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Costly Participation and The Evolution of Cooperation in the Repeated Public Goods Game

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Abstract

In real life, individuals often need to pay some costs to build and maintain long-termed relationships as well as interactions among them. However, previous studies of the repeated public goods game have focused almost exclusively on the cost-free participation. Here we introduce costly participation to the repeated public goods game in a conditional manner, and study the evolution of cooperation in both deterministic and stochastic dynamics for well-mixed populations. In the limit of an infinite population size, the deterministic dynamics can lead to either a stable coexistence between cooperators and defectors or even a complete dominance of cooperators over defectors if the initial frequency of cooperators is larger than some invasion barrier. In general, defectors are always able to resist invasion by cooperators. However, in finite populations, we show that natural selection can favor the emergence of cooperation in the stochastic dynamics. In the limit of weak selection and large populations, we derive a critical condition required for a cooperator to replace a population of defectors with a selective advantage by using several approximation techniques. Theoretical analysis of the critical condition reveals that participation cost determines the impacts of the number of game round in the emergence of cooperation. Interestingly, there exists an intermediate value of the participation threshold of cooperators leading to the optimal condition for a cooperator to invade and fixate in a population of defectors if the participation cost is smaller than a critical value. Numerical calculations confirm that the validity of the analytical approximations extends to much wider ranges of the selection strength as well as of the population size.

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Acknowledgements

We thank the editor and two anonymous reviewers for constructive comments, which helped us to improve this work.

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Authors and Affiliations

  1. Department of Automation, School of Information Science and Technology, Donghua University, Shanghai, 201620, China

    Xiaofeng Wang

  2. Engineering Research Center of Digitized Textile and Apparel Technology, Donghua University, Ministry of Education, Shanghai, 201620, China

    Xiaofeng Wang

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  1. Xiaofeng Wang
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Correspondence to Xiaofeng Wang.

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X.F.W acknowledges gratefully financial support from the NSFC (Grants No. 61903077), the Shanghai Sailing Program (Grants No. 19YF1402500) as well as the Fundamental Research Funds for the Central Universities (Grants Nos. 2232019D3-56 and 2232018G-09).

Appendices

A Detailed Analysis of Payoff Difference

In “Appendix A”, we study in detail the direction of the deterministic dynamics by analyzing the payoff difference \({f_{{C_{T_C}}}}\left( x \right) - {f_{{D_{T_D}}}}\left( x \right) \). In particular, we analyze the shape of \({f_{{C_{T_C}}}}\left( x \right) - {f_{{D_{T_D}}}}\left( x \right) \), assuming \({\bar{T}} \in \left\{ {2, \cdots ,N - 1} \right\} \), and later on followed by the analysis for the degenerate cases \({\bar{T}} = N\) and \({\bar{T}} = 1\).

Analysis for \({\bar{T}} \in \left\{ {2, \cdots ,N - 1} \right\} \). Let the polynomial \({G_1}\left( x \right) \) be

$$\begin{aligned} \begin{aligned} {G_1}\left( x \right)&= \left( {\delta {\bar{T}} - c - 1} \right) \left( {\begin{array}{*{20}{c}} {N - 1} \\ {{\bar{T}} - 1} \\ \end{array}} \right) {x^{{\bar{T}} - 1}}{\left( {1 - x} \right) ^{N - {\bar{T}}}} \\&\quad + \left( {\delta - 1} \right) \sum \limits _{j = {\bar{T}}}^{N - 1} {\left( {\begin{array}{*{20}{c}} {N - 1} \\ j \\ \end{array}} \right) {x^j}{{\left( {1 - x} \right) }^{N - 1 - j}}} . \\ \end{aligned} \end{aligned}$$
(20)

Note that \({G_1}\left( x \right) \) has the same shapes of function with the payoff difference in Eq. (13), and thus determines the non-trivial equilibria of the deterministic dynamics described by Eq. (10). Before presenting formally Proposition 2, we first introduce a necessary Lemma 1.

Lemma 1

Let \({\bar{T}} \in \left\{ {2, \cdots ,N - 1} \right\} \). The polynomial \({G_1}\left( x \right) \) satisfies following properties:

  1. 1.1

    \({G_1}\left( 0 \right) = 0\);

  2. 1.2

    \({G_1}\left( 1 \right) = \delta - 1\);

  3. 1.3

    If \(c = \bar{c} = \frac{{\left( {{\bar{T}} - 1} \right) \left( {N\delta - 1} \right) }}{{N - 1}}\), \({G_1}\left( x \right) \) decreases monotonically in \(x \in \left( { - \infty , + \infty } \right) \); Else if \(0 \le c < \bar{c}\), \({G_1}\left( x \right) \) attains a maximum at \({\bar{x}} = \frac{{\left( {{\bar{T}} - 1} \right) \left( {\delta {\bar{T}} - c - 1} \right) }}{{\left( {{\bar{T}} - 1} \right) \left( {\delta N - 1} \right) - c\left( {N - 1} \right) }}\). Moreover, \({G_1}'\left( x \right) = 0\) at \(x = {\bar{x}}\), \({G_1}'\left( x \right) > 0\) for \(x \in \left( { - \infty ,{\bar{x}}} \right) \) and \({G_1}'\left( x \right) < 0\) for \(x \in \left( {{\bar{x}}, + \infty } \right) \); Else if \(c > \bar{c}\), \({G_1}\left( x \right) \) attains a minimum at \({\bar{x}} = \frac{{\left( {{\bar{T}} - 1} \right) \left( {\delta {\bar{T}} - c - 1} \right) }}{{\left( {{\bar{T}} - 1} \right) \left( {\delta N - 1} \right) - c\left( {N - 1} \right) }}\). Moreover, \({G_1}'\left( x \right) = 0\) at \(x = {\bar{x}}\), \({G_1}'\left( x \right) < 0\) for \(x \in \left( { - \infty ,{\bar{x}}} \right) \) and \({G_1}'\left( x \right) > 0\) for \(x \in \left( {{\bar{x}}, + \infty } \right) \).

Proof

The first two properties follow immediately from Eq. (20). We prove the third property now. From Eq. (20), we obtain the following expression for the first order derivative \({G_1}'\left( x \right) \):

$$\begin{aligned} \begin{aligned} {G_1}'\left( x \right)&= \left[ {\delta \left( {{\bar{T}} - 1} \right) - c} \right] \left( {{\bar{T}} - 1} \right) \left( {\begin{array}{l} {N - 1} \\ {{\bar{T}} - 1} \\ \end{array}} \right) {x^{{\bar{T}} - 2}}{\left( {1 - x} \right) ^{N - {\bar{T}}}} \\&\quad - \left[ {\delta \left( {{\bar{T}} - 1} \right) - c} \right] \left( {N - {\bar{T}}} \right) \left( {\begin{array}{c} {N - 1} \\ {{\bar{T}} - 1} \\ \end{array}} \right) {x^{{\bar{T}} - 1}}{\left( {1 - x} \right) ^{N - {\bar{T}} - 1}} \\&\quad + \left( {\delta - 1} \right) \sum \limits _{j = {\bar{T}} - 1}^{N - 1} {j\left( {\begin{array}{c} {N - 1} \\ j \\ \end{array}} \right) } {x^{j - 1}}{\left( {1 - x} \right) ^{N - 1 - j}} \\&\quad - \left( {\delta - 1} \right) \sum \limits _{j = {\bar{T}} - 1}^{N - 1} {\left( {N - 1 - j} \right) \left( {\begin{array}{*{20}{c}} {N - 1} \\ j \\ \end{array}} \right) } {x^j}{\left( {1 - x} \right) ^{N - 2 - j}}. \\ \end{aligned} \end{aligned}$$
(21)

For \(j \in \left\{ {0, \cdots ,N - 2} \right\} \), the following recurrence relation is satisfied, i.e.,

$$\begin{aligned} \left( {N - 1 - j} \right) \left( {\begin{array}{c} {N - 1} \\ j \\ \end{array}} \right) = \left( {j + 1} \right) \left( {\begin{array}{c} {N - 1} \\ {j + 1} \\ \end{array}} \right) , \end{aligned}$$
(22)

so that

$$\begin{aligned} \begin{aligned} {G_1}'\left( x \right)&= \left( {\begin{array}{c} {N - 1} \\ {{\bar{T}} - 1} \\ \end{array}} \right) {x^{{\bar{T}} - 2}}{\left( {1 - x} \right) ^{N - {\bar{T}} - 1}} \\&\quad \times \left\{ \begin{array}{l} \left[ {\left( {{\bar{T}} - 1} \right) \left( {1 - \delta N} \right) + c\left( {N - 1} \right) } \right] x + \left( {{\bar{T}} - 1} \right) \left( {\delta {\bar{T}} - c - 1} \right) \\ \end{array} \right\} . \\ \end{aligned} \end{aligned}$$
(23)

Following Eq. (23), we prove the third property of Lemma 1. \(\square \)

Based on Lemma 1, we are able to prove the following Proposition 2:

Proposition 2

Let \({\bar{T}} \in \left\{ {2, \cdots ,N - 1} \right\} \). The function of the payoff difference \({f_{{C_{T_C}}}}\left( x \right) - {f_{{D_{T_D}}}}\left( x \right) \) satisfies the following properties:

  1. 2.1

    If \({{\left( {c + 1} \right) } /{{\bar{T}}}}< \delta < 1\) and \(0 \le c < \bar{c}\), there exists a critical number of rounds \(\bar{R} = 1 + {{\left( {1 - \delta } \right) } /{{G_1} (\bar{x})}}\), which determines the behavior of \({f_{{C_{T_C}}}}\left( x \right) - {f_{{D_{T_D}}}}\left( x \right) \):

    1. 2.1.1

      If \(1 \le R < \bar{R}\), \({f_{{C_{T_C}}}}\left( x \right) - {f_{{D_{T_D}}}}\left( x \right) < 0\) for \(x \in \left[ {0,1} \right] \).

    2. 2.1.2

      If \(R = \bar{R}\), \({f_{{C_{T_C}}}}\left( x \right) - {f_{{D_{T_D}}}}\left( x \right) \) has a double real root at \({x^*} = \bar{x} \in \left( {0,1} \right) \). \({f_{{C_{T_C}}}}\left( x \right) - {f_{{D_{T_D}}}}\left( x \right) < 0\) for \(x \in [0,{x^*})\) and \(x \in ({x^*},1]\).

    3. 2.1.3

      If \(R > \bar{R}\), \({f_{{C_{T_C}}}}\left( x \right) - {f_{{D_{T_D}}}}\left( x \right) \) has two real roots, \(x_L^*\) and \(x_R^*\), with \(x_L^* \in \left( {0,\bar{x}} \right) \) and \(x_R^* \in \left( {\bar{x},1} \right) \). \({f_{{C_{T_C}}}}\left( x \right) - {f_{{D_{T_D}}}}\left( x \right) < 0\) for \(x \in [0,{x_L^*})\) and \(x \in ({x_R^*},1]\), whereas \({f_{{C_{T_C}}}}\left( x \right) - {f_{{D_{T_D}}}}\left( x \right) > 0\) for \(x \in ({x_L^*},{x_R^*})\).

