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Game Theory : Matrix, Population Formulation (Lions and Lambs)

In a population, there are two kinds of individuals, LIONS and LAMBS. Whenever two individuals meet, 40 yen is at stake. When two LIONS meet, they fight each other until one of them is seriously injured. While the winner gets all the money, the loser has to pay 120 yen to get well again. If a LION meets a LAMB then the LION takes all the points without a contest. When two LAMBS meet, they debate endlessly until one side gives up. The loss of time means they are each fined 10 yen. You may assume that the probability of winning against an individual of the same kind is ½.

(i) Give a matrix that describes the game and argue that a population consisting solely of LIONS isn't stable, nor is one consisting solely of LAMBS.
(ii) Give an example of a population that is stable.

Solution Preview

We can describe the game by putting the possible outcomes of every meeting in a matrix.

When a Lion meets another Lion, he will win on average 0.5*40+0.5*(-120)=-40 yens.

When a Lion meets a Lamb, he will win always 40 yens.
When a Lamb meets a Lion, he will gain nothing.
When a Lamb meets a Lamb he will gain on average 0.5*(40-10)+0.5*(-10)=10 yens.

So the game is described by the matrix:


Solution Summary

A game is analyzed in terms of population formulation. The solution is detailed and well presented.