Cooperative behavior is commonly observed in animal populations and between human beings. This behavior in wild animals can be largely explained by kin selection , . For human beings, cooperative behavior is not only directed toward kin but also frequently found toward non-kin. Therefore, from an evolutionary viewpoint, understanding cooperative behavior has been problematic because cooperators are often inferior in fitness to defectors who do not pay costs to cooperate and benefit from the cooperators’ actions. The Prisoner’s Dilemma game describes this situation. Two players, who cannot bargain with each other, can either cooperate or defect. A cooperative player gives benefit b to the other player and pays the cost of cooperation, −c (b>c >0). A player who defects never gives anything to the other player. If both cooperate, they both score b − c. If one cooperates and the other defects, the cooperator’s score is −c and the other’s is b. If both choose defection, both score 0. In the classic Prisoner’s Dilemma game, both players choose to defect and receive a score of 0 although the score (b − c) for both players would be higher if they both cooperate. Therefore, the Prisoner’s Dilemma game describes the failure of mutual cooperation between two players.
The conditions or mechanisms that influence people’s cooperation have been studied using the Prisoner’s Dilemma game, public goods game, and other approaches . The public goods game describes social dilemma during collective action. Each player invests his/her resources in public goods. The pool of investments from all players is multiplied by a benefit factor and divided equally among all players. The payoff function suggests that when all players invest in public goods, the payoff is higher than when they do not invest at all. However, if one player changes from cooperation to defection, the player gains a higher payoff than what she/he would have gain under cooperation. Thus, all players would choose defection. This is known as a social dilemma. These investigations have identified a variety of factors that affect the evolution of cooperation, such as direct reciprocity , indirect reciprocity , group selection , , and spatial structure -. Among these mechanisms, costly punishment has attracted the most attention -. This idea introduces another stage, a punishment stage, into the prisoner’s dilemma game or the public goods game. In the punishment stage, a punisher, who can be either a cooperator or defector would attack an opponent as retribution if the opponent defected in the first stage. In the early model of Axelrod , a punisher had to be a cooperator, but a paradoxical strategy that allows defectors to punish is now considered -. Obviously, defections are less rewarded if the opponent is a punisher.
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Evolutionary game studies have demonstrated that cooperation can evolve or be maintained if punishment evolves , . However, we must ask whether punishing behavior itself evolves because while a punisher must pay a cost, a non-punisher does not, and thus often outperforms the punisher. This problem is called the second-order dilemma and mechanisms have been suggested to solve it. Fehr and Gächter argued that people often punish defectors whom they will never meet again; this is altruistic or cooperative behavior . They also stated that the primary reason for punishment in this case stems from a negative reaction toward defectors . If so, punishment might be, as much as cooperation, a product of evolution and natural selection. We then must ask why and how these behaviors evolve.
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One solution for the evolution of costly punishment is to adopt a setting that allows punishment to have fitness advantages , , . As Nakamaru and Iwasa demonstrated, if spiteful behavior which is defined as a behavior that decreases an opponent’s score by reducing one’s own score can have relatively high fitness, it may evolve with an updating rule called the score-dependent viability model or simply the viability model , . In this model, an individual’s score is inversely proportional to the probability of that individual’s death. When the individual dies, a randomly chosen player procreates and fills the empty site. With this action, spiteful behavior results in some advantage in relative fitness despite its cost because, if a spiteful individual can decrease the opponent’s score, the death probability is high enough to empty the opponent’s site, and then the individual has the chance to colonize the empty site. Spiteful behavior works effectively in a spatially structured population such as a lattice if a focal individual interacts with the neighbors and has the chance to colonize an empty site, which is created after the neighbor dies. As a mechanism, punishment can be considered spiteful, as it reduces both the opponents’ and one’s own scores. Therefore, punishment can evolve in the same manner as spiteful behavior. As a result, punishment can increase cooperation in the viability model if an individual behaves as both a punisher and cooperator.
The other widely studied concept is that of spatial structure. In a spatially structured population, players are distributed within a structure, such as a lattice, or other networks or subpopulations, and they interact locally only with their neighbors. Brandt et al. assumed that each player plays the public goods game with two other players in a population with a hexagonal lattice structure and showed that cooperation and punishment can evolve together . They also showed that cooperation can increase without punishment if a spatial structure exists, but the existence of punishment more strongly increases cooperation. Other studies have shown that a spatially structured population can resolve a second-order dilemma , .
Although numerous models of punishment have been developed, most assume discrete or binary strategies for both cooperation (cooperator or defector) and punishment (punisher and non-punisher). Brandt et al., for example, assumed four strategies: cooperate and punish, defect and punish, cooperate but do not punish, and defect but do not punish . These simple settings are convenient for mathematical analysis and are thus helpful for the discussion of basic mechanisms. However, in reality, the cooperation level or the amount of contribution to the public goods in the public goods game is continuous , . The cooperation level in certain collective actions is continuous, and there may be several discrete cooperation levels, e.g., levels 1, 2, …,10. We may consider the example of house cleaning by several housemates. Some housemates help clean the house completely, some do so almost completely, some do a bit, whereas others do not help at all. Thus, a continuous cooperation level can encompass a situation in which there are more than two choices. The punishment level is also continuous depending on the level of continuous cooperation. If we adopt these realistic assumptions, we can investigate the punishment level in response to the opponent who contributes a certain level of cooperation to the public goods.
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Nakamaru and Dieckmann investigated the effect of strictness of punishment on the evolution of the continuous cooperation level . Strict punishment means a policy of “zero tolerance”: a punisher severely punishes an opponent whose cooperation level is below the threshold and weakly punishes an opponent whose cooperation level is above the threshold. Nakamaru and Dieckmann concluded that, in a population with a lattice structure, the rule “the stricter, the better” had to be applied to punishment if cooperation were to evolve. They also mathematically proved that if each individual interacts with a randomly chosen individual from the population, then neither cooperation nor punishment can evolve. Gao et al. also studied the evolution of the continuous cooperation level and punishment in a spatially structured population  and implicitly assumed strict punishment. They discussed the effect of social tolerance corresponding to the threshold of punishment  on the evolution of cooperation and punishment.
If punishment is graduated, a punisher gradually changes the severity, adjusting to the cooperation level. This principle is followed by the criminal laws of most western countries . Other examples include the following. Ostrom found that graduated punishment is one of seven design principles for long-enduring common-pool resource institutions and graduated punishments for violators are likely to be assessed depending on the seriousness and context of the violation . Cox found that graduated punishments progress on the basis of either the severity or repetition of violations to deter participants from excessive violations of community rules . In many legal systems, repeat offenders are punished more severely than first-time offenders, and theoretical studies of criminal sanctions have shown that the erroneous conviction of innocent offenders and learning contribute toward making this sanction system optimal -.
Therefore, in this study, we investigate whether graduated punishment depending on the cooperation level increases cooperation in the continuous public goods game. We also investigate how spatial structure affects the results; this outcome depends on an updating rule that prescribes how the game score affects fitness and generational changes. Updating rules can dramatically change evolutionary dynamics. For example, Nakamaru and Iwasa ,  investigated whether the conditions for the evolution of cooperation differ between two different updating rules: the score-dependent viability model and the score-dependent fertility model, which is the same as “the death-birth” model . They found that punishment and cooperation can evolve in both the completely mixed population and the spatially structured population when using the viability model, whereas the coevolution of cooperation and punishment is impossible without the spatial structure when using the fertility model. Therefore, in contrast to Nakamaru and Dieckmann , we expect that the viability model can foster the coevolution of cooperation and punishment even in a completely mixed population.
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