We start within the framework of a two-person symmetric matrix game with two possible strategies: ‘cooperation’ and ‘non-cooperation’ as given in the pay-off matrix.

This is a cost–benefit game where if neither individual cooperates, then there are no benefits and costs. However, if one person cooperates, then there are both benefits and costs distributed according to the matrix. A non-cooperative focal individual versus a cooperative opponent receives a benefit

*b* from the opponent at a fraction 0≤

*β*≤1 of the opponent's cost

*c*. A cooperating focal individual versus a non-cooperating opponent pays the full cost

*c* and receives only a fraction 0≤

*α*≤1 of the benefit

*b*. The terms

*α* and

*β* introduce and scale the degree of ‘selfishness’ in the game (

West *et al*. 2007). If both the focal individual and the opponent cooperate, then each receive their own benefit plus a fraction of the opponent's benefit at their own cost plus a fraction of the opponent's cost. To ensure that this is a game of possible cooperation, we require that

.

It is easy to show that cooperation is a global ESS so long as

*αb*−

*c*>0 or

*α*>(

*c*/

*b*), otherwise non-cooperation is the global ESS. This game does not permit an ESS with a mix of cooperators and non-cooperators. Interestingly, the conditions for cooperation to evolve correspond with Hamilton's rule (

Hamilton 1963;

Queller 1985) where the coefficient of relatedness must be greater than the cost–benefit ratio, and with reciprocal altruism (

Trivers 1971;

Axelrod & Hamilton 1981;

Brown *et al*. 1982) where the coefficient of familiarity must be greater than the cost–benefit ratio. As a general result, cooperation can evolve if a sufficiently large fraction of the benefit is rebounded onto the cooperator either through the public good, non-random interactions (relatedness), or through reciprocity (

Brown 2001;

Nowak 2006).

We use this cost–benefit matrix game as a guide to form a continuous game of cooperation in which the individuals are able to scale the level of cooperation between total cooperation and total non-cooperation. Starting with the linear form of the cost–benefit matrix game, we add a nonlinear quadratic term to the benefits and costs. Let 0≤

*u*_{i}≤1 be a continuous variable describing the degree of cooperation of individual

*i*, where

*u*_{i}=0 represents an absence of any cooperation and

*u*_{i}=1 represents a maximal level of cooperation. Individuals within the population are assumed to interact randomly in a pairwise fashion resulting in linear/quadratic benefit and cost functions. These assumptions lead us to the following fitness-generating function or

*G*-function (

Vincent & Brown 2005):

where

*v* is a virtual variable, with the property that replacing

*v* by

*u*_{i} in

*G* results in the fitness function for a focal individual using the strategy

*u*_{i}. The vector of all

*n*_{s} strategies in the population is given by

, and

is the frequency vector of players using these strategies. The fitness for an individual using strategy

*u*_{i} is its expected pay-off from playing pairwise against all other players using strategies

*u*_{j} including

*j*=

*i* weighted by the probability

*p*_{j} of playing someone with strategy

*u*_{j}. The pay-off from an interaction has four terms. The two terms with parameters

*b*_{1} and

*b*_{2} represent the benefits bestowed from the interaction, and the two terms preceded by

*c*_{1} and

*c*_{2} represent the costs of cooperation. If the quadratic terms are removed (

*b*_{2}=0 and

*c*_{2}=0), then the resulting linear

*G*-function is a continuous representation of the cost–benefit matrix game above. A method for converting a matrix game into a continuous game is given in Vincent & Brown (

2005, ch. 9).

The level of cooperation by an individual's opponent using a strategy *u*_{j}≥0 always contributes a benefit to the individual. We let the term 0≤*α*≤1 determine the degree to which the individual derives any direct personal benefit from its own degree of cooperation, *αv*. Through *b*_{1}>0, benefits increase linearly with the level of each individual's cooperation. Through *b*_{2}, benefits increase at either a diminishing (*b*_{2}<0) or an increasing (*b*_{2}>0) rate with the individuals' levels of cooperation. If *b*_{2}<0, there may be some threshold value of *v* above which total benefits actually decline with increasing *v*.

The level of cooperation by an individual, *v*>0, always incurs a cost to the individual. We let the term 0≤*β*≤1 determine the degree to which the individual experiences an additional cost from its opponent's degree of cooperation, *βu*_{j}. Thus, *β* introduces an externality by which some of the cost of a cooperative act becomes public and unavoidable to the recipient. Through *c*_{1}>0 costs increase linearly with the level of each individual's cooperation. Through *c*_{2}, costs increase at either a diminishing (*c*_{2}<0) or an increasing (*c*_{2}>0) rate with the individuals' levels of cooperation. If *c*_{2}<0, there may be some threshold value of *v* above which total costs actually decline with increasing *v*.

While

*b*_{2} and

*c*_{2} can take on any value, they must remain sufficiently large (not too negative) to ensure that the benefits remain positive and the costs remain negative. The following constraints must be satisfied:

for

*i*=1,

…,

*n*_{s}, where

and

*j*=1,

…,

*n*_{s}.

The parameters

*α* and

*β* determine the extent to which the benefits are public and the costs are private. Setting

*α*=1 and

*β*=0 results in the Snowdrift game presented by

Doebeli *et al*. (2004). This game involves two individuals clearing a road blocked by a snowdrift. Each individual benefits equally from the digging effort, but the cost to an individual depends only on one's own level of effort. In this formulation, the sum of the strategies of the two interacting individuals represents the public good and the strategy of the focal individual solely determines the private cost.

Setting

*α*=0 and

*β*=0 results in a nonlinear variant of a game of altruism (

Killingback & Doebeli 2002). In this case, there are no direct benefits to the focal individual from cooperating, and the costs of cooperating are entirely private. The structure of this game is similar to Prisoner's Dilemma in which an individual always benefits from the cooperation of its partner, and the individual always avoids costs by defecting. Therefore, while both individuals would be better off if their opponent cooperated rather than defected, each individual has defect (or

*u*=0) as a dominating strategy. No matter what one's opponent does, one's own best strategy is to play

*u*=0.

Setting *α*=1 and *β*=1 creates a game of complete public benefits and costs. The benefits and costs to an individual are simply the sum of each individual's level of cooperation. It is this sum of cooperative behaviours that matters, not the source of the cooperative acts. Similarly, the costs of cooperation represent a complete externality. Both individuals experience the same costs regardless of which partner instigated the costs via some level of cooperative behaviour.

Setting *α*=0 and *β*=1 represents a fourth possible combination of public versus private costs and benefits. This represents a costlier form of altruism in which an individual contributes a benefit to its partner and also imposes a public cost to oneself and the opponent.

When *β*=0, an individual would like an opponent to be as cooperative as possible. With *β*=1, there may be limits to the level of cooperation one desires from their opponent. The maximum pay-off to an opponent may occur at an intermediate level of cooperation from the cooperating individual. However, the strategy of *u*=0 still represents a dominating strategy.

All of these four games can be extended by allowing *α* and/or *β* to take on values between 0 and 1. As a last specific example, we will consider the case of *α*=*β*=0.5. There is a public element to benefits and costs, but being cooperative still costs the cooperator more than the recipient, and the individual garners a smaller benefit from cooperating than the recipient.