A knockout tournament involving n competitors, numbered 1 through 11, starts by randomly choosing two oif the competitors to play a game, with the...

A knockout tournament involving n competitors, numbered 1 through n,

starts by randomly choosing two of the competitors to play a game, with

the loser of the game departing the tournament and the winner getting to

play another game against a randomly chosen remaining competitor. This

continues through n − 1 games, and the player who wins the final game is

declared the winner of the tournament. Whenever players i and j play against

each other, suppose that i wins with probability Pi, j , where Pi, j , i 

= j,

are specified probabilities such that Pi, j + Pj,i = 1. Let Wi denote the

probability that i is the winner of the tournament. A simulation study has

been developed to estimate the probabilities W1, . . . ,Wn. Each simulation

run begins by generating a random permutation of 1, . . . , n. If the random

permutation is I1, . . . , In, then contestants I1 and I2 play the first game, with

the winner being I1 if a generated random number is less than PI1,I2 , and

being I2 otherwise. The winner of the first game then plays I3, with the winner

of that game decided by the value of another random number, and so on. If

J is the winner in a simulation run, then the estimates of Wi from that run

are 0 for all i 

= J, and 1 for i = J.

(a) Explain how conditional expectation can be used to improve the

estimator ofWi . Hint: Condition on the permutation and whatever other

information is needed to be able to determine the conditional probability

that i is the winner of the tournament.

(b) Explain how post-stratification, relating to the random permutation, can

be employed to further improve the estimator of Wi .

29. A knockout tournament involving n competitors, numbered 1 through 11,starts by randomly choosing two oif the competitors to play a game, withthe loser of the game departing the tournament and the winner getting toplay another game against a randomly chosen remaining competitor. Thiscontinues through It — 1 games, and the player who wins the final game isdeclared the winner of the tournament. Whenever players :5 and j play againsteach other, suppose that i wins with probability PM, where PLPI.’ 75 j, are specified probabilities such that PM" + PL,- 2 1. Let W,- denote the probability that i is the winner of the tournament. A simulation study hasbeen developed to estimate the probabilities W1, . . . , W”. Each simulationrun begins by generating a random permutation of l, . . . , u. If the randompermutation is I], . . . , I”, then contestants 11 and 12 play the first game, withthe winner being 11 if a generated random number is less than P1121? andbeing 12 otherwise. The winner of the first game then plays 13 , with the winnerof that game decided by the value of another random number, and so on. If J is the winner in a simulation run, then the estimates of W,- from that runareOforalli 75 J, and l fori = J. (a) Explain how conditional expectation can be used to improve theestimator of W,- . Hint: Condition on the permutation and whatever otherinformation is needed to be able to determine the conditional probabilitythat i is the winner of the tournament. (b) Explain how post—stratification, relating to the random permutation, canbe employed to further improve the estimator of W,-.