1. Week 7 2. The dynamics of games Consider the following questions:
Which of the game’s EPs are also ESPs?
How does a game approach an ESP (if any)?
What happens if a game doesn’t have an EP?
To answer these questions, examine the game’s dynamics, i.e. how the population evolves in time.

2. The relative growth rate of a population is
(1)
where ni is the number of individuals of the ith species (strategy) and t is the time variable.
Let’s assume that the payoffs Pij are defined in such a way that the r.-h.s. of (1) is equal to the fitness of the ith species, i.e. or
(2)

3. Let’s replace the absolute numbers ni with proportions xi – i. e
Let’s replace the absolute numbers ni with proportions xi – i.e. let ni = N xi, where N is the no. of individuals in the population.
Then (2) becomes or
(3)
To eliminate N and have an equation just for ni, recall that
and differentiate this to obtain...

4. then use Eq. (2) to obtain
hence,
Now, equation (3) becomes

5. Cancel N from the previous equation and rearrange it:
then recall the definition of the fitness of the population as a whole,
(4)
This equation can be illustrated by a ‘box diagram’...

6. Remark:
It was implied that both birth and death rates are proportional to the fitness of the species. This isn’t a good model: the death rate should rather be the reciprocal of the fitness!

7. Putting games in normal form
Theorem 1:
Equation (4) is invariant (doesn’t change) with respect to adding a constant to a column of the payoff matrix P.
Remark:
Adding the same constant to all entries in the ith column of P implies changing by the same amount the payoffs to all species when they compete with the ith species. This shouldn’t change the fitnesses of the species relative to each other.
Theorem 2:
By adding appropriate constants to the columns of P, its diagonal entries can be eliminated.

8. ۞ If the diagonal entries of a payoff matrix P are all zeros, P is said to be in normal form.
Example 1:
The original payoff matrix of the hawk–dove gave is
It can be put in normal form by adding ? to the first column and ? to the second column, which yields...

9. Let’s derive the explicit form of the dynamical equation (4) for the hawk–dove game. To do so, we’ll need FH FD
and
Then, for i = H and i = D, Eq. (4) yields
(5a)
(5b)

10. To find an EP (if any), assume that xH,D do not change with time – hence, their derivates w.r.t. t are zero, and Eqs. (5) become
hence,
The solutions of these equations are...
(6)
(7)
(8)

11. (6)-(7) describe the endpoints, whereas (8) describes the EP we have calculated before.
Examining the stability of EPs using the dynamical equations
To examine the stability of EP (8), eliminate xD from (5a) by letting xD = 1 – xH:
hence,
(9)
Remark:
Alternatively, we could’ve substituted xH = 1 – xD into (5b).

12. Introduce the deviation from the EP, z = xH – 2/3, and rewrite (9) in terms of z. To do so, let
(10)
and substitute into (9):
hence, expanding the expressions in brackets,
(11)

13. If xH is close to EP, then z is small (recall that it’s the deviation of xH from the EP).
Since z is small, then z2 and z3 are even smaller and can be omitted [this procedure is called the “linearisation” of Eq. (11)]:
We’ve now got a linear homogeneous equation, which can be easily solved:
Observe that

14. Since z vanishes and since xH = 2/3 + z [see (10)], then
Evidently, the game homes onto the EP – hence, it’s stable (as expected).