In a 1935 paper on economic prediction, Oskar Morgenstern posed the following problem:
Sherlock Holmes, pursued by his opponent, Moriarty, leaves London for
Dover. The train stops at a station on the way, and he alights there
rather than travelling on to Dover. He has seen Moriarity at the
railway station, recognizes that he is very clever and expects that
Moriarity will take a faster special train in order to catch him in
Dover. Holmes' anticipation turns out to be correct. But what if
Moriarity had been still more clever, had estimated Holmes' mental
abilities better and had foreseen his actions accordingly? Then,
obviously, he would have travelled to the intermediate station.
Holmes, again, would have had to calculate that, and he himself would
have decided to go on to Dover. Whereupon, Moriarity would again have
"reacted" differently. [...] One may be easily convinced that there
lies an insoluble paradox.
The conclusion Morgenstern drew was that prediction is essentially impossible in economics. Later he met John von Neumann, who has written a paper on parlor games in 1928 in which he proved the minmax theorem: In every finite two-player zero-sum game (such as the one in the question), there exists a pair of strategies and a number $v$ such that one players strategy guarantees here a minimum payoff of $v$ and the other player has a strategy that guarantees here a minimum payoff of $-v$. These guarantees are independent of what the other players do. Since it is a zero-sum game, nobody can do better either, so this pair of strategies form what would later be known as a Nash equilibrium. The proof relied on two things: Players can use randomized strategies and players evaluate their payoff by taking expectations. Morgenster and von Neumann then collaborated and the rest is history.
There is a conceptual problem with randomized strategies, the mixed strategies. If you play two pure strategies $s$ and $s'$ both with positive probability, but $s$ gives a higher payoff than $s'$, then you could gain in expectation by increasing the probability of $s$ and reducing the probability of $s'$. So rational people only randomize when they cannot gain by randomization.
There is a solution to this problem that is now quite popular among game theorists and is championed by Robert Aumann. In his view, players do not really randomize. The randomizing is done in the head of the other players. Holmes doesn't have to believe that Moriarty is randomizing, he just has to form an probabilistic assesment of what Moriarty is doing. Under this view, Nash equilibrium can be interpreted as a profile of assesment satisfying certain conditions: For each player, all other players have the same assesment of her strategies. Nobody would be forced to change their assesment upon hearing the assesment of the others.