Economists who buy lottery tickets: Reflections on Less Wrong

Posted on 09/20/11 12 Comments

Faithful reader Robert asked me to respond to an article by Eliezer Yudkowsky of “Less Wrong” called “Outside the Laboratory.” The article deals with the rationality (and irrationality) of scientists. Robert Gressis added that he too would like to see my response, though “it seems like a massive undertaking.”

Always up for a challenge, I read through the article and then began writing a response. However, I found myself getting hung up on a point only a third of the way through the article. That point concerned whether economists can rationally buy lottery tickets. As we will see, this simple example requires some unpacking before we consider the rest of the article in a subsequent post.

Yudkowsky begins:

“‘Outside the laboratory, scientists are no wiser than anyone else.’  Sometimes this proverb is spoken by scientists, humbly, sadly, to remind themselves of their own fallibility.  Sometimes this proverb is said for rather less praiseworthy reasons, to devalue unwanted expert advice.  Is the proverb true?  Probably not in an absolute sense.  It seems much too pessimistic to say that scientists are literally no wiser than average, that there is literally zero correlation.

“But the proverb does appear true to some degree, and I propose that we should be very disturbed by this fact.”

The reader’s interest is piqued. What might this less-than-rational stance of a scientifically minded individual look like? Yudkowsky then provides the following example and it is here that we shall focus:

“Now suppose we discover that a Ph.D. economist buys a lottery ticket every week.  We have to ask ourselves:  Does this person really understand expected utility, on a gut level?  Or have they just been trained to perform certain algebra tricks?”

Yudkowsky seems to assume that there is some inconsistency between the purchase of a lottery ticket and knowledge of expected utility. But what precisely is that assumption? Let’s try formulating “Yudkowsky’s Assumption”:

 (YA) “If an individual really understands expected utility s/he will not buy a lottery ticket.”

I think it is quite clear that (YA) is not in fact Yudkowsky’s assumption because his article is motivated by the very problem that some scientists hold prima facie irrational beliefs and engage in prima facie irrational behavior precisely like that described in the economist example. 

Let’s try a second version of the assumption:

(YA-2) “If an individual really understands expected utility and is psychologically integrated and rationally consistent s/he will not buy a lottery ticket.”

This formulation allows for the economist who understands expected utility and still purchases a lottery ticket due to a lack of psychological integration or other irrational forces.

Unfortunately (YA-2) is false. You see it is very easy to think of cases where an economist understands expected utility, is psychologically integrated and still rationally purchases a lottery ticket. For example, that economist may have an ailing mother in the retirement home who has asked him to purchase a ticket for her. The economist recognizes that the anticipation of the draw provides much needed excitement for his mom’s otherwise dreary life and so he willingly purchases the ticket.

Let’s try another formulation of Yudkowsky’s assumption to see if we can make it come out true.

(YA-3) “If an individual really understands expected utility and is psychologically integrated and rationally consistent s/he will not buy a lottery ticket with the expectation that s/he has a reasonable chance of winning.”

(YA-3) allows for the purchase of a lottery ticket so long as the economist does not believe that ticket has a reasonable chance of winning. While this might seem to provide a true principle, unfortunately one can readily envision situations where (YA-3) is false. And here I won’t bother to mention cases where, for example, only ten tickets were sold. Instead I’m assuming that we’re dealing with a typical lottery where millions of tickets have been purchased. Even in that case (YA-3) is not true. For example, consider the following scenario:

Don the economist’s old high school friend Mack is now President and CEO of the Lottery Corporation. One day Don sees Mack at a coffee shop. Don is unloading on Mack the need to secure capital to invest in his micro-lending agency which is working in Bangladesh. Suddenly Mack chuckes and replies with a twinkle in his eye: “Don’t worry about it Don. Just make sure you purchase a ticket for the ‘Lucky 9′ draw this Thursday.” Don replies incredulously: “Mack, I’m an economist. I understand expected utility.” Mack’s trademark mischievous grin flashes across his face. “Trust me Don. Just buy your ticket for the ‘Lucky 9′ draw.”

Based on this exchange would it be rational for Don to purchase a lottery ticket? Assuming that Don has good reason to trust Mack, it would indeed be fully rational for Don to purchase the ticket. But this means (YA-3) too is false.

So now we come to a fourth and final revision:

(YA-4) “If an individual really understands expected utility and is psychologically integrated and rationally consistent s/he will not buy a lottery ticket with the expectation that s/he has a reasonable chance of winning unless s/he has some overriding reason to believe it is likely that s/he will win the lottery.”

Finally, we have arrived at a principle which seems likely to be true. Of course, I have no idea whether (YA-4) was in fact Yudkowsky’s assumption or not. But this little exercise in finding a defensible assumption does have an important payoff. You see, it reminds us that it is actually more difficult to identify a belief or action as being irrational than we might think. A person may hold a belief or engage in an action which would appear on the surface to strike us as irrational or otherwise inconsistent, but which may be fully rational based on all the knowledge, reasonable beliefs and experiences to which that individual has access. We will have to keep this in mind as we consider the rest of Yudkowsky’s article.

 

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  • Mike Fischer

    When I read that part of the article, the first thing that popped into my head was about those mathematicians that deduce the random number generation algorithms and use it to game the system. :) Obviously that’s not what the author meant, so the very next thing that popped into my head was pretty much (YA-4).

    For me, this isn’t so much about giving the author the benefit of the doubt, as it is about engaging with what the author most likely intended to say. This saves me time because I avoid dealing with what are probably just straw men anyway.

    Of course, as you (Randall) always show, this won’t do when you’re undertaking a formal critique of an argument. Thanks for being precise.

  • http://ingles.homeunix.net/ Ray Ingles

    We will have to keep this in mind as we consider the rest of Yudkowsky’s article.

    Hmmm. Let’s try formulating “Randal’s Admonition”:

    RA-1: We shall have to continually recite the words “it is actually more difficult to identify a belief or action as being irrational than we might think” over and over while we consider the rest of Yudkowsky’s article.

    I think it is quite clear that (RA-1) is not in fact Randal’s command, as few if any readers can both mentally rehearse a phrase and read pr think about an article at the same time.

    Let’s try a second version of the admonition:

    RA-2: We shall have to periodically recite the words “it is actually more difficult to identify a belief or action as being irrational than we might think” from time to time while we consider the rest of Yudkowsky’s article.

    This formulation allows for the reader to have some hope of processing the article while still keeping the concept in mind. Unfortunately, it doesn’t produce careful reflection.

    Let’s try another formulation of Randal’s admonition to see if we can make it work:

    RA-3: We shall have to interpret Yudkowsky’s article in light of the fact that “it is actually more difficult to identify a belief or action as being irrational than we might think”.

    While this might seem to provide a useful principle, unfortunately one can readily envision situations where (RA-3) is unhelpful. Interpreting an article solely in light of a single preconceived intention or attitude can lead the reader to miss important points, or to ignore the author’s intentions.

    So now we come to a fourth and final revision:

    RA-4: Among the concepts we should remember when interpreting Yudkowsky’s article, one of them should be “it is actually more difficult to identify a belief or action as being irrational than we might think”.

    Finally, we have arrived at a principle which seems likely to be useful. Of course, I have no idea whether (RA-4) was in fact Radal’s intention or not. But this little exercise in finding a defensible maxim does have an important payoff.

    It shows that, if one is determined to finely parse a statement in a legalistic manner, rather than accept the author’s obvious intent, it’s possible to make oneself look kind of like a jerk. :)

    • Robert

      I almost didn’t read your comment because it was long. I’m glad that I did.

    • randal

      Ray, I love the work you put into your comment. I just wish it were as analytically penetrating as you think it is. (Okay, to be honest, I’m glad it isn’t as analytically penetrating as you think it is, because if it were I’d be wrong somehow. And nobody likes to be wrong.)

