Three smart people A, B, and C are having a trio duel. The goal of each person is to survive and meanwhile to kill as many rivals as possible at the end of the duel. The duel consists of multiple rounds at each of which the three rivals shoot at one another at the same time and each person has only one bullet. Suppose that in all cases the probabilities of killing a person by one shot by A, B, and C are always 0.8, 0.7, and 0.5, respectively.

(a) WhatarethebeststrategiesforA,B,andC,respectively,forthefirstround?

(b) FindthecorrespondingprobabilitiesforA,B,andC,respectively,tosurvivethefirstround.

(c) Assumingthatapersonhasnocluewhetheranyofhisrivalswasshotornotinthefirstround,

what are the best strategies for A, B, and C, respectively, for the second round?

(d) Find the corresponding probabilities for A, B, and C, respectively, to survive the second

round.

(e) Assuming that a person has no clue whether any of his rivals was shot or not in the previous

rounds, what are the best strategies for A, B, and C, respectively, for the third round?

(f) FindthecorrespondingprobabilitiesforA,B,andC,respectively,tosurvivethethirdround.

(g) Whohasthelargestprobabilitytosurvive?Doesyourresultsmakesense?Discusstheimpli-

cation of your results.