# How to Analyze Models of Nonlinear Public Goods

## Abstract

**:**

## 1. Introduction

#### 1.1. Public Goods

#### 1.2. Multiplayer Games and Nonlinear Benefits

#### 1.3. Rationale of the Paper

## 2. How to Analyze Nonlinear Public Goods Games

#### 2.1. The Replicator Dynamics with Two Strategies

_{C}− W

_{D}is written in the form $\beta $(x), and

#### 2.2. Nonlinear Benefits

#### 2.3. Bernstein Polynomials

- The Bernstein polynomial basis of degree n on x in [0, 1] is defined, for i = 0, …, n, by$${p}_{i,n}\left(x\right)=\left(\begin{array}{c}n\\ i\end{array}\right){x}^{i}{\left(1-x\right)}^{n-i}$$
- A polynomial in Bernstein form of degree n associated with any continuous function f
_{i}on [0, 1] is defined for each positive integer n as$$F\left(x\right){{\displaystyle \sum}}_{i=0}^{n}{p}_{i,n}\left(x\right){f}_{i}$$ - f
_{i}is called the Bernstein coefficient

- End-point values: The initial and final values of f and F are the same: F(0) = f
_{0}; F(1) = f_{n}. - Shape preservation: F and f have the same shape (monotonicity, convexity, concavity).
- Variation-diminishing property: The number Z of real roots of F in (0,1)is less than the number S of sign changes of f by an even amount: Z = S − 2j for some integer j greater or equal to 0 and lower than the integer part of n/2 [53] (each root contributes according to its multiplicity).

#### 2.4. Characterizing the Dynamics

_{i}is its Bernstein coefficient. Hence, based on the properties defined above, it is straightforward to characterize the dynamics of a nonlinear public goods game based on the number of sign changes of the Bernstein coefficient Δb

_{i}instead of inspecting the gradient of selection of the Bernstein polynomial $\beta $. This Bernstein approach was introduced to study public goods games with sigmoid benefits in the context of cancer biology [39,40]. Games with concave or convex benefits can be analyzed [54] without resorting to the properties of Bernstein polynomials, although in that case the proof is unnecessarily long—the proof is trivial using the Bernstein approach (the results of this exercise have been published in [55], which also replicated the sigmoid case analyzed in [39]).

_{n}

_{−1}< c < Δb

_{0}; if the initial sign is -, then x* is unstable and x = 0 and x = 1 are stable: an example of this occurs in games with convex benefits when Δb

_{0}< c < Δb

_{n}

_{−1}. Games with concave or convex benefits can also have no sign change. Hence, with concave benefits, if c ≥ Δb

_{0}then x = 0 is the unique stable equilibrium and x = 1 is the unique unstable equilibrium; if Δb

_{n}

_{−1}< c < Δb

_{0}then there is a unique interior stable equilibrium and x = 0, x = 1 are unstable equilibria; if c ≤ Δb

_{n}

_{−1}then x = 1 is the unique stable equilibrium and x = 0 is the unique unstable equilibrium. With convex benefits, if c ≥ Δb

_{n}

_{−1}then x = 0 is the unique stable equilibrium and x = 1 is the unique unstable equilibrium; if Δb

_{0}< c < Δb

_{n}

_{−1}then there is a unique interior unstable equilibrium and x = 0, x = 1 are stable equilibria; if c ≤ Δb

_{0}then x = 1 is the unique stable equilibrium and x = 0 is the unique unstable equilibrium. In summary, games with concave or convex benefits can have at most one interior equilibrium, either stable or unstable (Figure 2).

_{i}is increasing for i < h, decreasing for i > h, and satisfies Δb

_{i}≠ 0 ∀i, then Δb

_{i}is single-peaked and can have at most two sign changes (Figure 3); thus, there are either two interior rest points, or one (if they coincide) or zero, and the dynamics can be characterized as follows:

