Is it possible to transfer information through strategic games?
What you are implying is thoroughly studied by what is called algorithmic game theory. There you can find the concept of a learning algorithm. In game theoretical scenarios where the mathematical equilibrium is difficult to evaluate, those algorithms are used to get an approximation.

What they do (in a simplified way)
1) Player A starts with an array of available strategies, each to be played with equal possibility.
2) In every iteration he randomly chooses one strategy and plays against his opponent.
3) The result comes as a feedback and according to an update function/algorithm, the array of strategies is re-evaluated/recalculated.
4) This is repeated many times, hopefully to converge to either one strategy which will be a nash equilibrium, or to a mixed-strategy profile.

It is a very interesting topic, connected a lot with machine learning and AI.
Here are some links to get an idea:

1) Game Theory &amp; Learning for wireless networks 

2) On Learning Algorithms for Nash Equilibria