A Three-Player Singled Out Game

The three-player singled out game is played in a series of rounds as follows: In each round, a coin is flipped and the result is kept secret. The players A, B, and C, in this order and one at a time, announce their guesses so that the others can hear them. After all players have guessed, the coin is revealed and all players who guessed correctly earn one point; incorrect guesses earn no points. The game continues until a player reaches n points and is declared the winner. For n = 5 and real questions instead of flipped coins, this game was first aired on an MTV show in 1995. We model the decision-making choices for players B and C. We determine Nash equilibria of the game and show that players B and C can always select the equilibrium with maximal payoffs for both of them. We also study the long-term behavior of the game and show that for large n, both B and C will win with probability almost 1/2.