$\left[\begin{array}{l}\text { Game Theory_(Miscellaneous) } \\ \text { [ Rational and Irrational Numbers ] }\end{array}\right]$

Two players play the following game. They take turns. One player names two numbers, which are the endpoints of an interval. The next player must name two other numbers, which are the endpoints of an interval nested within the previous one. The game continues indefinitely. The first player aims to ensure that the intersection of all named intervals contains at least one rational number, while the second player aims to prevent this. Who wins?
With each of his moves, the second can avoid certain rational numbers from falling into the intersection of all segments.

## Solution

The winning strategy for the second is as follows. On his first move, he chooses his segment so that it does not contain any integer points. On his second move, he chooses his segment so that it does not contain any points of the form $m / 2$, where $m$ is an integer. Continuing in this manner, on his $n$-th move, he chooses his segment so that it does not contain any points of the form ${ }^{m} / n$, where $m$ is an integer. Note that, regardless of the first player's moves, the second player can always choose segments according to the rules outlined above.

Suppose a rational number ${ }^{p} / q$ (for some integer $p$ and natural number $q$) lies in the intersection of all segments. But this contradicts the fact that on his $q$-th move, the second player chose a segment that does not contain rational numbers representable as a fraction with denominator $q$.

## Answer

The second.