})) \ge U_i(\varvec{\lambda }^*(N))$ for all $i\in N$ and $W(\varvec{\lambda }^*(N\cup \{n+1\}))) \ge W(\varvec{\lambda }^*(N))$, respectively. It is sufficient to observe that at equilibrium,

$$\begin{aligned} U_i^*(N\cup \{n+1\})&= h(G^*(N\cup \{n+1\}))-p_i(\lambda _i^*(N\cup \{n+1\})) \\&\ge h(G^*(N))-p_i(\lambda _i^*(N)) \\&= U_i^*(N). \end{aligned}$$

1.3 Proof of Proposition 3.1

We observe that the modified game $\Gamma (N,\eta )$ is still a potential game with potential function $\Phi$ but defined in the restricted domain $\big [\{0\}\cup [\eta ,1]\big ]^n$. In contrast to $\Gamma (N)$, the strategy sets of the modified game are not closed intervals of the real line, and the equilibria may not coincide with the global maxima of the potential function.

For a better reading of the following of the proof, we present it organized in 5 steps.

Step 1 Given a subset of the individuals $S\subseteq N$, with $s=|S|$, and for each $\eta \in [0,1]$, we define a new modified game $\Gamma '(S,\eta )=\left\langle S,[\eta ,1]^s, (U_i)_{i\in S} \right\rangle$, where the utility function $U_i$ is defined by (1) for each individual $i\in S$, but it is now restricted to the domain $[\eta ,1]$. $\Gamma '(S,\eta )$ is still a potential game, with potential function $\Phi$, defined on the restricted domain $[\eta ,1]^s$. As for the original game $\Gamma (N)$ and the modified game $\Gamma (N,\eta )$, we also observe that the game $\Gamma '(S,\eta )$ is a potential game with a concave potential function and where similar to $\Gamma (N)$ but different from $\Gamma (N,\eta )$, the strategy sets are closed intervals of the real line. It follows that a strategy profile is a Nash equilibrium if and only if it maximizes the potential function. Moreover, the potential function is strictly concave on the convex set on which it is defined; then, it has a unique global maximum $\varvec{\lambda }^M(S,\eta )\in [\eta ,1]^s$, which coincides with the unique Nash equilibrium of $\Gamma '(S,\eta )$. In particular, for each $\eta \in (0,1]$, $\varvec{\lambda }^M(N,\eta )$ is the unique Nash equilibrium of $\Gamma '(N,\eta )$.

Step 2 We define $\eta ^*$ as in Definition 1. We show that $\varvec{\lambda }^M(N,\eta ^*)$ is a Nash equilibrium of $\Gamma (N,\eta ^*)$. First, we observe that no individual has incentives to deviate to a quantity in $[\eta ^*, 1]$ because $\varvec{\lambda }^M(N,\eta ^*)$ is a Nash equilibrium of the game $\Gamma '(N,\eta ^*)$. It remains to be shown that no individual has incentives to go to zero. Individual n does not have incentives by Definition 1. For any other individual $i\ne n$, such that $\lambda _i^M(N,\eta ^*) = \eta ^*$, if agent n, who is the most privacy concerned, does not have incentives to deviate from $\eta ^*$, that is still valid for i. For any other agent $i\ne n$, such that $\lambda _i^M(N,\eta ^*) > \eta ^*$, as i does not have incentives to deviate to $\eta ^*$, then, because of the concavity of the utility function, she cannot have incentives to deviate to 0.

Step 3 We observe that for each $\eta \in [0, \eta ^*)$, $\varvec{\lambda }^M(N,\eta )$ is a Nash equilibrium of $\Gamma (N,\eta )$. Moreover, when $\eta \in (\lambda _n^*,\eta ^*)$, we can repeat the same reasoning as in Step 3, with the only difference being that individual n (and any other individual contributing $\eta$) is always strictly better off by contributing rather than free riding, and then, the inequality is always strict.