- If c ≥ ${\beta}_{MAX}$ (the maximum value of $\beta $) there is a unique stable equilibrium x = 0 (producers go extinct) (c
_{1}in Figure 3) - If Max[Δb
_{0}, Δb_{n}_{−1}] ≤ c < ${\beta}_{MAX}$ there are two stable equilibria: a pure equilibrium x = 0 made of all non-producers and a mixed equilibrium x*; the basins of attraction of these two equilibria are separated by a mixed unstable equilibrium x^; if the initial frequency of producers x < x^ the population will evolve to x = 0; if x > x^ it will evolve to x* (c_{2}in Figure 3). - If Δb
_{0}≤ c < Δb_{n}_{−1}(this case can exist only for h > 0.5) there is a unique interior unstable equilibrium x^ separating the basins of attraction of two pure stable equilibria: x = 0 and x = 1; if x < x^ the population will evolve to x = 0; if x > x^ it will evolve to x = 1 (c_{3}in Figure 3). - If Δb
_{n}_{−1}≤ c < Δb_{0}(this case can exist only for h < 0.5) there is a unique interior stable equilibrium x*, to which the population will converge irrespective of the initial frequencies of the two types (c_{3}in Figure 3). - If c < Min[Δb
_{0}, Δb_{n}_{−1}] there is a unique stable equilibrium x = 1 (non-producers will go extinct) (c_{4}in Figure 3).

_{i}increases for i < h and decreases for i > h. Figure 4 shows two examples of a more complex benefit function, and a case in which the benefits of the two strategies are different functions.

_{1}and steepness s

_{1}; for i > dn, the function is monotonically decreasing and has an inflection point at h

_{2}and steepness s

_{2}(with 0 < h

_{1}, h

_{2}≤ 1 and s

_{1},s

_{2}> 0); the additional parameter y measures the maximum damage of self-poisoning. While analyzing the gradient of selection with benefits given by (13) appears hopeless, the Bernstein approach enables us to characterize the dynamics simply by looking at the number and type of sign changes of the Bernstein coefficient, as shown in Figure 4A.

#### 2.5. Comparison with Pairwise and Linear Games

#### 2.6. Finding the Equilibria

_{i}, resorting to an additional property of Bernstein polynomials–Bernstein theorem: A Bernstein polynomial F(x) converges uniformly to its coefficient f(x) in [0, 1], that is, $\underset{n\to \infty}{\mathrm{lim}}F\left(x\right)=f\left(x\right)$. The approximation is rather slow. By Voronovskaya’s theorem [56], for functions that are twice differentiable:

_{0}, Δb

_{n}

_{−1}] (otherwise x = 1 is the only stable equilibrium) and c < s/4n (otherwise x = 0 is the only stable equilibrium). A qualitatively equivalent result holds for the normalized version in (9), because b(i) differs from l(i) only by the inclusion of the normalizing term. For the normalized version, the equilibria are given by

#### 2.7. Goodness of the Approximation

#### 2.8. Approaches to Calculate the Equilibria

- Using the exact formula from (8), by setting $\beta $(x) = 0. This is a reasonable approach for n not too large (the “Reduce” command in Mathematica 11, on a laptop with a 2.9 GHz processor and 16 GB of memory, enables this for n up to about 50) but the results become inaccurate as n grows and it is hopeless for very large n, when numerical methods become necessary.
- The equilibrium can also be approximated by setting Δb = 0, but similar computational problems arise.
- Using the Bernstein approximation outlined above, the equilibrium can be easily calculated for any n from Equations (21)–(24). The accuracy of this approach, as we have seen (Figure 6), improves with n and declines with s.
- For large n and large s, an alternative method is to find the equilibria of a Heaviside step function with the same inflection h using the following equation (25) (where k = hn):$$\left(\begin{array}{c}n-1\\ k-1\end{array}\right){x}^{k-1}{\left(1-x\right)}^{n-k}=c$$

#### 2.9. More Than Two Strategies

_{A}= γ

_{B}(j) = r[L

_{B}(j) − L

_{B}(0)]/[L

_{B}(n) − L

_{B}(0)]

_{C}(j) = 1 − [L

_{C}(j) − L

_{C}(0)]/[L

_{C}(n) − L

_{C}(0)]

_{1}and h

_{2}are analogous to h and the parameters s

_{1}and s

_{2}are analogous to s, thus (32) and (33) are sigmoid functions that can range from linear to step functions. In an infinite well-mixed population, if x

_{C}is the frequency of cells of type C, the fitness of the three types are:

## 3. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

- Hardin, J. The tragedy of the commons. Science
**1968**, 162, 1243–1248. [Google Scholar] [CrossRef] [PubMed] - Burt, A.; Trivers, R. Genes in Conflict; Harvard University Press: Cambridge, MA, USA, 2006. [Google Scholar]
- West, S.A.; Diggle, S.P.; Buckling, A.; Gardner, A.; Griffin, A.S. The social lives of microbes. Annu. Rev. Ecol. Evol. Syst.
**2007**, 38, 53–77. [Google Scholar] [CrossRef] - Jouanneau, J.; Moens, G.; Bourgeois, Y.; Poupon, M.F.; Thiery, J.P. A minority of carcinoma cells producing acidic fibroblast growth factor induces a community effect for tumor progression. Proc. Nat. Acad. Sci. USA
**1994**, 91, 286–290. [Google Scholar] [CrossRef] [PubMed] - Axelrod, R.; Axelrod, D.E.; Pienta, K.J. Evolution of cooperation among tumor cells. Proc. Nat. Acad. Sci. USA
**2006**, 103, 13474–13479. [Google Scholar] [CrossRef] [PubMed] - Packer, C.; Scheel, D.; Pusey, A.E. Why lions form groups, food is not enough. Am. Nat.
**1990**, 136, 1–19. [Google Scholar] [CrossRef] - Maynard Smith, J.; Szathmáry, E. The Major Transitions in Evolution; Freeman: San Francisco, CA, USA, 1995. [Google Scholar]
- Tucker, A. A two-person dilemma (1950). In Readings in Games and Information; Rasmusen, E., Ed.; Blackwell: Oxford, UK, 2001; pp. 7–8. [Google Scholar]
- Rapoport, A.; Chammah, A.M. The game of chicken. Am. Behav. Sci.
**1966**, 10, 10–28. [Google Scholar] [CrossRef] - Maynard Smith, J.; Price, G.R. The logic of animal conflict. Nature
**1973**, 246, 15–18. [Google Scholar] [CrossRef] - Sugden, R. The Economics of Rights, Cooperation and Welfare; Blackwell: Oxford, UK, 1986. [Google Scholar]
- Hamburger, H. N-person Prisoners Dilemma. J. Math. Sociol.
**1973**, 3, 27–48. [Google Scholar] [CrossRef] - Fox, J.; Guyer, M. Public Choice and cooperation in N-person Prisoner’s Dilemma. J. Confl. Resolut.
**1978**, 22, 469–481. [Google Scholar] [CrossRef] - Rankin, D.J.; Bargum, K.; Kokko, H. The tragedy of the commons in evolutionary biology. Trends Ecol. Evol.
**2007**, 12, 643–651. [Google Scholar] [CrossRef] [PubMed][Green Version] - Kollock, P. Social dilemmas, the anatomy of cooperation. Ann. Rev. Sociol.
**1998**, 24, 183–214. [Google Scholar] [CrossRef] - Cornish-Bowden, A. Fundamentals of Enzyme Kinetics, 4th ed.; Wiley Blackwell: Hoboken, NJ, USA, 2012. [Google Scholar]
- Frank, S.A. Input-output relations in biological systems, measurement, information and the Hill equation. Biol. Dir.
**2013**, 8, 31. [Google Scholar] [CrossRef] [PubMed] - Archetti, M.; Scheuring, I. Evolution of optimal Hill coefficients in nonlinear public goods games. J. Theor. Biol.
**2016**, 406, 73–82. [Google Scholar] [CrossRef] [PubMed][Green Version] - Tomlinson, I.P. Game-theory models of interactions between tumour cells. Eur. J. Cancer
**1997**, 33, 1495–1500. [Google Scholar] [CrossRef] - Tomlinson, I.P.; Bodmer, W.F. Modelling consequences of interactions between tumour cells. Br. J. Cancer
**1997**, 75, 157–160. [Google Scholar] [CrossRef] [PubMed] - Bach, L.A.; Bentzen, S.; Alsner, J.; Christiansen, F.B. An evolutionary-game model of tumour cell interactions, possible relevance to gene therapy. Eur. J. Cancer
**2001**, 37, 2116–2120. [Google Scholar] [CrossRef] - Bach, L.A.; Sumpter, D.J.T.; Alsner, J.; Loeschke, V. Spatial evolutionary games of interaction among generic cancer cells. J. Theor. Med.
**2003**, 5, 47–58. [Google Scholar] [CrossRef] - Basanta, D.; Hatzikirou, H.; Deutsch, A. Studying the emergence of invasiveness in tumours using game theory. Eur. Phys. J.
**2008**, 63, 393–397. [Google Scholar] [CrossRef] - Basanta, D.; Simon, M.; Hatzikirou, H.; Deutsch, A. Evolutionary game theory elucidates the role of glycolysis in glioma progression and invasion. Cell Prolif.
