Errata: Known Errors in "Heard on The Street" |
In fact, your opponent looks at your expected payoff -20pp'+9p+9p'-4 and sees that it may be written as p'(9-20p)+9p-4. For fixed p, your opponent sees a linear function b.p'+a, where the coefficient is b=(9-20p) and the intercept is a=9p-4.
If b>0, your opponent wants to minimize b.p'+a and chooses the lowest possible p' value (p'=0). In this case, the expected payoff is a=9p-4. Note, however, that b>0 implies 9-20p>0 implies p<9/20. So, the expected payoff is a=9p-4<81/20-4=$1/20 (So, I get an expected payoff strictly less than $1/20).
If b<0, your opponent wants to minimize b.p'+a and chooses the highest possible p' value (p'=1). In this case, your expected payoff is 5-11p. Note, however, that b<0 implies 9-20p<0 implies p>9/20. So, the expected payoff is 5-11p<5-99/20=$1/20 (So, you get an expected payoff strictly less than $1/20).
If, however, b=0 (i.e., p=9/20), then your expected payoff is 9p-4=$1/20 for every p' value, which is superior to all of the above. So, you want to choose p=9/20.