Sometimes, when purchasing some new item, we are offered an opportunity to buy a warranty. Is this a good idea? There are several interesting layers of rationality at play here, and it is fun to explore them, regardless of what conclusions we reach.

## Layer #1: Let's get the math out of the way

From a purely mathematical point of view, there are two concepts at play: expected value and variance.

### Expected Value

Typically, warranties have negative expected value. The expected value of a warranty, roughly, is (the cost of repair or replacement) times (the probability of needing that repair or replacement) minus (the cost of the warranty).

This number is usually negative. Meaning, if you buy many of these objects, then you spend less money paying for repairs or replacements as needed than you would buying the warranties.

### Variance

But you don't buy the same object many times. So it may not be relevant to you to know what happens if you did. Imagine that an oracle makes you the following offer: You flip a fair coin. If it comes up heads, the oracle gives you $200. If it comes up tails, you give the oracle $100. Do you accept?

You probably should, at least if you believe the oracle to be trustworthy, because the expected value is positive: your expected value is .5 x 200+.5 x (-100)=50.

The problem is that not all games with the same expected value are created equal: some are riskier than others. If the oracle offers you a chance to play a "game" in which you just get $50, then you will surely accept, since there's no risk involved. With the one above, you might accept, or you might not. If instead the oracle offers to give you $100000000200 for heads and you give the oracle $100000000100 for tails, then the expected value is still the same $50, but now you have to decline, because you cannot afford to lose. At some point, variance considerations are more important than expected value considerations.

### The Gist of Warranties

What a warranty allows you to do is to pay a cost in expected value in order to reduce your variance. That *might* be a good deal.

### Is it a good deal?

Now, even though we typically don't buy lots of instances of any particular item with warranties, we do over time buy many different such things. And they break rougtly independently of one another: if my computer breaks, that doesn't make my microwave any more or less likely to break. An important phenomenon in probability theory is that **the variance of a sum of independent random variables is the sum of the individual variances.**

The point is that the variance of cost of repairs increases when you buy more products, *but not that much!* In fact, in order to measure everything on the same scale, what matters is the *square root* of the variance. As you throw in more and more items, the standard deviation of the cost of all repairs gets small with respect to the cost of the items.

**From the math alone, it makes sense to create a policy whereby you never buy warranties, unless you happen to find one with a positive expected value.**

## Layer #2: Prospect Theory

One crucial assumption is that we are the value of money as linear: the value increases proportionally with the amount. However, it is well known that this is not true, and like most goods, money suffers from diminishing marginal utility. As the pioneer of probability and statistics, Daniel Bernoulli writes:

"There is no doubt that a gain of one thousand ducats is more significant to the pauper than to a rich man though both gain the same amount."

If gaining two thousand dollars feels less than twice as good as gaining one thousand dollars, what about the same relationship for *losing* money?

### Diminishing Sensitivity

Turns out, the function for losing money has the same concavity as for winning money. Prospect Theory notes that for both wins and losses, each incremental dollar provides less of an effect than the last. That is, losing two thousand dollars feels less than twice as bad as losing one thousand dollars. This phenomenon is known as diminishing sensitivity.

This is very bad news for warranties. The expected value of the warranty, once again, is (the cost of repair or replacement) times (the probability of needing that repair or replacement) minus (the cost of the warranty). Since we now know that the pain of paying for a repair is proportionally smaller relative to the cost of the repair, **the expected utility of the warranty is even worse than the expected value.**

However, this may not *feel* like the case because of a confounding effect.

### The Endowment Effect

Just by owning something, you now attribute more value to it. The endowment effect is supported by the fact that it costs way more to get someone to part with something that they own than to get someone to buy something that they don't.

One study shows that touching something already increases its perceived value. The study notes increased willingness-to-pay for warranties after touching an item, with study participants citing greater pain upon loss of the item. One subconscious rationalization for purchasing warranties could be the discrepancy between an item's actual cost and its perceived cost, leading one to believe that the warranty has positive expected value when it really doesn't.

## Layer #3: Beyond the math

The math is all pretty standard, but there's more to warranties. There's another layer of rationality to be uncovered:

*What if having a warranty changes your actions?*

### No-brainer repairs

Imagine that you buy a computer, and after a few months, something goes wrong with it. Not something that makes it unusable, but just something that makes it a little less fun to use. You could get it repaired, or you could continue using it as is. What do you do?

If you have a warranty, you should just send it back and have the manufacturer repair or replace it. It doesn't cost you anything other than the annoyance of being without it for a short period of time.

If, on the other hand, you have to pay to get your computer fixed, one needs to make a mental calculation to see if the repair is worth it. Even in the event that it is, maybe you will start rationalizing why the problem isn't so bad and do nothing. This will save in expenses, but it won't be as fun to use the slightly broken computer.

In this case, the warranty is a bad investment financially, but it can be a good investment in terms of framing future choices in a way that will make you happy.

### Risk compensation

You're probably more likely to take more liberties on items with warranties.

The important consideration here is something known as risk compensation. The key here is that people are more careful where they sense greater risk and are more at ease when protected. The phoneomenon is well documented in a wide range of contexts. Drivers with seatbelts are more reckless, football players with helmets tackle more, and skydivers with better equipment take more risks.

The same applies for warranties. Subconsciously, knowing that a warranty is in effect allows one to take more liberties. Regardless of whether the warranty actually covers manual damage, one typically spends less time worrying about an item after having bought a warranty.

For those who are more apt to enjoy things when less worried, the warranty can be a good behavioral choice.

## What's the takeaway?

Mathematically, you lose money on warranties. Even though it is offset partially by decreased variance, the variance decrease becomes small quickly once one starts buying many items during a lifetime.

The loss is not saved by utility considerations. In fact, they make it worse because the pain of losing money decreases as more is lost.

The primary benefit for most warranties is psychological, and some of them stem from fallacious reasoning. There still remain good reasons for buying warranties, *but only if you are a specific type of person*. If you frequently rationalize your way out of good decisions, warranties might be the way to go.

More generally, buy a warranty if and only if:

- You somehow find one with positive expected utility, or
- The cost in expected value is worth to you the warranty's behavioral benefits of framing and risk compensation

Otherwise, you are better off forgoing the warranty and always paying for repairs and replacements when they arise.