What's the name of the logical fallacy all lotteries are based on?

People choose a gamble, where they are at disadvantage, downplaying or outright ignoring the cost blinded by desirability of the (unlikely) positive results?

Or a variation in the "undesirable" direction: requiring full-body x-ray scanners in all airports, to prevent terrorist attacks; the number of people dying from cancer as result of the x-ray exposure exceeding the expected toll of terrorists, but the scanners being perceived a necessity due to the "increased security" they provide - highly-visible terrorist attacks vs "silent" deaths by cancer.

Edit: As Dave notes in the comment, these by themselves aren't fallacies because they are not arguments, but fallacious arguments may be made building upon that irrationality:

  • by presenting a false claim: "If you want to be you rich, you should buy our lottery ticket" or
  • by presenting a technically true claim while downplaying or hiding its actual cost: "The body scanners will significantly reduce the chance of a terrorist bringing a bomb or weapon onto a plane, and they are almost completely harmless". In this case the actual, false argument ("we should implement them") is not stated explicitly - it's implied by desirability of the stated outcome.
  • These examples are not fallacies; they are aspects of the irrationality of typical human decision making. – Dave Sep 24 '15 at 12:37
  • @Dave: See my edit. – SF. Sep 24 '15 at 12:51
  • "If you want to be rich, you should buy a lottery ticket." This is not irrational at all. To buy one ticket increases your odds of becoming rich from zero to a real number, as I note below. The contrary assertion: "If you want to be rich, you should not buy a lottery ticket" is an irrational non sequitur. – Nelson Alexander Sep 24 '15 at 20:21
  • @NelsonAlexander: Which is more likely to be true: "Spending your whole income on lottery tickets will make you a millionaire" or "Spending your whole income on lottery tickets will make you a beggar"? – SF. Sep 24 '15 at 22:35
  • Certainly the latter is, if not necessarily "true" (winning is not impossible) far more probable. I am only arguing, as below, that precisely one ticket is rational and each subsequent ticket is increasingly irrational. As you suggest. – Nelson Alexander Sep 24 '15 at 22:41

Hypothetical example: "the scanners look good but cause more harm than they prevent."

This is a "triumph of style over substance".


Noting about gambling, humans have a propensity to "discount the future". This isn't a fallacy though.

The human instinct to choose instant gratification — known as future or delay discounting

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  • I'm not entirely convinced. The "style over substance" asserts that good presentation of X asserts X is true. In this case, we have something else: "In order to obtain X, you must do Y" (which is true, but value/probability of X and cost of Y are misrepresented or obfuscated). Unless one boils it down to argument of "X is worth more than Y" in which case I'd agree - but often that claim doesn't make any explicit appearance, depending on audience's inherent inaccuracy in appraising the costs/chances instinctively. – SF. Sep 24 '15 at 12:27
  • I don't think the examples in the wiki link are helpful. I included the link only to show that "style over substance" is a recognised fallacy. I have heard "triumph of style over substance" applied many times when someone has despaired at appearance winning out over the truth of some matter. It fits fine as an answer to your question. – Chris Degnen Sep 24 '15 at 12:47

There is no fallacy, just two perverse incentives in Capitalism, deriving from the fact that the utility of money is not fixed, and its marginal rate is not constant. (The same is true of power, safety, etc. when they are conceived of in terms of fungibility and rates of exchange.)

(Responding to the nature of the comments, let me be clear about what I mean by perverse incentive:

Capitalism, since it can state the value of objects only at the point of sale, has to use approximations of value that can be expressed a that point. The resulting gap between the approximation and the real value creates cognitive dissonance (as it should, being false). This in turn generates alternate psychological values. Those then create their own artificial utility by extension. And that gets served directly in spending decisions, contrary to the actual efficiency of money.

Modern politics is capitalist. We 'pay' attention, we 'spend' time, we 'save' lives, there are 'trade-offs'. There is a capitalist construction of value throughout our planning logic. It has the same difficulty with perverse incentive that all instances of capitalism have.

We assume constantly that the capitalist construction of exchange rate itself is the real representation of value. This is a fallacy. And that is far more interesting than this question. But it is not really germane to the answer, which is tied directly to a context of rating value in terms of exchange.)

Capitalism often values labor according to the time invested in earning it. Of course, that is not realistic, as productivity varies hugely, and real discovery or artistry is largely independent of the time invested. But it encourages us to value our time according to the rate of earning.

