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My son says my rain forecasts could be “Dutch-booked”. What does that mean?

Akosua Mensah Contributor
· 4 replies · 158 views
Rationality & probabilityAnalyticBruno de FinettiFrank Ramsey#credence#dutch-book#probability#weather
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My son is in his second year at university and sends me his course readings, I think to keep his mother humble. This week it was something about probability. On the phone I told him I was sixty percent sure the rains would come early this year. When he asked about the dry season holding on instead, I said I was also fairly sure of that, maybe sixty percent, because the radio and the old men at the cooperative do not agree.

He laughed and said a bookmaker could take the farm off me with numbers like that. He called it a “Dutch book” and then his credit ran out.

I do not bet. I grow cocoa on eleven acres and I plant when the soil tells me to. So I would like two things explained, plainly if possible:

  1. What exactly is the trick the bookmaker plays with my sixty and sixty?
  2. Why should someone who never places a bet care about it at all?

If it is only a lesson for gamblers, my son can keep it. But if it says something about how a sensible person ought to hold her opinions, I want to know, because I have been holding opinions about the weather for forty years and nobody has ever sent me a bill for them.

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4 answers

Accepted answer

  1. Wanjiru Kamau

    ✓ Accepted answer Contributor

    Here is the trick, with small money. If you are really 60% sure of rain, then a ticket that pays 1 cedi if it rains should look fair to you at 60 pesewas. You are also 60% sure of no rain, so a ticket that pays 1 cedi if it stays dry also looks fair at 60 pesewas. The bookmaker sells you both. You pay 1 cedi 20. Whatever the weather does, exactly one ticket pays, so you get back 1 cedi. You have lost 20 pesewas for certain, in every possible season.

    The repair is simple: your confidence in something and in its opposite must add up to 100. Sixty for rain means forty for no rain.

    Why care if you never bet? Ramsey and de Finetti used betting as a way of measuring how confident someone is, not because life is a casino. The real lesson is about planning. If you buy seedlings as though rain is 60% likely and also pay for extra watering as though dry is 60% likely, you have paid twice for one season. That is the farm version of the bookmaker.

    Helpful · 2
    1. Akosua Mensah

      Contributor

      Thank you both. Wanjiru, the seedlings and the watering is exactly the sort of mistake I have watched neighbours make, so now I believe it.

      But I think my honest answer to my son was not really sixty and sixty. It was that I did not know, and “sixty” was my way of saying so twice. The old men say one thing, the radio another, and I have no way of deciding between them. Is there a number for not knowing? Or must I always pretend to a precise figure?

      1. Vikram Sethi

        Fellow

        There is a whole literature on exactly your question, Akosua. Some philosophers say a reasonable person may hold a range rather than a point: “somewhere between forty and sixty for early rain”. That is called imprecise probability. It keeps the rule that rain and no-rain must fit together (forty to sixty for one means forty to sixty for the other) while admitting you have not got enough evidence to pick a single number.

        I teach this to my Class 12 students with a coin I refuse to show them. They always want to say fifty, and I ask whether fifty means “I know it is fair” or “I know nothing”. Those are very different states, and one number hides the difference.

        Helpful · 1

Other answers

  1. Tomasz Wójcik

    Fellow

    Wanjiru's answer is the standard one and it is right as far as it goes. The fair objection, which teachers sometimes skip, is that the argument only bites if your confidence just is your willingness to bet at those odds. Plenty of careful people refuse bets they think are fair, because losing hurts more than winning pleases, or because they suspect the bookmaker knows something.

    That is why some philosophers moved to a different defence. James Joyce argued in the late 1990s that incoherent degrees of belief are always less accurate than some coherent alternative, however the world turns out. No bookmaker needed: the incoherent forecaster is beaten on accuracy alone. Akosua's sixty-sixty would be beaten by fifty-fifty whatever the rains do.

    Helpful · 3