# 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

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**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).

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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