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Ezekiel Price
Ezekiel Price

Game Theory: Decisions, Interaction And Evoluti... REPACK


Game theory is the study of mathematical models of strategic interactions among rational agents.[1] It has applications in all fields of social science, as well as in logic, systems science and computer science. Originally, it addressed two-person zero-sum games, in which each participant's gains or losses are exactly balanced by those of other participants. In the 21st century, game theory applies to a wide range of behavioral relations; it is now an umbrella term for the science of logical decision making in humans, animals, as well as computers.




Game Theory: Decisions, Interaction and Evoluti...



These are games prevailing over all forms of society. Pooling games are repeated plays with changing payoff table in general over an experienced path, and their equilibrium strategies usually take a form of evolutionary social convention and economic convention. Pooling game theory emerges to formally recognize the interaction between optimal choice in one play and the emergence of forthcoming payoff table update path, identify the invariance existence and robustness, and predict variance over time. The theory is based upon topological transformation classification of payoff table update over time to predict variance and invariance, and is also within the jurisdiction of the computational law of reachable optimality for ordered system.[40]


Evolutionary ecologists aim to understand the complex behavioral relationships between organisms as they interact to obtain resources. In general, these interactions range from antagonistic to cooperative, but cases of exploitation and altruism also occur. Interactions are costly: energy is invested by each organism in both confrontation and cooperation as a means to acquire the resource. Indeed, even avoidance can be costly if energy is expended and no resources are obtained. The energy spent is a cost to the organism, and the resources are benefits. Comparing the relative cost to the benefits obtained following an interaction determines the net gain or loss incurred by the organism, and this value is referred to as the payoff. Different interaction strategies, such as combative or cooperative, result in different payoffs based on nature of the interaction. Evolutionary ecologists treat these strategies as phenotypes. The most successful organisms maximize their payoff and increase their ability to reproduce. In short, the organism with the best interaction strategy has the highest fitness. Since the interaction strategy (phenotype) can directly relate to fitness, the optimum strategy will be favored under natural selection. Interactions between organisms with either competing or identical strategies can be treated as games with multiple players. Since biological interactions involve two or more decision makers (i.e., individuals with strategies), biologists utilize game theory to elucidate evolutionary consequences of interactions1. Game theory is a branch of mathematics that investigates the outcomes of multi-individual interactions, in which the payoff for any one individual depends not only on its own strategy, but the strategies of the other individuals involved. An individual does not control the decisions of others. While much of game theory is based on the interactions of rational decision makers (humans), in evolutionary game theory the payoff is a surrogate for fitness and strategies evolve over many repeated iterations of the game2,3. Estimating fitness for strategies based on the payoff matrix reveals stable payoff equilibriums may be achieved through natural selection, an important feature of evolutionary game theory. Furthermore, stable payoff equilibriums have been shown for cooperation and altruism, behaviors that seem contradictory to the strongly supported individualistic, survival of the fittest mode of evolution3.


Evolutionary games are often visually represented as a payoff matrix (Fig. 1a), in which the payoff for the individual involved is shown for all possible interactions. In the matrix, the payoffs typically are those assigned to the player depicted in the row playing against the strategy in the column; thus, in Figure 1a, if player 1 is a hawk, and encounters a hawk, the payoff for player 1 is (B-C), the value in the upper left quadrant. If player 1 is a hawk and encounters a dove, the payoff for player 1 is B, the value in the upper right quadrant. The payoffs for player 1 adopting the dove strategy against a hawk and against a dove are in the lower left and lower right quadrants, respectively.


The examples so far have used two-player, fixed payoff games to demonstrate the evolution of different interaction strategies. Biological interactions and strategies are often more complex. Strategies are considered pure if the behavior is constant for all interactions, or they may be considered mixed if the behavior expressed is conditional on the organisms involved4,6. For instance, a mixed strategy may be size related: avoid larger individuals and attack smaller individuals. The value or quality of the benefit also influences strategies. From an evolutionary standpoint, the reward for competitive behavior must outweigh the risk and cost of competition. Thus, low quality resources are unlikely to drive intense combat. However, the higher the quality or importance of the benefit, the more intense the competition will be.


In the prior examples, we assume a uniform environment, and resources are available everywhere. In natural environments, spatial and temporal variability in resource abundance and quality influence biotic interactions7,8; moreover, organisms modify the environment by utilizing resources. Payoff matrices may be either symmetric or asymmetric. In symmetric payoff matrices, payoffs are identical for each individual based on the interaction, whereas payoffs may be different for each individual in an asymmetric payoff matrix. A common asymmetric game occurs between organisms competing for territories in which one player occupies the territory and the other is the intruder. The payoff in the game is the territory, and the asymmetry is that the territory holder may either maintain or lose the territory, while the intruder may or may not acquire the territory. The territory holder is more likely to escalate the conflict via an aggressive strategy to protect the territory, while the intruder is less likely to escalate9. A second asymmetry that arises in animal conflicts is the different abilities to acquire or hold a resource such that the cost of escalating the interaction differs due to combative capability10. Here, the superior competitor is more likely to employ an aggressive strategy.


Evolutionary game theory in which cooperation is rewarded with fidelity provides rationale for altruistic behavioral strategies seemingly incongruent with natural selection. In evolutionary terms, altruism occurs when an individual reduces its own immediate fitness probability (i.e. its likelihood to reproduce in the short term) to improve the fitness of another individual. Altruism is most common in animals that exhibit social behavior, such as warning calls or food sharing, and colony forming behavior. The benefit of warning calls and food sharing is based on reciprocal altruism, namely that the benefits provided to others will be returned in the future13. A bird that provides a warning call is temporarily more apparent to predators, thus reducing its immediate fitness probability. However, the same bird may benefit in the future from warning calls from other birds. Likewise, a vampire bat that regurgitates its food to feed hungry bats will benefit from the same service when it is unable to find food14. Reciprocal altruism relies on organisms interacting multiple times, and game theory provides a framework for the evolution of reciprocal altruism via the prisoners' dilemma described earlier. The iterated prisoners' dilemma typically involves repeated interactions between small numbers of prisoners; however, altruism has been shown to evolve even in large groups through a combination of direct and indirect altruistic reciprocity15. Direct reciprocity occurs when the same two individuals interact repeatedly, while indirect reciprocity is when subsequent interactions are between different partners. In sum, while altruistic behavior may temporarily reduce the fitness of the individual, reciprocal altruism increases fitness over the lifetime of the individual.


In the Results Section, I begin by a static analysis of the two-stage game followed by studying the evolutionary dynamics of both direct interaction model and reputation-based model in a well-mixed population, and end by studying the evolutionary dynamics in a structured population.


As we have seen, when game B is an anti-coordination game, starting from a rather broad range of initial conditions, a set of cooperation-supporting moral norms emerge through the dynamics and help the maintenance of cooperators in the population. To see the nature of the dynamical phenomena through which this happens, in Fig 6A and 6B, I plot the density of different strategies, in the direct interaction model, as a function of time. Here, game B is a Snow Drift game. The dynamic is similar for other anti-coordination games, and in the reputation-based model. As can be seen in Fig 6C (replicator-mutator dynamics) and Fig 6D (simulations in a population of size 10000), where the density of cooperators in the PD and soft strategy in the SD are plotted, starting from the center initial condition in which all the strategies are found in the same density, the population rapidly goes to a state where the density of strategies in both games is close to its Nash equilibrium value. This is the case because, initially, no coupling between the two games exists, and strategies of the individuals evolve in such a way that the frequency of strategies in each game follows its Nash equilibrium value. Consequently, the densities of all the strategies are close to their value in the defective fixed point, such that it might appear that the system has settled in the defective phase. However, this only sets the stage for the second phase of the evolution, during which a set of cooperation-supporting moral norms evolve and give rise to an outburst of cooperation. As cooperators are found in a very small frequency during this transient phase, strategies that play softly with cooperators do not impose a high cost on their bearer and grow in number. A cooperation favoring moral system is established when such strategies accumulate enough. At this point, the system shows a rapid dynamical transition to the cooperative fixed point where cooperators emerge in large numbers. In this regime, cooperators always play a hard strategy with defectors, and defectors always play the soft strategy with cooperators. This compensates for the cost of cooperation that they pay. On the other hand, both cooperators and defectors play a combination of soft and hard strategies among themselves. This phenomenology shows, when a game has an asymmetric equilibrium, as is the case in anti-coordination games, individuals can use information about the strategy of their opponent in a social dilemma to efficiently coordinate in an asymmetric equilibrium and avoid paying the cost of coordination failure. Consequently, a set of behavioral or moral norms emerges, according to which cooperators are allowed to play hard and deserve to be played soft with. This supports cooperation by compensating for the cost of cooperation. Importantly, by facilitating anti-coordination, this mechanism also increases the cooperative behavior in the second game. 041b061a72


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