      You think I am engaged in a “legalistic” parsing which ignores “the author’s obvious intent”. I’m not sure what evidence you have to identify what the author’s obvious intent was. After all, Yudkowsky provides no evidence in the article of being aware of all the subtlely nuanced neworks of beliefs and circumstances that can make a belief (e.g. I ought to buy a lottery ticket) or action (e.g. a person’s buying a lottery ticket) to be not only fully rational but completely justified.

      This may be a relatively trivial problem when it comes to an economist purchasing a lottery ticket. It becomes orders more important when the matter concerns an economist (or a biologist, or a physicist) who believes in, say, the God of Christian theism. If Yudkowsky had applied to those “religious” beliefs the same level of nuance I’ve suggested he apply to his analysis of the purchase of a lottery ticket, you would have had a completely different article, and one completely innocuous to a Christian theist.

      • http://ingles.homeunix.net/ Ray Ingles

        Ray, I love the work you put into your comment.

        A little cut-n-paste isn’t so much. :)

        You think I am engaged in a “legalistic” parsing which ignores “the author’s obvious intent”.

        Inventing ‘buying a lottery ticket for his mother’ out of whole cloth is not reading charitably, sorry.

        My own mother occasionally buys a lottery ticket when the jackpot gets large (and I’ve been known to do so as well, while fully grasping the odds involved – I’ve also been known to call the lottery “a tax on people who are bad at math”). She calls it “a license to dream”. It’s purchased for entertainment value, a chance to think about what one would or could do with the money – not as part of a retirement plan, for example.

        That’s the kind of counterargument you could have engaged in, but didn’t.

        After all, Yudkowsky provides no evidence in the article of being aware of all the subtlely nuanced neworks of beliefs and circumstances that can make a belief (e.g. I ought to buy a lottery ticket) or action (e.g. a person’s buying a lottery ticket) to be not only fully rational but completely justified.

        Again, though, you’re not reading the article charitably. I only buy a lottery ticket when the jackpot rises to roughly near the odds of winning (~135 million or so). It’s still not a great bet, but it’s not indefensible. Buying a lottery ticket “every week” has significantly lower utility from an economic perspective, however – roughly 365 times less – and that’s the case Yudkowsky tackles explicitly.

        • randal

          “Inventing ‘buying a lottery ticket for his mother’ out of whole cloth is not reading charitably, sorry.”

          Huh? Are you joking? I provided different examples in which an economist could rationally purchase a lottery ticket. I can easily adjust the example for you so that it is every week. Mom asks her economist son to purchase the ticket every week. The head of the Lottery Corp instructs his economist friend to purchase a ticket every week. In both cases the economist rationally purchases a ticket.

          • http://ingles.homeunix.net/ Ray Ingles

            I provided different examples in which an economist could rationally purchase a lottery ticket.

            All of which posit entities or conditions beyond the situation as stated, and contrary to the plain intent of the thought experiment.

  • Robert

    The massive undertaking is not so much in responding to ‘Outside the Laboratory’, but to grok the sequences and write a theistic response.

    Yudkowsky is better than most at building up a long argument and the sequences capture that. Outside the Laboratory builds on the Map a Territory sequence (which has changed my perception of the world more than anything else I’ve read in the past 2 or 3 years).

    So, a theist would need to devote a lot of time to build a charitable and thorough rebuttal of the any of the sequences, but I think this would be a worthy goal for someone to accept.

    • randal

      Thanks. I’ll finish my critique of this essay first and then back up at your request.

  • http://lesswrong.com/lw/hl/lotteries_a_waste_of_hope/ Eliezer Yudkowsky

    Of which I would reply that counterargument requires probability, not just possibility. If all the possible exceptions are improbable, the argument carries.

    • randal

      Eliezer, thanks for joining us.

      We judge what is probably true based on what we believe is actually true.

      • Robert

        Randal, you added counterfacturals that would change the probability calculation if we believe them true and count them as evidence. Absent these counterfactuals (as the story is told), it’s rational to wonder if this person understands expected utility on a gut level.