3-person duel
A particular basketball shootout game consists of a number of duels. In each duel, one player is the challenger. The challenger chooses another player to challenge, and then gets one chance to shoot the hoop. If the player makes the shot, the playing being challenged is out. If the player does not make the shot, or if the player chooses to skip his turn, then the game continues with the next duel. A player wins when only that player remains.
One day, this game is played by three players: A, B, and C. Their skill levels vary considerably: player A makes every shot, player B has a 50% chance of making a shot, and player C has a 30% chance of making a shot. Because of the difference in skill levels, they decide to let C begin, then B, then A, and so on (skipping any player who is out of the game) until there is a winner. If everyone plays to win, what strategy should each player follow?
[For this follow-up question, it will be helpful to have a paper and pen–not because the calculations are hard, but because it helps in remembering the numbers.]
If A, B, and C follow their winning strategies (as determined above), which player has the highest chance of winning the game?
To determine the optimal strategy for each player, we can use backwards induction. We start with the assumption that each player will play to win, and work backwards to determine the best move for each player given the possible moves of the other players.
When it is C’s turn, she has a 30% chance of eliminating one of the other players. If C chooses to challenge A, then A will always win the duel and C will be eliminated. If C chooses to challenge B, then B has a 50% chance of missing the shot and being eliminated, and a 50% chance of making the shot and eliminating C. Therefore, C should always challenge B.
When it is B’s turn, he has a 50% chance of eliminating one of the other players. If C is still in the game, then B should challenge C, since C is the weakest player. If C is already out, then B should challenge A, since A is the strongest player.
When it is A’s turn, he will always eliminate one of the other players, since he makes every shot. Therefore, A should choose to challenge the player who has the highest chance of winning, which is B.
Using this strategy, the game will proceed as follows:
- C challenges B: 30% chance of eliminating B.
- B challenges A: 50% chance of eliminating A.
- C challenges B: 30% chance of eliminating B.
- B challenges C: 50% chance of eliminating C.
- B challenges A: 50% chance of eliminating A.
- B wins the game.
Therefore, following the optimal strategies, B has the highest chance of winning the game, with a 50% chance of winning. A has a 100% chance of winning any duel he participates in, but he will only participate in one duel. C has a 30% chance of winning the first duel, but is unlikely to win subsequent duels.
Your Answer
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