  2. 2.2

    If \({1 /N} < \delta \le {{\left( {c + 1} \right) } /{{\bar{T}}}}\) and \(0 \le c < \bar{c}\), \({f_{{C_{T_C}}}}\left( x \right) - {f_{{D_{T_D}}}}\left( x \right) < 0\) for \(x \in \left[ {0,1} \right] \).

  3. 2.3

    If \(c \ge \bar{c}\), \({f_{{C_{T_C}}}}\left( x \right) - {f_{{D_{T_D}}}}\left( x \right) < 0\) for \(x \in \left[ {0,1} \right] \).

Proof

2.1 If \({{\left( {c + 1} \right) } /{{\bar{T}}}}< \delta < 1\) and \(0 \le c < \bar{c}\), Lemma 1 states that \({G_1}\left( x \right) \) attains a maximum at \(\bar{x} \in \left( {0,1} \right) \).

2.1.1 Let \(1 \le R < \bar{R}\). We obtain the following inequality by applying Lemma 1:

$$\begin{aligned} \begin{aligned} {f_{{C_{T_C}}}}\left( x \right) - {f_{{D_{T_D}}}}\left( x \right)&= \delta - 1 + \left( {R - 1} \right) {G_1}\left( x \right) \\&< \delta - 1 + \left( {\bar{R} - 1} \right) {G_1}\left( {\bar{x}} \right) = 0, \\ \end{aligned} \end{aligned}$$
(24)

for \(x \in \left[ {0,1} \right] \).

2.1.2 For \(R = \bar{R}\), we have

$$\begin{aligned} {f_{{C_{T_C}}}}\left( {\bar{x}} \right) - {f_{{D_{T_D}}}}\left( {\bar{x}} \right) = \delta - 1 + \left( {\bar{R} - 1} \right) {G_1}\left( {\bar{x}} \right) = 0. \end{aligned}$$
(25)

To prove that \(\bar{x}\) is a double root, we show that \({\left[ {{f_{{C_{T_C}}}}\left( {\bar{x}} \right) - {f_{{D_{T_D}}}}\left( {\bar{x}} \right) } \right] ^\prime } = 0\) and \({\left[ {{f_{{C_{T_C}}}}\left( {\bar{x}} \right) - {f_{{D_{T_D}}}}\left( {\bar{x}} \right) } \right] ^{\prime \prime }} \ne 0\). From Eq. (23), we obtain \({\left[ {{f_{{C_{T_C}}}}\left( {\bar{x}} \right) - {f_{{D_{T_D}}}}\left( {\bar{x}} \right) } \right] ^\prime } = \left( {R - 1} \right) {G_1}'\left( {\bar{x}} \right) = 0\). Besides, the second order derivative \({\left[ {{f_{{C_{T_C}}}}\left( {\bar{x}} \right) - {f_{{D_{T_D}}}}\left( {\bar{x}} \right) } \right] ^{\prime \prime }}\) at \(x = \bar{x}\) is given by

$$\begin{aligned} \begin{aligned} {\left[ {{f_{{C_{T_C}}}}\left( {\bar{x}} \right) - {f_{{D_{T_D}}}}\left( {\bar{x}} \right) } \right] ^{\prime \prime }} =&\, \left( {R - 1} \right) \left( {\begin{array}{c} {N - 1} \\ {{\bar{T}} - 1} \\ \end{array}} \right) {{\bar{x}}^{{\bar{T}} - 2}}{\left( {1 - \bar{x}} \right) ^{N - {\bar{T}} - 1}} \\&\times \left\{ \begin{array}{l} {\left( {N - {\bar{T}}} \right) ^2}\left[ {c - \delta \left( {{\bar{T}} - 1} \right) } \right] + \left( {{\bar{T}} - 1} \right) \left( {c + 1 - \delta {\bar{T}}} \right) \\ \end{array} \right\} . \\ \end{aligned} \end{aligned}$$
(26)

As \(\delta > {{\left( {c + 1} \right) } /{{\bar{T}}}}\), we have \(c + 1 - \delta {\bar{T}} < 0\) and \(c - \delta \left( {{\bar{T}} - 1} \right)< {{\left( {c + 1} \right) } / {{\bar{T}}}} - 1 < {{\left( {\bar{c} + 1} \right) } /{{\bar{T}}}} - 1\). Since \(\bar{c} = {{\left( {{\bar{T}} - 1} \right) \left( {N\delta - 1} \right) } /{\left( {N - 1} \right) }} < {\bar{T}} - 1\), we further obtain \(c - \delta \left( {{\bar{T}} - 1} \right)< {{\left( {\bar{c} + 1} \right) } /{{\bar{T}}}} - 1 < 0\). Together with \(c + 1 - \delta {\bar{T}} < 0\), we conclude that \({\left[ {{f_{{C_{T_C}}}}\left( {\bar{x}} \right) - {f_{{D_{T_D}}}}\left( {\bar{x}} \right) } \right] ^{\prime \prime }} < 0\). Thus \(\bar{x}\) is a double root for \(x \in \left( { - \infty , + \infty } \right) \).

From Lemma 1, we know that \({G_1}^\prime \left( x \right) > 0\) for \(x \in \left( { - \infty ,\bar{x}} \right] \) and \({G_1}^\prime \left( x \right) < 0\) for \(x \in \left[ {\bar{x}, + \infty } \right) \) when \(0 \le c < \bar{c}\), and thus \({\left[ {{f_{{C_{T_C}}}}\left( x \right) - {f_{{D_{T_D}}}}\left( x \right) } \right] ^\prime } > 0\) for \(x \in \left( { - \infty ,\bar{x}} \right] \) and \({\left[ {{f_{{C_{T_C}}}}\left( x \right) - {f_{{D_{T_D}}}}\left( x \right) } \right] ^\prime } < 0\) for \(x \in \left[ {\bar{x}, + \infty } \right) \) when \(0 \le c < \bar{c}\). Hence, \({{f_{{C_{T_C}}}}\left( x \right) - {f_{{D_{T_D}}}}\left( x \right) }\) increases monotonically in \(\left( { - \infty ,\bar{x}} \right] \) and decreases monotonically in \(\left[ {\bar{x}, + \infty } \right) \). As a result, together with the condition that \(\delta > {{\left( {c + 1} \right) } / {{\bar{T}}}}\) and \({f_{{C_{T_C}}}}\left( 1 \right) - {f_{{D_{T_D}}}}\left( 1 \right) = R\left( {\delta - 1} \right) \le {f_{{C_{T_C}}}}\left( 0 \right) - {f_{{D_{T_D}}}}\left( 0 \right) = \delta - 1 < 0\), we can conclude that \(\bar{x} \in \left( {0,1} \right) \).

2.1.3 For \(R > \bar{R}\), we have

$$\begin{aligned} \begin{aligned} {f_{{C_{T_C}}}}\left( {\bar{x}} \right) - {f_{{D_{T_D}}}}\left( {\bar{x}} \right)&= \delta - 1 + \left( {R - 1} \right) {G_1}\left( {\bar{x}} \right) \\&> \delta - 1 + \left( {\bar{R} - 1} \right) {G_1}\left( {\bar{x}} \right) = 0. \\ \end{aligned} \end{aligned}$$
(27)

As shown in 2.1.2, if \(\delta > {{\left( {c + 1} \right) } /{{\bar{T}}}}\) and \({f_{{C_{T_C}}}}\left( 1 \right) - {f_{{D_{T_D}}}}\left( 1 \right) = R\left( {\delta - 1} \right) \le {f_{{C_{T_C}}}}\left( 0 \right) - {f_{{D_{T_D}}}}\left( 0 \right) = \delta - 1 < 0\), we can obtain that \(\bar{x} \in \left( {0,1} \right) \). On the other hand, since \({f_{{C_{T_C}}}}\left( 0 \right) - {f_{{D_{T_D}}}}\left( 0 \right) = \delta - 1 < 0\) and \({f_{{C_{T_C}}}}\left( {\bar{x}} \right) - {f_{{D_{T_D}}}}\left( {\bar{x}} \right) > 0\), the Intermediate Value Theorem predicts that \({{f_{{C_{T_C}}}}\left( x \right) - {f_{{D_{T_D}}}}\left( x \right) }\) has a root \(x_L^*\) in \(\left( {0,\bar{x}} \right) \). Similarly, since \({f_{{C_{T_C}}}}\left( {\bar{x}} \right) - {f_{{D_{T_D}}}}\left( {\bar{x}} \right) > 0\) and \({f_{{C_{T_C}}}}\left( 1 \right) - {f_{{D_{T_D}}}}\left( 1 \right) = R\left( {\delta - 1} \right) < 0\), the Intermediate Value Theorem predicts that \({{f_{{C_{T_C}}}}\left( x \right) - {f_{{D_{T_D}}}}\left( x \right) }\) has a root \(x_R^*\) in \(\left( {\bar{x},1} \right) \).

2.2 If \(0 \le c < \bar{c}\), \({{f_{{C_{T_C}}}}\left( x \right) - {f_{{D_{T_D}}}}\left( x \right) }\) increases monotonically in \(\left( { - \infty ,\bar{x}} \right] \) and decreases monotonically in \(\left[ {\bar{x}, + \infty } \right) \). On the other hand, for \({1 / N} < \delta \le {{\left( {c + 1} \right) } /{{\bar{T}}}}\), \(\bar{x} \le 0\). Therefore, \({{f_{{C_{T_C}}}}\left( x \right) - {f_{{D_{T_D}}}}\left( x \right) }\) decreases monotonically for \(x \in \left[ {0,1} \right] \). As a result, together with the condition that \({f_{{C_{T_C}}}}\left( 0 \right) - {f_{{D_{T_D}}}}\left( 0 \right) = \delta - 1 < 0\), \({f_{{C_{T_C}}}}\left( x \right) - {f_{{D_{T_D}}}}\left( x \right) < 0\) for \(x \in \left[ {0,1} \right] \).

2.3 Lemma 1 states that \({G_1}^\prime \left( x \right) < 0\) for \(x \in \left( { - \infty , + \infty } \right) \) when \(c = \bar{c}\), and thus \({\left[ {{f_{{C_{T_C}}}}\left( x \right) - {f_{{D_{T_D}}}}\left( x \right) } \right] ^\prime } < 0\) for \(x \in \left[ {0,1} \right] \). Hence, \({{f_{{C_{T_C}}}}\left( x \right) - {f_{{D_{T_D}}}}\left( x \right) }\) decreases monotonically in \(\left[ {0,1} \right] \). As \({f_{{C_{T_C}}}}\left( 0 \right) - {f_{{D_{T_D}}}}\left( 0 \right) = \delta - 1 < 0\), \({f_{{C_{T_C}}}}\left( x \right) - {f_{{D_{T_D}}}}\left( x \right) < 0\) for \(x \in \left[ {0,1} \right] \).

