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I have recently been involved in a number of discussions with people with little or no background in mathematics when we considered a problem of the following shape.

A random event is going to happen in the future (e.g. a number $X$ will be chosen uniformly at random from the range $\{1,2,\dots, N\}$). We have no control over the outcome of this event, but we can choose how we prepare for it. For simplicity, suppose that we have two options $A$ and $B$ and that if we choose $A$ then after the event we get $f_A(X)$ £, and likewise if we choose $B$ we get $f_B(X)$ £, where $f_A$, $f_B$ are two functions that are known to us. For practical purposes, assume that $f_A,f_B$ take relatively small values (so that the utility of money is approximately linear), but that the event only happens once. It feels natural to me (and presumably to most mathematicians) that to determine the better option, one needs merely compute the expected values $\mathbb{E}(f_A(X)), \mathbb{E}(f_B(X))$; if $\mathbb{E}(f_A(X)) > \mathbb{E}(f_B(X))$ then go with A, else go with B.

To make things concrete, consider a lottery where we can win $10$£ with probability $1/10$, and a lot costs $x$ £. Then the expected value associated to buying a lot is $1 - x$ and the expected value associated to not buying none is $0$. Putting aside any other factors (like the bother of actually doing to buy a lot, the thrill of waiting for the result, etc.), I would say that it's worth it to buy a lot as long as $x < 1$ (with situation perhaps being slightly non-obvious close to $x=1$), and I would say that this is obvious.

However, some of my non-mathematical friends would disagree. The reasons strike me as fallacious, but I have not good counterarguments. Here are some of them:

  1. Then expected value is a mathematical, abstract construct. Why would I care about it for making everyday decisions?

  2. If I buy a lottery ticket, I will be $x$ £ behind, unless I win. Most probably I won't win, and $10$£ isn't that much money in the first place. Hence, most probably it's better not to buy a lot.

  3. Spending $x$ £ is not significant, and I can win $10$ £, which is significant. It's unlikely that I'll win, but you never know. Hence, I should play (The opposite of the above)

(The two latter arguments loosely follow the pattern: "It's random, therefore you never know. Therefore, you should rely on gut instinct/argument only involving payoffs but not probabilities.")

What is a good argument showing that expected value is the right way of comparing different options? (Unless of course it's not, in which case I should ask for an argument to correct my mistaken belief.) The ideal answer should be accessible to a layman, and not rely on the assumption that the process could be repeated.

P.S. Apologies if this is too soft a question for this site, or too opinion based. Feel free to close or delete if such is the will of the community.

EDIT: To clarify, assume that the payoffs $f_A, f_B$ already cover all consequences, e.g. one would be indifferent between getting the outcome if $X$ happens and strategy $A$ is used and receiving $f_A(X)$ pounds. In the lottery, suppose that all other factors are negligible, or that the payoffs are shifted to account for other factors (e.g. if the lot costs $1$ £, and time and energy needed to buy it are worth $.5$ £, then take $x = 1.5$ above). I'd be speaking about utility instead of money, but that involves defining what utility is, and how it can (in principle) be quantified.

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  • $\begingroup$ Take a look at a good book on decision theory. I would recommend Notes On The Theory Of Choice by David Kreps and Theory of Decision under Uncertainty by Itzhak Gilboa. $\endgroup$ – Michael Greinecker Jun 15 '14 at 23:00
  • $\begingroup$ Do not assume that utility is the only component of desirability. $\endgroup$ – Eric Towers Jun 15 '14 at 23:14
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One of the potential problems with expected value is that it's easy to take the expected value of the wrong thing, and get information that isn't really helpful. For example, one hundred dollars means a lot more to me if I only if I only have one hundred dollars than if I have a billion dollars. It's the "utility" that I care most about, rather than its numerical value.

Risking everything I own for a 50/50 chance to receive three times as much sounds like a great deal if you only look at the expected quantity of "stuff", but it's a terrible deal if you look at the expected value of the utility of stuff.

