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Chapter 4: Problem 6

There are 2 coins in a bin. When one of them is flipped it lands on heads with probability 6 , and when the other is flipped it lands on heads with probability .3. One of these coins is to be randomly chosen and then flipped. Without knowing which coin is chosen, you can bet any amount up to 10 dollars and you then either win that amount if the coin comes up heads or lose it if it comes up tails. Suppose, however, that an insider is willing to sell you, for an amount \(C\), the information as to which coin was selected. What is your expected payoff if you buy this information? Note that if you buy it and then bet \(x\), then you will end up either winning \(x-C\) or \(-\dot{x}-C\) (that is, losing \(x+C\) in the latter case). Also, for what values of \(C\) does it pay to purchase the information?

Short Answer

Expert verified
The expected payoff with insider information is given by \(0.2 - 0.1C\). The expected profit without information is $0.20. It is not worth buying the insider information for any values of C, as the negative value of C would not make any sense in this context.

Step by step solution

01

To calculate the optimal bet without insider information, we know that for one coin the probability of heads is 0.6, and for the other coin the probability of heads is 0.3. In order to maximize payoff, we can use the Kelly Criterion, which is given by the formula \(f = \frac{bp - q}{b}\), where \(f\) is the fraction of the current bankroll to wager, \(b\) is the odds received on the wager, \(p\) is the probability of winning, and \(q\) is the probability of losing. Here, since the odds are 1:1 (either win the amount bet or lose the amount bet), both \(b\) and \(q\) will be 1. For the first coin with probability of heads 0.6: \(f = \frac{1(0.6) - (1-0.6)}{1} = \frac{0.2}{1} = 0.2\) For the second coin with probability of heads 0.3: \(f = \frac{1(0.3) - (1-0.3)}{1} = \frac{-0.4}{1} = -0.4\) Since we cannot bet a negative amount, the optimal bet for the second coin without insider information is 0. Thus, the optimal bet for the first coin without insider information is 0.2 of our bankroll, which in this case is 10 dollars. #Step 2: Calculate optimal bet for each coin with insider information#

Now that we have insider information, we know which coin will be flipped. So we can directly calculate the optimal bet for each coin. For the first coin, it remains 0.2 of our bankroll (0.2 * \(10 = \)2), as calculated in Step 1. For the second coin, since we know it has a 0.3 probability of getting heads, we can estimate the optimal bet using the Kelly Criterion as $0, because maximising with 0.3 probability of heads returns a negative value as calculated in Step 1. #Step 3: Calculate expected payoff with insider information#
02

The expected payoff for the first coin can be calculated as follows: \((0.6)(\$2 - C) - (1-0.6)(\$2 + C)\) The expected payoff for the second coin with insider information is \$0, as calculated in Step 2. So the overall expected payoff with the insider information would be: \[\frac{1}{2}((0.6)(\$2 - C) - (1-0.6)(\$2 +C)) + \frac{1}{2}(0) = 0.3(2 - C) - 0.2(2 + C)\] Simplifying: \[0.6 - 0.3C - 0.4 + 0.2C = 0.2 - 0.1C\] #Step 4: Determine the values of C for which it's worth buying the information#

First, let's calculate the expected profit without information: \[\frac{1}{2}(0.6(\$2) - (1 - 0.6)(\$2)) + \frac{1}{2}(0) = 0.3(\$2) - 0.2(\$2) = 0.1(\$2) = \$0.20\] Now, in order for the expected profit with the information (\(0.2 - 0.1C\)) to be greater than the expected profit without the information (\(0.20\)), the following inequality must hold true: \[0.2 - 0.1C > 0.20\] Solving for C: \[-0.1C > 0\] \[C < 0\] Since the value of C cannot be negative, it's not worth buying the insider information.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability
Understanding probability is crucial when dealing with scenarios that involve risk and uncertainty, such as flipping a coin or betting on a game. Probability is a way of quantifying the chances of a particular event occurring. It's expressed as a number between 0 and 1, with 0 indicating the event cannot occur, and 1 indicating certainty that the event will occur.

For example, if a fair coin is flipped, it has a probability of 0.5 (or 50%) to land on heads, and the same probability to land on tails. When we have a situation like the one in our exercise, with different probabilities for two coins (0.6 and 0.3 for landing on heads), we have to assess the risk and the potential reward for each coin separately before making a bet.
Kelly Criterion
The Kelly Criterion is a mathematical formula used to determine the optimal size of a series of bets. Created by John L. Kelly Jr. in 1956, it calculates the percentage of one's bankroll to wager on a bet with a positive expected value to maximize the logarithm of wealth. The principle is to balance the growth against the risk of losses.

In the formula \(f = \frac{bp - q}{b}\), \(f\) represents the fraction of the bankroll to bet, \(b\) stands for the odds, \(p\) is the probability of winning, and \(q\) is the probability of losing. Applying this to our exercise, we find that with a 0.6 probability of winning, the optimal betting fraction for the first coin is 0.2, or 20% of the bankroll. For the second coin with a 0.3 chance, the Kelly Criterion advises not betting, as a negative fraction would not be viable.
Optimal Betting Strategy
An optimal betting strategy aims to increase the player's bankroll while controlling risk, usually via the Kelly Criterion. This strategy takes into account the player's available funds, the odds of winning, and the payouts to determine the best amount to wager. Without insider information, one should only bet a fraction of their bankroll on positive expectancy outcomes.

In reference to our exercise, without knowing which coin will be flipped, we balance the potential profit against the risk and wager only the amount suggested by the Kelly Criterion for the first coin (20% of the bankroll), while opting not to bet on the second coin due to its negative expected value. When insider information is available, we reassess our strategy by betting according to the chances of success for each specific coin.
Expected Value
Expected value (EV) is a predicted value of a variable, calculated as the sum of all possible values each multiplied by the probability of its occurrence. It provides a measure of the center of a probability distribution and in gambling, it represents the average amount one can expect to win or lose per bet if the bet were repeated many times.

In the context of our exercise, by determining the EV for each scenario, we can compare the gains from bets placed without insider information to those placed with the additional cost of insider information. The EV helps us decide whether purchasing insider information adds any real value to our betting strategy. In this case, unless the cost is zero or negative (which isn't possible in reality), buying the insider information does not offer a positive expected return, and hence doesn't justify the expenses.

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Most popular questions from this chapter

In the game of Two-Finger Morra, 2 players show 1 or 2 fingers and simultaneously guess the number of fingers their opponent will show. If only one of the players guesses correctly, he wins an amount (in dollars) equal to the sum of the fingers shown by him and his opponent. If both players guess. correctly or if neither guesses correctly, then no money is exchanged. Consider a specified player and denote by \(X\) the amount of money he wins in a single game of two-finger Morra. (a) If each player acts independently of the other, and if each player makes his choice of the number of fingers he will hold up and the number he will guess that his opponent will hold up in such a way that each of the 4 possibilities is equally likely, what are the possible values of \(X\) and what are their associated probabilities? (b) Suppose that each player acts independently of the other. If each player decides to hold up the same number of fingers that he guesses his opponent will hold up, and if each player is equally likely to hold up \(\mathrm{I}\) or 2 fingers, what are the possible values of \(X\) and their associated probabilities?

From a set of \(n\) randomly chosen people let \(E_{i j}\) denote the event that persons \(i\) and \(j\) have the same birthday. Assume that each person is equally likely to have any of the 365 days of the year as his or her birthday. Find (a) \(P\left(E_{3,4} \mid E_{1,2}\right)\); (b) \(P\left(E_{1,3} \mid E_{1,2}\right) ;\) (c) \(P\left(E_{2,3} \mid E_{1,2} \cap E_{1,3}\right)\). What can you conclude from the above about the independence of the \(\left(\begin{array}{l}n \\ 2\end{array}\right)\) events \(E_{i j} ?\)

Suppose that 10 percent of the chips- produced by a computer hardware manufacturer are defective. If we order 100 such chips, will the number of defective ones we receive be a binomial random variable?

To determine whether or not they have a certain disease, 100 people are to have their blood tested. However, rather than testing each individual separately, it has been decided first to group the people in groups of \(10 .\) The blood samples of the 10 people in each group will be pooled and anayzed together. If the test is negative, one test will suffice for the 10 people; whereas, if the test is positive each of the 10 people will also be individually tested and, in all, 11 tests will be made on this group. Assume the probability that a person has the disease is .1 for all people, independently of each other, and compute the expected number of tests necessary for each group. (Note that we are assuming that the pooled test will be positive if at least one person in the pool has the disease.)

An urn initially contains one red and one blue ball. At each stage a ball is randomly chosen and then replaced along with another of the same color. Let \(X\) denote the selection number of the first chosen ball that is blue. For instance, if the first selection is red and the second blue, then \(X\) is equal to 2 . (a) Find \(P\\{X>i\\}, i \geq 1\). (b) Show that with probability 1 , a blue ball is eventually chosen. (That is, show that \(P\\{X<\infty\\}=1 .\) ) (c) Find \(E[X]\).
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