Certain Equivalent

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In this workshop, we will gain more familiarity and experience with the Certain Equivalent Trick for determining utility valuations of outcomes. We will also practice the art of thinking in terms of certain equivalents.

Instructional Material [15 minutes]

Watch the videos above during the workshop session, if you didn't watch them beforehand.

Utility In Money [20 minutes]

Calculate your utility-in-money curve. Do not share the results with your cohort unless you want to. Go to the spreadsheet Utility in Money, make a copy, and fill it out as follows:

  1. Fill in the blue cells manually to get a sense of your certain equivalent values for a variety of deals.
  2. Then choose a value for the r coefficient (in the red cell) that you feel comfortable with and that leads to numbers you feel are reasonable in the calculated green cells.

Note that this equation isn't magic, and there's no particular reason to believe that your utility in money strictly follows a logarithmic curve. But it is generally the shape that is most useful for representing different risk-attitudes toward money, and it is easy to calculate. Small positive values of r indicate moderate risk aversion, which is the typical attitude for humans.

Use this equation for the subsequent questions if you need it.

U(x)=x1r1r U(x) = \frac{x^{1-r}}{1-r}

Note that the price of equivalence for a deal should also be thought of as the price of ambivalence. You shouldn't care whether you're trading, say, $200 in cash for a 50% shot at $500, or a 50% shot at $500 for $200; whichever side of the deal you're on, you're ambivalent at that price, or the given odds. Your sale price would also be your purchase price if you were in the opposite position.

Thought Experiments [50 minutes]

The following list of exercises involves thinking in terms of certain equivalents. In each example you will be offered a certainty of X in exchange for a probability of something greater than X, or vice versa. These exercises will get your mind limbered up and thinking about how to treat probability and utility. Use your previously-calculated utility in money curve if you feel confident with the calculations, or simply pretend you have linear utility in money. Building intuition is the goal here.

If you feel that the question is ambiguous, make a reasonable assumption. Answer keys are only provided for some problems to help guide your progress.

Question 1: the Lottery Ticket

You have in your hand a lottery ticket for a $1,000,000 jackpot. Your numbers are 3 8 19 25 65 93. You are at a bar, listening to the broadcast as the lottery officials read out the winning numbers. So far they have read out "3 8 19 25 65…" As of that instant, you have a 1 in 100 chance of winning the jackpot. The person on the stool next to yours abruptly leans over and quickly offers to buy your lottery ticket for $5,000 on the spot, before the last number is read.

  1. Do you accept the deal?
  2. At what price would you be ambivalent between holding onto your ticket and taking the money?

Question 2: the Insurance Deductible

You have been given the option of a policy that costs a $4,000 yearly premium for full insurance coverage, or a policy with a $2,000 yearly premium and a $5,000 annual deductible. You are extremely bad at driving so the odds of an accident in the next year are 50%.

  1. Which policy do you prefer?
  2. At what odds-of-an-accident would you be ambivalent between the two options?

Question 3: Car Repairs

Your car has been making ominous noises, so you take it to the mechanic, who tells you that the transmission probably needs to be replaced, which will cost $2,000. However, it's possible that the problem could be resolved by simply flushing and replacing the transmission fluid for only $300. Bear in mind that if the fluid flush fails, you'll need to pay for the full transmission replacement.

  1. At what probability of the transmission fluid flush actually solving the problem would you be ambivalent between the flush option and the full replacement option? This is a tricky one; check the solution if you get stuck.

Question 4: Job Offer

You receive two job offers. One position has a salary of $80,000 while the second job offers a $65,000 salary with a one-time hiring bonus of $30,000.

  1. Which would you prefer?
  2. At what size of hiring bonus would you be ambivalent between the two options?

Question 5: Entrepreneurship

You have been running a small business for a few years. Your annual revenue is $300,000. A competitor offers to buy your business for $1,000,000.

  1. Do you take the deal?
  2. At what price would you sell this business?

Question 6: Medical Treatment

You can choose between two treatments for a medical condition. The first treatment has a 90% chance of partial recovery. The second treatment has a 60% chance of complete recovery.

  1. I haven't given you enough information to choose one or the other. What other sorts of questions would you want to ask about this problem to gain more clarity and make a better decision?

Question 7: Scholarship Opportunity

You have been offered a partial scholarship for a prestigious graduate program that covers 50% of the tuition. The remaining tuition cost would be $20,000 per year. You have the option to defer enrollment for a year to apply for additional scholarships, with a 40% chance of securing a full scholarship, but a 60% chance of not receiving any additional funding.

  1. Do you accept the partial scholarship and start the program now, or defer and try for a full scholarship?
  2. At what probability of securing a full scholarship would you be ambivalent between the two options?

Wrap Up [10 minutes]

If there's any time left, suggest situations in your own life that could be considered as choices between "deals", which could be understood in terms of their certain equivalents.

Community Notes

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