The expected utility of betting when the probability of winning is p is given by below formula.
The expected utility of not placing a bet is:
The utility function is a logarithmic function. The expected utility of betting should be equal to expected utility of not betting.
The above equation yields the value of p as.525, which is the minimum probability that the person placing the bet must believe that he would win.
It is required to be shown that a person whose utility curve is convex would prefer a fair gamble to certain income and may sometimes even prefer an unfair gamble.
A convex utility curve is shown in Figure 1.
If a gamble gives wealth w 1 and wealth w 4 with equal probability of 0.50, then the expected wealth under this gamble would be w 2 , as represented by the mid-point C of the line segment AE. Point C also shows that expected utility of this gamble is u 4.
If the person gets the wealth of w 2 with certainty, his utility would be u 2 , as represented by point D. Since this utility is lower than his expected utility under the gamble, he will prefer the gamble.
This shows that the person will prefer fair gamble to certain income.
The figure shows another point B. This point shows that if the person gets a wealth of w 3 with certainty, he gets a utility of u 3. Although wealth w 3 is higher than his expected wealth under the gamble, his utility from this wealth is still lower than his expected utility under the gamble. Therefore, the person may even prefer the gamble over a certain income that is even higher than the expected wealth under the gamble.
This shows that this person may even prefer unfair gambles.
Preferring unfair gambles is irrational behavior, but most real-life people do not exhibit such behavior. Most real-life people are risk averse. The case considered in this question is of a risk-loving person.
a) In strategy 1, individual transports all eggs at once and there is a 50% chance that all eggs would break during a trip. So, there is a 50% chance that 12 eggs would be safely transported and a 50% chance that zero eggs would be left.
In strategy 2, individual transports 6 eggs at once and there is 50% chance that all eggs would break during a trip. In trip 1, there is a 50% chance that 6 eggs would be safely transported and a 50% chance that zero eggs would be left. Similar logic applies to trip 2. Overall there is a 25% chance that all eggs would be safely transported and 25% chance that none would be.
b) The graph given below in Figure 1 shows the expected utility of different strategies. It is evident that the strategy 2 is preferable as its utility is higher.
c) The higher the number of trips, the higher the expected utility of that strategy. If one trip incurs a cost (c), then the possibility that more trips would give higher utility would be reduced. For example, if the second strategy requires one more trip like in the above example than the individual would prefer the second strategy only when: