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2×2 games:

Structured populations

by Christoph Hauert, Version 2.1, June 2003.

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In the case of the prisoner's dilemma it is well known that including spatial extensions enables cooperators to thrive by forming clusters and thereby reducing exploitation through defectors. This contrasts with results for well-mixed populations, i.e. with randomly matched interaction partners, where defectors will always outperform cooperators and drive the latter to extinction. Quite naturally, the inclusion of spatial dimensions (or any type of population structure) which limits interactions of all individuals to a small neighborhood, also has pronounced effects on all the other 2×2 games.

In this tutorial you find interactive Java applets that let you explore the rich and fascinating dynamics of such simple models. In addition there are several sample scenarios described in more detail which emphasize the most important findings of this research work.

Let us begin with a nice example which is admittedly only of limited scientific relevance but has quite some entertainment value. If a symmetrical initial configuration of a regular lattice is prepared and if the players follow a deterministic update rule - such as imitating the strategy of the best performing neighbor - then the initial symmetry is preserved and the lattice resembles a dynamically changing persian carpet or an evolutionary kaleidoscope.

Clicking either on one of the pictures below or the corresponding link to the right will open a new window with a running applet illustrating the respective scenario. You can use this as a starting point to study effects of variations of the parameters.

Legend	Time evolution of cooperators and defectors in well-mixed populations where individuals interact according different 2×2 games. Color code: Cooperators Defectors   New cooperator New defector
	Moore neighborhood The Moore neighborhood limits interactions of an individual to its eight neighbors reachable by a chess-kings-move. The rectangular lattice is initialized with a single defector in the center surrounded by cooperators.
	von Neumann neighborhood The von Neumann neighborhood is smaller than the Moore neighborhood and each player interacts only with its four nearest neighbors to the north, east, south and west. This neighborhood puts a stronger emphasis on the lattice structure because spreading velocities are twice as fast in the lattice orientation than in diagonal directions.
	Hexagonal and triangular lattices Apart from rectangular lattices there are other regular lattices, i.e. hexagonal or honeycomb lattices where each player has six neighbors and triangular lattices with three neighbors. Do you find suitable parameter values for S and T to produce more kaleidoscopes? - If you find one, please let me know.

A simple analysis of 2×2 games in well-mixed populations (or in the mean-field approximation) reveals four regimes with different dynamics. The spatial extension or more generally the population structure leads to several important modifications but essentially the four dynamical domains still exist.

Legend	Time evolution of cooperators and defectors in well-mixed populations where individuals interact according different 2×2 games. Color code: Cooperators Defectors   New cooperator New defector
	Defection For T > 1 and S < 0 cooperators go extinct and defection reigns in well-mixed populations. This is generally equally true in spatially extended systems. However, some exceptions apply for small T and large S. This is particularly important because the prisoner's dilemma falls into this category. With a deterministic update rule for the players, clusters of cooperators can survive roughly as long as T < S + 5/3 holds. Adding noise to either the update rule of the players or the lattice tends to further decrease this area. Outside this small parameter region the system quickly relaxes into homogenous states with all defectors.
	Co-existence In well-mixed populations co-existence requires that T > 1 and S > 0. The equilibrium frequency of cooperators is given by fc = S/(S + T - 1). In spatially extended systems almost independently of the initial configuration similar equilibrium frequencies are observed but interestingly, generally spatial extensions tends to favour defectors. The most prominent representative of this category is the hawk-dove or chicken game (they require additionally S < 1).
	Cooperation For T < 1 and S > 0 a well-mixed population eventually relaxes into a homogenous state of cooperators regardless of the initial configuration. In spatially extended systems this is generally equally true. However, for small S it becomes increasingly unlikely that an isolated cooperator is able to expand - obviously depending on the update rule of the players. When sparse, such cooperators or small clusters of cooperators act as seeds for the future success of cooperators.
	Bi-stability Well-mixed populations with T < 1 and S < 0 are bi-stable, i.e. depending on the initial configuration the system approaches either a state of homogenous cooperation or homogenous defection. In this parameter range spatial extension has the most pronounced effect on the long term fate of the strategies. The parameter values S, T determine the stability i.e. average lifetime of certain local configurations. The probability for their creation at initialization time sensitively depends on the initial fraction of cooperators.

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 panel that allows to set and change various parameters concerning the game as well as the population structure, while 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.

Note: The yellow and green 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.

The list below describes only the few parameters relevant for specifying the 2×2 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.

Reward:

reward for mutual cooperation.

Temptation:

temptation to defect, i.e. payoff the defector gets when matched with a cooperator. Without loss of generality two out of the four traditional payoff values R, S, T and P can be fixed and set conveniently to R = 1 and P = 0. This means mutual cooperation pays 1 and mutual defection zero. For example for the prisoner's dilemma T > R > P > S must hold, i.e. T > 1 and S < 0.

Sucker:

sucker's payoff which denotes the payoff the cooperator gets when matched with a defector.

Punishment:

punishment for mutual defection.

Init Coop, init defect:

initial fractions of cooperators and defectors. If they do not add up to 100%, the values will be scaled accordingly. Setting the fraction of cooperators to 100% and of defectors to zero, then the lattice is initialized with a symmetrical configuration suitable for observing evolutionary kaleidoscopes.