Tuesday, March 08, 2011

Why big organizations don’t take risks proportionate to their size (a theoretical argument)

TLDR version: People optimize E(Log(x)). Big organizations should optimize E(Log(Σ x)), but because they are composed of many people, they optimize ΣE(log(x)).

Logarithmic Utility Curves

Suppose someone offers you a 50% chance of a $1M prize, or a 10% chance of a $10M prize. Most people who don’t already have a lot of money would take the first, more certain option, even though the expected value of the second is double that of the first. ($0.5M vs $1M). $1M would improve my life a lot. $10M would also improve my life a lot, but not so much more than $1M that I’m willing to risk getting nothing. Economists call this “risk aversion.” and model it using a “utility curve.” Though quite abstract, it corresponds pretty well to intuition. The idea is that $10M isn’t “worth” 10x as much to me as $1M, so I have to account for that before I take the expected value. The most common utility curve used in simple models is logarithmic, both because it’s mathematically simple, and close enough to reality for most purposes. Intuitively, no matter how much money I make, I’m willing to put in a linear amount of additional effort in order to raise my income level by 10%, so you end up with logarithmic utility.

To compare these two offers, before we take the expected value, we’ll take the log of the outcomes.
log(1M) ~ 6,
log(10M) ~ 7,
0.5 * 6 > 0.1 * 7, so we take the first offer as observed.

Rather than optimizing E(x), we optimize E(log(x)) and get behavior somewhat like what a real person would do.


It makes sense that an organization should have a utility curve looking like a log. Going out of business (log 0) is -infinity as observed, and companies care about percentage improvements rather than absolute improvements just like individuals do. I apologize for not having better arguments that this is the way organizations should behave. This is a blog post, not a research paper. :)

This means that if the output of each employee is x, the outcome for the company as a whole is Σ x, and the company should be optimizing E(log(Σ x)).

How do they actually behave? The company will behave in the way that the aggregate of its individuals behave. If individuals are rewarded in proportion to their personal outcomes rather than the company’s outcomes, they’ll behave by optimizing their personal utility function. As a result, the company as a whole will optimize Σ E(log(x)).

Big organizations apply the risk-aversion of an individual to losses that are teeny relative to the size of the organization, because people care more about their individual risk than the risk to the company. Ideally, a company should have many people working on huge improvements with a small probability of success. In reality, it's almost impossible to structure incentives so that doing things with a small chance of success is a good personal strategy. It’s hard to reward competence and not outcomes. If you reward outcomes, the company will be risk averse. Having a few engineers work for years on something that doesn’t finally work is a teeny risk to the company, but potentially a huge risk to the careers of those engineers.