Bias-Variance Tradeoff in Everything

The bias-variance tradeoff says that, when choosing which machine learning algorithm to use, there is a tradeoff between two desirable properties: 1) Become more accurate as you add more data, and 2) Produce similar models when you feed the algorithm slightly different data points (i.e., don't overfit to the particular samples that you've given the model). For example, linear regression falls on the "high bias and low variance" side of the tradeoff: It cannot capture nonlinear relationships, but it is very robust to the particular data points that you feed it. In contrast, many modern deep neural networks fall on the "low bias and high variance" side: They are very flexible in the types of relationships that they can capture and hence tend to improve with more data, but they are prone to overfitting.

Since modern neural networks overfit so easily, lots of tricks are used to reduce the variance (and thereby tend to increase the bias): Weight decay explicitly encourages small weights, dropout reduces the ability of the network to overfit to some small subset of the inputs, and better architecture design (e.g., using a CNN instead of a fully-connected network) creates an "inductive bias" against certain types of functions.

This core idea of "increasing average cost in exchange for a decrease in the variance" can be useful for understanding lots of other behaviors that are counterintuitive on the surface:

Insurance: In order for an insurance company to be profitable, they need to make money on average. If the insurance company is making money on average, then the customer must be losing money on average: Why do we still find it worthwhile to accept a losing deal? What gives?

We accept such a deal because it decreases the variance of our outcomes; instead of risking a large negative outcome (e.g., paying entirely for a car crash), we can hedge our risk with a fixed monthly payment.

Bitcoin Mining Pools: Transactions on the blockchain are validated by a large pool of computers, each of which works to solve a hard computational problem (in particular, finding an integer that, when concatenated with the new state of the blockchain, has a hash that contains a certain number of leading zeroes). When an individual computer in the pool solves the problem, they are rewarded with Bitcoin.

Oftentimes, individual miners will aggregate their compute resources together, splitting the mined Bitcoin in proportion to the amount of compute that each person contributes. Of course, the mining pool infrastructure (to tally compute effort, do the splitting of rewards, etc.) costs money, so the pool organizer takes a percentage of all revenue from the pool. Once again, if the pool organizer makes money on average, then the miners must be losing money on average. We again have the bias-variance tradeoff: Miners exchange a percentage of their average profits for a more consistent stream of revenue.

Futures: A future is a financial contract that obligates a buyer and a seller to transact at a future date, for some agreed-upon good at an agreed-upon price. (In contrast to more-commonly-known options, where one party merely has the option of executing the deal).

Contracting for future transactions allows for producers to hedge against risk: If the value of orange juice concentrate is expected to be $1.30 per pound in six months, orange growers can guarantee this revenue by purchasing a futures contract at $1.25 per pound. This loses the orange growers money on average, but takes the variance to zero.

Startup Acquisitions: Most acquisitions fail. Quantitatively defining and estimating failure is difficult, but some studies suggest between 70 and 90 percent https://hbr.org/2011/03/the-big-idea-the-new-ma-playbook failure rates. Shouldn't businesses be more diligent about what companies they acquire? Isn't profit maximization the name of the game for corporations?

One explanation of this seemingly-irrational behavior of corporations is that the incentives of the executives are not perfectly aligned with the incentives of the corporation itself: CEOs can be praised for "reasonable" behavior that doesn't necessarily work out, and might be ridiculed for more erratic moves that would actually maximize profits (e.g., Zuckerberg acquiring Instagram without explicit discussion/approval from other board members).

However, it can also be rational for a corporation to pursue a losing bet, so long as it reduces variance: By purchasing a potentially-competitive startup, the corporation is paying a fee for reduced risk of total elimination.

Equity Premium: Over the past century, the S&P 500 has grown ~10% per year on average, whereas government bonds have returned ~5% per year. Since equities (i.e., stocks) have such a higher return than bonds, why would anyone invest in bonds? The answer is the "equity premium", the idea that since stocks have such a higher variance than bonds, risk-averse investors (retirees, pension funds, etc.) will prefer a reduction in variance in exchange for a lower return.

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Returning to the core idea behind these behaviors, why do there seem to be so many situations in which we do not maximize expected return? Fundamentally, all these risk-reduction strategies are rational due to the distinction between monetary utility and money itself.

The idea of "monetary utility" refers to how much real value we can get out of a dollar. Though it increases as we have more dollars (you are never sad about an additional dollar), the rate of increase will decrease (i.e., it is a concave function). We would be much happier if our yearly salary increased from $30k to $31k, versus from $500k to $501k, though both are increases of a thousand dollars. This is because we tend to spend our money on our most important purchases first (food, rent, etc.).

So, instead of maximizing our dollar amount, both individuals and companies are maximizing monetary utility. Of course, monetary utility is monotonic with respect to money, but the distinction between money and monetary utility matters when reasoning under uncertainty. Suppose you are offered either 1) A guaranteed 1000 dollars (which gives, say, 3 monetary utility points) or 2) A 50/50 chance of 10,000 dollars (4 utility points) or 10 dollars (1 utility point). Option 2 maximizes expected money, but option 1 maximizes expected monetary utility.

If we have "reduced variance for the price of a reduced mean" behavior with concave utility functions, what do we expect with convex utility functions? Increasing marginal utility is much less common than decreasing marginal utility, but we can oftentimes find increasing marginal utility in situations where we relieve unhappiness (rather than gaining more happiness): Debt, looking for a cure to a physical illness or negative emotional state, etc. For example, the difference between a small amount of chronic pain and no chronic pain provides much more utility than the difference between a large amount of chronic pain and slightly less pain.

In these convex utility situations, we actually expect "increased variance for the price of reduced mean" behavior. Seemingly-irrational behavior that makes the average outcome worse while increasing the variance of outcomes can surprisingly be utility-maximizing behavior.

In all, the shift in view from dollar amounts to monetary utility is a reminder that transactions are positive-sum: Everyone has a slightly different utility function, and mutually-beneficial transactions (and hence, progress of society in general) is only possible through these utility differences.