Evolutionary Game Theory - Who Is Playing The Game?

Who Is Playing The Game?

At first glance it may be surmised that the contestants of evolutionary games are the individuals present in each generation who directly participate in the game. On reflection however we see that individuals live only through one game cycle, and instead it is the strategies that really contest with one another over the full time span of these recursive games. So it is ultimately genes that play out a full contest – genes of STRATEGY (i.e. selfish genes). The contesting genes are not just present in an individual and his/her direct linage; they are also present to a relational degree in all of the individual’s kin. This can sometime profoundly affect the kinds of strategies that will survive, and nowhere is this more pertinent than in issues of cooperation and defection. William Hamilton, whose work defined kin selection, was involved in much of the groundbreaking mathematical work in this area and Hamilton treated many of these cases using game theoretic models. Kin related treatment of game contests help to explain many aspects of the behaviours of eusocial insects, the altruistic behaviour in parent/offspring interactions, mutual protection behaviours, and co-operative care of offspring. For such games Hamilton defined an extended form of fitness which is operative – Inclusive fitness, which extends the fitness measure to include an individual’s offspring as well as any other “offspring equivalents” found in kin.

The Mathematics of Kin Selection
The concept of Kin Selection can be formulated into a mathematical relationship by expressing the basic definition that: inclusive fitness = own contribution to fitness + contribution of all relatives. It is worth pointing out that fitness is measured relative to the average population at large, e.g. a fitness=1 means growth at the same average rate of the population, fitness<1 means decreasing percentage-wise in the population (dying out), fitness>1 means increasing percentage-wise in the population (taking over). The inclusive fitness of an individual w_i is the sum of its specific fitness of itself a_i plus the specific fitness of each and every relative weighted by the degree of relatedness which equates to the summation of all *r_jb_j....... where r_j is relatedness of a specific relative and b_j is that specific relatives fitness - producing: Now if individual a_i sacrifices his "own average equivalent fitness of 1" by accepting a fitness cost C, and then to "get that loss back", w_i must still be 1 (or greater than 1)...AND if we use RB* to represent the summation we get: 1< (1-C)+RB ....or rearranging..... R>C/B.

The Mathematics of Kin Selection

The concept of Kin Selection can be formulated into a mathematical relationship by expressing the basic definition that:

inclusive fitness = own contribution to fitness + contribution of all relatives.

It is worth pointing out that fitness is measured relative to the average population at large, e.g. a fitness=1 means growth at the same average rate of the population, fitness<1 means decreasing percentage-wise in the population (dying out), fitness>1 means increasing percentage-wise in the population (taking over).

The inclusive fitness of an individual w_i is the sum of its specific fitness of itself a_i plus the specific fitness of each and every relative weighted by the degree of relatedness which equates to the summation of all r_j*b_j....... where r_j is relatedness of a specific relative and b_j is that specific relatives fitness - producing:

Now if individual a_i sacrifices his "own average equivalent fitness of 1" by accepting a fitness cost C, and then to "get that loss back", w_i must still be 1 (or greater than 1)...AND if we use R*B to represent the summation we get:

1< (1-C)+RB ....or rearranging..... R>C/B.

Hamilton then went on beyond areas of kin relatedness to work with Robert Axelrod to analyze games that involved co-operation under conditions not involving kin where reciprocal altruism comes into play.

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