The Briefcase You Didn't Pick

A game host sets two locked briefcases in front of you; one contains exactly twice as much cash as the other. You grab one at random. Before opening it you reason: "Call my amount $X$. The other case holds $2X$ or $X/2$ with equal probability, so its expected value is $1.25X$ — I should trade." But the identical logic would then tell you to trade back, and so on forever. Pinpoint exactly what is wrong with the argument.