Step 4 We observe that for any $\eta \in (0,\eta ^*]$, $\varvec{\lambda }^M(N,\eta )$ is the unique Nash equilibrium of $\Gamma (N,\eta )$ such that each individual has a nonzero contribution, i.e., such that $\lambda ^M_i(N,\eta )>0$ for each $i\in N$. To show that, it is sufficient to observe that an equilibrium of $\Gamma (N,\eta )$ such that each individual has a nonzero contribution, is also an equilibrium of the game $\Gamma '(N,\eta )$ (as if an individual does not have incentives to deviate in $\{0\}\cup [\eta ,1]$, she does not have incentives to deviate in the restricted strategy set $[\eta ,1]$ either), and the equilibrium of $\Gamma '(N,\eta )$ is unique.

Step 5 We show that any other NE $\bar{\varvec{\lambda }}$ of $\Gamma (N,\eta )$ is such that $\bar{\lambda }_i=\lambda _i^M(S,\eta )$ for each $i\in S$, where $S=\{i\in N \mid \bar{\lambda }_i \ne 0\}$. Let $\eta \in (\lambda _n^*,\eta ^*]$, and we let $\bar{\varvec{\lambda }}=\bar{\varvec{\lambda }}(N)$ be a Nash equilibrium of $\Gamma (N,\eta )$ such that $\bar{\lambda }_i(N)=0$ for at least one $i\in N$. We define S as the set of individuals who contribute nonzero values at this equilibrium. As, by definition, of NE, none of the individuals has incentives to deviate, and, in particular, none of the individuals in S has incentives to deviate in $[\eta , 1]$. It follows that $\bar{\varvec{\lambda }}(S)$, i.e., the vector of the nonzero elements of the vector $\bar{\varvec{\lambda }}(N)$, is a Nash equilibrium of the game $\Gamma ^\prime (S,\eta )$ defined in step 2 of this proof. As we have seen, this equilibrium is unique and coincides with the unique Nash equilibrium $\varvec{\lambda }^M(S,\eta )$ of the game $\Gamma (S,\eta )$ such that no individual is free riding. Trivially, we can conclude that $\varvec{\lambda }^*(N,\eta )$ is the unique NE of $\Gamma (N,\eta )$ s.t., $\lambda ^*_i(N,\eta ) > 0$ for all $i\in N$.

1.4 Proof of Theorem 3.2

Let us recall that we assumed that the equilibrium contribution $\varvec{\lambda }^*(N)$ of $\Gamma (N)$ is such that $\lambda ^*_n(N) < 1$, i.e., it is such that the total level of contribution is not optimal.

When $\eta \in [0,\lambda _n^*(N)]$, $\varvec{\lambda }^*(N,\eta )=\varvec{\lambda }^*(N)$; consequently, $G^*$ is constant, i.e., $G^*(N,\eta )=G^*(N)$.

When $\eta \in (\lambda _n^*(N),\eta ^*]$, individual n contributes to equilibrium $\lambda _n^*(N,\eta ) = \eta > \lambda _n^*(N)$. We now show that for $\eta ^* \ge \eta _2 > \eta _1 \ge \lambda _n^*$, $G^*(N,\eta _2) > G^*(N,\eta _1)$. By contradiction, we assume that $G^*(N,\eta _2) \le G^*(N,\eta _1)$. It follows that

$$\begin{aligned} h^\prime (G^*(N,\eta _2)) \ge h^\prime (G^*(N,\eta _1)), \end{aligned}$$

(A3)

because of the concavity of the public good utility function h. Moreover, as the total level of contribution with $\eta _2$ is less than or equal to the level of contribution with $\eta _1$, we know that individual n has a strictly larger level of contribution; it follows that there exists $i\in N$ who contributes strictly less, i.e., such that,

$$\begin{aligned} \eta _2 \le \lambda _i^*(N,\eta _2) < \lambda _i^*(N,\eta _1) \le 1. \end{aligned}$$

(A4)

By equation (A4), because of the convexity of the cost of contribution function $p_i$ and from the KKT conditions for the potential function of the modified game $\Gamma ^*(N,\eta )$, it follows that

$$\begin{aligned} h^\prime (G^*(N,\eta _2)) \le p_i^\prime (\lambda _i^*(N,\eta _2)) < p_i^\prime (\lambda _i^*(N,\eta _1)) \le h^\prime (G^*(N,\eta _1)) \end{aligned}$$

and this contradicts equation (A3).

