Guess the Number?

Suppose we have a game with three players, each secretly choosing a specific number from a countably infinite set of natural numbers. In the first phase, each player tries to guess another player’s number: α guesses β, β guesses γ, and γ guesses α. When a player’s number is correctly guessed, they are eliminated, and the remaining players advance to the second phase. The game is then played under the rules of misère: the eliminator in the first round starts second, and the first player whose number is correctly guessed wins.