### 数学代写|计算复杂度理论代写Computational complexity theory代考|Order-Disorder Phase Transitions in the Agent Population

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## 数学代写|计算复杂度理论代写Computational complexity theory代考|Order-Disorder Phase Transitions in the Agent Population

As discussed before, in the “information domain,” we can study the system by mapping strategies to spins. In addition, we can map the difference between winning

probabilities, of cooperators and defectors, to an external magnetic field: $h=$ $p_{c}^{b}-p_{d^{b}}^{b}$. In doing so, by the Landau theory, we can analytically identify an orderdisorder phase transition. Notably, we analyze the free energy $F$ of the spin system on varying the control parameter $m$ (corresponding to the magnetization $M$ )
$$F(m)=-h m \pm \frac{m^{2}}{2}+\frac{m^{4}}{4}$$
where the sign of the second term depends on the temperature, i.e., positive for $T_{s}>$ $T_{c}$ and negative for $T_{s}<T_{c}$; we remind that $T_{c}$ represents the temperature beyond which it is not possible to play the PD due to the high particle speed (according to our assumption). For the sake of clarity, we want to emphasize that the free energy is introduced in order to evaluate the nature of the final equilibrium achieved by the system. In particular, looking for the minima of $F$ allows to investigate if our population reaches the Nash equilibrium, or different configurations (e.g., full cooperation). Figure $3.5$ shows a pictorial representation of the phase transitions that can occur in our system, on varying $T_{s}$ and the external field $h$. Then, the constraints related to the average speed of particles, and to the distance between each group and the permeable wall, can be in principle relaxed, as we can imagine to extend this description to a wider system with several groups, where agents are uniformly distributed in the whole space. Now, it is worth to highlight that our results are completely in agreement with those achieved by authors who studied the role of motion in the PD and in addition are able to explain why clusters of cooperators emerge in these conditions. At the same time, we remind that, in this model, agents are “memory-aware,” while usually investigations consider agents that reset their payoff at each step.

## 数学代写|计算复杂度理论代写Computational complexity theory代考|The Role of the Temperature in the Spatial Public Goods Game

In this section, we aim to analyze the role of the temperature in the spatial PGG. Before to proceed, it is important to remind the reader that, in this section, the terms “temperature” and “noise” refer to the same concept. As discussed in Chap. 1 , the dynamics of this game are affected by a number of parameters and processes, namely, the topology of interactions among the agents, the synergy factor, and the strategy revision phase. We remind that the latter is a process that allows agents to change their strategy. Notably, rational agents tend to imitate richer neighbors, in order to increase the probability to maximize their payoff. By implementing a stochastic revision process, it is possible to control the level of noise in the system, so that even irrational updates may be observed. In particular, we study the effect of noise on the macroscopic behavior of a finite structured population. We consider both the case of a homogeneous population, where the noise in the system is controlled by tuning a parameter representing the level of stochasticity in the strategy revision phase, and a heterogeneous population composed of a variable proportion of rational and irrational agents. In both cases numerical investigations show that the PGG has a very rich behavior, which strongly depends on the amount of noise in the system and on the value of the synergy factor. In doing so, we aim to provide a description of the PGG by the lens of statistical physics, focusing in particular on the impact of noise in the population dynamics. Saying that rational agents are those that tend to imitate their richer neighbors, we can state that irrational agents are those that randomly change their strategy. In the case of a homogeneous population, the intensity of noise in the system is controlled by tuning the level of stochasticity of all agents during the SRP, by means of a global parameter (indicated by $K$ ) that represents the noise/temperature. Instead, in the case of a heterogeneous population, the noise is controlled by tuning the density of irrational agents in the population. Results indicate that tuning the level of noise to interpolate between configurations where agents fully utilize payoff information (low noise) to those where they behave at random (high noise) strongly affects the macroscopic behavior of a population.

## 数学代写|计算复杂度理论代写Computational complexity theory代考|Model

In the case of well-mixed populations of infinite size, the behavior of the system can be predicted as a function of the synergy factor $r$ by studying the related Nash equilibria. In particular, when agents play in groups of $G$ players, two different absorbing states appear separated at a critical point $r_{\mathrm{wm}}=G$. The population falls into full defection for $rr_{\mathrm{wm}}$. Conversely, when agents are arranged in the nodes of a network, surprisingly some cooperators can survive for values of $r$ lower than $r_{\text {wm }}$. This effect, discussed in Chap. 1 , is known as network reciprocity. At the same time, the network structure allows a limited number of defectors to survive also beyond $r=r_{\mathrm{wm}}$. We refer to the two critical values of $r$ at which cooperators first appear and defectors eventually disappear from the population, respectively, as $r_{c 1}$ and $r_{c 2}$. It is worth mentioning that most investigations in EGT are performed by numerical simulations, and an analytical definition of the critical thresholds (i.e., $r_{c 1}$ and $r_{c 2}$ ) identified in networked topologies is missing. As a result, when studying EGT models by arranging agents in different spaces, the values of critical thresholds are achieved by Monte Carlo simulations (see Chap. 2). In a networked population, depending on the values of $r$ and on how agents are allowed to update their strategy, it is possible to observe different regimes: two ordered equilibrium absorbing phases, where only one strategy survives (either cooperation or defection), and an active but macroscopically stable disordered phase corresponding to the coexistence between the two species/strategies.

F(米)=−H米±米22+米44

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## MATLAB代写

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