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What is the lottery paradox?




 My recent reading has included a book on epistemology,  It is an anthology of essays on the "lottery paradox," edited by Igor Douven, a professor at the Sorbonne. 

What is the lottery paradox? That takes some explaining. Consider two propositions: "I will be in Chicago next week" on the one hand and "this lottery ticket will lose" on the other. 

Make some reasonable stipulations here in two sets First, I believe I will be in Chicago next week because I have made plans. I have both the airline ticket and hotel reservations.  I have a great deal of interest in an event that will take place there and I have enough money to make the trip and come back without difficulties. 

In this case, we generally do say without hesitation, "I know that I will be in Chicago next week." We are not bothered by the possibility that, say, a fire in Chicago could destroy the hotel and cause the cancellation of the event. In that case, I will presumably not take the flight. And given that not-impossible scenario, my present confidence that I will be in Chicago might turn out next week to have been false. 

Now, consider my lottery ticket and a second set of stipulations. I have entered an honestly run lottery sponsored by a very solvent organization. I have one ticket out of, say, five thousand. One of them will win a considerable prize. Let us say the sponsor charged $1 for each ticket. Half of that money goes to some worthy cause, the other half, $2,500, goes to a prize. 

In this case, I might well say, "I probably have a losing ticket." We would not say simply "I do have a losing ticket" or "I know that I have a losing ticket."  Why not?  Yes, this is presumably related to the psychological question why intelligent people who understand probability ever enter lotteries. After all, if your real concern was contributing to the worthy cause you could simply do so directly and ignore the lottery middlemen.  You surely buy the ticket in part for the same reason that you would NOT say simply "I know that this ticket I just bought is a losing ticket."

So here is the problem: do we really have any "lottery knowledge" or not? If we can be said to know that this ticket is a loser we have lottery knowledge. If we can't, we don't. That seems a binary choice. Yet there are powerful arguments against either the "yes" or the "no." 

Arguing against the No answer

No, you might say, we don't have lottery knowledge -- we don't know the ticket is a loser. That is merely a probable judgment which for somebody is going to turn out to be wrong. 

One of the counterarguments to that, though, is that we DO claim to have knowledge of things like our impending trips to Chicago given the above stipulations. The difference can't merely be a matter of probability. For all I know the probability that I will actually get to Chicago may be less, as judged by some reasonable metric, then the probability that my  ticket turns out to be a loser. If you don't think so, simply increase the number of tickets you imagine  to have been sold. Ten Thousand tickets? A million tickets? The odds that I will actually get to Chicago next week are surely not as good as the 999,999/1 Million chance of me having a losing ticket in the latter case. 

Arguing against the Yes answer

Look at it from the other side. Suppose I say, consistent with my understanding that I have knowledge of my travel plans next week, that I KNOW my ticket in the million-ticket lottery is a loser. Can't I also be said by parity of reasoning to know that about any other ticket you might ask me about?  Yes, ticket 000572 is a loser. But what about ticket 010725? Well, that is a loser too. If we keep going by iteration we end up with the conclusion that every ticket is a loser -- meaning there will be no winner. But this seems to require us to believe both (a) there will be one winner and (b) there won't be any winner.

Both the case against lottery knowledge and the case in favor of lottery knowledge seem very strong. THAT is the lottery paradox.   

Its resolution requires, I think, acceptance of the idea that probability, knowledge, works somewhat differently in the lottery case than the way it works in the case of anticipated travel. Working out why they are different and whether as philosophers we can accept that as rational: THAT is worth a few essays. 

The title of the book is LOTTERIES, KNOWLEDGE, AND RATIONAL BELIEF: ESSAYS ON THE LOTTERY PARADOX (2021). I may have more to say about it tomorrow.    

Comments

  1. HenryJanuary 25, 2023 at 7:01 AM

    I see no lottery paradox. No one believes "both (a) there will be one winner and (b) there won't be any winner." Everyone believes there will be one winner (they rarely consider the possibility that a disaster might occur that prevents a winner from being selected). And no lottery ticket holder would say "this lottery ticket will lose," except as an expression of pessimism not to be taken literally. Everyone knows that his or her lottery ticket has a slim chance of winning. The so-called paradox is completely contrived.

    The reason that some intelligent people who understand probability enter lotteries is that it gives them an opportunity to fantasize about what they would do with their winnings. They know that their fantasy has a high probability of not coming true, but fantasizing about what they would do with their winnings gives them pleasure, and how many pleasurable experiences can you buy for only two bucks?

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    Replies
    1. HenryJanuary 25, 2023 at 7:16 AM

      Another reason that some intelligent people who understand probability enter lotteries may be that they are desperate for money and are willing to gamble $2 even though they know that the possibility of winning is next to nothing. That may not be an intelligent decision, but the person making it may be sufficiently intelligent to know that it is not an intelligent decision, because, being desperate for money, he should not throw away even $2. But even intelligent people do not always act rationally.

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  2. ChristopherJanuary 25, 2023 at 10:06 AM

    We do seem to be comfortable making statements like "I will be going to Chicago next week." (The book uses an example of a trip to Munich -- but it is written by a professor at the Sorbonne. Not a very tricky issue of transportation. Here in the northeastern US I'm using a trip to Chicago as an analog.) Anyway, my trip to Chicago next week is something I would normally describe as knowledge, even while being aware of the possibilities that it won't happen. So it seems to me prima facie that there is some difficulty reconciling this with what I think I know, or want to DENY that I know, about my lottery ticket. I will write a bit more about this for tomorrow's post. Stay tuned for my personal favored resolution of the paradox you don't acknowledge. (Are you familiar with the "preface paradox"? They are somewhat similar.)

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