Pick the Least Chosen Response Pick the least chosen response. You have 4 identical options. What is the optimal strategy in this game?
I have a feeling that it ends with infinite loop. Otherwise if there is a an optimal strategy and if all players are perfect logicians then they will always choose the most popular answer and lose the game?

 A: To make matters precise. Suppose there are $n$ players choosing between $m$ options. An action is to choose one of the options. A player wins the game if his action is least chosen. If more than one player choose the winning option the prize is allocated at random between all those players. The same is also true if more than one option is the one chosen the least. An option that is not chosen by anyone cannot be the winning option.
First, it is easy to see that there is no dominant strategy (I suppose that is what you mean by optimal strategy) for $n\geq 3$: If a player chooses an option $j$ and $n-2$ players choose $j$ and one player another option, there is a deviation incentive. So we are looking for a Nash equilibrium.
If $n\leq m$, there are many Nash equilibria in pure strategies, such that each player choses a distinct option. All players then win with the same probability according to the rules specified above.
Similarly, if $n=km$ for some integer $k$, there are many Nash equilibria in pure strategies, such that each group of $k$ players choose one option and the option of the groups do not overlap.
For all other cases the Nash equilibrium is in mixed strategies, such that each player chooses each option with probability $1/m$. This equilibrium also exists for the second case above.
Overall, if your goal is to actually outsmart the game you posted as an image, game theory provides only little guidance.