That means the utility of money is related to how hard you have worked for it. And in that frame of reference, the lottery can makes sense, especially for those whose labor is otherwise poorly paid.

There is a small likelihood of a big payoff with minimal effort (so the utility is a small number, times a big number, then divided by a very small number, which can still be a large number.) Winning would affirm a high value for your personal effort, contradicting, perhaps, negative messages that your time is not valuable. (God rewards you even if your boss doesn't.)

Capitalism also binds many of us to endless debt. Again, this is not too logical, as it injects a lot of fake money into the system disguising genuine value, and enables catastrophic default (witness Sept 2007). But it allows for money to be created out of nothing when it is needed, and can be used to stabilize markets. So it is the basis of the entire Federal Reserve system.

Given that frame of reference, the utility of money may take a sharp upward turn when it enables you to free yourself completely from some given obligation -- say, your mortgage. So there again, there is logic in your investment. Even though it is not a good investment in sheer monetary terms, the utility of the money is higher than the utility of the cost. (Being independently wealthy is far better than daily living, in the imaginations of many.)

Capitalism also pays premiums to compensate suffering.

More intricate forms of gambling involve getting paid to do something enjoyable. So if your value for money is related to how unpleasant it was to get ahold of it, this is a reasonable motivation to spend time and money in this way.

The same kind of utility computations apply to the scanners. Our culture claims that dying at the hands of criminals is far worse than simply dying. There is some extra negative value to being wronged by the criminal intent. So we keep the police armed with guns even though they kill innocent people regularly by mistake. It is somehow better to be shot by accident than on purpose, and the sense of safety seems worth the risk (to the culture in general, if not to me).

It also exaggerates the value of deaths that happen together. A war is somehow worse than the equal number of people who might die slowly of a pollution problem. This is not an error, it is a choice of values. So preventing 200 deaths that would happen together can be worth causing 2000 deaths that are not grouped. Because grouped deaths subjectively cause more disquiet in the population.

Disagreements about relative situational or marginal utility are not fallacies.

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    Comments are not for extended discussion; this conversation has been moved to chat. – virmaior Sep 27 '15 at 12:12
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    I've moved the comments to chat. Please don't argue in the comments. Instead post answers of your own. If you find yourself arguing with the person who asked a question, ask whether there's a problem with the question but don't waste your keystrokes in comments. – virmaior Sep 27 '15 at 12:14

As Dave has pointed out in his comment to your OP, it is not technically the case that your examples represent fallacies. Such beliefs/"fallacies" are related to The Law of Truly Large Numbers which states that, give a large enough sample size, anything, no matter how outrageously unlikely, will likely happen.

In the case of the lottery, if the odds of winning are, say, fourteen million to one, then if fourteen million tickets are sold, each using a different combination, then all combinations are used and therefore a winner will occur with certainty. This does not in anyway enhance the outrageously unlikely event that your ticket is the winner.

The odds of a particular instance of an X-ray scan causing cancer are remote, however given enough such scans, the instances of resulting cancers would appear to be inevitable.

Similarly, the preposterously large odds that all the necessary events occur in order to create intelligent life in the universe may be matched by the vast size and age of the universe. Yet many naively insist that the fact that intelligent life exists on earth implies that the universe must be teaming with life.

