Special Topics 2 The Truel Three gunfighters meet for a truel, a three-person duel. Gunfighter A hits his target 50% of the time, gunfighter B hits 75% of the time, and gunfighter C hits 100% of the time The gunfighters take turns shooting in the order A, B, C, A, B, C, etc. Of course, a dead gunfighter misses his turn. The last one standing is the winner. What is A's best strategy? If A kills C, then B will probably kill A on the next shot. On the other hand, if a kills b then c will certainly kill a on the next shot. This does not look good. But there is a another possibility: A could intentionally miss, let B and C shoot it out, and then try to kill the winner Let's evaluate that strategy, assuming that B and c actually try to hit each other C WNS A kills c From the tree diagram, we have 1 Pr wins 8 12.5% Now let a be the probability that b eventually wins in the situation where C is dead and a has the next shot. This situation arises at two different points in our tree diagram. We can exploit that fact to obtain an equation expressing c in terms of itself 1311 Solving this equation, we find that r=3/7. The probability that b wins overall is4 Special Topics 2 The Truel Three gunfighters meet for a truel, a three-person duel. Gunfighter A hits his target 50% of the time, gunfighter B hits 75% of the time, and gunfighter C hits 100% of the time. The gunfighters take turns shooting in the order A, B, C, A, B, C, etc. Of course, a dead gunfighter misses his turn. The last one standing is the winner. What is A’s best strategy? If A kills C, then B will probably kill A on the next shot. On the other hand, if A kills B, then C will certainly kill A on the next shot. This does not look good. But there is a another possibility: A could intentionally miss, let B and C shoot it out, and then try to kill the winner! Let’s evaluate that strategy, assuming that B and C actually try to hit each other. x = prob. that B wins from this state A miss 1/2 1/2 B kills C 3/4 1/4 B miss C kills B 1 A miss 1/2 A kills C 1/2 C kills A 1 B miss 1/4 B kills A 3/4 A kills B . . . A WINS B WINS A WINS C WINS From the tree diagram, we have: Pr (C wins) = 1 4 · 1 · 1 2 · 1 = 1 8 = 12.5% Now let x be the probability that B eventually wins in the situation where C is dead and A has the next shot. This situation arises at two different points in our tree diagram. We can exploit that fact to obtain an equation expressing x in terms of itself: x = 1 2 · 3 4 + 1 2 · 1 4 · x Solving this equation, we find that x = 3/7. The probability that B wins overall is: