Reward & Punishment:
by Christoph Hauert, Version 2.2, February 2005.
Punishment opportunities can create high incentives to cooperate in social dilemmas such as the Prisoner's Dilemma or the more general public goods game. However, punishment alone is unable to establish persistent cooperative behavior. Additional prerequisites must be met such as structured populations with limited local interactions or in the present case that individual carry some sort of reputation which indicates whether they will readily punish non-cooperative actions.
The assumption that the cooperative stage is immediately followed by a punishment stage results in four distinct strategic patterns: G1 social types that cooperate and punish defectors; G2 paradoxical or bully type that defects but nevertheless punishes defectors (this strategy is called paradoxical because it fares poorly against its own kin); G3 asocial type that neither cooperates nor punishes; and G4 mild types that cooperate but avoid the costly punishment. Through reputation, individuals may occasionally learn about the punishing type of the opponent. If the opponent does not punish, then G1 and G4 can temporarily switch to defection because they know that they can get away with it. Similarly, if the opponent does punish, then G2 and G3 can momentarily opt for cooperation to avoid punishment. Interestingly, it turns out that the former mechanism is much more important.
In well-mixed populations, punishment and reputation lead to a bi-stable situation where the initial configuration determines whether the population ends up in a social or asocial homogenous state. By adjusting the parameter values, the social state, in which individuals cooperate and punish defectors, can always become risk-dominant which means it has the larger basin of attraction than the asocial state where everybody defects and does not punish.
The applet below illustrates the different components. Along the bottom there are several buttons to control the execution and the speed of the simulations. Of particular importance are the Param button and the data views pop-up list on top. The former opens a new panel that allows to set and change various parameters concerning the game as well as the population structure. The latter displays the simulation data in different ways. Clicking on the examples below opens a new window with a larger applet and all parameters preset accordingly.
|New social||New bully||New asocial||New mild|
Note: The pale strategy colors are very useful to get an intuition of the activitiy in the system. The shades of grey of the payoff scale are augmented by blueish and reddish shades, which indicate the payoffs for mutual cooperation and defection, respectively.
|Params||Pop up panel to set various parameters.|
|Views||Pop up list of different data presentations.|
|Slider||Idle time between updates. On the right your CPU clock determines the update speed while on the left updates are made roughly once per second.|
|Mouse||Mouse clicks on the graphics panels generally start, resume or stop the simulations.|
|Structure - Strategy||Snapshot of the spatial arrangement of strategies. Mouse clicks cyclically change the strategy of the respective site for the preparation of custom initial configurations.|
|Mean frequency||Time evolution of the strategy frequencies.|
|Simplex S4||Frequencies plotted on a manifold of the simplex S4. Mouse clicks set the initial frequencies of strategies (the manifold k is determined by the initial frequencies set on the parameter panel).|
|Structure - Fitness||Snapshot of the spatial distribution of payoffs.|
|Mean Fitness||Time evolution of the mean payoff of each strategy together with the average population payoff.|
|Histogram - Fitness||Histogram of payoffs for each strategy.|
The list below is restricted to the few parameters particularly related to punishment and reputation in public goods game. Follow the link for a complete list and descriptions of all other parameters e.g. referring to update mechanisms of players and the population.
- multiplication factor r of public good.
- cost of cooperation c (investment into public good).
- fine imposed on defecting co-player through punishment.
- punishment is costly and the punisher has to bear these costs.
- Rep. Mu:
- reputation - probability to learn that all co-players are non-punishers and taking advantage of this knowledge by temporarily switching to defection.
- Rep. Nu:
- reputation - probability to learn that at least one co-players punishes and taking advantage of this information to avoid punishment by temporarily switching to cooperation.
- Init coop/punish, init coop/none, init defect/punish, init defect/none:
- initial fractions of the social (cooperate and punish, G1), mild (cooperate but do not punish, G4), bully/paradoxical (defect and punish, G2) and rational (defect, don't punish, G3) strategies. If this does not add up to 100%, the values are scaled accordingly.