1.5 Proof of the results at the beginning of Sect. 3.3

First, we observe that

$$\begin{aligned}&h(G^*(N,\eta ^*))-p_j(\lambda _j^*(N,\eta ^*)) \ge h(G^*(N,\eta ^*) - \lambda _j^*(N,\eta ^*))-p_j(0)\ \ \forall j\in N \end{aligned}$$

(A5)

because, as shown in Proposition 3.1, $\varvec{\lambda }^*(N,\eta ^*)$ is a NE of $\Gamma (N,\eta ^*)$, and thus, no individual has incentives to go to zero. Now, we suppose by contradiction that the vector $\varvec{\nu }$ such that, $\nu _i=\lambda _i^*(N,\eta ^*)=\lambda _i^*(N\setminus \{n\},\eta ^*)$ for each $i\ne n$ and $\nu _n=0$ is not a NE of $\Gamma (N,\eta ^*)$. This means that there exists an individual $j\in N\setminus \{n\}$ for whom it is convenient to deviate, i.e., such that, it holds that

$$\begin{aligned} h(G^*(N,\eta ^*)-\eta ^*)-p_j(\lambda _j^*(N,\eta ^*)) < h(G^*(N,\eta ^*) - \eta ^* - \lambda _j^*(N,\eta ^*))-p_j(0). \end{aligned}$$

(A6)

From Equations (A5) and (A6), it follows that

$$\begin{aligned}&h(G^*(N,\eta ^*))-h(G^*(N,\eta ^*) - \lambda _j^*(N,\eta ^*)) \ge p_j(\lambda _j^*(N,\eta ^*))-p_j(0) \\&\quad > h(G^*(N,\eta ^*)-\eta ^*) - h(G^*(N,\eta ^*) - \eta ^* - \lambda _j^*(N,\eta ^*)) \end{aligned}$$

This cannot hold because of the concavity of h. It follows that $\varvec{\nu }$ is a NE of $\Gamma (N,\eta ^*)$.

Second, we observe that $\varvec{\nu }$ provides the same value of the potential function as $\varvec{\lambda }^*(N,\eta ^*)$. Then, if the second one is a potential maximizer, the first one is as well.

1.6 Proof of Proposition 3.3

Let $\bar{\eta }$ be defined as in Definition 2, $\eta \in (\lambda _n^*,\bar{\eta })$, and $\varvec{\lambda }^*(N,\eta )=\varvec{\lambda }^M(N,\eta )$ be the unique nonzero component Nash equilibrium of the modified game $\Gamma (N,\eta )$, which is the unique maximizer of $\Phi$ on $[\eta ,1]^n$. As $\eta < \bar{\eta }$, it follows that

$$\begin{aligned} \Phi (\varvec{\lambda }^M(N,\eta )) > \Phi \mid _{\lambda _n=0}(\varvec{\lambda }^{M0}(N\setminus \{n\}),\eta ). \end{aligned}$$

(A7)

As $\varvec{\lambda }^{M0}(N\setminus \{n\})$ is the potential maximizer of function $\Phi$ restricted to domain $[\bar{\eta }, 1]^{n-1}\cup \{0\}$, it also holds that

$$\begin{aligned} \Phi \mid _{\lambda _n=0}(\varvec{\lambda }^{M0}(N\setminus \{n\}),\eta ) \ge \Phi \mid _{\lambda _i=0, \forall i\in N\setminus S}(\varvec{\lambda }^{M0}(S,\eta )). \end{aligned}$$

(A8)

(A7) and (A8) show that $\varvec{\lambda }^*(N,\eta )=\varvec{\lambda }^M(N,\eta )$ is the unique PMNE of the modified game $\Gamma (N,\eta )$.

1.7 Proof of the results of Section  3.4

When the game $\Gamma (N)$ is homogeneous, the potential function $\Phi$ is a symmetric function on a symmetric domain. As a consequence, the unique maximum is also symmetric, i.e., $\lambda _i^*=\lambda ^*$ for each $i\in N$. In particular, as the equilibrium vector is a nonzero vector, we have $\lambda ^*>0$.

