There is a small remark in
Rational Choice in an Uncertain World: The Psychology of Judgment and Decision Making about insurance saying that all insurance has negative expected utility, we pay too high a price for too little a risk, otherwise insurance companies would go bankrupt. If this is the case should we get rid of all our insurances? If not, why not?
There is a small remark in Rational Choice in an Uncertain World: The Psychology of Judgment and Decision Making about insurance saying that all insurance has negative expected utility, we pay too high a price for too little a risk, otherwise insurance companies would go bankrupt.
No—Insurance has negative expected monetary return, which is not the same as expected utility. If your utility function obeys the law of diminishing marginal utility, then it also obeys the law of increasing marginal disutility. So, for example, losing 10x will be more than ten times as bad as losing x. (Just as gaining 10x is less than ten times as good as gaining x.)
Therefore, on your utility curve, a guaranteed loss of x can be better than a 1/1000 chance of losing 1000x.
ETA: If it helps, look at a logarithmic curve and treat it as your utility as a function of some quantity. Such a curve obeys diminishing marginal utility. At any given point, your utility increases less than proportionally going up, but more than proportionally going down.
(Incidentally, I acutally wrote an embarrasing article arguing in favor of the thesis roland presents, and you can still probably find on it the internet. That exchange is also an example of someone being bad at explaining. If my opponent had simply stated the equivalence between DMU and IMD, I would have understood why that argument about insurance is wrong. Instead, he just resorted to lots of examples of when people buy insurance that are totally unconvincing if you accept the quoted argument.)
I voted this up, but I want to comment to point out that this is a really important point. Don’t be tricked into not getting insurance just because it has a negative expected monetary value.
I voted Silas up as well because it’s an important point but it shouldn’t be taken as a general reason to buy as much insurance as possible (I doubt Silas intended it that way either). Jonathan_Graehl’s point that you should self-insure if you can afford to and only take insurance for risks you cannot afford to self-insure is probably the right balance.
Personally I don’t directly pay for any insurance. I live in Canada (universal health coverage) and have extended health insurance through work (much to my dismay I cannot decline it in favor of cash) which means I have far more health insurance than I would purchase with my own money. Given my aversion to paperwork I don’t even fully use what I have. I do not own a house or a car which are the other two areas arguably worth insuring. I don’t have dependents so have no need for life or disability coverage. All other forms of insurance fall into the ‘self-insure’ category for me given my relatively low risk aversion.
Risk is more expensive when you have a smaller bankroll. Many slot machines actually offer positive expected value payouts—they make their return on people plowing their winnings back in until they go broke.
Citation please? A cursory search suggests that machines go through +EV phases, just like blackjack, but that individual machines are -EV. It’s not just that they expect people to plow the money back in, but that pros have to wait for fish to plow money in to get to the +EV situation.
The difference with blackjack is that you can (in theory) adjust your bet to take advantage of the different phases of blackjack. Your first sentence seems to match Roland’s comment about the Kelly criterion (you lose betting against snake eyes if you bet your whole bankroll every time), but that doesn’t make sense with fixed-bet slots. There, if it made sense to make the first bet, it makes sense to continuing betting after a jackpot.
This comes up frequently in gambling and statistics circles. “Citation please” is the correct response—casinos do NOT expect to make a profit by offering losing (for them) bets and letting “gambler’s ruin” pay them off. It just doesn’t work that way.
The fact that a +moneyEV bet can be -utilityEV for a gambler does NOT imply that a -moneyEV bet can be +utilityEV for the casino. It’s -utility for both participants.
The only reason casinos offer such bets ever is for promotional reasons, and they hope to make the money back on different wagers the gambler will make while there.
The Kelly calculations work just fine for all these bets—for cyclic bets, it ends up you should bet 0 when -EV. When +EV, bet some fraction of your bankroll that maximizes mean-log-outcome for each wager.
Some casinos advertise that they have slots with “up to” a 101% rate of return. Good luck finding the one machine in the casino that actually has a positive EV, though!
Because slot machines are designed to hook you in, you’re going to get some return on investment from them if you hold yourself to a specific amount. At the Casino de Lac Leamy, up in Canada (run, I would add, by the Quebec provincial government. Now that’s a lottery system), the slots are ‘loose.’ They pay out relatively often. In fact, when Weds and I have played twenty dollars worth of slots together, we’ve never failed to leave the casino floor with more money than we had entering the floor. That twenty dollars has been anything from thirty to sixty-five dollars, the three or four times we’ve done this.
I’ll give you that “many” is almost certainly flat wrong, on reflection, but such machines are (were?) probably out there.
That move was full of falsehoods. For example, people named Silas are actually no more or less likely than the general population to be tall homicidal albino monks—but you wouldn’t guess that from seeing the movie, now, would you?
That twenty dollars has been anything from thirty to sixty-five dollars, the three or four times we’ve done this.
I’m pretty sure it’s not that unlikely to come up ahead ‘three or four’ times when playing slot machines (if it weren’t so late I’d actually do the sums). It seems much more plausible that the blog author was just lucky than that the machines were actually set to regularly pay out positive amounts.
That’s definitely a related result. (So related, in fact, that thinking about the +EV slots the other day got me wondering what the optimal fraction of your wealth was to bid on an arbitrary bet—which, of course, is just the Kelly criterion.)
I’d like to pose a related question. Why is insurance structured as up-front payments and unlimited coverage, and not as conditional loans?
For example, one could imagine car insurance as a options contract (or perhaps a futures) where if your car is totaled, you get a loan sufficient for replacement. One then pays off the loan with interest.
The person buying this form of insurance makes fewer payments upfront, reducing their opportunity costs and also the risk of letting nsurance lapse due to random fluctuations. The entity selling this form of insurance reduces the risk of moral hazard (ie. someone taking out insurance, torching their car, and then letting insurance lapse the next month).
Except in assuming strange consumer preferences or irrationality, I don’t see any obvious reason why this form of insurance isn’t superior to the usual kind.
Well, look at a more extreme example. Imagine an accident in which you not just total a car, but you’re also on the hook for a large bill in medical costs, and there’s no way you can afford to pay this bill even if it’s transmuted into a loan with very favorable terms. With ordinary insurance, you’re off the hook even in this situation—except possibly for the increased future insurance costs now that the accident is on your record, which you’ll still likely be able to afford.
The goal of insurance is to transfer money from a large mass of people to a minority that happens to be struck by an improbable catastrophic event (with the insurer taking a share as the transaction-facilitating middleman, of course). Thus a small possibility of a catastrophic cost is transmuted into the certainty of a bearable cost. This wouldn’t be possible if instead of getting you off the hook, the insurer burdened you with an immense debt in case of disaster.
(A corollary of this observation is that the notion of “health insurance” is one of the worst misnomers to ever enter public circulation.)
Alright, so this might not work for medical disasters late in life, things that directly affect future earning power. (Some of those could be handled by savings made possible by not having to make insurance payments.)
But that’s just one small area of insurance. You’ve got housing, cars, unemployment, and this is just what comes to mind for consumers, never mind all the corporate or business need for insurance. Are all of those entities buying insurance really not in a position to repay a loan after a catastrophe’s occurrence? Even nigh-immortal institutions?
I wouldn’t say that the scenarios I described are “just one small area of insurance.” Most things for which people buy insurance fit under that pattern—for a small to moderate price, you buy the right to claim a large sum that saves you, or at least alleviates your position, if an improbable ruinous event occurs. (Or, in the specific case of life insurance, that sum is supposed to alleviate the position of others you care about who would suffer if you die unexpectedly.)
However, it should also be noted that the role of insurance companies is not limited to risk pooling. Since in case of disaster the burden falls on them, they also specialize in specific forms of damage control (e.g. by aggressive lawyering, and generally by having non-trivial knowledge on how to make the best out specific bad situations). Therefore, the expected benefit from insurance might actually be higher than the cost even regardless of risk aversion. Of course, insurers could play the same role within your proposed emergency loan scheme.
It could also be that certain forms of insurance are mandated by regulations even when it comes to institutions large enough that they’d be better off pooling their own risk, or that you’re not allowed to do certain types of transactions except under the official guise of “insurance.” I’d be surprised if the modern infinitely complex mazes of business regulation don’t give rise to at least some such situations.
Moreover, there is also the confusion caused by the fact that governments like to give the name of “insurance” to various programs that have little or nothing to do with actuarial risk, and in fact represent more or less pure transfer schemes. (I’m not trying to open a discussion about the merits of such schemes; I’m merely noting that they, as a matter of fact, aren’t based on risk pooling that is the basis of insurance in the true sense of the term.)
I wouldn’t say that the scenarios I described are “just one small area of insurance.” Most things for which people buy insurance fit under that pattern—for a small to moderate price, you buy the right to claim a large sum that saves you, or at least alleviates your position, if an improbable ruinous event occurs.
Intrinsically, the average person must pay in more than they get out. Otherwise the insurance company would go bankrupt.
Since in case of disaster the burden falls on them, they also specialize in specific forms of damage control (e.g. by aggressive lawyering, and generally by having non-trivial knowledge on how to make the best out specific bad situations).
No reason a loan style insurance company couldn’t do the exact same thing.
I’d be surprised if the modern infinitely complex mazes of business regulation don’t give rise to at least some such situations.
‘Rent-seeking’ and ‘regulatory capture’ are certainly good answers to the question why doesn’t this exist.
For one thing, insurance makes expenses more predictable; though the desire for predictability (in order to budget, or the like) does probably indicate irrationality and/or bounded rationality.
What’s unpredictable about a loan? You can predict what you’ll be paying pretty darn precisely, and there’s no intrinsic reason that your monthly loan repayments would have to be higher than your insurance pre-payments.
Obviously if you know your utility function and the true distribution of possible risks, it’s easy to decide whether to take a particular insurance deal.
The standard advice is that if you can afford to self-insure, you should, for the reason you cite (that insurance companies make a profit, on average).
That’s a heuristic that holds up fine except when you know (for reasons you will keep secret from insurers) your own risk is higher than they could expect; then, depending on how competitive insurers are, even if you’re not too risk-averse, you might find a good deal, even to the extent that you turn an expected (discounted) profit, and so should buy it even if you have zero risk aversion. Apparently in California, auto insurers are required to publish the algorithm by which they assign premiums (and are possibly prohibited from using certain types of information).
Conversely, you may choose to have no insurance (or extremely high deductible) in cases where you believe your personal risk is far below what the insurer appears to believe, even when you’re actually averse to that risk.
Of course, it’s not sufficient to know how wrong the insurer’s estimate of your risk is; they insist on a pretty wide vig—not just to survive both uncertainties in their estimation of risk and the market returns on the float, but also to compensate for the observed amount of successful adverse selection that results from people applying the above heuristic.
I suppose it may also be possible that the insurer won’t pay. I don’t know what exactly what guarantees we have in the U.S.
to compensate for the observed amount of successful adverse selection that results from people applying the above heuristic.
Actually, I think that for voluntary insurance, the observed adverse selection is negative, but I can’t find the cite. People simply don’t do cost-benefit calculations. People who buy insurance are those who are terribly risk-averse or see it as part of their role. Such people tend to be more careful than the general population. In a competitive market, the price of insurance would be bid down to reflect this, but it isn’t.
Some insurances are not worth getting, obviously. Like insurance on laptops or music players. But that insurance in general has negative expected utility assumes no risk aversion. If you can handle the risks on your own—if you are effectively self-insuring—then you probably should do that. But a house burning down or getting a rare cancer that will cost millions to treat: these are not self-insurable things unless you are a millionaire.
Should we buy insurance at all?
There is a small remark in Rational Choice in an Uncertain World: The Psychology of Judgment and Decision Making about insurance saying that all insurance has negative expected utility, we pay too high a price for too little a risk, otherwise insurance companies would go bankrupt. If this is the case should we get rid of all our insurances? If not, why not?
No—Insurance has negative expected monetary return, which is not the same as expected utility. If your utility function obeys the law of diminishing marginal utility, then it also obeys the law of increasing marginal disutility. So, for example, losing 10x will be more than ten times as bad as losing x. (Just as gaining 10x is less than ten times as good as gaining x.)
Therefore, on your utility curve, a guaranteed loss of x can be better than a 1/1000 chance of losing 1000x.
ETA: If it helps, look at a logarithmic curve and treat it as your utility as a function of some quantity. Such a curve obeys diminishing marginal utility. At any given point, your utility increases less than proportionally going up, but more than proportionally going down.
(Incidentally, I acutally wrote an embarrasing article arguing in favor of the thesis roland presents, and you can still probably find on it the internet. That exchange is also an example of someone being bad at explaining. If my opponent had simply stated the equivalence between DMU and IMD, I would have understood why that argument about insurance is wrong. Instead, he just resorted to lots of examples of when people buy insurance that are totally unconvincing if you accept the quoted argument.)
I voted this up, but I want to comment to point out that this is a really important point. Don’t be tricked into not getting insurance just because it has a negative expected monetary value.
I voted Silas up as well because it’s an important point but it shouldn’t be taken as a general reason to buy as much insurance as possible (I doubt Silas intended it that way either). Jonathan_Graehl’s point that you should self-insure if you can afford to and only take insurance for risks you cannot afford to self-insure is probably the right balance.
