What are the ways in which you can find all set of Rationalizable strategies in a Two-Playe Normal Form Game , like the one given below:
(This is not a homework problem !) Hi. I am not clear on how to find a set of rationalizable strategy (pure and mixed) in a two player game. I have given the game payoff matrix below that I am trying to find all the sets of Rationalizable Startegies. I have also given my workout solution below. I would like to get additional input and feedback on my work.
Consider the following normal form game:
\begin{array}{|c|c|c|c|} \hline 1/2 & a & b & c \\ \hline A & 4,5 & 0,2 & 0,0 \\ \hline B & 3,2 & 3,0 & 3,2 \\ \hline C & 0,0 & 0,2 & 4,4 \\ \hline \end{array}
Workout:
1) To begin with , I try to find pure rationalizable strategy. We know that in a two player game Iterated Elimination of Strictly Dominated Strategies is equivalent to Rationalizable Strategies. So here, we see that none of the pure strategies for both players are strictly dominated and hence we can say that there are $\textit{no pure rationalizable startegies}$
2) Now we look at Mixed Rationalizable Strategy: Here I am reasoning that for player two "b" is dominated by "a" and "c" when Player 2 (P2) puts probability of 2/3 on "a" and 1/3 on "c" ..in this case b is strictly dominated by a and c.. Thus, eliminate "b" from P2 strategy profile and we get now a 3x2 game matrix [{A,B,C}x{a,c}]
\begin{array}{|c|c|c|} \hline 1/2 & a & c \\ \hline A & 4,5 & 0,0 \\ \hline B & 3,2 & 3,2 \\ \hline C & 0,0 & 4,4 \\ \hline \end{array}
Now I am confused as to how do I proceed further. Do I stop here and say that [{A,B,C}x{a,c}] are the set of rationalizable strategies or is there more I can do ?
I was thinking of plotting the new 3x2 game matrix in a graph: Probability Plot of Strategies.
But Then through this I see that When $P (probability) < 1/4$, Utility from playing C is better and when $P > 3/4$ then Utility from playing A is better.
So Can I say then P1's strategy of B is strictly dominated by A and C when p <1/4 and when p>3/4? Does this allow me to delete B from P1's strategy profile?
Let me know please how I can improve my understanding of finding Rationalizable Strategies..