The Friedman-Savage Utility Function

by on November 21, 2006 at 6:02 am in Economics | Permalink

For a long time I didn't think much of Milton's seminal 1948 article on expected utility theory, but it turns out the joke is on me...

The Friedman-Savage piece starts with an obvious puzzle: why do people both buy lottery tickets and insurance against losses?  That would seem to make them both risk-loving and risk-averse at the same time.  The proffered answer is simple: part of the utility function is concave, and part is convex.  Across the lower range we wish to play it safe, but above a certain margina we are willing to take gambles (by the way, here is some evidence, and why it might follow from market constraints).

For years this approach rubbed the "foundationalist Tyler" the wrong way.  "Surely there is a more general approach which will allow us to derive both behaviors from a few axioms concerning risk and utility.  We can’t just postulate arbitrary shifts in the curve across the utility space.  Maybe both parts of the curve follow from the "temporal resolution of uncertainty," that missing variable from so much of expected utility theory.  The Friedman-Savage approach will someday be seen as a diversion from the path which led to truth."

Many articles explored these routes, most notably Mark Machina’s 1982 piece on generalized expected utility theory.  None of them caught on.  A subsequent dose of empirical and experimental work indicated that behavior toward risk is strongly context-dependent.  Neuroeconomics implied that different decisions in fact may stem from different parts of our brain, thereby challenging the assumption of a unified agent.  Probably there is no overarching approach to all of the so-called violations of expected utility theory.  People simply behave differently toward risk in different situations.

In other words, Friedman and Savage were ahead of their time.  This is no accident, but rather it stems from Milton’s wise pragmatism, and from his general lack of interest in foundations.  He also never explained "why people hold money," or "what money really is," yet he charged ahead with monetary theory and indeed monetary policy.  The monetary foundationalists have been just as unsuccessful as the utility foundationalists.

1 S November 21, 2006 at 8:23 am

A lottery tickets cost $1.Your house $60ks.
Why is irrational to waste $1 and protect a $60ks
investment?.People is afraid of planes and not of cars, and the latter are riskiers.
In the first case the rule is de minimus non curat.In the second misinformation or misperception

2 Daniel Elstein November 21, 2006 at 9:18 am

I’d have thought that the Friedman-Savage approach is actually too foundationalist. People play lotteries because gambling is fun. Asking why people buy lottery tickets is like asking why they play chess. There’s no need to postulate a convexity in the utility function with respect to money in order to explain this behaviour; rather, we should say that the decrease in expected utility attributable to the purely monetary aspect of the gamble and the concavity of the utility function is outweighed by the gain in utility from enjoyment of the game (e.g. the excitement of watching the draw). Whilst there may be counterexamples to decreasing marginal utility, playing the lottery isn’t one of them.

3 theCoach November 21, 2006 at 10:11 am

He also never explained … “what money really is,”

I have been wondering about what a completely digitalized form of money might be able to accomplish. It strikes me as potentially very useful, and possibly much more efficient – taxes take on a different light. Is there any research going into the possibilities?

4 BDK November 21, 2006 at 10:41 am

This point has essentially been made by Keith and Daniel (some risk is fun, some isn’t, and fun has utility; my apologies if this is way off), but let me add my gloss.

The fun of buying a lottery ticket is made up of many parts, different for different people. For example, people like to fantasize. While we could certainly fantasize about winning a $200 million Powerball while watching the numbers drop on TV, paying the cost of entry, $1, makes it feel more concrete. It also adds suspense and gives people something to talk about with folks at work.

I don’t think many people fantasize about not losing the value of their house.

That said, presumably different sorts of gamblers have different brain structures and different motivations. The guy who buys $50 in lottery tickets a week probably gambles for a different reason than my friend who buys a single ticket every time he visits his country house upstate.

Really, we’re all big dumb animals programmed by millions of years of evolution to do things because they make little bubbles of hormones pop in our heads, without regard for convex and concave curves.

5 Daniel Elstein November 21, 2006 at 11:21 am

On Michael’s point that gambling isn’t always fun: probably what happens is that taking a risk gives you a buzz which initially outweighs the disutility of monetary risk; but then you get desensitized so that you can no longer get the buzz when risking only small amounts of money. Some people will by that stage have become addicted to the buzz so that they can’t stop and gamble greater and greater amounts, even when they can see that the pleasure to be got from gambling is insignificant compared to the potential losses, and indeed the fear of those losses overwhelms the pleasure.

There are two things to explain: why people gamble in the first place, and why they carry on gambling when they know that it’s pathological. I think that Friedman and Savage are only responding to the first question: they want to explain how gambling can make sense. In the pathological case the whole point is that the behaviour is irrational and is known to be so by the agent, so there’s no question of rationalisation. Instead we need a psychological (neurological?) explanation of addiction. It’s only in the case of the casual, non-addicted gambler that the shift-in-the-utility-function explanation is remotely plausible. So the addiction case seems like a red herring here.

6 Raging Ranter November 21, 2006 at 12:55 pm

Housing insurance allows for the removal of a small, though real, risk of sustaining a very large loss. The cost of removing this risk (i.e. the premium) is known and certain.
A lottery ticket allows for the introduction of a small, though real, opportunity for fantastic gains. Once again, costs are certain.

I may get enjoyment (utility) from trying to walk a tight-rope that is only two feet off the ground. However, the exact same activity performed 200 feet in the air with no safety net gives me no utility whatsoever, and in fact, entails considerable risk.
Fear of death (or losing one’s house) can alter one’s utility very quickly.