Personally I don’t directly pay for any insurance. I live in Canada (universal health coverage) and have extended health insurance through work (much to my dismay I cannot decline it in favor of cash) which means I have far more health insurance than I would purchase with my own money. Given my aversion to paperwork I don’t even fully use what I have. I do not own a house or a car which are the other two areas arguably worth insuring. I don’t have dependents so have no need for life or disability coverage. All other forms of insurance fall into the ‘self-insure’ category for me given my relatively low risk aversion.
Risk is more expensive when you have a smaller bankroll. Many slot machines actually offer positive expected value payouts—they make their return on people plowing their winnings back in until they go broke.
Citation please? A cursory search suggests that machines go through +EV phases, just like blackjack, but that individual machines are -EV. It’s not just that they expect people to plow the money back in, but that pros have to wait for fish to plow money in to get to the +EV situation.
The difference with blackjack is that you can (in theory) adjust your bet to take advantage of the different phases of blackjack. Your first sentence seems to match Roland’s comment about the Kelly criterion (you lose betting against snake eyes if you bet your whole bankroll every time), but that doesn’t make sense with fixed-bet slots. There, if it made sense to make the first bet, it makes sense to continuing betting after a jackpot.
This comes up frequently in gambling and statistics circles. “Citation please” is the correct response—casinos do NOT expect to make a profit by offering losing (for them) bets and letting “gambler’s ruin” pay them off. It just doesn’t work that way.
The fact that a +moneyEV bet can be -utilityEV for a gambler does NOT imply that a -moneyEV bet can be +utilityEV for the casino. It’s -utility for both participants.
The only reason casinos offer such bets ever is for promotional reasons, and they hope to make the money back on different wagers the gambler will make while there.
The Kelly calculations work just fine for all these bets—for cyclic bets, it ends up you should bet 0 when -EV. When +EV, bet some fraction of your bankroll that maximizes mean-log-outcome for each wager.
Some casinos advertise that they have slots with “up to” a 101% rate of return. Good luck finding the one machine in the casino that actually has a positive EV, though!
On the scale from “saw it in The Da Vinci Code” to “saw it in Nature”, I’d have to say all I have is an anecdote from a respectable blogger:
I’ll give you that “many” is almost certainly flat wrong, on reflection, but such machines are (were?) probably out there.
That move was full of falsehoods. For example, people named Silas are actually no more or less likely than the general population to be tall homicidal albino monks—but you wouldn’t guess that from seeing the movie, now, would you?
That’s why it represents the bottom end of my “source-reliability” scale.
The only relevant part of the quote seems to be:
I’m pretty sure it’s not that unlikely to come up ahead ‘three or four’ times when playing slot machines (if it weren’t so late I’d actually do the sums). It seems much more plausible that the blog author was just lucky than that the machines were actually set to regularly pay out positive amounts.
Ahh, Kelly criterion, correct?
...
*looks up Kelly criterion*
That’s definitely a related result. (So related, in fact, that thinking about the +EV slots the other day got me wondering what the optimal fraction of your wealth was to bid on an arbitrary bet—which, of course, is just the Kelly criterion.)
I’d like to pose a related question. Why is insurance structured as up-front payments and unlimited coverage, and not as conditional loans?
For example, one could imagine car insurance as a options contract (or perhaps a futures) where if your car is totaled, you get a loan sufficient for replacement. One then pays off the loan with interest.
The person buying this form of insurance makes fewer payments upfront, reducing their opportunity costs and also the risk of letting nsurance lapse due to random fluctuations. The entity selling this form of insurance reduces the risk of moral hazard (ie. someone taking out insurance, torching their car, and then letting insurance lapse the next month).
Except in assuming strange consumer preferences or irrationality, I don’t see any obvious reason why this form of insurance isn’t superior to the usual kind.
Well, look at a more extreme example. Imagine an accident in which you not just total a car, but you’re also on the hook for a large bill in medical costs, and there’s no way you can afford to pay this bill even if it’s transmuted into a loan with very favorable terms. With ordinary insurance, you’re off the hook even in this situation—except possibly for the increased future insurance costs now that the accident is on your record, which you’ll still likely be able to afford.
The goal of insurance is to transfer money from a large mass of people to a minority that happens to be struck by an improbable catastrophic event (with the insurer taking a share as the transaction-facilitating middleman, of course). Thus a small possibility of a catastrophic cost is transmuted into the certainty of a bearable cost. This wouldn’t be possible if instead of getting you off the hook, the insurer burdened you with an immense debt in case of disaster.
