Model 1

Interpreting Results

If you read the section on suggested parameters (or you may have figured it out yourself), you know that in this model there are three different scenarios that can occur. These are:

  1. The cooperators completely take over.
  2. The defectors completely take over.
  3. There is a mix of cooperators and defectors (which can be dominated by one type or the other).

Under the constraints defined for this situation, the relationships among the parameters that determines which of these scenarios will occur are reasonably straightforward. If the different payoffs are defined a -> d as follows:

 

 Player 2
Player 1 
   Defect Cooperate
Defect a  b
Cooperate  c  d

then the winning type is determined solely be the relationships between a & c and b & d. Corresponding to the situations defined above, these conditions are:

  1. a < c and b < d (or a > c and b < d, see below)
  2. a > c and b > d (or a > c and b < d, see below)
  3. a < c and b > d

The first parts of situations 1 and 2 stand by themselves (i.e. if a < c and b < d then the cooperators win, and if a > c and b > d then the defectors win). However, situation 3 and the parenthetical portions of 1 and 2 require further explanation.

If conditions are such that the parameters fall in situation 3, even though there are both cooperators and defectors present, the system still may be dominated by one or the other. The equilibrium (final) proportion of cooperators (p) can be determined from the expression

p=( a - c ) / [ ( a + d ) - ( b + c ) ]

and the number of defectors is 1 - p. A heuristic explanation for the existence of this internal equilibrium is that defectors do well against cooperators, but not against other defectors, while cooperators do well against defectors but not against other cooperators. This means that when cooperators are abundant in the environment they start to do worse, and defectors do better thus increasing the proportion of defectors. The situation is reversed when defectors are abundant. The success of the less prevalent type insures the persistence of both types.

Finally (this idea is sort of complicated so don't worry about it too much), when the parameters are chosen such that a > c and b < d then either the cooperators will win, or the defectors will win, depending on the proportion of each at the beginning (due to an unstable equilibrium). Since there is always a 50/50 mix of cooperators and defectors then you expect cooperators to win when

d - b > a - c

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