**2008**, 41, 980–987. [Google Scholar] [CrossRef] [PubMed] - Basanta, D.; Scott, J.G.; Rockne, R.; Swanson, K.R.; Anderson, A.R. The role of IDH1 mutated tumour cells in secondary glioblastomas, an evolutionary game theoretical view. Phys. Biol.
**2011**, 8, 015016. [Google Scholar] [CrossRef] [PubMed] - Basanta, D.; Scott, J.G.; Fishman, M.N.; Ayala, G.; Hayward, S.W.; Anderson, A.R. Investigating prostate cancer tumour-stroma interactions, clinical and biological insights from an evolutionary game. Br. J. Cancer
**2012**, 106, 174–181. [Google Scholar] [CrossRef] [PubMed][Green Version] - Dingli, D.; Chalub, F.A.; Santos, F.C.; Pacheco, J. Cancer phenotype as the outcome of an evolutionary game between normal and malignant cells. Br. J. Cancer
**2009**, 101, 1130–1136. [Google Scholar] [CrossRef] [PubMed] - Gerstung, M.; Nakhoul, H.; Beerenwinkel, N. Evolutionary games with affine fitness functions, applications to cancer. Dyn. Games Appl.
**2011**, 1, 370–385. [Google Scholar] [CrossRef] - You, L.; Brown, J.S.; Thuijsman, F.; Cunningham, J.J.; Gatenby, R.A.; Zhang, J.S.; Stankova, K. Spatial vs. non-spatial eco-evolutionary dynamics in a tumor growth model. J. Theor. Biol.
**2017**, 435, 78–97. [Google Scholar] [CrossRef] [PubMed] - Zhang, J.S.; Cunningham, J.J.; Brown, J.S.; Gatenby, R.A. Integrating evolutionary dynamics into treatment of metastatic castrate-resistant prostate cancer. Nat. Commun.
**2017**, 8, 1816. [Google Scholar] [CrossRef] [PubMed] - Broom, M.; Cannings, C.; Vickers, G.T. Multiplayer matrix games. Bull. Math. Biol.
**1997**, 59, 931–952. [Google Scholar] [CrossRef] [PubMed] - Archetti, M.; Scheuring, I. Review: Game theory of public goods in one-shot social dilemmas without assortment. J. Theor. Biol.
**2012**, 299, 9–20. [Google Scholar] [CrossRef] [PubMed] - Archetti, M. The volunteer’s dilemma and the optimal size of a social group. J. Theor. Biol.
**2009**, 261, 475–480. [Google Scholar] [CrossRef] [PubMed] - Archetti, M. Cooperation as a volunteer’s dilemma and the strategy of conflict in public goods games. J. Evol. Biol.
**2009**, 22, 2192–2200. [Google Scholar] [CrossRef] [PubMed] - Archetti, M.; Scheuring, I. Coexistence of cooperation and defection in public goods games. Evolution
**2011**, 65, 1140–1148. [Google Scholar] [CrossRef] [PubMed] - Boza, G.; Számadó, S. Beneficial laggards, multilevel selection, cooperative polymorphism and division of labour in threshold public good games. BMC Evol. Biol.
**2010**, 10, 336. [Google Scholar] [CrossRef] [PubMed] - Archetti, M. Dynamics of growth factor production in monolayers of cancer cells. Evol. Appl.
**2013**, 6, 1146–1159. [Google Scholar] [CrossRef] [PubMed][Green Version] - Archetti, M. Evolutionarily stable anti-cancer therapies by autologous cell defection. Evol. Med. Public Health
**2013**, 1, 161–172. [Google Scholar] [CrossRef] [PubMed] - Archetti, M. Evolutionary game theory of growth factor production, implications for tumor heterogeneity and resistance to therapies. Br. J. Cancer
**2013**, 109, 1056–1062. [Google Scholar] [CrossRef] [PubMed][Green Version] - Archetti, M. Evolutionary dynamics of the Warburg effect, glycolysis as a collective action problem among cancer cells. J. Theor. Biol.
**2014**, 341, 1–8. [Google Scholar] [CrossRef] [PubMed][Green Version] - Archetti, M. Stable heterogeneity for the production of diffusible factors in cell populations. PLoS ONE
**2014**, 9, e108526. [Google Scholar] [CrossRef] [PubMed][Green Version] - Archetti, M. Heterogeneity and proliferation of invasive cancer subclones in game theory models of the Warburg effect. Cell Prolif.
**2015**, 482, 259–269. [Google Scholar] [CrossRef] [PubMed][Green Version] - Archetti, M. Cooperation among cancer cells as public goods games on Voronoi networks. J. Theor. Biol.
