Consider a two-player game with the following strategy spaces: S1 = [0, 5] and S2 = [0, 5]. Suppose the players’ best-response functions, s1 = BR1(s2) and s2 = BR2(s1), are as pictured here. What is the set of rationalizable strategy profiles for this game?
The process to compute the rationalizable sets for the two players is iteratively computing k-rationaliable sets let's denote them with $S_1^k$ and $S_2^k$ for the two players respectively. The full set of strategies are the 0-rationalizable sets, i.e. $S_1^0 = S_1$ and $S_2^0 = S_2$ the update formulas are:
$$ S_1^{k+1} = BR_1(S_2^k) \\ S_2^{k+1} = BR_2(S_1^k) $$
As an example in the 1st iteration you will get something like $$ S_1^1 = BR1(S_2^0) = BR1([0;5]) = [1;4] \\ S_2^1 = BR2(S_1^0) = BR2([0;5]) = [0;4] $$
The interpretation is that it can not be rational for player 1 to play anything outside [1;4] and for player 2 to play anything outside [0;4].
In the second iteration you will get (by visual inspection of the graphs): $$ S_1^2 = BR1(S_2^1) = BR1([0;4]) = [1;3.5] \\ S_2^2 = BR2(S_1^1) = BR2([1;4]) = [0.6;3.2] $$
The interpretation is that knowing that it can not be rational for player 2 to play anything outside [0;4] it can not be rational for player 1 to play anything outside [1, 3.5]. And knowing that it can not be rational for player 1 to play anything outside [1;4] it can not be rational for player 2 to play anything outside [0.6;3.2].
Iterating further the sets will converge to: $$ S_1^\infty = [2;3] \\ S_2^\infty = [1;3] $$ which are the rationalizable strategies for the two players.