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
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
- A polynomial in Bernstein form of degree n associated with any continuous function fi on [0, 1] is defined for each positive integer n as
- fi is called the Bernstein coefficient
- End-point values: The initial and final values of f and F are the same: F(0) = f0; F(1) = fn.
- 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
- If c ≥ (the maximum value of ) there is a unique stable equilibrium x = 0 (producers go extinct) (c1 in Figure 3)
- If Max[Δb0, Δbn−1] ≤ c < 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* (c2 in Figure 3).
- If Δb0 ≤ c < Δbn−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 (c3 in Figure 3).
- If Δbn−1 ≤ c < Δb0 (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 (c3 in Figure 3).
- If c < Min[Δb0, Δbn−1] there is a unique stable equilibrium x = 1 (non-producers will go extinct) (c4 in Figure 3).
2.5. Comparison with Pairwise and Linear Games
2.6. Finding the Equilibria
2.7. Goodness of the Approximation
2.8. Approaches to Calculate the Equilibria
- Using the exact formula from (8), by setting (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):
2.9. More Than Two Strategies
3. Conclusions
Acknowledgments
Conflicts of Interest
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Archetti, M. How to Analyze Models of Nonlinear Public Goods. Games 2018, 9, 17. https://doi.org/10.3390/g9020017
Archetti M. How to Analyze Models of Nonlinear Public Goods. Games. 2018; 9(2):17. https://doi.org/10.3390/g9020017
Chicago/Turabian StyleArchetti, Marco. 2018. "How to Analyze Models of Nonlinear Public Goods" Games 9, no. 2: 17. https://doi.org/10.3390/g9020017