Straffin, P. D. (1993): Game Theory and Strategy. How is the security level on p. 103 calculated?
Suppose Colin plays A with probability c and B with probability 1-c, while Rose plays A with probability r and B with probability 1-r. Then Rose's payoff is 

$2rc+4(1-r)c+10r(1-c)+0(1-r)(1-c)$

Rose's security level is calculated by taking the max-mini payoff: that is, for each value of r, find the minimum payoff over all c, then find the maximum such value.

So, first, we can simplify the above expression:

$2rc+4c-4rc+10r-10rc = 4c+10r-12rc$

As a function of c, this is a line with intercept $10r$ and slope $4-12r$. 

Case I: $r = \frac13$, this line is a constant $10r$. Plug $r =\frac13$ in and you get $\frac{10}3$

Case II: $r &gt; \frac13$, then it's negatively sloped, and the minimum occurs at $c = 1$, with a payoff of $4-2r$. Since $r &gt; \frac13$, we have that the payoff is less than $4-2/3 = \frac{10}3$.

Case II: $r &lt; \frac13$, then it's positively sloped, and the minimum occurs at $c = 0$, with a payoff of $10r$. Since $r &lt; \frac13$, the payoff is less than $\frac{10}3$.

In other words, if Rose plays A $\frac13$ of the time, then she gets an average of $\frac{10}3$ regardless of what Colin does. If she plays A less than $\frac13$ of the time, then if Colin plays B, then she gets 10 less than $\frac13$ of the time and 0 the rest of the time, getting her less than  $\frac{10}3$. If she plays A more than $\frac13$ of the time, then if Colin plays A, she gets 2 more than $\frac13$ of the time and 4 the rest of the time, getting her less than $\frac{10}3$.

So Rose can guarantee herself $\frac{10}3$ by playing A $\frac13$ of the time, and every other strategy has a possibility of giving her less than that. Thus, $\frac{10}3$ is her security level.

The reason BA should be played at least $\frac13$ of the time is that if BA is played  $\frac13$ of the time  and AB is played  $\frac23$ of the time, then Colin's payoff is  $8*\frac13+ 5*\frac23 =  \frac83 +  \frac{10}3 = \frac{18}3=6$, which is his security level. If BA is played less than  $\frac13$ of the time, then his payoff goes below his security level, so he would be better off not cooperating.