## Risk vs. Reward: Expectation Value of Utility

I’m **you must choose between two possible jobs, let’s call them "Big Company" and "Startup."** Big Company will pay you $100k per year. Startup can only pay you $50k per year, but with a 10% chance of paying you a $3 million bonus in 3 years. **Which one do you take?** I’ll present an analytical economic framework for making this decision which shows why **this decision is ultimately very personal**. For the purposes of this discussion, I’m going to ignore the time value of money — let’s pretend the appropriate discount rate is 0%. Let’s also make up some excuse why only the next 3 years matters like you’re determined to stop working at that point.

### Expectation Value of Money

A basic analysis looks at what your expected income will be over the next 3 years. Big company will clearly pay $300k. Startup will pay $150k in salary. A 10% chance of $3m is worth $300k in terms of its "expected value" following standard statistical methods. So by this logic, startup is a better bet at $450k over the 3 years. This analysis is correct and totally appropriate for how a large institution would analyze the trade-off. But it has a problem when applied to individuals: it ignores the differences between utility and money.

### Utility != Money

**Different quantities of money have different relative values to people.** Economists express this in terms of "utility." Most people will find $20 about twice as useful as $10. Because you can do twice as much with $20 vs $10, we say that $20 has twice the "utility" of $10. My favorite econ professor refers to utility as "jollies." Utility is in arbitrary units, but here let’s say $1 is worth about 1 jolly, more or less. So far this should be totally intuitive.

Things get a little stranger when we’re talking about large quantities of money. For example, most people would not find $2 billion twice as useful as $1 billion. For me, getting a gift of one billion dollars or two billion dollars would have almost exactly the same effect on my life — either way I’d be rich beyond my dreams. The marginal utility of that second billion dollars is worth far less than 1 billion jollies — maybe only another million jollies over the first billion? So for me, the graph of utility vs. money becomes nearly flat at these extremely high dollar values.

The shape of this curve at "intermediate" dollar values varies a lot from person to person, as does what intermediate would mean. For example, consider two people — one who is working a minimum wage

job and the other who just made their first million dollars. A $1,000

bonus would be extremely useful to the minimum wage worker, and almost

ignored by the millionaire. But to the millionaire, the difference

between $400k and $800k is much more meaningful than it is to the

minimum wage worker. For them, that much money sounds more like a billion dollars would to me. **Most everybody has a range of money that is too small to matter to them
and a range of money which is so large that differences don’t really
matter either.** Here are some factors that can affect the shape of a person’s utility curve:

- How much disposable income they are used to having
- How much money they have saved
- What expectations they have about future earnings
- Career growth path
- Are they expecting inheritance?
- Pension plan or 401k?

- How much money would be needed to reach financial goals that can markedly affect their quality of life, like
- Buying a car or house
- Getting out of debt
- Putting children through college

As promised, figuring out the shape of this curve is extremely personal. I’ll describe a technique for doing so numerically at the end, and then I’ll duck. Here is a spreadsheet *{link fixed}* that unscientifically attempts to graph what a utility curve might look like for somebody who is financially comfortable, but not exactly wealthy — like somebody who might be considering such a career decision. And here’s the graph:

### Applying Utility to the job choice

You might think that the next step in the analysis is to figure out how many jollies you’d get for the $300k from Big Company and compare that to the $450k from Startup. Using the above data, this works out to about 649 kilojollies for Big Company and 907 killojollies for Startup, and Startup clearly wins. These values are the *utilities of the expectation values of money*, which sounds impressive, but really isn’t very useful. **For uncertain outcomes, you should figure out the utility value of each financial possibility before calculating the expectation value. **

Using the above utility curve, the correct analysis involves figuring out how many jollies you would get for each of these three possible outcomes:

- Work at Big Company ($300k = 649 kilojollies)
- Work at Startup with no bonus ($150k = 351 kilojollies)
- Work at Startup with big bonus ($3,150k = 3.25 megajollies)

**The appropriate comparison is between **649 kilojollies for Big Company and (0.90 * 351kj) + (0.10 * 3246kj) = 641 kilojollies for Startup, which is comparing * the expectation values of utility*. Expectation values are calculated by adding up the products of the probability of each event with the utility of that event. See how this is different from the utility of the expectation value of money? You apply the utility transformation before applying the probability weights. Because the utility transform is non-linear, the operations are not associative, and order matters. So, the important question is how valuable would that $3 million dollar bonus be to you, and is a shot at it worth the loss of guaranteed salary?

By this analysis, it’s not worth it for our hypothetical job seeker to take the chance. The 649kj from Big Company is slightly higher than the 641kj from Startup, although it’s really close. The closeness shouldn’t invalidate the result though. You might want to check the assumptions used to derive this, but ultimately **we have to make choices based on limited information**. We can understand this result intuitively because the sweet-spot on the money-utility graph is in the hundreds of thousands of dollars, not the millions. This is the order of magnitude of money that would change this person’s life in the most meaningful way.

### Tautological Conclusion

This kind of "what if" exercise is an accepted technique for calculating the shape of a person’s utility curve. Would you rather have $10 or a 10% chance at $100? Would you rather have $100 or a 10% chance at $1k? Maybe I’ll put together a spreadsheet that actually uses this technique to find an accurate utility curve. But you can see that this entire analysis boils down to a realization that **you must decide for yourself if the reward is worth the risk.**

Haha! ;^)

/me ducks

(Hopefully this discussion gives a framework for thinking about such decisions.)

Josh — If I were doing this for real, I would draw out a probability distribution for several possible payouts from the startup. Maybe there'd be a 1% chance of $10m and a 3% change of $3m and a 6% chance at $1m. But that would just clutter what I'm trying to express in this example.

Geoff — To clarify, I didn't say this was my own personal utility curve. It's just hypothetical. After all, I'm a full-time grad-student and not looking for a job right now.

Interesting post. How did you come up with 10% success rate at a startup? This seems to be the variable with the largest weight. 10% seems either too optimistic or too over confident depending on your point of view. I've seen enough startups go under to think that the true success rate is closer to 1%…

That said, I'm confident that my chance of success is higher than the average… just like everyone else. :)

Interesting!

This does help explain that gambling sense people have to at least *try* to get that big payout, even if the odds are set up such that the expected value (without converting to Jollies) seems significantly lower than the cost of paying to play — like the lottery.

Like you say, your personal curve of Jollies/$ sure makes a big difference!

But, I suspect your curve flattens off a bit too early for most people.

My personal curve would accelerate (more and more Jollies/$) and then wouldn't flatten off until I reach a $ value for which everything I can reasonably dream of buying is satisfied and I can live on the interest from the remaining money forever. After that value the curve levels off quickly. So, maybe a peak at about $2 million for me.

With the peak of Jollies/$ moved out to about the $2 million point, the decision in the big company vs. startup question with your pay numbers isn't nearly so close.

— Geoff