From Equation (A1), recalling that we are now in a homogeneous case, we observe that $\lambda ^*>0$ is the unique solution of the following fixed-point problem

$$\begin{aligned} \lambda =g(n,\lambda ), \end{aligned}$$

(A9)

where the function $g: {\mathbb {N}}_+ \times (0,1] \rightarrow [0,+\infty ]$ is defined for each $\lambda \in (0,1]$ and for each $n\in {\mathbb {N}}_+$ as

$$\begin{aligned} g(n,\lambda ) = \min \left\{ (p^\prime )^{-1} (h^\prime \left( n \lambda \right) ), 1\right\} . \end{aligned}$$

(A10)

We consider the problem with the parameter n defined on the real interval $[1,+\infty ]$. For each $n\in [1,+\infty ]$, g is continuous in $\lambda$. Moreover, the function g is monotonic and nonincreasing in n. Indeed,

$$\begin{aligned} \frac{\partial {g}}{\partial {n}}=&\frac{\lambda }{p^{\prime \prime }((p^\prime )^{-1}(h'\left( n \lambda \right) ))}h^{\prime \prime }\left( n \lambda \right) < 0 \end{aligned}$$

for each $\lambda$ internal solution, and g is identically 1 otherwise. Applying Corollary 1 of Milgrom and Roberts (1994), the unique fixed point $\lambda ^*(n)>0$ is nonincreasing in n; consequently, $\lim _{n\rightarrow +\infty }\lambda ^*(n)\ge 0$ is well defined.

Second, for each $\lambda >0$,

$$\begin{aligned} \lim _{n\rightarrow +\infty }g(n,\lambda )=0 \end{aligned}$$

pointwise. Indeed, $\lim _{x\rightarrow +\infty }h^\prime (x)=0$, $p^\prime (0)\ge 0$ and $p^\prime$ are strictly monotonic.

If we suppose by contradiction that $\lim _{n\rightarrow +\infty }\lambda ^*(n) = a > 0$, then, from Equation (A9), it follows that

$$\begin{aligned} 0 < \lim _{n\rightarrow +\infty }\lambda ^*(n) = \lim _{n\rightarrow +\infty }g(n,\lambda ^*(n)) = \lim _{n\rightarrow +\infty }g(n,a) = 0. \end{aligned}$$

Now, we know that $G^*(n+1) \ge G^*(n)$ for each $n\in N$. In particular, we know that the inequality holds strictly at equilibrium in the homogeneous case where any new individuals are not free riders. We suppose by contradiction that $\lim _{n\rightarrow + \infty }G^*(n)<+\infty$. It follows that $\lim _{n\rightarrow + \infty }h^\prime (n\lambda ^*(n))>0$; then, by (A10), the solution to the fixed-point problem is such that $\lim _{n\rightarrow + \infty } \lambda ^*(n)>0$, which contradicts the fact that we already know that this limit goes to zero.

1.8 Proof of Theorem 3.4

For each $\eta \in (0,1]$, let $\varvec{\lambda }^M(n,\eta )$ be the unique global maximum of $\Phi$ on $[\eta , 1]^n$. When the individuals are homogeneous, the potential function $\Phi$ is a symmetric function that we are maximizing on a symmetric domain. As a consequence, the maximum is $\lambda _i^M(n,\eta )=\lambda ^M(n,\eta )$ for each $i\in N$, with

$$\begin{aligned} \lambda ^M(n,\eta )=\left\{ \begin{aligned}&\lambda ^*(n)&\text { if } 0< \eta \le \lambda ^*(n) \\&\eta&\text { if } \lambda ^*(n) < \eta \le 1. \\ \end{aligned} \right. \end{aligned}$$

(A11)

• (i) For each $\eta \in (0,\eta ^*(n)]$, where $\eta ^*(n)$ is defined as in Definition 1, $\varvec{\lambda }^*(n,\eta )=\varvec{\lambda }^M(n,\eta )$ is the unique NE of the modified game $\Gamma (n,\eta )$ such that no individual is free riding. As a particular case of Proposition 3.3, for each $\eta \in (0,\bar{\eta }(n)]$, where $\bar{\eta }(n)$ is defined as in Definition 2, $\varvec{\lambda }^*(n,\eta )$ is the unique PMNE of the modified game $\Gamma (n,\eta )$.

• (ii) This result follows Definition 1 for the homogeneous case.

• (iii) For each $\eta \in (\eta ^*(n),1]$, we show that there does not exist a NE $\bar{\varvec{\lambda }}$ of $\Gamma (n,\eta )$ such that $\bar{\lambda }_i > 0$ for each $i\in N$. We assume that $\eta ^*(n)<1$. First, we observe that, with reasoning similar to that in the proof of Proposition 3.1, A.3-Step 5, whenever we have a nonzero component NE of $\Gamma (n,\eta )$, this has to be a NE of the corresponding game $\Gamma ^{\prime }$ and then a maximum of the potential function $\Phi$ restricted to the domain $[\eta ,1]^n$. Because of this symmetry, the only possible candidate nonzero component NE is the vector $\bar{\varvec{\lambda }}=\varvec{\eta }=(\eta , \ldots , \eta )$. Second, we observe that by definition, $\eta ^*(n)$ is the smallest solution of the following fixed-point problem:

  $$\begin{aligned} p(\eta )&= h(n\eta )-h((n-1)\eta ) \end{aligned}$$

  (A12)

  or, equivalently, of

  $$\begin{aligned} \frac{p(\eta )}{\eta }&= \frac{h(n\eta )-h((n-1)\eta )}{\eta } \end{aligned}$$

  (A13)

  where n is a fixed parameter. Then, we show that $\eta ^*(n)$ is, in fact, the unique solution of (A13). Indeed, because of the concavity of h, we have that

  $$\begin{aligned} \frac{h(y)-h(x)}{y-x} \end{aligned}$$

  is a decreasing function both in y and in x; in particular, the right term of (A13) is decreasing in $\eta$. Moreover, because of the convexity of p, we have

  $$\begin{aligned} \frac{p(y)-p(x)}{y-x} \end{aligned}$$

  is an increasing function both in y and in x; in particular, the left term of (A13) is increasing in $\eta$ (when $y=\eta$, $x=0$ and p(0) are constants). It follows that the fixed-point problem in (A13) has at most one solution.

  In particular, applying Corollary 1 of Milgrom and Roberts (1994), the unique (i.e., smallest) fixed point $\eta ^*(n)>0$ is nonincreasing in n. Moreover, if we assume that $\eta ^*(n) = \eta ^*(n-1) = \eta < 1$, we obtain from (A12) that

  $$\begin{aligned} h(n\eta ) - h((n-1)\eta ) = h((n-1)\eta ) - h((n-2)\eta ), \end{aligned}$$

  which contradicts the fact that h is strictly concave. Hence, we conclude that $\eta ^*(n)$ is strictly decreasing in n.

  Third, we know that $\varvec{\lambda }^*(n)$ is an SNE of $\Gamma (n)$; i.e., we know that $p(\lambda ^*(n)) < h(n\lambda ^*(n))-h((n-1)\lambda ^*(n))$. Moreover, we observe that $\eta ^*(n)$ verifies the equality by definition when $\eta ^*(n)<1$. As the solution of the previous fixed-point problem is unique when it exists, it follows that

  $$\begin{aligned} p(\eta ) > h(n\eta )-h((n-1)\eta ) \end{aligned}$$

  (A14)

  for each $\eta \in (\eta ^*(n),1]$. Equation (A14) translates to the fact that when individuals adopt the strategy $\varvec{\eta }=(\eta , \ldots , \eta )$, with $\eta \in (\eta ^*(n),1]$, each individual in N is strictly better able to go to zero, and then, the only candidate nonzero component NE cannot be a NE.

  Now, we assume that $\eta \le \eta ^*(1)$ and let $m_{\eta } \in \{1, \cdots , n-1\}$ be the integer such that $\eta \in (\eta ^*(m_{\eta }+1), \eta ^*(m_{\eta })]$. Such an integer exists and is unique because $\eta ^*(n)$ is strictly decreasing in n.

  Let us note that by definition of $m_\eta$, we see that it is possible that $\eta ^*(m_{\eta })=1$ (in which case $\eta ^*(m) = 1$ for all $m\le m_{\eta }$ but those values of m never correspond to any $m_{\eta ^{\prime }}$ for any $\eta ^{\prime }$); however, we necessarily have $\eta ^*(m_{\eta }+1)<1$.