**2016**, 396, 191–203. [Google Scholar] [CrossRef] [PubMed][Green Version] - Archetti, M.; Ferraro, D.A.; Christofori, G. Heterogeneity for IGF-II production maintained by public goods dynamics in neuroendocrine pancreatic cancer. Proc. Nat. Acad. Sci. USA
**2015**, 112, 1833–1838. [Google Scholar] [CrossRef] [PubMed][Green Version] - Kaznatcheev, A.; Velde, R.V.; Scott, J.G.; Basanta, D. Cancer treatment scheduling and dynamic heterogeneity in social dilemmas of tumour acidity and vasculature. Br. J. Cancer
**2018**, in press. [Google Scholar] - Sartakhti, J.S.; Manshaei, M.H.; Bateni, S.; Archetti, M. Evolutionary dynamics of tumor-stroma interactions in multiple myeloma. PLoS ONE
**2016**, 1112, e0168856. [Google Scholar] [CrossRef] [PubMed] - Hofbauer, J.; Sigmund, K. Evolutionary Games and Population Dynamics; Cambridge University Press: Cambridge, UK, 1998. [Google Scholar]
- Bernstein, S. Démonstration du théorème de Weierstrass fondée sur le calcul des probabilities. Comm. Soc. Math. Kharkov
**1912**, 13, 1–2. [Google Scholar] - Farouki, R.T. The Bernstein polynomial basis, a centennial retrospective. Comput. Aided Geom. Des.
**2012**, 29, 379–419. [Google Scholar] [CrossRef] - Lorentz, G.G. Bernstein Polynomials; University of Toronto Press: Toronto, ON, Canada, 1953. [Google Scholar]
- DeVore, R.A.; Lorentz, G.G. Constructive Approximation; Springer: Berlin, Germany, 1993. [Google Scholar]
- Phillips, G.M. Interpolation and Approximation by Polynomials; Springer: Berlin, Germany, 2003. [Google Scholar]
- Schoenberg, I.J. On variation diminishing approximation methods. In On Numerical Approximation; Langer, R.E., Ed.; University of Wisconsin Press: Madison, WI, USA, 1959. [Google Scholar]
- Motro, U. Cooperation and defection, playing the field and the ESS. J. Theor. Biol.
**1991**, 151, 145–154. [Google Scholar] [CrossRef] - Pena, G.; Lehmann, L.; Noeldeke, G. Gains from switching and evolutionary stability in multi-player matrix games. J. Theor. Biol.
**2014**, 346, 23–33. [Google Scholar] [CrossRef] [PubMed][Green Version] - Voronovskaya, E. Détermination de la forme asymptotique d’ approximation des fonctions par les polynômes de M. Bernstein. CR Acad. Sci. URSS
**1932**, 79, 79–85. [Google Scholar] - Mabry, R. Problem 10990. Am. Math. Mon.
**2003**, 110, 59. [Google Scholar] [CrossRef] - Hamilton, W.D. The genetical evolution of social behaviour. J. Theor. Biol.
**1964**, 7, 1–52. [Google Scholar] [CrossRef] - Frank, S.A. Foundations of Social Evolution; Princeton University Press: Princeton, NJ, USA, 1998. [Google Scholar]
- Nowak, M.A. Evolutionary Dynamics; Harvard University Press: Cambridge, MA, USA, 2006. [Google Scholar]
- Axelrod, R.; Hamilton, W.D. The Evolution of cooperation. Science
**1981**, 211, 1390–1396. [Google Scholar] [CrossRef] [PubMed] - Greig, D.; Travisano, M. The Prisoner’s Dilemma and polymorphism in yeast SUC genes. Proc. R. Soc. Lond. Ser. B
**2004**, 27, S25–S26. [Google Scholar] [CrossRef] [PubMed] - Gore, J.; Youk, H.; van Oudenaarden, A. Snowdrift game dynamics and facultative cheating in yeast. Nature
**2009**, 459, 253–256. [Google Scholar] [CrossRef] [PubMed] - MacLean, R.C.; Fuentes-Hernandez, A.; Greig, D.; Hurst, L.D.; Gudelj, I. A Mixture of “Cheats” and “Co-Operators” Can Enable Maximal Group Benefit. PLoS Biol.
**2010**, 8, e1000486. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Sigmoid benefits. The benefit b(i) of a public good as a function of the number of producers (i), described by a normalized logistic function (Equation (9)) with inflection point h; h→1 gives increasing returns and h→0 diminishing returns. Multiple curves are shown, with increasing steepness s (increasing opacity).

