Suppose that a decision-maker's preferences can be represented by the expected value of a utility function u. Find another utility function u' whose expected value represents the decision-maker's preferences and assigns a payoff of 1 to the best outcome and a payoff of 0 to the worst outcome.
Hint: Try to use this fact - Suppose there are at least 3 possible outcomes. The expected values of the Bernoulli payoff functions u and v represent the same preferences over lotteries (and certain outcomes) if and only if there exist numbers d and c, with c>0 such that v(x) = d + cu(x)
To find the utility function u' that represents the decision-maker's preferences and assigns a payoff of 1 to the best outcome and a payoff of 0 to the worst outcome, we can follow these steps:
Step 1: Define the outcomes
Identify the set of possible outcomes. Let's denote the best outcome as "B," the worst outcome as "W," and any other intermediate outcome as "X."
Step 2: Assign payoffs
Assign payoffs to each outcome. In this case, we want the best outcome (B) to have a payoff of 1 and the worst outcome (W) to have a payoff of 0. Assign any intermediate outcomes (X) with values between 0 and 1, depending on their relative desirability compared to B and W.
Step 3: Apply the hint
Apply the hint provided: "Suppose there are at least 3 possible outcomes. The expected values of the Bernoulli payoff functions u and v represent the same preferences over lotteries (and certain outcomes) if and only if there exist numbers d and c, with c > 0, such that v(x) = d + cu(x)."
In this case, we have the payoff function u, which represents the decision-maker's preferences. We want to find a new utility function u' that satisfies the given conditions. Let's set v(x) as the new utility function u'.
We know that v(B) = 1, v(W) = 0, and v(X) should be between 0 and 1.
By using the equation v(x) = d + cu(x), we can solve for d and c.
Since v(B) = 1, we have 1 = d + c * u(B).
Since v(W) = 0, we have 0 = d + c * u(W).
Given that c > 0, we can solve these two equations simultaneously to find the values of d and c.
Step 4: Substitute back into the equation
Once we have determined the values of d and c, substitute them back into the equation v(x) = d + cu(x) to obtain the new utility function u'.
Step 5: Verify the properties
Verify that the expected values of the utility functions u and u' represent the same preferences over lotteries by comparing their respective expected values for a set of lotteries.
Following these steps should help you find the utility function u' that represents the decision-maker's preferences and assigns a payoff of 1 to the best outcome and a payoff of 0 to the worst outcome.