# An Evolutionary Game Theoretic Approach to Multi-Sector Coordination and Self-Organization

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## Abstract

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## 1. Introduction

## 2. Evolutionary Game Theory

_{C}(fitness of C), and similarly, f

_{D}stands for the success of D. Both quantities depend on the frequency of strategies in the population. How does the fraction x evolve? The famous replicator equation [35,43] translates the following intuition: Strategies that do better than average will grow, whereas those that do worse than average will dwindle. Denoting by F the average fitness of the population, the evolution of x is thereby summarized in the gradient of selection $g(x)=\dot{x}=x({f}_{C}-F)$. By noting that F = xf

_{C}+ (1 − x)f

_{D}and using Δf

_{C,D}(x) = f

_{C}(x) − f

_{D}(x) the gradient can be rewritten in the form:

^{*}satisfying g(x*) = 0 are the fixed points of the evolutionary dynamics, which in the simplest scenario of one population and two strategies can be characterized as stable (g’(x*) < 0) or unstable (g’(x*) > 0). If g’(x*) > 0, x* is also called coordination point [8] providing an essential ingredient to unveil the coordination dynamics of the populating being studied. If, instead of having infinite populations, we consider a more realistic setting in which populations or sectors involve a finite number of individuals, then one must modify the dynamics in order to take into consideration stochastic effects that in some cases, may be important [44,45]. In this regime, one is typically concerned with the time evolution of the probability distribution over the phase space, which can be modeled by resorting to a master equation [46,47]. Under specific circumstances—large but finite population sizes—these equations can be approximated by a Fokker–Plank equation [48]. Notably, this approximation of a discrete model by a continuous counterpart is an exercise with particularities (e.g., the scaling between population size and time-scales) that may impact the resulting description of the system [49]. By focusing on infinite populations, in this paper we resort to dynamical portraits that leave stochastic effects out for two main reasons:

- (1)
- We are concerned with providing an example of multi-sector dynamics and as we will see, the analytical challenges of these dynamics can be grasped even considering the infinite population assumption and the analytical framework that this limit entails.
- (2)
- The coordination nature of the games under study puts special relevance in the comprehension of the resulting self-organizing dynamics, namely, the characterization of each equilibrium, the phase space that leads to them and the qualitative changes (in the dynamical portrait) that result from adding new sectors.

#### A Simple Game—One Population, Two Strategies

## 3. Models for Multi-Sector Populations

#### 3.1. Two Sectors, Two Strategies

#### 3.2. Three Sectors, Two Strategies

_{(C,C,C)}has three negative eigenvectors) whenever c < t + 2b and a > 0. In this way, when the condition 0 < 2b − c< −t is satisfied, (C,C,C) looses its stability in this 3-sector extension. Additionally, a > 0 results in a pernicious dynamics whereby the only asymptotically stable state [35] is (D,D,D), in which Consumers are impaired. In this case, a < 0 would result in the stability of (C,C,D). Naturally, the effect of parameter a in providing the needed stability to a state in which both Producers and Consumers are coordinated into (C,C) is completely extraneous in the 2-sector analysis. Interestingly, this 3-sector study allows us to observe that, on top of the common assumption about external entities (public) being needed to solve coordination failures of producers/consumers, the feedback provided by the consumers/public into the public sector also provides an important contribution to the overall dynamics. The difficulties in apprehending the resulting three sector dynamics would be more pronounced if we had extended the scenario assuming increasing benefits (of scale) of Producers (as in Equation (5)). Furthermore, we focus on one-shot interactions, not considering iterated plays or strategies conditioned on past information (that could be modeled as asymmetric, depending on the specific sector), as that would result in an extended strategy space that would prevent a tractable exposition of the global dynamics involved in a three-sector dynamics.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Gradient of selection for the game in Table 1. Assuming that Z is infinitely large (Equation (1)), there is an interior unstable fixed point (open circle) whose position (x*) depends on S as x* = S/(S − 0.5) [52]. (

**b**) Position of the interior unstable fixed point (dashed line) when S varies. S defines which equilibrium (C or D) is risk-dominant, i.e., has a basin of attraction larger than ½.

**Figure 2.**Gradient of selection (Equation (2)) of the game represented in Table 2, when b = 2, c = 1. State (C,C) corresponds to point (1,1) whereas state (D,D) corresponds to (0,0).

