ABSTRACT
We introduced the concept of a matrix game in Example 2.5 in Chapter 2. Recall that contests involve two players, each with n available pure strategies {1, 2, . . . n} where the payoff to an individual playing i against one playing j is aij , so all payoffs can be summarised by the matrix
A = (aij)i,j=1,...,n. (6.1)
The key property is that the payoffs are given by the function
E[x,y] = xAyT, (6.2)
which is continuous and linear in each variable. The players are matched randomly and the payoff to an individual using strategy σ in a population∑ j αjδpj is given by
E σ;∑
αjδpj
= ∑ j
αjE[σ,pj ]. (6.3)
In this chapter we show how to find all of the ESSs of any given matrix in a systematic way. We then look at the possible combinations of ESSs that can occur, through the study of patterns of ESSs. Finally we look at some specific examples of matrix games which are developments of the Hawk-Dove game that we looked at in Chapter 4.
