The role of competitiveness in the Prisoner’s Dilemma
 Marco A Javarone^{1, 2}Email author and
 Antonio E Atzeni^{3}
https://doi.org/10.1186/s4064901500245
© Javarone and Atzeni. 2015
Received: 26 February 2015
Accepted: 11 July 2015
Published: 31 July 2015
Abstract
Background
Competitiveness is a relevant social behavior and in several contexts, from economy to sport activities, has a fundamental role. We analyze this social behavior in the domain of evolutionary game theory, using as reference the Prisoner’s Dilemma.
Methods
In particular, we investigate whether, in an agent population, it is possible to identify a relation between competitiveness and cooperation. The agent population is embedded both in continuous and in discrete spaces, hence agents play the Prisoner’s Dilemma with their neighbors. In continuous spaces, each agent computes its neighbors by an Euclidean distancebased rule, whereas in discrete spaces agents have as neighbors those directly connected with them. We map competitiveness to the amount of opponents each agent wants to face; therefore, this value is used to define the set of neighbors. Notably, in continuous spaces, competitive agents have a high interaction radius used to compute their neighbors. Instead, since discrete spaces are implemented as directed networks, competitiveness corresponds to the outdegree of each agent, i.e., to the number of arrows starting from the considered agent and directed to those agents it wants to face.
Results and conclusions
Then, we study the evolution of the system with the aim to investigate if, and under which conditions, cooperation among agents emerges. As result, numerical simulations of the proposed model show that competitiveness strongly increases cooperation. Furthermore, we found other relevant phenomena as the emergence of hubs in directed networks.
Keywords
Background
In the last years, social and economic phenomena have attracted the interest of scientists belonging to hard sciences, as mathematics, physics and computer science. As result, the interdisciplinary fields of social dynamics [1, 2] and econophysics [3] have rapidly emerged. For instance, several analytical and computational approaches have been developed for studying behaviors such as homophily [4], conformity [5–8], and rationality [9, 10]. Furthermore, many social and economic phenomena can be studied in the context of Evolutionary Game Theory [11–13], which represents the attempt of describing the evolution of populations by Game Theory using famous models like the Prisoner’s Dilemma [14, 15] (PD hereinafter). Since the PD allows to analyze the phenomenon of cooperation [16–18], it is possible to study the evolutionary dynamics among agents whose interactions are based on this game. In doing so, we can evaluate if, and under which conditions, cooperation emerges. It is worth to highlight that simple games like the PD, implemented considering different social behaviors, contexts (see [19]), or topologies (e.g., [20–24]) to implement agent’s interactions, as sketched before, allow to investigate a wide variety of topics such as criminality [25], biological systems [26], imitation phenomena [27], and further social psychology aspects such as conformity [28, 29]. Here, we consider an important social character, i.e., the competitiveness, that strongly affects dynamics in animal herds and among individuals [4]. In particular, in this study, we aim to investigate if there is a relation between competitiveness and cooperation. To this end, we implement a population whose agents, provided with a parameter that represents their degree of competitiveness (see [30]), play the PD. The relevance of this work lays in the fact that, both in herds and in human communities, many contexts are defined as competitive, e.g., stock markets, athletic challenges, and job markets. Numerical simulations, of the proposed model, allowed to analyze parameters as the average outdegree over time and to define the TSdiagram; the latter constitutes a relevant tool to assess if, and in which extent, cooperation emerges among agents. As result, we found that competitiveness strongly affects these dynamics and, in particular, it increases the cooperation among agents. The remainder of the paper is organized as follows: “Model” introduces the model for studying the PD in continuous spaces and in discrete spaces. “Results” shows results of numerical simulations on varying the initial conditions. Eventually, "Discussion and conclusion” ends the paper.
Model
 1.
A randomly chosen agent, say the jth agent, computes the set of its neighbors in accordance with the interaction radius r (or with the network structure in the discrete space);
 2.
The jth agent faces its neighbors (note that each single challenge involves only two agents at time);
 3.
All agents, playing at this step (i.e., the jth agents and its neighbors), compute their new payoff;
 4.
The jth agent updates its strategy according to a revision rule.
Continuous space
As shown in [32], using low values of T and high values of S, cooperation among agents emerges only under particular conditions, i.e., when agents randomly move over time. It is worth to highlight that in [32], all agents have the same radius to compute the set of their neighbors. Furthermore, this radius depends on the average number of opponents agents face. Here, we consider the same geometrical framework (i.e., that defined in [32]) to implement the proposed model on continuous spaces, with two main differences: (1) agents are fixed (i.e., they cannot move) and (2) agents can vary their radius. Notably, agents have an interaction radius whose length depends on gained payoff: as their payoff increases/decreases their radius increases/decreases. Hence, agents with high payoff become more competitive and, as result, they face a higher number of opponents than agents with a small payoff. At time \(t=0,\) all agents have the same radius computed according to the average number of opponents they can face (if selected). In particular, the radius \(r(t=0)\) is computed as \(r(0) = \sqrt{\bar{k(0)}/(\pi N)}.\) Then, considering that each radius varies in accordance with agent’s payoff, and that agents face a number of opponents in the range \([1,N1],\) the radius is computed as \(r= \alpha r_0,\) where \((\sqrt{{1/\bar{k}}}) \le \alpha \le \sqrt{(}N/\bar{k}).\) Thus, at \(t = 0,\) the value of \(\alpha \) is \(\alpha _0 = 1.\) In general, after n time steps, each agent plays an average number of times equal to \(\bar{n} = n/N.\) Since best agents (i.e., those with high payoff) should get the maximum radius in \(\bar{n}\) steps, every time agents play, their value of \(\alpha \) increases to \(\delta \alpha = (\alpha _{\rm max}\alpha _0)/\bar{n}.\) Hence, the radius is modified to \(\pm \delta r,\) where \(\delta r = r_0 \delta \alpha,\) depending on which the considered agent obtains a positive or a negative payoff.
Discrete space
The discrete space is implemented by a directed network, i.e., a network whose connections can be represented by arrows. In the proposed model, an arrow from one agent to another one represents the challenger agent (i.e., the one that faces someone else) and the faced agent (i.e., agent identified as neighbor of the challenger one). In directed networks, the definition of neighbors is not immediate as for undirected networks, where connections can be represented by simple lines. Notably, arrows represent links (or edges) and their direction represents the meaning of the relation. For instance, an arrow starting from node A, and ending to node B, codifies a relation from A to B, and not vice versa. Thus, neighbors of the jth node are those nodes connected to it by arrows starting from the jth node itself. In doing so, an arrow starts from the challenger and it ends on the faced agent. To analyze the structure of these networks, using the degree distribution, we have to consider both the “indegree” distribution and the “outdegree” distribution. The former represents the distribution of links ending in nodes, whereas the latter those of links starting from nodes. Then, competitiveness can be mapped to the outdegree of each node. As for the continuous space, at \(t=0\) all agents begin to play in the same conditions, i.e., all nodes have the same outdegree and the same indegree. On the other hand, as the population evolves (i.e., agents play the PD over time), winning agents increase their outdegree (randomly selecting new opponents) and loosing agents do the opposite, i.e., they reduce their outdegree (randomly selecting nodes to remove from their neighborhood). As before, the increment/reduction of the outdegree has as constraint that each agent cannot play with more than \(N1\) agents nor less then 1 agent. Furthermore, the increasing and the decreasing is unitary, i.e., the \(k_{\rm out}\) can vary at each time step of \(\pm 1.\) Finally, we recall that in both domains we adopted the ‘imitation of the best’ strategy revision rule, and in all simulations we consider an equal initial distribution of strategies, i.e., at the beginning the \(50\%\) of the population is composed of cooperators and the remaining \(50 \%\) of defectors.
Results

Meanfield approximation

Continuous spaces

Discrete spaces
The first case represents a classical generalization of the studied system, as we introduce the trivial hypothesis that all agents interact with all the others, at each time step. In terms of networks theory, this scenario corresponds to a fully connected network, hence complex interaction patterns are not considered nor the competitiveness is represented. Notably, competitiveness is mapped to the number of opponents each agent faces; therefore, in the event everyone faces everyone, competitiveness vanishes. Anyway, when studying complex systems, before focusing on complex scenarios it is often useful to analyze results coming from simple or trivial configurations. Then, once we performed the first analysis, we proceed on analyzing results related to the continuous space and to the discrete space.
Meanfield approximation
Anyway, it is possible that, if we observe the evolution for a time longer than \(10^4\) time steps, all agents of the population become defectors. In general, this first result confirms that in absence of particular behaviors (e.g., movements and social characters) the defection strategy dominates, according to the expected Nash equilibrium. Hence, we can go ahead studying the population by introducing the competitiveness.
Simulations on the continuous space
Observations of these diagrams in Figures 2 and 3 let emerge that when agents have a higher initial average degree the final density of cooperators decreases. Furthermore, it is relevant to emphasize that by arranging agents in a regular lattice, with 4 and 8 neighbors, when they increase/decrease their radius the variation of faced opponents is equal to their initial average degree, i.e., \(\pm 4\) and \(\pm 8,\) respectively.
Simulations on the discrete space
It is worth to see how the indegree distributions vary much lesser than the outdegree distributions, although both are involved in the evolution of the system.
Discussion and conclusion
In this study, we aim to investigate if there are relations between two social behaviors, i.e., cooperation and competitiveness, when an agent population evolves playing the Prisoner’s Dilemma. In particular, we map the competitiveness to a parameter embedded in the model, so that competitive agents face many opponents, whereas noncompetitive ones do the opposite. In the proposed model, becoming a noncompetitive agent entails to loose challenges, while playing the Prisoner’s Dilemma. After performing a brief meanfield analysis of our model, where the population reached the expected Nash equilibrium, agents have been arranged in two different domains: a continuous space and a discrete space. The former is represented by a bidimensional square, whereas the latter has been modeled by a directed network. First of all, we highlight the main differences between our work and those performed by previous authors (e.g., [31, 32, 34]): we focus our attention on fixed agents and we provide them with a social character, i.e., the competitiveness. Due to the computational cost of our model, we were able to perform simulations up to \(t = 10^4\) time steps, with \(N = 100\) agents in the continuous space and with \(N = 1,000\) agents in the discrete space. In general, the main result of numerical simulations shows that competitiveness allows the emergence of cooperation areas in the TSplane, in both domains. Moreover, in the continuous domain, we investigated the outcomes on varying the initial conditions: the spreading of agents in the bidimensional square (i.e, random vs regular lattice) and the average degree (i.e., \(\bar{k(0)} = 4\) and \(\bar{k(0)} = 8\)). Notably, when agents are randomly spread, several intermediate phases are obtained, indicating an equal presence of cooperators and defectors, instead by an ordered distribution (i.e., lattice) we found more neat areas of cooperation and defection. On the other hand, the initial average degree seems to have a strong influence on these dynamics, as for \(\bar{k(0)} = 4\) the cooperation area in the TSplane is greater than for \(\bar{k(0)} = 8,\) using the two spreading strategies. This difference can be explained by the fact that, as for each agent the number of neighbors increases (at \(t = 0\)), the probability that the related social circle be composed of cooperators (i.e., be a cluster of cooperative agents) reduces. In the discrete domain, the scenario is a bit different as only for very low T values and high S values, a full cooperation emerges. An analysis related to the influence of the initial arrangements of agents, in both domains, performed to understand why some of them appear more advantageous to obtain more cooperation is important and it will constitute the argument for future investigations. Finally, we analyzed the degree distributions (i.e., the indegree and the outdegree distributions) of directed networks. This analysis is relevant as agents can vary their indegree distribution and outdegree distribution as result of their behavior (more competitive or not). It is important to note that the indegree distribution has low variations over time, whereas the opposite happens for the outdegree distribution. Notably, this latter represents the competitive parameter, i.e., the number of opponents that competitive agents face as their payoff increases. Analyzing networks related to cooperation areas, in the TSplane, we found that the outdegree distribution is characterized by the presence of more hubs (i.e., many competitive agents appear, even if they tend to cooperate among themselves). On the other hand, considering networksrelated noncooperative areas, of the TSplane, we found only few variations of the outdegree distribution. In our view, this difference between the two areas, considering the outdegree distributions, means that when agents cooperate the network loses its homogeneous structure (recall that at \(t= 0\) all agents have the same values of \(k_{\rm in}\) and \(k_{\rm out}\)); while when agents do not cooperate, the network structure has an exponential degree distribution (i.e., the homogeneous structure is conserved over time). In the light of these results, we can state that competitiveness strongly affects cooperation. Therefore, it is important trying to explain the underlying mechanism that leads to this result. Let us consider first the continuous case, where agents are fixed and, according to previous works, should not cooperate. Now, if only few of them have many cooperative agents in their neighborhood, they increase their interaction radius. Hence, they face more agents during next time steps, having the opportunity to face other cooperative agents. Now, according to the matrix 1, clusters of cooperators strongly increase their payoff, while clusters of defectors do not increase it in absence of cooperators. Since cooperators are randomly spread in the space, increasing the interaction radius the probability to find cooperators increases. On the other hand, defector agents, although never decrease their radius, may increase their payoff (and their radius) only for high values of T, otherwise they will have a constant small radius and, as a consequence, a small degree of competitiveness. Similar considerations hold also for the discrete domain, where defectors do not increase their outdegree, while cooperators have this opportunity. To conclude, we highlight that achieved results clearly indicate the existence of a relation between competitiveness, interpreted as an inclination to face many players, and the emergence of cooperation in the Prisoner’s Dilemma.
Declarations
Authors’ contributions
MAJ devised the research work. Both authors performed experiments and analyzed the outcomes. Both authors read and approved the final manuscript.
Acknowledgements
MAJ would like to thank Fondazione Banco di Sardegna for supporting his work.
Compliance with ethical guidelines
Competing interests The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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