From Lemma 1, we know that \({G_1}^\prime \left( x \right) < 0\) for \(x \in \left( { - \infty ,\bar{x}} \right] \) and \({G_1}^\prime \left( x \right) > 0\) for \(x \in \left[ {\bar{x}, + \infty } \right) \) when \(c > \bar{c}\). This means that \({f_{{C_{T_C}}}}\left( x \right) - {f_{{D_{T_D}}}}\left( x \right) \) attains a maximum either at \(x = 0\) or \(x = 1\) for \(x \in \left[ {0,1} \right] \). As \({f_{{C_{T_C}}}}\left( 0 \right) - {f_{{D_{T_D}}}}\left( 0 \right) = \delta - 1 < 0\) and \({f_{{C_{T_C}}}}\left( 1 \right) - {f_{{D_{T_D}}}}\left( 1 \right) = R\left( {\delta - 1} \right) < 0\), \({f_{{C_{T_C}}}}\left( x \right) - {f_{{D_{T_D}}}}\left( x \right) < 0\) for \(x \in \left[ {0,1} \right] \). This completes the proof. \(\square \)

Analysis for \({\bar{T}} = N\). When \({\bar{T}} = N\), we obtain the following expression for \({f_{{C_{T_C}}}}\left( x \right) - {f_{{D_{T_D}}}}\left( x \right) \):

$$\begin{aligned} {f_{{C_{T_C}}}}\left( x \right) - {f_{{D_{T_D}}}}\left( x \right) = \delta - 1 + \left( {R - 1} \right) \left( {\delta N - c - 1} \right) {x^{N - 1}}. \end{aligned}$$
(28)

From Eq. (28), we can obtain the following Proposition 3 for \({\bar{T}} = N\).

Proposition 3

Let \({\bar{T}} = N\). \({f_{{C_{T_C}}}}\left( x \right) - {f_{{D_{T_D}}}}\left( x \right) \) satisfies the following properties:

  1. 3.1

    If \(0 \le c < {\bar{c}} = \delta N - 1\), there exists a critical number of rounds \({\bar{R}}_1 = 1 + {{\left( {1 - \delta } \right) } /{\left( {\delta N - c - 1} \right) }}\), which determines the behavior of \({f_{{C_{T_C}}}}\left( x \right) - {f_{{D_{T_D}}}}\left( x \right) \):

    1. 3.1.1

      If \(1 \le R < {\bar{R}}_1\), \({f_{{C_{T_C}}}}\left( x \right) - {f_{{D_{T_D}}}}\left( x \right) < 0\) for \(x \in \left[ {0,1} \right] \).

    2. 3.1.2

      If \(R \ge {\bar{R}}_1\), \({f_{{C_{T_C}}}}\left( x \right) - {f_{{D_{T_D}}}}\left( x \right) \) has one simple root \({{x_1}^*}\) in \(\left[ {0,1} \right] \). \({f_{{C_{T_C}}}}\left( x \right) - {f_{{D_{T_D}}}}\left( x \right) < 0\) for \(x \in [0,{{x_1}^*})\) and \({f_{{C_{T_C}}}}\left( x \right) - {f_{{D_{T_D}}}}\left( x \right) > 0\) for \(x \in ({{x_1}^*},1]\).

  2. 3.2

    If \(c \ge {\bar{c}} = \delta N - 1\), \({f_{{C_{T_C}}}}\left( x \right) - {f_{{D_{T_D}}}}\left( x \right) < 0\) for \(x \in \left[ {0,1} \right] \).

Proof

3.1 When \(0 \le c < {\bar{c}} = \delta N - 1\), \({f_{{C_{T_C}}}}\left( x \right) - {f_{{D_{T_D}}}}\left( x \right) \) increases monotonously with x. The Intermediate Value Theorem tells us that \({f_{{C_{T_C}}}}\left( x \right) - {f_{{D_{T_D}}}}\left( x \right) \) has one single root if and only if \({f_{{C_{T_C}}}}\left( 0 \right) - {f_{{D_{T_D}}}}\left( 0 \right) < 0\) and \({f_{{C_{T_C}}}}\left( 1 \right) - {f_{{D_{T_D}}}}\left( 1 \right) \ge 0\). Because the condition \({f_{{C_{T_C}}}}\left( 0 \right) - {f_{{D_{T_D}}}}\left( 0 \right) = \delta - 1 < 0\) always holds for \(\delta < 1\), the sign of \({f_{{C_{T_C}}}}\left( 1 \right) - {f_{{D_{T_D}}}}\left( 1 \right) \) determines the existence of the root for \(x \in \left[ {0,1} \right] \).

3.1.1 If \(1 \le R < {\bar{R}}_1\), \({f_{{C_{T_C}}}}\left( 1 \right) - {f_{{D_{T_D}}}}\left( 1 \right) < 0\). Therefore, \({f_{{C_{T_C}}}}\left( x \right) - {f_{{D_{T_D}}}}\left( x \right) < 0\) for \(x \in \left[ {0,1} \right] \).

3.1.2 If \(R \ge {\bar{R}}_1\), \({f_{{C_{T_C}}}}\left( 1 \right) - {f_{{D_{T_D}}}}\left( 1 \right) \ge 0\). Thus \({f_{{C_{T_C}}}}\left( x \right) - {f_{{D_{T_D}}}}\left( x \right) \) has one simple root \({{x_1}^*}\) in \(\left[ {0,1} \right] \). \({f_{{C_{T_C}}}}\left( x \right) - {f_{{D_{T_D}}}}\left( x \right) < 0\) for \(x \in [0,{{x_1}^*})\) and \({f_{{C_{T_C}}}}\left( x \right) - {f_{{D_{T_D}}}}\left( x \right) > 0\) for \(x \in ({{x_1}^*},1]\).

3.2 If \(c \ge {\bar{c}} = \delta N - 1\), \({f_{{C_{T_C}}}}\left( x \right) - {f_{{D_{T_D}}}}\left( x \right) \) non-increases monotonously with x for \(x \in \left[ {0,1} \right] \). Hence we have \({f_{{C_{T_C}}}}\left( x \right) - {f_{{D_{T_D}}}}\left( x \right) \le {f_{{C_{T_C}}}}\left( 0 \right) - {f_{{D_{T_D}}}}\left( 0 \right) = \delta - 1 < 0\) for \(x \in \left[ {0,1} \right] \). \(\square \)

Analysis for \({\bar{T}} = 1\). For \({\bar{T}} = 1\), \({f_{{C_{T_C}}}}\left( x \right) - {f_{{D_{T_D}}}}\left( x \right) \) is given by

$$\begin{aligned} {f_{{C_{T_C}}}}\left( x \right) - {f_{{D_{T_D}}}}\left( x \right) = R\left( {\delta - 1} \right) - c\left( {R - 1} \right) {\left( {1 - x} \right) ^{N - 1}}. \end{aligned}$$
(29)

Eq. (29) proves the below Proposition 4 for \({\bar{T}} = 1\).

Proposition 4

Let \({\bar{T}} = 1\). \({f_{{C_{T_C}}}}\left( x \right) - {f_{{D_{T_D}}}}\left( x \right) < 0\) for \(x \in \left[ {0,1} \right] \).

B Analytical Approximation for Accumulated Difference of Payoff

In “Appendix B”, we provide technical details of analytical approximation for the accumulated difference of payoff \({\sum \nolimits _{i = 1}^{M - 1} {\sum \nolimits _{k = 1}^i {\left[ {{f_{{C_{{T_C}}}}}\left( k \right) - {f_{{D_{{T_D}}}}}\left( k \right) } \right] } } }\) in the limit of a large but finite population size \(M \gg N\). When \(M \gg N\), the hypergeometric probabilities in Eq. (15) can be approximated by binomial probabilities:

$$\begin{aligned} \frac{{\left( {\begin{array}{c} {k - 1} \\ j \\ \end{array}} \right) \left( {\begin{array}{c} {M - k} \\ {N - 1 - j} \\ \end{array}} \right) }}{{\left( {\begin{array}{c} {M - 1} \\ {N - 1} \\ \end{array}} \right) }} \approx \left( {\begin{array}{c} {N - 1} \\ j \\ \end{array}} \right) {\left( {\frac{{k - 1}}{{M - 1}}} \right) ^j}{\left( {\frac{{M - k}}{{M - 1}}} \right) ^{N - 1 - j}} \end{aligned}$$
(30)

and

$$\begin{aligned} \frac{{\left( {\begin{array}{c} k \\ j \\ \end{array}} \right) \left( {\begin{array}{c} {M - 1 - k} \\ {N - 1 - j} \\ \end{array}} \right) }}{{\left( {\begin{array}{c} {M - 1} \\ {N - 1} \\ \end{array}} \right) }} \approx \left( {\begin{array}{c} {N - 1} \\ j \\ \end{array}} \right) {\left( {\frac{k}{{M - 1}}} \right) ^j}{\left( {\frac{{M - 1 - k}}{{M - 1}}} \right) ^{N - 1 - j}}. \end{aligned}$$
(31)

Introducing the notation \(x = {i / M}\) and \({p_{{C_{{T_C}}}}} = {k / M}\) in the continuum, one can find that the sums in Eq. (14), \(\sum \nolimits _{i = 1}^{M - 1} {\sum \nolimits _{k = 1}^i {\left( \bullet \right) } }\), can be estimated as the the integral, \(\int _0^1 {dx\int _0^x {d{p_{{C_{{T_C}}}}}\left( \bullet \right) } }\). Using Eqs. (15), (30) and (31), the accumulated difference of payoff in Eq. (14) can be approximated as

$$\begin{aligned} \begin{aligned}&\sum \nolimits _{i = 1}^{M - 1} {\sum \nolimits _{k = 1}^i {\left[ {{f_{{C_{{T_C}}}}}\left( k \right) - {f_{{D_{{T_D}}}}}\left( k \right) } \right] } } \\&\quad \approx \frac{{M\left( {M + 1} \right) }}{6}\left[ {\delta \left( {M - N} \right) - \left( {M - 1} \right) } \right] \\&\qquad + {\left. {\left( {R - 1} \right) \left( {\begin{array}{c} {N - 1} \\ j \\ \end{array}} \right) \left[ {\delta \left( {j + 1} \right) - c - 1} \right] \int _0^1 {{p_{{C_{{T_C}}}}}^j{{\left( {1 - {p_{{C_{{T_C}}}}}} \right) }^{N - j}}d{p_{{C_{{T_C}}}}}} } \right| _{j = {\bar{T}} - 1}}\\&\qquad - \left( {R - 1} \right) \left( {1 - \delta } \right) \sum \limits _{j = {\bar{T}}}^{N - 1} {\left( {\begin{array}{c} {N - 1} \\ j \\ \end{array}} \right) \int _0^1 {{p_{{C_{{T_C}}}}}^j{{\left( {1 - {p_{{C_{{T_C}}}}}} \right) }^{N - j}}d{p_{{C_{{T_C}}}}}} } . \\ \end{aligned} \end{aligned}$$
(32)

