In Exercises $1-4,$ let P

trixfor a Markov chain with two states. Let x$0$ the initial state vector for the population.

this approach is not guaranteed to break all ties.

$1.$ Compute x: and x$2.$

Roddick.

be in

Exercises $3\mathrm{.}1$

Markov Chains

$0\mathrm{.}5$

What proportion of the state $1$ population will

$0\mathrm{.}30\mathrm{.}50\mathrm{.}7$

$2.$

In our case,we have

A$)$

following

$01011300111310010=20000110010$o $1$

which produces the

ranking: First:

Second: Third:

be the transition ma $0\mathrm{.}5$

state $2$ after two

$0\mathrm{.}5$

be

steps?

$3.$ What proportion of the state $2$ population will be in

state $2$ after two steps?

$4.$ Find the

$0$

$10$

$\backslash \backslash$Lo $0$

$010121001001$

$10010\_/\_1\_\backslash "8\backslash "$

Djokovic, Federer $($tie$)$ Nadal

Roddick, Safin $($tie$)$

Are the players who tied in this ranking equally strong? Djokovic might argue that since he defeated Federer, he deserves first place. Roddick would use the same type

Since in a group of ties there may not be a player who defeated all the others in the group, the notion of indirect wins seems more useful. Moreover, an indirect victory

corresponds to a $2-$path in the digraph, so we can use the square of the adjacency

matrix. To compute both wins and indirect wins for each player, we need the row sums

argument to break the tie with Safin. However, Safin could argue that he has two

of

$\backslash "$indirect$\backslash "$ victories because he beat Nadal, who defeated two others; furthermore, he might note that Roddick has only one indirect victory $($over Safin, who then defeated Nadal$).$

of the matrix A $+$ A$2,$

$($A $+$ A$2)$j

$1$

Thus, we would rank the players as follows: Djokovic, Federer, Nadal, Safin,

which are given by

$00212$~ $\backslash \backslash \backslash "$l$\backslash "0101111$

$31127$

$12=6012$

oJ J$\_$

steady

state vector