In the original "Guessing 2/3 of the average" game, guessing 0 is said to be the optimal strategy assuming that everyone is rational.
In this modified version of this game, players can input 2 answers, the first answer being any integer from 0 to 100 while the other answer must be a positive integer from 50 to 100.
The goal is to have your first answer to be as close as possible to 2/3 of the average of all answers.
The lowest possible 2/3 of average is 50/3 and the highest possible average is still at 200/3.
If we attempt to follow the optimal strategy considered in the original game, then one should guess 17 for the first answer and 50 for the second answer.
However, if everyone follows this strategy, the average would be 67/3 instead and a higher first guess is better so this strategy is not optimal.
If this repeats, the optimal value for the first answer should be 25. I assume that this would be the most optimal strategy.
However, for any positive integer from 50 to 100 for the 2nd answer, there is a "optimal" strategy such that if everyone else also chooses that integer for the 2nd answer, the player has the highest chance of winning as well.
Does that mean there are multiple possible optimal strategies? I highly doubt so and hence I am trying to figure out which strategy is truly optimal.
(My choice of word for optimal may be used imprecisely so I apologize if I did use it in an incorrect manner.)
Ok what you seem to want looks like a fixed point theorem. Let us denote $x_0,y_0$ the "optimal inputs", everybody would logically input the same if they want to win.
The condition is then $$\frac{2}{3}\frac{x_0+y_0}{2} = x_0$$ this gives you: $$x_0 = \frac{y_0}{2}$$ So you have a lot of "possible optimal strategies" Let's say now that every player (other than you) chose randomly (and independently i.e. they don't form alliances) its $x$ and then just chooses $y = 2x$. He has an uniform choice for its $x$ to make in $\{0,\dots ,50\}$ so you could tell that the average choice would be $24,5$, hence you should choose $(24,50)$ or $(25,49)$ these are the closest you could get