**Figure 3.**Gradient of selection (Equation (2)) of the game represented in Table 3, when b = 2, c = 1, t = 2.5. State (C,C) corresponds to point (1,1) whereas state (D,D) corresponds to (0,0).

**Figure 4.**Gradient of selection of the overall game played by the three sectors, represented only when all Public is C or D. The game is a synthesis of the games governed by payoff structures in Table 1, Table 2 and Table 3. Here, the three sectors co-evolve. Other parameters: b = 2, c = 1, t = 2.5.

**Figure 5.**Gradient of selection of the overall game played by the three sectors. The game is a synthesis of the games governed by payoff structures in Table 1, Table 2 and Table 3. Here, the three sectors co-evolve. For each possible configuration there is an associated vector that translates the most probable dynamic, given the gradient of selection (see Section 2). The configurations that are parte of paths leading to (0,0,0) are colored with gray; configurations that are part of paths leading to (1,1,1) are colored with blue (

**a**) b = 2, c = 1, t = 4, a = 2; (

**b**) b = 2, c = 4, t = 1, a = 2.

**Figure 6.**Relative size of the basins of attraction leading to full adoption of C. We calculate, numerically, the solutions of Equation (6) given a grid of initial conditions (x,y,z) such that 0 < x,y,z < 1 and x,y,z being multiples of 0.05. We compute the fraction of initial conditions that, at time 1000, lead to a fraction of C adopters higher than 0.9, thus, for a wide range of t and given (

**a**) a = 1, (

**b**) a = −1, we calculate an approximation for the fraction of the state space leading to the full adoption of C by each of the three sectors. As the differences in (

**a**) and (

**b**) show, the impact of Public policy (t) in routing Producers/Consumers to C is contingent on the feedback provided by Producers/Consumers to Public (parameter a). (

**a**) Whenever t < −3, the monomorphic state (C,C,C) is no longer stable, as we show below through a stability analysis. Notwithstanding, when both (C,C,C) and (D,D,D) are stable (t > −3), coordinating into C can be easier/harder given the role of the public sector through parameter t. (

**b**) when a < 0, the public sector is anti-coordinated with producers/consumers, which is here observable by the impact of parameter t in decreasing (increasing) the basin of attraction of public (producers/consumers) towards C. Other parameters: b = 2, c = 1.

**Table 1.**Generic payoff matrix of a two-person, two-strategy game within one population. Each cell dictates the payoff earned by the row player for each possible strategy profile. We test the specific coordination game occurring when R = 1, T = 0.5, P = 0 and S ranging from −1.5 to 0.

C | D | |
---|---|---|

C | R | S |

D | T | P |

**Table 2.**Producer-Consumer game. When a transaction occurs (strategies of Producer and Consumer are the same) there is a benefit for both. Producing a product C has an increased cost (c > 0) given the high quality of the product. The benefit earned by transacting a high quality product (2b > 0) is higher than in the low quality case (b > 0).

**Table 3.**Producer-Consumer game with external intervention. An extra-cost (t) is applied to Producers that assemble the low-quality product D. Interestingly, the case in which t < 0 models a corruptive external intervention, where low-quality products are perversely favored.

**Table 4.**Payoffs accruing to the Public sector while interacting with Producers and Consumers. A Public agent (with strategy C or D) will earn payoff a when interacting with individuals with the same strategy (respectively, C or D) from other sector. This creates an extra coordination, between Public and the other sectors.

© 2016 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/).

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**MDPI and ACS Style**

Santos, F.P.; Encarnação, S.; Santos, F.C.; Portugali, J.; Pacheco, J.M.
An Evolutionary Game Theoretic Approach to Multi-Sector Coordination and Self-Organization. *Entropy* **2016**, *18*, 152.
https://doi.org/10.3390/e18040152

**AMA Style**

Santos FP, Encarnação S, Santos FC, Portugali J, Pacheco JM.
An Evolutionary Game Theoretic Approach to Multi-Sector Coordination and Self-Organization. *Entropy*. 2016; 18(4):152.
https://doi.org/10.3390/e18040152

**Chicago/Turabian Style**

Santos, Fernando P., Sara Encarnação, Francisco C. Santos, Juval Portugali, and Jorge M. Pacheco.
2016. "An Evolutionary Game Theoretic Approach to Multi-Sector Coordination and Self-Organization" *Entropy* 18, no. 4: 152.
https://doi.org/10.3390/e18040152