  We assume that there exists a NE $\bar{\varvec{\lambda }}$ of $\Gamma (N, \eta )$ where s individuals in set S contribute positively, i.e., $\bar{\lambda }_i >0$ for all $i\in S$ and $\bar{\lambda }^*_i =0$ for all $i\in N{\setminus } S$, where $s = |S|$. First, following arguments from the proof of Proposition 3.1, $( \bar{\lambda }_i)_{i\in S}$ is equal to $\varvec{\lambda }^M(S, \eta )$ which is symmetric. We denote by $\bar{\lambda }$ the value of $\bar{\lambda }_i$ for $i\in S$ and we have $\bar{\lambda } = \lambda ^*(s)$ is $\eta < \lambda ^*(s)$ and $\bar{\lambda } =\eta$ otherwise.

  Next, we observe that we must have $s \le m_{\eta }$. Indeed, we assume that $s>m_{\eta }$. Then, $\lambda ^*(s) \le \eta ^*(s) \le \eta ^*(m_{\eta }+1) < \eta$, so that $\bar{\lambda } = \eta$. In that case, an individual $i\in S$ switching from $\bar{\lambda }$ to zero contribution gains $p(\bar{\lambda }) - \left( h(s\bar{\lambda })h((s-1)\bar{\lambda }) \right)$, which is strictly positive since, by using arguments similar to the ones introduced above, we can show:

  $$\begin{aligned} \frac{p(\bar{\lambda })}{\bar{\lambda }} = \frac{p(\eta )}{\eta }&> \frac{p(\eta ^*(m+1))}{\eta ^*(m+1)} \\&= \frac{h((m+1)\eta ^*(m+1)) - h(m\eta ^*(m+1))}{\eta ^*(m+1)} \\&\ge \frac{h(s\eta ^*(m+1)) - h((s-1)\eta ^*(m+1))}{\eta ^*(m+1)} \\&\ge \frac{h(s\bar{\lambda }) - h((s-1)\bar{\lambda })}{\bar{\lambda }}. \end{aligned}$$

  This contradicts the fact that $\bar{\varvec{\lambda }}$ is a NE.

  We now show that for $s=m_{\eta }$, the profile $\bar{\varvec{\lambda }}$ with $\bar{\lambda }_i = \max (\lambda ^*(s), \eta ) \in (\eta ^*(m_{\eta }+1), \eta ^*(m_{\eta })]$ for all $i\in S$ is indeed a NE. First, let us note that, by definition of $\eta ^*(m_{\eta }+1)$, individuals outside S (i.e., given zero) have no incentive to deviate. Indeed, if $i\in N\setminus S$ deviates to $\lambda ^{\prime } \ge \eta > \eta ^*(m_{\eta }+1)$, the gain is

  $$\begin{aligned} h((s+1)\lambda ^{\prime }) - h(s\lambda ^{\prime }) - p(\lambda ^{\prime })&= \lambda ^{\prime } \frac{h((s+1)\lambda ^{\prime }) - h(s\lambda ^{\prime })}{\lambda ^{\prime }} - \lambda ^{\prime } \frac{p(\lambda ^{\prime })}{\lambda ^{\prime }}\\&\le \lambda ^{\prime } \frac{h((s+1)\eta ^*(m_{\eta }+1)) - h(s\eta ^*(m_{\eta }+1))}{\eta ^*(m_{\eta }+1)} +\\&- \lambda ^{\prime } \frac{p(\eta ^*(m_{\eta }+1))}{\eta ^*(m_{\eta }+1)}\\&= 0, \end{aligned}$$

  Here, we again use the convexity of p and concavity of h, as derived above. Similarly, individuals in S have no incentive to deviate to zero. Hence, $\bar{\varvec{\lambda }}$ is a NE.

1.9 Proof of the additional results of Sect. 3.4

It remains to be observed that, because of the monotonicity, $\lim _{n\rightarrow +\infty }\eta ^*(n)\ge 0$ is well defined. Indeed, $\lim _{x\rightarrow +\infty }h(nx)-h((n-1)x)=0$, $p(0)\ge 0$ and p are strictly monotonic. If we suppose by contradiction that $\lim _{n\rightarrow +\infty }\eta ^*(n) = a > 0$, then, from Equation (A12), it follows that

$$\begin{aligned} 0&< p(a) \\&= \lim _{n\rightarrow +\infty }p(\eta ^*(n)) \\&= \lim _{n\rightarrow +\infty }h(n\eta ^*(n))-h((n-1)\eta ^*(n)) \\&= \lim _{n\rightarrow +\infty }h(na)-h((n-1)a) \\&= 0. \end{aligned}$$