(A corollary of this observation is that the notion of “health insurance” is one of the worst misnomers to ever enter public circulation.)
Alright, so this might not work for medical disasters late in life, things that directly affect future earning power. (Some of those could be handled by savings made possible by not having to make insurance payments.)
But that’s just one small area of insurance. You’ve got housing, cars, unemployment, and this is just what comes to mind for consumers, never mind all the corporate or business need for insurance. Are all of those entities buying insurance really not in a position to repay a loan after a catastrophe’s occurrence? Even nigh-immortal institutions?
I wouldn’t say that the scenarios I described are “just one small area of insurance.” Most things for which people buy insurance fit under that pattern—for a small to moderate price, you buy the right to claim a large sum that saves you, or at least alleviates your position, if an improbable ruinous event occurs. (Or, in the specific case of life insurance, that sum is supposed to alleviate the position of others you care about who would suffer if you die unexpectedly.)
However, it should also be noted that the role of insurance companies is not limited to risk pooling. Since in case of disaster the burden falls on them, they also specialize in specific forms of damage control (e.g. by aggressive lawyering, and generally by having non-trivial knowledge on how to make the best out specific bad situations). Therefore, the expected benefit from insurance might actually be higher than the cost even regardless of risk aversion. Of course, insurers could play the same role within your proposed emergency loan scheme.
It could also be that certain forms of insurance are mandated by regulations even when it comes to institutions large enough that they’d be better off pooling their own risk, or that you’re not allowed to do certain types of transactions except under the official guise of “insurance.” I’d be surprised if the modern infinitely complex mazes of business regulation don’t give rise to at least some such situations.
Moreover, there is also the confusion caused by the fact that governments like to give the name of “insurance” to various programs that have little or nothing to do with actuarial risk, and in fact represent more or less pure transfer schemes. (I’m not trying to open a discussion about the merits of such schemes; I’m merely noting that they, as a matter of fact, aren’t based on risk pooling that is the basis of insurance in the true sense of the term.)
Intrinsically, the average person must pay in more than they get out. Otherwise the insurance company would go bankrupt.
No reason a loan style insurance company couldn’t do the exact same thing.
‘Rent-seeking’ and ‘regulatory capture’ are certainly good answers to the question why doesn’t this exist.
For one thing, insurance makes expenses more predictable; though the desire for predictability (in order to budget, or the like) does probably indicate irrationality and/or bounded rationality.
What’s unpredictable about a loan? You can predict what you’ll be paying pretty darn precisely, and there’s no intrinsic reason that your monthly loan repayments would have to be higher than your insurance pre-payments.
You can’t predict when you’ll have to start paying.
It’s not predictable when you’ll have to start making payments.
Obviously if you know your utility function and the true distribution of possible risks, it’s easy to decide whether to take a particular insurance deal.
The standard advice is that if you can afford to self-insure, you should, for the reason you cite (that insurance companies make a profit, on average).
That’s a heuristic that holds up fine except when you know (for reasons you will keep secret from insurers) your own risk is higher than they could expect; then, depending on how competitive insurers are, even if you’re not too risk-averse, you might find a good deal, even to the extent that you turn an expected (discounted) profit, and so should buy it even if you have zero risk aversion. Apparently in California, auto insurers are required to publish the algorithm by which they assign premiums (and are possibly prohibited from using certain types of information).
Conversely, you may choose to have no insurance (or extremely high deductible) in cases where you believe your personal risk is far below what the insurer appears to believe, even when you’re actually averse to that risk.
Of course, it’s not sufficient to know how wrong the insurer’s estimate of your risk is; they insist on a pretty wide vig—not just to survive both uncertainties in their estimation of risk and the market returns on the float, but also to compensate for the observed amount of successful adverse selection that results from people applying the above heuristic.
I suppose it may also be possible that the insurer won’t pay. I don’t know what exactly what guarantees we have in the U.S.
Actually, I think that for voluntary insurance, the observed adverse selection is negative, but I can’t find the cite. People simply don’t do cost-benefit calculations. People who buy insurance are those who are terribly risk-averse or see it as part of their role. Such people tend to be more careful than the general population. In a competitive market, the price of insurance would be bid down to reflect this, but it isn’t.
We should form large nonprofit risk pools.
Some insurances are not worth getting, obviously. Like insurance on laptops or music players. But that insurance in general has negative expected utility assumes no risk aversion. If you can handle the risks on your own—if you are effectively self-insuring—then you probably should do that. But a house burning down or getting a rare cancer that will cost millions to treat: these are not self-insurable things unless you are a millionaire.