BIO150Y: Evolution of Cooperation Reference

Frequently Asked Questions
Expanded treatment of some of the tutorial concepts

How is my payoff matrix built?

In our prison example, each player has two possible moves (options): cooperate (with your partner) and remain silent or defect (betray your partner) and implicate your partner in the major crime. Since your fate depends both on your move and your partner's move, there are in fact four possible outcomes with different prison terms (payoffs):

  • Outcome 1: You both cooperate and remain silent. Police can only charge you with the minor crime. The reward for mutual cooperation is 2 years in prison.
  • Outcome 2: You both defect and betray each other. Police drop charges on the minor crime, but charge each of you with the major crime. The punishment for mutual defection is 8 years in prison.
  • Outcome 3: You defect and betray your partner who cooperates and remains silent. The temptation to defect is that you go free, while the other person has been suckered and gets 10 years in prison.
  • Outcome 4: You cooperate and remain silent, while your partner defects and betrays you. You are a sucker and go to prison for 10 years ( 2 years + 8 years, since you are charged for both crimes), while your partner goes free.

The outcomes can be summarised in a 2x2 payoff matrix as shown below.

Cooperate Defect
Cooperate 2 10
Defect 0 8


How is the cost-benefit payoff matrix built?

In the chimpanzee mutual grooming example, each chimp has two options: cooperate and groom the other chimp, or defect and refuse to reciprocate. It is assumed that it costs 2 points to groom a partner, and the benefit of being groomed is 4 points. There are four outcomes:

  • Outcome 1: Both chimps cooperate and groom each other. The reward for mutual cooperation is 2 points, since you receive the benefits of being groomed (4 points) less the cost of grooming the partner (2 points) for a net payoff of 2 points.
  • Outcome 2: You both defect and betray each other. The punishment for mutual defection is 0 points. By defecting you have paid no costs of grooming your partner, but you receive no benefits either, since they defected too.
  • Outcome 3: You defect and refuse to groom your partner, while they groom you. The temptation to defect is that you get 4 points; you received all the benefits and paid none of the costs.
  • Outcome 4: You cooperate and groom your partner, while they defect. You are a sucker and get -2 points. You have paid the cost of grooming and received no benefits in return.

The outcomes can be summarised in a 2x2 payoff matrix as shown below.

Raw cost-benefit payoff matrix

Cooperate Defect
Cooperate 2 -2
Defect 4 0

To simplify its use, the (raw) cost-benefit payoff matrix can be adjusted so that the lowest payoff is zero rather than negative. In this example, we can add 2 points to each cell so that the resulting (normalized) cost-benefit payoff matrix looks like this:

Normalized cost-benefit payoff matrix

Cooperate Defect
Cooperate 4 0
Defect 6 2


Some common strategies in the Prisoner's Dilemma Game

ALTERNATE: The player alternates between C and D, starting with a C.

ALWAYS COOPERATE: The player always plays C, no matter what their partner has played in the past. Also known as sucker.

ALWAYS DEFECT: The player always plays D. Also known as cheat.

GRUDGER: The player starts playing C and continues to do so until the other player plays D. After that it plays D for the rest of the game with that particular partner.

RANDOM: The player chooses either C or D with equal probability.

SNEAKER: The player starts with a C and then plays whatever its partner play in the previous move. However, at random intervals it plays D.

TIT FOR TAT: The player starts playing C and then plays whatever its partner did in the previous move.

TIT FOR TWO TATS: The player plays C in the first and second moves. After that, if its partner played D in the two previous moves they play D, otherwise they continue to play C.

TWO TITS FOR TAT: The player starts with C, and then if its partner plays D, then plays D in the following two moves, otherwise plays C.


Calculating relative success

In an imaginary evolutionary game, 24 players pair at random and play a multiple-moves game. Points are then totalled for all players in each strategy. In this example, the 12 suckers amass only 240 points compared to the 840 points amassed by the 12 cheats. The relative success of each strategy is expressed as a proportion of the total points amassed; for suckers this is a relative success of 0.22, which is 240 points divided by the total points of 1080.

The number of players for each strategy in the next round of games (generation) is simply the relative success of the strategy multiplied by the total number of players in the population (24). Note that this assumes that population size stays constant and just the proportion of players in each strategy changes. The new player numbers are rounded to the nearest integer. So in the next generation of the game, there will be 5 suckers and 19 cheats.

Worked example of relative success calculation

Suckers Cheats Total
Players 12 12 24
Points 240 840 1080
Relative success 240/1080 = 0.22 840/1080 = 0.78 1.0
New players 0.22*24 = 5.3 0.78*24 = 18.7 24
Players (rounded) 5 19 24



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