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  • The probability of winning the lottery can be calculated quite precisely. But this cannot be applied to the universe, I believe. Calculating the probability of "life" in the "universe" compares two single cases. I believe the inductive case that life here means life elsewhere is arguable as strong. The illogic is not that "someone will win a lottery" but that "I will win this lottery." So I would argue that Law of Large Numbers is not really a good candidate for an answer here. – Nelson Alexander Sep 24 '15 at 18:14
  • @NelsonAlexander Good point. The example of intelligent life elsewhere is not a good analogue in this context. I'm not sure I can agree with you about the applicability of the Law of Truly Large Numbers. Have a looks at this article scientificamerican.com/article/… which explains the role of this law as applied to the lottery. The "In Brief" box may suffice if you wish to avoid the entire article. Yes we can calc the odds precisely. The surprise is that anyone ever wins. – Nick Sep 24 '15 at 21:10
  • Interesting, but LOTLN only says "someone" will win the lottery. The fallacy would go like this: Someone will win the lottery. I am "someone." Therefore I will win the lottery. Not sure what that false identity is called. But "someone" needs to be unpacked. Even so, as I note above, according to probability, one lottery ticket is arguably rational, but two, etc. is not. – Nelson Alexander Sep 24 '15 at 21:31
  • @NelsonAlexander Yes, that is exactly what the fallacy says. Perhaps I have misunderstood the OP. On a personal note, my brother has mental health problems and I look after his money for him. I give him $200 a week in spending money and some weeks he'll spend $150 on scratch cards and lottery tickets!!! I say to him, "for f--- sake, you knob. Would you stop! Please!" and he will reply, "but I won $5.". When I say "Yes, but you spent $150", his response is a puzzled look accompanied by holding up the winning $5 card. It's a big problem. I want him to be happy. It's a tax on the dumb. – Nick Sep 24 '15 at 22:34
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    @NelsonAlexander Your comments about hope are very potent in my brother's case. Hope seems to make him happy. It's the Christmas morning crash he experiences each time that never seems to sink in. Honestly, how someone can spend $150 a week on the lottery and only $50 on food and still weigh 320lbs!!! is a mystery to me. I weigh 132lbs so it isn't a family issue. Anyway, the system is telling me to avoid chat so we'll have to leave it at that. – Nick Sep 24 '15 at 23:45

If you want to be you rich, you should buy our lottery ticket

Is marketing rhetoric like

Persil (a brand of washing powder) can wash clothes whiter than white

Is a fallacy only if taken literally; but generally it's not expected that one should do this.

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The reasoning behind buying a lottery ticket is not a fallacy, as others have pointed out. Nor is it mathematically "unreasonable."

If you buy one lottery ticket, you have increased your odds of winning, say, $10 million from zero (O) to, say, 1 in 10 million. This is a technically "infinite" increase in your odds of winning the money, versus a tiny decrease in your probable loss relative to your lifetime income.

It is the second, third, etc., lottery ticket that is "unreasonable." Your wagers and probable losses double each time, increasing exponentially, while your odds of winning increase at a minimal and calculable rate. The same would be true for multiple tickets in a single lottery, or for single tickets in multiple lotteries. Only your first ticket goes from zero to a real number.

I offer this merely by way of provocation. There are surely counterarguments once the dollar "value" of the ticket has been properly weighted, if possible, relative to some probable lifetime income. However, I do think it is a valid argument. Unfortunately, I do not know enough probability theory to formalize it properly or provide you with a covering term.

To conclude, in my reconfiguration, one answer might be the confusion of Exponential Versus Linear Growth, applied in this case to your odds of winning. This is a common error in personal financial decisions of all sorts... and in the political-economic compounding of "returns" on capital versus linear growth in wages.

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  • No, your odds of winning increase linearly with your investment, and the value of winning remains fixed. Until the odds times the utility of winning is exceeded by the utility of the cost, each additional ticket is as rational as the previous one. – user9166 Sep 24 '15 at 22:30
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    @jobermark: "is as rational as the previous one" - or as irrational. Because if you expand that up to all your available income - or (especially) to the total pool of the tickets, the expected outcome is definitely undesirable. Lottery is a game of negative sum, and not only that, comparing to many other methods of multiplying your income by a random factor, it's an especially lousy one, with expected value way below what you can make, say, playing Blackjack optimally. So even if you are in a situation where randomization of your wealth is desirable, lottery is a poor choice. – SF. Sep 24 '15 at 22:42
  • None of that makes the answer any more logical, so what is your point? I did not say this was a good idea, just that it is not outright fallacious. – user9166 Sep 24 '15 at 22:47
  • I agree basically. Your tiny odds increase linearly but your accumulating ticket purchases (exponential or not) can only improve the odds in inverse relation to your actual winnings (with ticket costs subtracted). Presumably the lottery takes some vig, so you never even reach 100 percent. My main point is that the "first ticket" is rational (odds leap from zero to some real number), the subsequent ones are increasingly irrational. – Nelson Alexander Sep 24 '15 at 22:54
  • @jobermark: "If you want to be a millionaire, you should buy our lottery ticket" is at the very least a False Dilemma fallacy (suggesting the choice is between playing the lottery and not playing the lottery). But the root of the problem reaches deeper: randomization of wealth through a negative sum game is hardly ever rational. You'll need a quite contrived situation where it would be a rational choice. – SF. Sep 24 '15 at 23:05

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