The probability distribution which describes the process of choosing an integer uniformly at random on the interval [0,N] converges

*as a function*(to the function which is identically zero), but it does not converge

*as a probability distribution*since the total mass of the limit is 0. Hence, that limiting function can not be used to define a probability.

Gees - since you did not get my hint, let me spell out the solution to you.

This is the 2x2 game.

France only choice is which turn to play the attacking move. When he plays the attacking move, the game will end (if it did not already end). Italy's only choice is which turn to play the counterattack. When he plays the counterattack, the game ends (if it did not already end).

That is, France choses a natural number k. Italy chodes a natural number j. The probability that Italy wins is the ”probabolity” that j=k. The probability that France wins is the "probability" that j != k.

These probabilities should be understood just like the probability you computed for that a number is even. The answer is that France's probability to win is 1.

Also in game theory, the notion ”strategy” is wider than just mixed strategies for games in normal form. This game is not in any of the standard forms ( that I know of). The reason is that the "payoff" matrix is not actually a payoff matrix - it includes the option ”play another round”. I might be wrong, but I would be very surprised if games like these have Nash equilibria.