I will boldly assert that the expected value of the right quantity is always the right thing to consider. But the right quantity is usually not the obvious quantity. (although the obvious one may be really close to right in the case where you have many decisions that combine additively)

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  • $\begingroup$ What if the right quantity is not quantifiable? For instance, if it represents a subjective experience. $\endgroup$ – Ryan Reich Jun 15 '14 at 22:28
  • $\begingroup$ I absolutely agree. That's why I mentioned the payoffs being small (presumably, close to $0$, the utility of $x$ pounds is proportional to $x$). $\endgroup$ – Jakub Konieczny Jun 15 '14 at 22:30
  • $\begingroup$ @Feanor I don't think this is affected by small payoffs. Gambling may be anywhere from "neutral" to "sort of fun" for you, depending on your mood at the time, which is neither definite nor quantifiable, but also not large. $\endgroup$ – Ryan Reich Jun 15 '14 at 23:17
  • $\begingroup$ @Ryan: Then you're not trying hard enough. :) I don't claim quantification is easy, just possible. e.g. as a trivial example, if you're faced with the problem of "Do I do X?", there are two outcomes Y and Z, and your decision method is "If the probability of getting outcome Z is greater than P, then I'll do it", then you can numerical assign quantities to Y and Z such that you'll get the same decision by looking at the expected values. $\endgroup$ – Hurkyl Jun 15 '14 at 23:26
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Is it the right way? If they want to buy just one lottery ticket, it hardly matters what their average gains are; if the price is acceptable to them and the jackpot is high enough, then buying one ticket in lieu of, say, a box of candy, is worth the cost (as long as they didn't want the candy badly). That is, the money they spend is essentially nothing to them and the money they'll get is either nothing (which is what they would get anyway) or huge, and either way, maybe they have a little fun.

If they want to make a habit of it, of course, then not to accept the significance of the average value is to deserve to pay the tax on the innumerate that the lottery tends to represent. Unless, that is, they just enjoy gambling.

In short, this kind of economic decision has more than just pure mathematics behind it.

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  • $\begingroup$ +1. Expectation is not a measure of how much one can expect to get from one trial or even a small number of trials, but from infinitely many trials, and for this reason, even when the number of trials is large but finite, with sufficiently large variance, one can play many times and still lose a game with positive expectation. $\endgroup$ – heropup Jun 15 '14 at 22:49
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Using expectation makes theoretical sense only for repeated games. Minimax is the more natural and standard decision rule when playing a single game.

Thus I would say that the approach number 2 in your question is the right one (from the point of view of a game theorist).

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  • $\begingroup$ But the outcome is random (not decided by a foe after we choose the strategy). Are you sure that minmax is the right way to go? $\endgroup$ – Jakub Konieczny Jun 15 '14 at 22:28
  • $\begingroup$ "The right way" depends on the utility of the player. However, minmax is the standard approach here (IIUC). $\endgroup$ – sds Jun 15 '14 at 22:44
  • $\begingroup$ Actually, von Neumann and Morgenstern invented von Neumann and Morgenstern expected utility theory... $\endgroup$ – Michael Greinecker Jun 15 '14 at 22:57
  • $\begingroup$ @MichaelGreinecker: I think expected utility is more useful for repeated games. $\endgroup$ – sds Jun 15 '14 at 23:08
  • $\begingroup$ That may well be, but expected utility is certainly standard in game theory. Minmax is a solution concept for zero-sum games. In decision theory, it has only really be advocated for by Abraham Wald in the context of statistical decision theory. $\endgroup$ – Michael Greinecker Jun 15 '14 at 23:16
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This is where utility theory and psychology comes into play. To expand a bit on what was previously said. Firstly, and most importantly, human beings are not purely rational (sorry, Spock) and different people have different perceptions of risk. In general, the pain at losing money is not a linear function. People are more than 1000 times as averse to losing \$5000 than \$5. As such, people are more willing to pay a small amount for a thrill–the ability to dream that maybe they'll win the lottery and by a jet and a private island in the Pacific–even though in expectation they are losing money. While in insurance and other cases when pooling risk, it makes sense for various reasons to look at the expectation as we are dealing with groups of people, when speaking about any one person, their own psychology of risk-reward will dominate the mathematical expectations. In other words, while basic probability may say one thing, any given person's actual response to the situation may not reflect that, and properly so depending on their own risk-reward valuation.

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