Show me appropriate theorem about game theory.

Question

"For every table with $A$ lines, $B$ columns and pair $(a_i, b_j)$ in every cell, exist sequential game with perfect information such as this table is a normal form".

I think it is obviously true. But how it can be prove?

Answer 1

Following Wikipedia, a sequential game with perfect information is given by a structure $G=\langle P,\mathbf S,\mathbf F\rangle$, and for the given situation, we can let $P=\{1,2\}$, $\mathbf S =\{S_1,S_2\}$ with $S_1=\{1,2,\ldots,A\}$ and $S_2=\{1,2,\ldots, B\}$, and $\mathbf F=\{F_1,F_2\}$, where $F_1\colon S_1\times S_2\to \Bbb R$ is given by $(i,j)\mapsto a_i$ and $F_2\colon S_1\times S_2\to \Bbb R$ is given by $(i,j)\mapsto b_j$. This matches the given table.