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2. Wobbly Utility

The Insurance/Gambling Paradox

The question:

Do you own an insurance policy? Is your car insured against accidental damages? Does your house have insurance against fire and hurricane damage? Is your property covered against theft and burglary? Do you have health insurance?

And do you, dear reader, occasionally buy a lottery ticket? Visit a casino? Enter a sweepstake?


The paradox:

Generally, people are risk averse. Most people are willing to pay a small amount of money to safeguard their property. Homeowners buy insurance for their personal belongings against most imaginable, if unlikely, events like fires, earthquakes and hurricane damage. Car owners insure their vehicles against theft and accidents. Of course, most people also pay monthly premiums for health insurance, in order to avoid the enormous costs of hospitalization. One is willing to pay small amounts of money in order to avoid the risks of losing, or having to pay, much larger sums, even though the insurance premiums are higher than the expected losses. It is the prudent thing to do.

However, very many of the same people buy raffle tickets, put coins into slot machines, go to the horse races, in exchange for the unlikely event of winning the grand prize. They are willing to pay small amounts of money to assume risks even though the costs of participating in lotteries or playing with slot machines are higher than the expected wins.

So, here’s the paradox:  since one must pay above the expected loss to avoid risk, and more than the expected win in order to assume risk, why do many of the same people who insure their belongings, also visit casinos and place bets? On the one hand, they pay insurance premiums in order to avoid the risks of unlikely events; on the other hand, they buy lottery tickets in order to assume risks for unlikely events.


People value additional money differently, depending on whether they are poor or rich. (See the chapter on the St. Petersbug Paradox.)

Alice is a well-to-do businesswoman; additional money increases her utility but it does so at a decreasing rate. On a graph, this can be depicted by the utility curve which always rises but bends downwards. This shape of this utility curve is the underlying reason why people are risk-averse and why they buy insurance. (See Figure 1.) Hence, she does the prudent thing and buys insurance against losses. She does not gamble, never.

But many people do gamble. What’s going on?

The economist Milton Friedman and his colleague at the University of Chicago, the statistician Leonard Savage were perplexed by the paradox.[1] “Offhand, it seems inconsistent for the same person both to buy insurance and to gamble: he is willing to pay a premium, in the one case, to avoid risk, in the other, to bear risk,” they wrote in 1948. Moreover, the paradoxical phenomenon is by no means rare, they remarked, but so pervasive that “[m]any governments find…lotteries [to be] an effective means to raise revenue.”


Friedman and Savage concluded that the fact that people gamble can be explained if their utility curve has a ‘rather special shape’. Now, what is this rather special shape?

We saw that well-to-do Alice has a regular, upward-sloping, downward-bending utility curve. And as was shown in Figure 1, the fact that the utility of money increases but at a decreasing rate, explains why people avoid risk and why they buy insurance. But another shape is also conceivable.

Let’s take a look at the behavior of Bob, a workman who has 30,000$. Like all of his colleagues, he is risk averse and has purchased insurance against the loss of his possessions. But he is not happy. If only he had another 5,000$, he could make the downpayment for a home in a desirable neighborhood, thus jumping into the middle-class. So, sober, risk-averse Bob decides to go to the casino to try and ‘make’ more money. He knows full well that on average the patrons of casinos lose money. Nevertheless, he gambles. Why?

Well, between a wealth of zero and about 30,000$, Bob’s utility curve has the the normal, upward-sloping, downward-bending shape which indicates that he buys insurance to safeguard the value of his possessions. However, between 30,000 and 35,000, every single dollar gets him closer to his aim of reaching middle-class; every additional dollar gives him more utility than the previous one. Hence, in that region, the utility curve increases at an increasing rate; it bends upwards. And at wealth levels where the utility curve bends upward, people seek risk. (See Figure 2.)

 Beyond $35,000, having reached his goal, Bob is risk-averse again. So, there’s a wobble in his utility curve! Bob, the lower middle-class worker insures his belongings in order not to descend into poverty in case of a calamity, but, simultaneously participates in a lottery so as to rise to middle-class if lucky. Friedman and Savage claim that wobbly utility curves are the reason for the simultaneity of gambling and insurance.

Paradox solved!


Technical supplement:

Figure 1:

Alice has assets worth $100,000. A gamble provides a fifty-fifty chance of winning either $2000 or $3000. /// to follow


Figure 2:

Bob gambles ////to follow

[1] Friedman would win the Nobel Prize in Economics in 1976.

Corrections, comments, observations: