The following equation doesn't seem to be working under the laws of Algebra so I wonder what the rules are here.
"Solving 2x + 0(1 − x) = 0x + 1(1 − x) for x, we get x = 1/3"
The problem is from this paper on the third page near the middle of the page.
Why is x = 1/3?
I'll include the whole paragraph in case you would like more context.
"Monica is the row player, and Gary is the column player. Suppose Monica and Gary are going out on a date and need to choose an activity (e.g. movie, restaurant, etc.). Gary would like to go to a football game (G) and Monica wants to see a movie (M). They both prefer going together to the same activity, yet each feels less rewarded for choosing the other’s preference. Suppose Monica always chooses M. Gary is better off choosing M and has no incentive to unilaterally deviate from that pure strategy. Likewise, if Gary always chooses G, Monica has no incentive to unilaterally deviate from her pure strategy. The utility is always (2, 1) or (1, 2). So, (M, M) and (G, G) are two pure Nash equilibria profiles. However, there is a mixed strategy Nash equilibrium as well. An equilibrium can be reached when each player, seeing other strategies, is indifferent to the choice of action, i.e. all are equally good. What would have to be the case for Monica to be indifferent to the Gary’s choice? Let σGary(M) = x be the probability of Gary choosing the movie. Then the utility that Monica expects is 2x + 0(1 − x) and 0x + 1(1 − x) respectively. For Monica to be indifferent between G and M, these two expected utilities would need to be equal. Solving 2x + 0(1 − x) = 0x + 1(1 − x) for x, we get x = 1/3."
BONUS QUESTION: It also says "Let σGary(M) = x be the probability of Gary choosing the movie." Is this math from statistics? Is that sigma for standard deviation?
We're looking at the equation:
$$2 \cdot x + 0 \cdot (1 − x) = 0 \cdot x + 1 \cdot (1 − x).$$
Assuming we are working exclusively with real numbers, the equation can be rewritten as:
$$2 \cdot x = 1 - x,$$
so we have $3 \cdot x = 1$. It follows that $x = \frac{1}{3}$, as desired.
As for your bonus question, game theorists typically use $\sigma$ to refer to a strategy. See, for example, Fudenberg and Tirole's textbook on the subject.