3. Suppose we play the following game. I give you $100 for your initial bankroll. At each time n, you decide how much of your current wealth to bet. You cannot borrow money. You can only play with the money I gave you in the beginning or any money that you have won so far. The game is simple. At each time n ≥ 1, you decide the amount to bet. I will roll a fair die. If the die comes up 1,2,3,..., or 5, you win; if the die comes up 6, then you lose. IOW, if you bet $10 on the first roll, you will either have $90 or $110 after the first roll. (a) Suppose you wish to maximize your profit on the first roll. How much should you bet? (Most of you will get this wrong.) (b) What is the expected profit on the first roll if your bet is b with 0 ≤ bs 100? (c) Suppose you wish to maximize your expected profit on the first roll. How much should you bet? (d) Suppose you wish to maximize your expected profit betting on the nth roll. How much of your current wealth do you bet? (e) Let X, be your wealth after n plays; thus, Xo = $100. What is your ex- pected wealth after n plays if you always bet to maximize your expected profit? (f) Assuming you maximize your expected profit each time, your expected wealth converges to what as n → ∞o. (Sounds good, no?) (g) What happens to your wealth as n → ∞o? This problem and the insurance problem suggest that looking at the mean when making financial decisions can lead you astray when losses or gains may be large relative to your current financial situation. When losses and 2 gains are relatively small, the mean is an excellent way of deciding whether to play and how much to bet/invest. In the solution, I will try to remember to give information about what's a good way to play the game described in this problem.
3. Suppose we play the following game. I give you $100 for your initial bankroll. At each time n, you decide how much of your current wealth to bet. You cannot borrow money. You can only play with the money I gave you in the beginning or any money that you have won so far. The game is simple. At each time n ≥ 1, you decide the amount to bet. I will roll a fair die. If the die comes up 1,2,3,..., or 5, you win; if the die comes up 6, then you lose. IOW, if you bet $10 on the first roll, you will either have $90 or $110 after the first roll. (a) Suppose you wish to maximize your profit on the first roll. How much should you bet? (Most of you will get this wrong.) (b) What is the expected profit on the first roll if your bet is b with 0 ≤ bs 100? (c) Suppose you wish to maximize your expected profit on the first roll. How much should you bet? (d) Suppose you wish to maximize your expected profit betting on the nth roll. How much of your current wealth do you bet? (e) Let X, be your wealth after n plays; thus, Xo = $100. What is your ex- pected wealth after n plays if you always bet to maximize your expected profit? (f) Assuming you maximize your expected profit each time, your expected wealth converges to what as n → ∞o. (Sounds good, no?) (g) What happens to your wealth as n → ∞o? This problem and the insurance problem suggest that looking at the mean when making financial decisions can lead you astray when losses or gains may be large relative to your current financial situation. When losses and 2 gains are relatively small, the mean is an excellent way of deciding whether to play and how much to bet/invest. In the solution, I will try to remember to give information about what's a good way to play the game described in this problem.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Question
Transcribed Image Text:3. Suppose we play the following game. I give you $100 for your initial bankroll.
At each time n, you decide how much of your current wealth to bet. You
cannot borrow money. You can only play with the money I gave you in the
beginning or any money that you have won so far. The game is simple. At
each time n ≥ 1, you decide the amount to bet. I will roll a fair die. If the
die comes up 1,2,3,..., or 5, you win; if the die comes up 6, then you lose.
IOW, if you bet $10 on the first roll, you will either have $90 or $110 after
the first roll.
(a) Suppose you wish to maximize your profit on the first roll. How much
should you bet? (Most of you will get this wrong.)
(b) What is the expected profit on the first roll if your bet is b with 0 ≤ b ≤
100?
(c) Suppose you wish to maximize your expected profit on the first roll.
How much should you bet?
(d) Suppose you wish to maximize your expected profit betting on the nth
roll. How much of your current wealth do you bet?
(e) Let X₂, be your wealth after n plays; thus, Xo = $100. What is your ex-
pected wealth after n plays if you always bet to maximize your expected
profit?
(f) Assuming you maximize your expected profit each time, your expected
wealth converges to what as n→ ∞o. (Sounds good, no?)
(g) What happens to your wealth as n → ∞o?
This problem and the insurance problem suggest that looking at the mean
when making financial decisions can lead you astray when losses or gains
may be large relative to your current financial situation. When losses and
2
gains are relatively small, the mean is an excellent way of deciding whether
to play and how much to bet/invest. In the solution, I will try to remember to
give information about what's a good way to play the game described in this
problem.
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