# Set of rationalizable strategies

Consider a guessing game with ten players, numbered 1 through 10. Simultaneously and independently, the players select integers between 0 and 10. Thus player i's strategy space is $\mathbf{S}_i$ $=$ $\begin{Bmatrix} 0, &1, &2, &3, &4, &5, &6, &7, &8, &9, &10 \end{Bmatrix}$ for $i=1, 2, \dots , 10$. The payoffs are determined as follows: First, the average of the players' selections is calculated and denoted $a$. That is, $a=\frac{s_1+s_2+\dots +s_{10}}{10}$, where $s_i$ denotes player i selection, for $i=1, 2, \dots , 10$. Then, player i's payoff is given by $u_i=(a-i-1)s_i$. What is the set of rationalizable strategies for each player? I am trying to figure out a faster way of doing it, if there is one (an alternative is to write out a matrix representation for 10 players with 10 choices each, which would be a lot of time).

I claim each player should choose $0$. Player $10$ reasons as follows: $a \le 10$, so $(a-11) \lt 0$, so my payoff is negative unless I choose $0$, in which case I break even. If player $9$ thinks player $10$ is rational and will therefore pick $0$ (or will pick $0$ for some other reason), he can say, I know $a \le 9$, so $(a-10) \lt 0$, so I should choose $0$. It continues down the line to player $1$, who knows the others all picked $0$, so $a \le 1 \dots$