7 jim November 21, 2006 at 4:29 pm

The Friedman-Savage article was an extremely innovative one without a doubt, but was fundamentally flawed. Through a series of significant modifications, however, it led eventually led to a model: 1) devoid of clear “deficiencies”; and 2)that provided operational predictions. Here is the story:

In his 1952 article in the JPE entitled “The Utility of Wealth”, Harry Markowitz obeserved that the Friedman-Savage model was deficient because its perdictinos for individuals with wealth endowments outside a small neighborhood of the first inflection point were inconsistent with reality: 1) a person endowed roughly midway between the two inflection points must be willing to take an actuarially fair bet with a large variance; and 2) a person endowed close to the second inflection point (“almost rich” by Friedman-Savage perspective) must be unwilling to buy insurance with an assured, actuarially unfair small loss to avoid a potentially large loss. To resolve these anomalies, Markowitz postulated a thrice inflected utility of wealth function and postulated that whenever an individual’s wealth was at its “customary” level it would be at the second inflection point. The problem with the Markowitz model was well explained by Alchian in his 1953 AER paper(“The Meaning of Utility Measurement”); Alchian explained:

“Markowitz recognized that until an unambiguous procedure is discovered for determining when and to what extent current income deviated from customary income, the hypothesis will remain nonverifiable because it is not capable of denying any observed behavior.”

To operationalize the Markowitz model, Coelho and McClure in their 1998 JEBO article (“Social context and the utility of wealth: Addressing the Markowitz challenge”) augmented the Markowitz model by casting decisions in terms of a utility function not only in terms of own wealth, but also: 1)the wealth of one’s peers; and 2) the individual’s status within his peer group.

8 Jason Briggeman November 22, 2006 at 12:16 am

I know economists hate putting things in the utility function, for fear of logical circularity problems, but hey, something’s gotta be in there, right?

If economists would concern themselves with magnitudes, this problem would go away. It’s okay to say everything is or can be in the utility function if you say something about the extent or magnitude with which it matters. There’s no logical circularity there.

9 aaron November 22, 2006 at 9:15 am

I don’t think you argument is correct. A person who buys insurance and plays the lotto is not conflicted, they are both low risk. A better example of conflict is the uninsured who plays the lotto. You have a person who is not risk averse, but who chooses to play a game with a near certain outcome.

10 jim November 22, 2006 at 11:07 am

People who play lotteries pay actuarially unfair prices that increase their exposure to risk. People who insure pay actuarially unfair prices that reduce risk exposure. To explain a person who engages in both types of activities with an expected utility theory, Friedman-Savage recognized that an inflected utility function was the key; Harry Markowitz greatly improved upon the inflected utility approach and challenged future researchers to make this approach operational by addressing the issues regarding the re-ranking of utility that follow changes that cause “customary wealth” to deviate from current wealth. Were these Nobel prize winners (Friedman and Markowitz) on the right track? Yes, I think they were.

11 michael webster November 23, 2006 at 3:57 pm

jim wrote:
michael wrote:

“People don’t gamble because gambling is fun: have you ever been in a casino and smelt the overpowering scent of fear?”

This is silly. The waitresses who are serving “free” drinks are seductively clad in heels, short skirts, and cleavage revealing tops. Casinos reek of sex and booze.

As P.J. O’Rourke put it “Las Vegas has sleeze down to an art form”.

Of course people go to Vegas to have fun: Why else are casinos so lavish? Why else are there so many elaborate shows? Why else are there so many great places to eat, drink, dance, swim, tan, etc., etc.?”

1. Have you been in a casino and observed the “fun” gamblers? My personal observation is that they are not having fun.

2. Of course the trapping of fun would have to be present if people a) were not having fun, and b) could not admit.

3. Losing, which is what most people do, is not fun.

12 jim November 23, 2006 at 9:57 pm

Michaels wonders:

“Have you been in a casino and observed the “fun” gamblers? My personal observation is that they are not having fun.”

Yes, Michael, I have been in many casinos, many times. I am one of what I persoanlly would deem “fun” gamblers. I enjoy playing cards, just like billions of people worldwide. What I am doing in a casino is paying for the right to play cards. I typically play blackjack for low stakes at a casino offering single deck play. It is fun to place initial wagers of varying size, double down bets, split bets, etc. depending upon my best guesses about the “richness” of the remaining deck held by the dealer. My objective at the casino is entertainment. As I said, I enjoy cards, and as I implied in my previous post, I think that the many compliments (“free” drinks, attractive waitresses, etc.) to gambling that casinos provide are entertaining to lots of people. It is possible that you do not enjoy, for example, playing cards in a casino. Many do not share my preferences; myy wife certainly doesn’t. But so what? Preferences are diverse. We don’t all like the same movies, or foods. So what? The fact that you do not have fun in casinos doesn’t at all mean that lots of people are not. I think that you are vastly overgeneralizing from your personal preferences in suggesting that no one is having any fun in casinos.

A state lottery, where a modestly clad person on TV pulls numbered balls to determine winners is much closer to pure gambling than is a casino because it is much less laden with entertainment margins.

Hence, I do not think that the entertainment theory of gambling applies with equal force (or much of any force really) to lotteries.

13 jim November 24, 2006 at 8:19 am

Stan,

Thanks, I’ll have a look at the reference; but it sounds a lot like Alchian’s article “The Meaning of Utility Measurement”. I am not unsympathetic to Austrian perspectives on a variet of points.

However, again, I see no error in Friedman’s methodological view that assumptions about unobservables can be useful in deriving positive propositions in economics.

14 Anonymous October 14, 2008 at 2:34 am

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