**Figure 2.**The gradient of selection $\beta $ (bottom panels, continuous blue line) and its Bernstein coefficient Δb

_{i}(bottom panels, dotted line) for different types of benefit functions (top panels): concave (n = 9, h = 0.1, s = 10, c = 0.1); convex (n = 9, h = 0.9, s = 10, c = 0.1); sigmoid (n = 10, h = 0.4, s = 9, c = 0.1); step function (n = 9, h = 0.4, s = 1000, c = 0.1). Arrows show the direction of the dynamics (determined by the sign of the gradient of selection); squares show the equilibria (full: stable; empty: unstable).

**Figure 3.**Types of dynamics when the benefit is a sigmoid function. The gradient of selection $\beta $ (continuous blue line) and its Bernstein coefficient Δb

_{i}(dotted line); squares show the equilibria (full: stable; empty: unstable) for different values of c (gray lines); arrows show the direction of the dynamics (determined by the sign of the gradient of selection); s = 5; n = 9, h = 0.4 or 0.6.

**Figure 4.**The gradient of selection $\beta $ (continuous blue line) and its Bernstein coefficient Δb

_{i}(dotted line) for more complex functions; squares show the equilibria (full: stable; empty: unstable); arrows show the direction of the dynamics (determined by the sign of the gradient of selection); A: The double inverse sigmoid function described in (13) (n = 10, h

_{1}= 0.4, h

_{2}= 0.2, s

_{1}= 10, s

_{2}= 10, c = 0.1, y = 2, d = 0.5). B: Two different benefit functions described by (16) and (17) (n = 10, h

_{1}= 0.6, h

_{2}= 0.2, s

_{1}= 5, s

_{2}= 20, c = 0.1).

**Figure 5.**Effect of s and n. The gradient of selection $\beta $ (continuous blue line) and its Bernstein coefficient Δb

_{i}(dotted line) in public goods games with sigmoid benefits, for different values of s (n = 10, h = 0.4, c = 0.05) or for different values of n (s = 10, h = 0.4, c = 0.05); squares show the equilibria (full: stable; empty: unstable); arrows show the direction of the dynamics (determined by the sign of the gradient of selection).

**Figure 6.**Error [Δb − ${\beta}^{*}\left(x\right)$] at the stable equilibrium, as a function of s, for different values of n; c = 0.01, h = 0.5.

**Figure 7.**Comparison between the values of the stable equilibria (as a function of n) calculated using different methods, with c = 0.04, h = 0.5 and different values of s.

**Figure 8.**Three-strategy game. The dynamics of the three strategies A, B, and C with fitness given by Equations (33)–(35) with n = 20, h

_{1}= 0.2, h

_{2}= 0.4, r = 0.5 with pairwise interactions, linear benefits (s

_{1}= s

_{2}= 0.01) or nonlinear benefits (s

_{1}= 50, s

_{2}= 20).

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Archetti, M. How to Analyze Models of Nonlinear Public Goods. *Games* **2018**, *9*, 17.
https://doi.org/10.3390/g9020017

**AMA Style**

Archetti M. How to Analyze Models of Nonlinear Public Goods. *Games*. 2018; 9(2):17.
https://doi.org/10.3390/g9020017

**Chicago/Turabian Style**

Archetti, Marco. 2018. "How to Analyze Models of Nonlinear Public Goods" *Games* 9, no. 2: 17.
https://doi.org/10.3390/g9020017