Three people are playing paintball and are engaged in a three-way duel. The play

Question

Three people are playing paintball and are engaged in a three-way duel. The players are Juan, Emma,
and Madison. There are two rounds. In the first round, Juan goes first. If Emma was not hit by Juan,
then she gets to have a shot. If Madison is not hit by Juan or Emma, then Madison gets to have a shot.
After the first round, anyone not hit by a paintball goes to the second round.
In the second round the order is as follows. If Juan and Emma are the only ones left, then Juan goes
first. If Juan and Madison are the only ones left, then Juan goes first and if Emma and Madison are the
only ones left then Emma goes first.
Juan is a poor shot, with only a 30 percent change of hitting a person at whom he aims. Emma is a much
better shot, achieving 80 percent accuracy. Madison is a perfect shot, never missing. Assume that each
party engages in optimal behavior-which is to maximize the probability of surviving the two rounds.
(a) If both Emma and Madison are in the game after Juans first shot who will Emma aim at?
(b) If both Emma and Madison are in the game after Emmas first shot who will Madison aim at?
(c) In Round One what is Juan's optimal strategy? Explain why that strategy makes sense.
(d) T/F. Explain. It is impossible for all three players to make it to the second round.
(e) Assume that this game is played 100,000 times. Calculate the number of times that Juan has not
been hit after two rounds, the number of times that Emma has not been hit after two rounds,
and the number of times that Madison has not been hit after two rounds. Show all your work.

Explanation / Answer

Emma will aim at Madison. Madison will aim at Emma Juans optimal strategy is to aim Madison who if eliminated will reduce his probability of loosing. True, because Madison will hit for sure.

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