| Errata: Known Errors in "Heard on The Street" |
In many cases, an error in a later edition also appeared in earlier editions. So, look at the errors listed for your edition and also look at errors listed for any later editions.
Thank you to Ethan Utley for the corrections below.
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.