I'm trying to understand what is the optimal strategy for a mixed game.
I can illustrate the game as a trading system where you can go Long or Short. Going Long will give 80% win rate and going short will give the remaining 20%. Does the optimal strategy is to go always long or a mixed strategy is more optimal(a mix between going Long/Short with a random percentage)?
How do you approach such games?
The game as I understand it is:
If I go long, I will win 1 with probability $0.8$. If I go short, I will win 1 with probability $0.2$. Suppose I play the game $n$ times. What is my optimal strategy for those $n$ plays?
This is called a Poisson distribution. For each $i$ with $1\le i\le n$, I can either choose $p_i=0.8$ or $p_i=0.2$. My expected winnings is given by the sum $p_1+p_2+\cdots+p_n$. To maximize my expected winnings, I should always go long, in which case my expected winnings are $0.8n$.