Let \(B\left( {j,N - j} \right) = \int _0^1 {{p_{{C_{{T_C}}}}}^j{{\left( {1 - {p_{{C_{{T_C}}}}}} \right) }^{N - j}}d{p_{{C_{{T_C}}}}}}\), we have

$$\begin{aligned} B\left( {j,N - j} \right)= & {} \int _0^1 {{p_{{C_{{T_C}}}}}^j{{\left( {1 - {p_{{C_{{T_C}}}}}} \right) }^{N - j}}d{p_{{C_{{T_C}}}}}} \nonumber \\= & {} \frac{1}{{j + 1}}\int _0^1 {{{\left( {1 - {p_{{C_{{T_C}}}}}} \right) }^{N - j}}d{p_{{C_{{T_C}}}}}^{j + 1}} \nonumber \\= & {} \frac{1}{{j + 1}}\left[ {\left. {{{\left( {1 - {p_{{C_{{T_C}}}}}} \right) }^{N - j}}{p_{{C_{{T_C}}}}}^{j + 1}} \right| _0^1 - \int _0^1 {{p_{{C_{{T_C}}}}}^{j + 1}d{{\left( {1 - {p_{{C_{{T_C}}}}}} \right) }^{N - j}}} } \right] \nonumber \\= & {} \frac{{N - j}}{{j + 1}}\left[ {\int _0^1 {{p_{{C_T}}}^{j + 1}{{\left( {1 - {p_{{C_T}}}} \right) }^{N - j - 1}}d{p_{{C_T}}}} } \right] \nonumber \\= & {} \frac{{N - j}}{{j + 1}}B\left( {j + 1,N - j - 1} \right) . \end{aligned}$$
(33)

From above recursive equation, one can obtain

$$\begin{aligned} B\left( {j,N - j} \right) = \frac{1}{{\left( {N + 1} \right) \left( {\begin{array}{*{20}{c}} N \\ j \\ \end{array}} \right) }}. \end{aligned}$$
(34)

Substituting Eq. (34) into Eq. (32) leads to the analytical approximation of the accumulated payoff difference in the limit of \(M \gg N\):

$$\begin{aligned}&\sum \nolimits _{i = 1}^{M - 1} {\sum \nolimits _{k = 1}^i {\left[ {{f_{{C_{{T_C}}}}}\left( k \right) - {f_{{D_{{T_D}}}}}\left( k \right) } \right] } } \nonumber \\&\quad \approx \left[ {\frac{{M\left( {M + 1} \right) \left( {M - N} \right) }}{6} + \frac{{\left( {R - 1} \right) \left( {N - {\bar{T}} + 1} \right) \left( {N + {\bar{T}}} \right) }}{{2N\left( {N + 1} \right) }}} \right] \delta \nonumber \\&\qquad - \frac{{M\left( {M + 1} \right) \left( {M - 1} \right) }}{6} - \frac{{\left( {R - 1} \right) \left( {N - {\bar{T}} + 1} \right) \left( {N - {\bar{T}} + 2c + 2} \right) }}{{2N\left( {N + 1} \right) }}. \end{aligned}$$
(35)

C Proof of Proposition 1

In “Appendix C”, we prove formally Proposition 1.

Proof

1.1 :

Following Eq. (17), we directly prove the first property of Proposition 1.

1.2 :

By simplifying Eq. (17), we have

$$\begin{aligned} \begin{array}{l} {\delta _{th}} = \frac{{M\left( {M + 1} \right) \left( {M - N} \right) N}}{{3\left( {N - {\bar{T}} + 1} \right) \left( {N + {\bar{T}}} \right) }} \times \frac{{\frac{{M - 1}}{{M - N}} - \frac{{N - {\bar{T}} + 2c + 2}}{{N + {\bar{T}}}}}}{{\frac{{M\left( {M + 1} \right) \left( {M - N} \right) N}}{{3\left( {N - {\bar{T}} + 1} \right) \left( {N + {\bar{T}}} \right) }} + \frac{{R - 1}}{{N + 1}}}} + \frac{{N - {\bar{T}} + 2c + 2}}{{N + {\bar{T}}}}. \\ \end{array} \end{aligned}$$
(36)

From Eq. (36), one can obtain that

1.2.1:

\({\delta _{th}}\) is decreased with \(R \in \left\{ {1, \cdots , + \infty } \right\} \) if \(c \in \left[ {0,{{\bar{c}}_1}} \right) \);

1.2.2:

\({\delta _{th}}\) is increased with \(R \in \left\{ {1, \cdots , + \infty } \right\} \) if \(c \in \left( {{{\bar{c}}_1}, + \infty } \right) \);

1.2.3:

\({\delta _{th}}\) is invariant with \(R \in \left\{ {1, \cdots , + \infty } \right\} \) if \(c = {{\bar{c}}_1}\), where the critical value \({{\bar{c}}_1}\) is given by Eq. (18). This proves the second property of Proposition 1.

1.3:

If replacing the discrete variable \({\bar{T}} \in \left\{ {1, \cdots ,N} \right\} \) with a continuous variable \(z \in \left[ {1,N} \right] \) in Eq. (17), we obtain the following expression for the first order partial derivative of \({\delta _{th}}\) with respect to z:

$$\begin{aligned} \frac{{\partial {\delta _{th}}}}{{\partial z}} = \frac{{\frac{{{{\left( {R - 1} \right) }^2}\left( {N + c + 1} \right) }}{{2{N^2}{{\left( {N + 1} \right) }^2}}}{G_2}\left( z \right) }}{{{{\left[ {\frac{{M\left( {M + 1} \right) \left( {M - N} \right) }}{6} + \frac{{\left( {R - 1} \right) \left( {N - z + 1} \right) \left( {N + z} \right) }}{{2N\left( {N + 1} \right) }}} \right] }^2}}}, \end{aligned}$$
(37)

where the polynomial \({{G_2}\left( z \right) }\) is given by

$$\begin{aligned} {G_2}\left( z \right)= & {} - {z^2} + \left[ {\frac{{M\left( {M + 1} \right) N\left( {N + 1} \right) \left( {2M - N - 1} \right) }}{{3\left( {R - 1} \right) \left( {N + c + 1} \right) }} + 2\left( {N + 1} \right) } \right] z \nonumber \\&- {\left( {N + 1} \right) ^2} - \frac{{M\left( {M + 1} \right) N\left( {N + 1} \right) \left[ {2\left( {M - N} \right) \left( {N + c + 1} \right) + 2M - N - 1} \right] }}{{6\left( {R - 1} \right) \left( {N + c + 1} \right) }}.\nonumber \\ \end{aligned}$$
(38)

