Each of m + 2 players pays 1 unit to a kitty in order to play the
Question:
Since each of the coin flips is equally likely to land on either heads or tails, m of the players have decided to make their predictions in a totally random fashion. Specifically, they will each flip one of their own fair coins n times and then use the result as their prediction. However, the final 2 of the players have formed a syndicate and will use the following strategy: One of them will make predictions in the same random fashion as the other m players, but the other one will then predict exactly the opposite of the first. That is, when the randomizing member of the syndicate predicts an H, the other member predicts a T. For instance, if the randomizing member of the syndicate predicts (H, H, T), then the other one predicts (T, T, H).
(a) Argue that exactly one of the syndicate members will have more than n/2 correct predictions. (Remember, n is odd.)
(b) Let X denote the number of the m nonsyndicate players that have more than n/2 correct predictions. What is the distribution of X?
(c) With X as defined in part (b), argue that
E[payoff to the syndicate] = (m + 2) × E[1/X + 1]
(d) Use part (c) of Problem 59 to conclude that
E[payoff to the syndicate] = 2(m + 2)/m + 1 × [1 – (1/2)m+1]
and explicitly compute this number when m = 1, 2, and 3. Because it can be shown that
2(m + 2)/m + 1 [1 – (1/2)m+1] > 2
it follows that the syndicate’s strategy always gives it a positive expected profit.
Distribution
The word "distribution" has several meanings in the financial world, most of them pertaining to the payment of assets from a fund, account, or individual security to an investor or beneficiary. Retirement account distributions are among the most...
Fantastic news! We've Found the answer you've been seeking!
Step by Step Answer:
Related Book For
Question Posted: