(a) A player wins if she takes the total to 100 and additions of any value from 1 through 10 are allowed. Thus, if you take the sum to 89, you are guaran- teed to win; your opponent must take the sum to at least 90 but can take it no higher than 99. In either case you can get to 100 on the next move. Using rollback, you can show that you can win if you can get the sum to 78 or to 67 . . . or to 12 or to 1. Thus, being the first mover and using a strategy that entails choosing 1 on the first move and then saying 11 minus whatever your opponent says allows you to win; you take the sum successively to 12, 23, . . ., 78, 89, and 100.
Technically, the full equilibrium strategy is
(i) if you are the first player, start with 1;
(ii) if the current total is not (100 – 11n) for some n, then choose the number that will bring the total to this form; or
(iii) if the current total is of the form (100 – 11n), then choose any number (all choices are equally bad).
(b) In this version, you lose if you force the total to equal or exceed 100, so you can win if you take the total to 99. Using the same type of analysis as above, you see that you can win if you can get the sum to 88, 77, . . ., 22, or 11. This time you want to be the second mover. Your strategy should be to say 11 minus whatever your opponent says; this strategy takes you successively to 11, 22, . . ., 77,88, 99, and a win.
The full equilibrium strategy is
(i) if you are the first player, choose any number (all choices are equally bad);
(ii) if the current total is a multiple of 11, choose any number (all choices are equally bad); or
(iii) if the current total is not a multiple of 11, choose the number that will make the total a multiple of 11 (this is equivalent to choosing 11 minus the number just chosen by your opponent).