From Eq. (38), one can observe that \({G_2}\left( z \right) \) increases with z in the interval \(\left[ {1,N} \right] \). Furthermore,

$$\begin{aligned} {G_2}\left( 1 \right) = - \frac{{M\left( {M + 1} \right) N\left( {N + 1} \right) \left[ {2\left( {M - N - 1} \right) \left( {c + N} \right) + N + 2c + 1} \right] }}{{6\left( {R - 1} \right) \left( {c + N + 1} \right) }} - {N^2} < 0.\nonumber \\ \end{aligned}$$
(39)

The Intermediate Value Theorem predicts that \({G_2}\left( z \right) \) has one single root in the interval \(\left( {1,N} \right] \) if and only if

$$\begin{aligned} {G_2}\left( N \right)= & {} \frac{{M\left( {M + 1} \right) N\left( {N + 1} \right) }}{{6\left( {R - 1} \right) \left( {c + N + 1} \right) }}\nonumber \\&\times \left[ { - 2\left( {M - N} \right) c + 2MN - 4M + N + 1} \right] - 1 \ge 0, \end{aligned}$$
(40)

which leads to the critical value \({{\bar{c}}_2}\) given by Eq. (19). Otherwise, there is no root for \({G_2}\left( z \right) \) in the interval \(\left( {1,N} \right] \). Furthermore, one can observe from Eq. (37) that \({G_2}\left( z \right) \) has the same sign and root with \({{\partial {\delta _{th}}} /{\partial z}}\). Therefore, the optimal discrete solution \({\bar{T}}^* \in \left\{ {1, \cdots ,N} \right\} \) resulting in the minimum \({{\delta _{th}}}\) is given by

$$\begin{aligned} \begin{array}{l} {\bar{T}}^* = \left\{ \begin{array}{l} \left\lfloor {z'} \right\rfloor \; {\mathrm{if}}\; {\delta _{th}}\left( {\left\lfloor {z'} \right\rfloor \; } \right) \le {\delta _{th}}\left( {\left\lceil {z'} \right\rceil \; } \right) \; {\mathrm{and}}\; c \le {{\bar{c}}_2}, \\ \left\lceil {z'} \right\rceil \; {\mathrm{if}}\; {\delta _{th}}\left( {\left\lfloor {z'} \right\rfloor \; } \right) > {\delta _{th}}\left( {\left\lceil {z'} \right\rceil \; } \right) \; {\mathrm{and}}\; c \le {{\bar{c}}_2}, \\ N\; \; {\mathrm{otherwise,}} \\ \end{array} \right. \\ \end{array} \end{aligned}$$
(41)

where \(z'\) corresponding to the left root of \({G_2}\left( z \right) = 0\) is given by

$$\begin{aligned} z'= & {} \frac{{M\left( {M + 1} \right) \left( {2M - N - 1} \right) N\left( {N + 1} \right) }}{{6\left( {R - 1} \right) \left( {c + N + 1} \right) }} + N + 1 \nonumber \\&- \sqrt{\frac{{M\left( {M + 1} \right) \left( {2M - N - 1} \right) N\left( {N + 1} \right) }}{{6\left( {R - 1} \right) \left( {c + N + 1} \right) }}\left[ {\frac{{M\left( {M + 1} \right) \left( {2M - N - 1} \right) N\left( {N + 1} \right) }}{{6\left( {R - 1} \right) \left( {c + N + 1} \right) }} - \frac{{2\left( {M - N} \right) \left( {c + N + 1} \right) }}{{2M - N - 1}} + 2N + 1} \right] }.\nonumber \\ \end{aligned}$$
(42)

If \(c \le {{\bar{c}}_2}\), \({\delta _{th}}\) is decreased with \({\bar{T}}\) in the set \(\left\{ {1, \cdots ,{{\bar{T}}^*}} \right\} \) and is increased with \({\bar{T}}\) in the set \(\left\{ {{{\bar{T}}^*}, \cdots ,N} \right\} \) since \({{\partial {\delta _{th}}} /{\partial z}} < 0\) for \(\left[ {1,z'} \right) \) and \({{\partial {\delta _{th}}} /{\partial z}} > 0\) for \(\left( {z',N} \right] \). If \(c > {{\bar{c}}_2}\), \({\delta _{th}}\) is decreased with \({\bar{T}}\) in the set \(\left\{ {1, \cdots ,N} \right\} \) since \({{\partial {\delta _{th}}} /{\partial z}} < 0\) for \(\left[ {1,N} \right] \). Therefore, for any given \({{T_D}}\), one can obtain that the optimal solution \({{T_C}^*}\) leading to the minimum \({\delta _{th}}\) satisfies

1.3.1:

\({T_C}^* = {\bar{T}}^*\) if \(c \le {{\bar{c}}_2}\);

1.3.2:

\({T_C}^* = N\) if \(c > {{\bar{c}}_2}\), which proves the third property of Proposition 1. \(\square \)

D Summary of Critical Parameters: Notations and Mathematical Expressions

In “Appendix D”, we describe the critical parameters used in this study for the purpose of clarifying their definitions and mathematical expressions (see Table 3).

Table 3 Summary of critical parameters: notations and mathematical expressions
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Wang, X. Costly Participation and The Evolution of Cooperation in the Repeated Public Goods Game. Dyn Games Appl 11, 161–183 (2021). https://doi.org/10.1007/s13235-020-00352-1

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