# Car Insurance and an Ear Full of Cider

Thursday, June 30, 2011
By dreeves

Yesterday Decision Science News asked “Should you buy car insurance?” By which they mean collision insurance, liability insurance being required by law in these parts.

I’m shocked and appalled that decision scientists could even ask such a thing. Actually, it’s a good question with some legitimate subtleties. I just happen to have an unreasonably strong opinion about this, which is that you should sooner invest your life savings in lottery tickets than buy collision insurance for your car, no matter how new or fancy it is.

There are just two reasons you should ever buy insurance:

1. You know more than the insurance company. Like you’re about to burn down your house for the insurance money. Wait, that’s illegal. Maybe you’re about to let your teenager start driving your car. Actually, the insurance company probably knows that too. [1] Hint: you probably don’t know more than the insurance company. You could maybe make an argument on these grounds that you should get the extended laptop warranty because you like to be crazy reckless with your electronics. But those warranties are so overpriced that, really, you’re probably not that reckless.

2. You’re insuring against the loss of something so expensive that paying for it yourself would change the very value of a dollar for you. Like it would literally change your lifestyle. If you’re pretty poor (by first world standards) then maybe a couple thousand dollars could change your lifestyle a bit, temporarily. For the supernerds reading either Decision Science News or Messy Matters it probably realistically has to be in the tens of thousands — often much more. Collision insurance on your car is not in this category.

You probably don’t even need any fancy econ concepts like risk aversion and expected values. You know what slimeballs insurance companies are. They raise your rates when you make a claim and manipulate the deductibles. Not to mention weaseling out of paying altogether whenever they can get away with it. They pretty much guarantee that for any claim you make for smashing up your car you’ll pay for it many times over in premiums (if you haven’t already).

#### If you could write a check and go on with your life as if nothing had happened, do not insure against it.

But I claim that even when it feels tempting, have faith in the math (which the insurance company’s actuaries have carefully worked out) and JUST SAY NO. They wouldn’t offer you the insurance if you could, in expectation, come out ahead from it. Think about the worst case: if you could write a check [2], beat your forehead against the wall a few times, and then go on with your life as if nothing had happened, do not insure against it.

### An Ear Full of Cider

Marlon Brando’s character, Sky Masterson, has a great monologue in “Guys and Dolls”, relating advice from his father:

One of these days in your travels a guy is going to show you a brand-new deck of cards, on which the seal is not yet broken. Then this guy is going to offer to bet you that he can make the jack of spades jump out of this brand-new deck of cards and squirt cider in your ear. But, son, you do not accept this bet. Because as sure as you stand there you’re going to wind up with an ear full of cider.

Nice life lesson. We can call it the Ear Full of Cider Principle. [3] What does it have to do with insurance? Buying insurance is, quite literally, accepting a wager with the insurance company. You’re betting that you will crash your car and the insurance company is betting that you won’t. The ratio of your premiums to the payout for the possible claim establishes the odds that the insurance company is giving you. They wouldn’t offer you those odds unless the bet was a savvy one for them.

Which makes you the sucker.

### Efficient Markets

Of course, “sucker” might be hyperbolic. After all, insurance is ostensibly a competitive industry so market efficiency suggests that their profits not be too obscene overall, which means the effective odds you’re offered shouldn’t be too skewed. Fundamentally, it’s an empirical question though, hinging on claims paid vs. premiums collected. I could well be proved wrong: buying insurance may only be slightly stupid, not ear-full-of-cider stupid.

### Epilogue

Since this is apparently heresy to everyone I know, I’ll address some counter-arguments. (Scroll ahead to the bonus puzzle if you’re already convinced.) Some of these are pretty reasonable and do leave some wiggle room to rationalize your crazy risk aversion if we also suppose that the efficient markets argument is true and the odds are only slightly skewed against you.

Simplicio: First of all, that was quite a concession you made in admitting that insurance is a competitive industry. If they’re only making an expected few bucks profit off me (and assuming the company’s overhead is not a big factor) then your whole tirade reduces to ranting about a trivial “waste” of money. (And why it’s not really a waste I’ll get to shortly!)

Salviati: My glib response (I really hate insurance companies, if that wasn’t clear yet) is that they stay competitive by coming up with creative ways to deny claims. But, actually, it’s true this is an excellent argument and I admit my characterization of buying insurance as a sucker bet may be a bit unfair. Maybe it’s more like a slot machine that just very gradually sucks money out of you.

But to make my glib response more serious, one way the argument from competition can break down is if they lose money on the people who watch them like hawks, raise a stink when they try to weasel, etc, while making obscene profits on the people who don’t have time for that shit. And of course they lose money on crazy reckless people which they have to make up for on average people (like you, you presume — if not, see my reason #1 that insurance can be rational).

Another potential flaw in the argument from competition: When the cost to enter the market is so high, efficiency can’t be taken for granted.

Finally, peace of mind is so rampantly overvalued (or just blatant failure to understand the underlying math) that I’m not entirely sure they don’t all make obscene profits on collision insurance, despite the competition. They do have a million ways to avoid competing on raw price, after all. (Kind of like how laptop extended warranties are drastically overpriced because they tack it on to the laptop price and you shrug it off because, hey, what’s another couple hundred bucks?)

Simplicio: You argue that insurance companies wouldn’t offer you the bet unless the odds were skewed in their favor, but insurance companies fail all the time by underpricing risk. Look at AIG in 2008!

Salviati: Red herring. You can identify that in hindsight but in foresight you have to presume that they’ve priced the risk more carefully than you.

Simplicio: Just because in expectation you’ll lose some money doesn’t mean there isn’t a significant chance of you coming out ahead.

Salviati: Also beside the point. We certainly agree that you might come out ahead buying insurance.

Simplicio: Isn’t peace of mind worth something?

Salviati: Sure, I get it by having faith in the math and the calculated risk I’m taking.

Simplicio: It’s not just about the money when you buy insurance. There’s all the hassle of replacing your car and negotiating for repairs that they handle for you.

Salviati: This is a lame rationalization. It’s mostly about the money. In theory, sure, you might value that service highly enough, but in practice you’re probably drastically overpaying for it. (Not to mention that dealing with the insurance company can be a bigger nightmare than just buying a new car.)

Of course I’m going by hearsay here, never having bought collision insurance, but I also hear that they’ll blatantly screw you over time by raising your rates. So you have to keep shopping around and switching companies every few years. Maybe that doesn’t compare to the hassle of replacing or fixing your car but remember, that’s a certain hassle compared to a possible hassle.

Simplicio: The insurance companies may have a bargaining advantage and can negotiate better deals on repairs or replacement cars than you can. If they can repair cars much cheaper than individuals then the effective odds could actually be skewed in your favor, making collision insurance fully rational.

Salviati: Maybe. But mostly you’re just rationalizing again.

You could even imagine this being the other way around. If you’re paying the mechanic yourself they might offer to do a 90% repair — perfectly adequate unless you’re super fussy about your car — for much less money. But if insurance is paying then that option is off the table. They’re either doing the 100% repair or they’re not paying at all. If that’s typical then it’s actually more expensive for insurance companies to pay for repairs than it is for individuals. Which means more cider in your ear if you buy insurance.

Simplicio: The downside protection against significant losses just makes me sleep better. In other words, I just have inherent value for it. You can’t argue with that.

Salviati: Sure I can! Your utility function is all wrong! I’m actually kind of serious here. I think the reason this makes you happy is that you have disutility for experiencing regret.

But regret is not rational!

#### In hindsight I should’ve gotten the insurance, but there’s no such thing as making decisions in hindsight.

You should retrain yourself to not experience it (I’m still being serious). When you forgo the insurance and smash your brand new car the next day you need to think to yourself, “Sure, in hindsight I should’ve gotten the insurance but that’s actually a meaningless statement! There’s no such thing as making decisions in hindsight. In foresight, I made the optimal decision. Or, if not (maybe — though this is likely hindsight bias — I should’ve actually known how accident prone I was), then, well, I was dumb and there’s not much I can do about that either, except be less dumb from now on.”

Simplicio: Hey, you’re arguing that you’re necessarily getting screwed (in expectation) because the insurance company is making a profit. You could apply that reasoning to everything you buy!

Salviati: So you might think! But the difference is quite fundamental. It’s that insurance is fundamentally zero-sum — the only transaction is money going back and forth. When you buy a refrigerator, say, your value for it is huge. (If refrigerators cost $50k you’d probably suck it up and buy one — how the hell are you going to live without a refrigerator?) So you come out way ahead when you get it for [I haven’t actually the faintest clue what a refrigerator costs]. And Maytag or whoever does too. Win-win! Insurance is fundamentally either win-lose or lose-win. Simplicio: I admit that I’ve spent more in premiums than I’ve gotten back in claims [necessarily true for most people!] but I wouldn’t have actually had the cash to cover the damages that one time I crashed my car. Salviati: You could set aside the money you otherwise spend on premiums into an emergency fund. Simplicio: Ha, as if I have that kind of self-discipline. Salviati: Ah, so you need a commitment device! Get a friend to accumulate the money for you and not let you access it unless you crash your car. Think about the myriad advantages that has over giving the money to an insurance company. Like your friend doesn’t just keep your money if you don’t crash your car. Not to mention greater flexibility in what counts as valid grounds to tap the fund. The one disadvantage is the risk that you’ll crash your car before there’s enough money in the fund. Again: calculated risk. ## Bonus Puzzle Two people are driving along and they collide, totaling their cars. By bizarre coincidence they were driving identical fancy Italian custom-built cars — the only two such cars ever made! By further coincidence they have the same insurance company (and are both dumb enough to have collision insurance). The insurance company has a cap of$100,000 on their coverage, but of course wants to pay as little as possible.

So here’s what they do: They separate the two parties and ask them to write down the value of the car — any number from $2k to$100k, rounded to the nearest thousand. If they write down the same number then the insurance company will treat that as the true value and reimburse them both that amount. But if one writes down a smaller number then of course that amount will be taken as the true value. And to punish the one who inflated the value and reward the presumably honest one, the insurance company will pay $2k less to the former and$2k extra to the latter.

The question: What’s the optimal number to write down? (Assume all the usual ridiculous things: common knowledge of rationality, risk neutrality, pure selfishness, etc.)

## Addendum

This article was discussed on Hacker News. I also discussed it with a real-live economist and we distilled the contention down to my claim that people are or should be essentially risk neutral for amounts of money that wouldn’t change their lifestyle. They pointed out that this claim is sloppy and thus not very convincing. I’m working on that!

### Puzzle Answer

Congratulations to Arthur Breitman for solving the puzzle. It was a disguised instance of the Travelers’ Dilemma. Preposterously, both drivers should write down $2k as the value of their fancy custom cars. Here’s how it theoretically plays out (key word theoretically): It’s impossible to rationally write down 100 because if your opponent writes down 100 then your optimum is 99. If your opponent writes down something less then 100 then you’re still better off writing down 99 than 100 (99 is either better or equally good). So 99 dominates 100. Writing down 100 is right out. You know that your opponent has figured that out as well. But given that 100 is off the table, the same reasoning as above rules out 99 too! And you can see where this is going. Every single value is logically off the table until you get to 2. And given that you’ve logically deduced that your opponent will be writing down 2, it really is easy to see that it’s optimal for you to also write down 2. Write down anything bigger and you’re hit with the penalty and get nothing. So you’re best off taking the$2k.

What’s funny is that that’s so hyperrational that it’s insanely and literally idiotic. An actual person could never possibly do that. They could at most carry that backward induction a few steps. Like “I’ll write down 99 so if he writes 100 I’ll get 101. He’ll probably think that too so I’ll put 98 and get 98+2=100. But he’ll probably think that so I’ll put 97! At this point I really don’t know where he’ll stop so I guess I’ll just go with 97 or 98 and hope I luck out and end up just below him.”

Illustration by Kelly Savage.
Thanks to Dan Goldstein, not only for posing the question but for a fascinating debate on this topic. Thanks also to Sharad Goel, Bethany Soule, and Martin Reeves for contributions.

## Footnotes

[1] If they don’t they’ll claim they should have and weasel out of paying.

[2] Or sell some stock, or even borrow some money.

[3] Economists call this the no-trade theorem. Economists are very boring.

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• Willem

Fipo. Disclaimer: Working for an insurer.

Your 2 is very true. Why you want mandatory insurance for (motor) third party liability (and in Europe, we have, but only for motor):

You cannot ever afford the claim. If you hit someone by car and that person has serious trauma, expect claims in six figures. Same thing goes for other personal accidents. We see dozens of those.

Who is most likely to hit a person or do grave bodily harm trying out something usually very very stupid? The person most unlikely to get insurance. So, everybody owning a car chips in. And everybody alive should chip in.

We make a huge profit of some 15% on capital. That’s pretty small considering we pay some 6% interest. Who get’s the profit? The shareholders. Who own our shares? Our clients. That’s all there is to it.

Caveat: I don’t buy all the products we sell. So practice what you preach, but only within boundaries. No need for legal aid, since I know my way around law. For the rest: stock up, it’s a bargain.

Might be that our firm is totally different to your average US insurer. Our main lifeline is a happy customer, not profit (customerowned, remember). I feel we try our best to make that succeed.

• Frank

This is surprising: “this is apparently heresy to everyone I know.” I agree with everything here except the bargaining-power argument, which I think does hold water for medical insurance, and probably more broadly.

#2 is the only reason I would/do buy insurance. As a student, my bank account could easily hit rock bottom from stuff like this. #1′s not very reliable or sustainable if the industry keeps tabs on your behavior; and it’s a lot of work staying one step ahead.

• http://dreev.es dreeves

@Frank, I guess by “everyone I know” I meant everyone who I both know and from whom the draft of this tirade prompted indignant replies.

I may have introduced a bias in my sampling. :)

It’s true that a lot of us infuriatingly hyperrational folks find this painfully obvious. It even seems to be conventional wisdom on the internets, as Sharad found some evidence of:

(Or at least that you should get the highest possible deductible if you do buy collision insurance, which is a step in the right direction and which most people infuriatingly refuse to do.)

• Arthur B.

I find $2k for the answer to the puzzle… It’s the obvious result though, so I must be missing something. • Manoel Galdino I’d agree with, except: 1. I live in Brazil and I don’t care to be protected against collision. I care to be protected against people stealing my car. And no, credit in Brazil isn’t an option. Interest rates in Brazil are too high to borrow money to pay for a new car (the Government pay 12% year, credit Card is around 110%+ year, for cars is around 20% – 30% year). So, my simple calculations are: If someone steal my car and I have to borrow money to buy a new car, then I’ll pay more of interest rates than what I saved by not having insurance! 2. By your argument, we shouldn’t have health insurance either. 3. “Regret isn’t rational”. But if you know you will regret, is it rational to not buy insurance and then regret later? THIS isn’t rational. • Manoel Galdino Forgot to say: nice reference to Galileo! • http://dreev.es dreeves @Manoel, I don’t understand your leap to health insurance. That falls solidly under reason #2 that it *is* rational to buy insurance. Though rationally you should (if possible) just insure against getting limbs hacked off or getting cancer or whatnot, not normal doctors visits. As for regret, your proposal is one hyperrational way to deal with your irrationality. I think you can do better though! • http://dreev.es dreeves @Arthur, your answer is correct! But to get credit you’ve got to explain why. If you know enough game theory it’s probably obvious but I thought it was surprising and interesting the first time I saw something like that happen. It might’ve even seemed like an infuriating paradox. Now I view it as just an example of how an elegant theoretical model need not be remotely close to reality. Reminds me of the ludic fallacy. • Arthur B. Ah, well the answer is that both player being symmetric, they have to come to the same number. However, if that number weren’t$2k, you could benefit by undercutting him by $1k.$2k is the only fixed point in the game matrix.

Nash proved that such fixed point always exist in games using Brouwer’s fixed point theorem (also called Brouwer’s not unfixed point theorem*). It’s been recently proven (Daskalakis, Goldberg, Papadimitriou) that finding these points can be PPAD-complete, which implies it’s at least as hard as solving problems known to be very difficult. This makes these seemingly paradoxical results even more unrealistic.

*puzzle: why?

• http://www.cs.berkeley.edu/~kevin/ Kevin Canini

@Arthur and @dreeves: Your $2k analysis of the puzzle seems to assume that each knows what the other plans to write down. Without that knowledge, it’s impossible to undercut the other guy by$1k. Or is there something about the common knowledge of rationality that allows you to ignore that?

• http://dreev.es dreeves

@Kevin, you’ve opened up a philosophic can of worms! But first to try to answer your question directly, we don’t explicitly assume that the players know what the other will write down. But like you suggest, common knowledge of rationality implies that they know.

It’s impossible to rationally write down 100 because if your opponent writes down 100 then your optimum is 99. If your opponent writes down something less then 100 then you’re still better off writing down 99 than 100 (99 is either better or equally good). So 99 dominates 100. Writing down 100 is right out. You know that your opponent has figured that out as well.

But given that 100 is off the table, the same reasoning as above rules out 99 too! And you can see where this is going. Every single value is logically off the table until you get to 2. And given that you’ve logically deduced that your opponent will be writing down 2, it really is easy to see that it’s optimal for you to also write down 2. Write down anything bigger and you’re hit with the penalty and get nothing. So you’re best off taking the $2k. What’s funny is that that’s so hyperrational that it’s insanely and literally idiotic. An actual person could never possibly do that. They could at most carry that backward induction a few steps. Like “I’ll write down 99 so if he writes 100 I’ll get 101. He’ll probably think that too so I’ll put 98 and get 98+2=100. But he’ll probably think *that* so I’ll put 97! At this point I really don’t know where he’ll stop so I guess I’ll just go with 97 or 98 and hope I luck out and end up just below him.” Now the philosophic can of worms, I think, is this: By symmetry you know that your opponent will write down the same number as you. So you should write down the number x such that if both players write down x your payoff is maximized. That x is 100! So write down 100. Your opponent will do so as well and you get the highest payoff that’s possible without one person screwing the other, which, like we said, by symmetry is impossible! • Liam Carton Utterly, utterly unconvinced by answers of the “2k” kind. The maximum payout that anyone can get is 100K. If I bet 100 and the other party bets 100 we both get 100. If I bet 100 and the other party bets 99 s/he gets 100 and I get 98. If I assume that the other party is as logical as I am it is clear that 100 is the correct figure. Even if I assume that s/he could be some kind of greedy vulture who will try to screw me out of 2k for no extra value for themselves (In other words they could bet 99 just to screw me over) then my best option is still 100 as I cannot “know” that they will be this dumb! The problem with the 2k bet answers is that the person setting this question have got themselves caught by the fixed maxima (no one can gain more than 100k). If the difference where to be more than 1k, say 5k, (so that one person *could* get more than 100k) then the logical regression to minimum might be more believable. As this isn’t the case 100k can be the only logical bet. • http://dreev.es dreeves @Liam, definitely$100k is the right answer for the game you’re thinking of! But the game in the puzzle has a twist you missed: if I write down the lower number I get a $2k bonus. So if my opponent writes down$100k and I write down $99k, I get$101k.

The $2k answer is still ridiculous, though, I agree with that! • http://www.aerowingroup.com/ รับจดทะเบียนบริษัท I just happen to have an unreasonably strong opinion about this, which is that you should sooner invest your life savings in lottery tickets than buy collision insurance for your car, no matter how new or fancy it is. • Teresa PUZZLE Assuming that the other person is also hyperrational is critic (same type of person, same rare car.. starting to believe in fate!). Just out of curiosity… wouldn’t it be rational for the hyperrational person also to consider that the hyperrationalisation is idiotic and none would actually do that? Another point is maximisation by comparison or absolte vs. relative utility (probably not the correct term). I would still write 100k because the worst it could happen is that the other person writes 2k and I get nothing. There is almost no difference between 2 and 0 when you could get 100. Even if the other person write 99, i still get 97, and so on. (I know this is a serious discussion, but I like this clip i’ve been sent on overanalysing situations and find it pertinent here: http://youtu.be/zQoJOAbAsRM • Unknown I know someone who bought temporary health insurance after several other people in the same building came down with a serious illness that required hospital stays costing thousands of dollars. The day after he bought the insurance, he came down with the illness. I guess this falls under reason 1. • Army1987 In my country, civil liability insurance on cars is mandatory. • http://messymatters.com Daniel Reeves @Army1987, same in the US. Liability insurance is mandatory. This article is just about collision insurance, namely, insuring against damage to your own car. • JoeK The optimal algorithm I can use, under the assumption that the opponent is equally rational and will cooperate, is to choose$100K. If I calculate the expected mutual value of each possible choice given the assumption of a cooperating counter-party, total utility is maximized at $100K. Further, “Not being a dick” is an important term in my personal utility function. So I really don’t care if I get 97 and she gets 101, or whatever. • JoeK Sorry, I guess I’m not maximizing utility for the insurance company. Poor dears. I expect they’ll survive. • Liam Carton @dreeves. Sorry, but I don’t agree. the satement in the original puzzel was that there was a 100K cap on the payment. Therefore my assertion that 100K was the maximum payout is true. So if person a says 100K and person b says 99K then a will get 98K and b will get 100K, NOT 101K. So they go for 99K for absolutely no benefit! Therefore the logical choice is for both a and b to quote 100K. Sorry, but that is what the puzzel says. Of course b could go 98K, but again they gain nothing over quoting 100K, except to screw a out of 4,000. Still not rational. Cheers Liam • http://dreev.es dreeves @Liam, I see, yeah, I guess the way to disambiguate it is to say$100k is the max *coverage* but the $2k is a bonus or penalty on top of whatever the coverage is (I think that was the natural interpretation!). Or scrap the “max coverage” part.$100k is just the max you can write down.

Oh, and note that even if the insurance company pays $99k and$101k they are in fact paying the max of $100k *per claim*. So just think of the$2k not as an adjustment to the coverage but as a transfer between the two claimants.

• http://stoneglasgow.blogspot.com Stone Glasgow

Your line of reasoning is valid, but change is only an option for people who own their cars free and clear. Most people in the United States buy their car using a bank loan, and they are required to carry collision insurance by the bank.