I apologize for the vague and potentially misleading title, I am very new to statistics and do not yet have a handle on the jargon. Essentially, I have the table below:
A B
C 0.925 1.008
D 1.033 0.871
This is a table of the payoff when a given player plays option A or B versus what the opponent plays (C or D). For those of you familiar with game theory, this is simply a standard action and output table. Solving for the Nash Equilibrium, I assign option A to be played at probability "p", and thus B is played at 1-p. Multiplying the values by the probability at which they are played and setting row C equal to row D, I have found that it is optimal to play option A 55.95% of the time. However, assuming a self-interested opponent, they would catch on to this pattern, thus forcing an adjustment to volume at which each option is played. How would I adjust the values within the matrix when accounting for the volume at which each is played to find a continuous equilibria? Please let me know if I wasn't clear about anything.
$55.92\%$ would be marginally closer to the equilibrium point but it makes little difference
The point about a mixed strategy with this value is that no matter whether the opponent plays $C$ or $D$ or some mixture of the two, the expectation of the outcome will always be about $0.9616$ since this is $55.95\% \times 0.925 + 44.05\% \times 1.008$ and is also $55.95\% \times 1.033 + 44.05\% \times 0.871$
So it does not matter whether the opponent catches on or not. There is nothing the opponent can do, good or bad, to make the expected outcome worse for the original player.