Diachronic Dutch Books

A diachronic Dutch Book argument uses the fact that if you engage in a certain cognitive process, then there is a series of bets across different times that you will each find acceptable, but whose net consequence is that you lose money in every possibility. For instance, say that your current credence in p is 0.5, but you plan to have your credence in p tomorrow be 0.8. Now consider a bet that pays $1 if p, and nothing otherwise, and assume the marginal utility of money is constant enough. You’ll happily sell such a bet for 60 cents today. And you’ll happily buy it back tomorrow for 70 cents. So you’ll have lost 10 cents, whether the bet pays out or not. That’s bad, so you shouldn’t have arbitrary, and planned, credal jumps like that. A generalisation of this argument shows that any planned updating strategy that is not conditionalisation leads to sure losses, and, it is concluded from that, is bad.

But there’s something very odd about the argument here. There’s nothing wrong per se with a trading strategy that leads to a sure nominal loss. If there was, there would be something wrong with ever borrowing money at a positive interest rate. In the example above, you do end up with 10 cents less than you start with. But you also have the use of 60 cents for a day. Now 17% per day is probably a high price to pay for the use of that money. But we think having money to use is worth something. A non-zero liquidity preference is not irrational.

So what, exactly, is worse about the trading strategy the non-conditionaliser uses, and which leads to sure nominal loss than the trading strategy someone uses when they borrow money at a positive interest rate?

Part of the answer has to do with expected inflation, but presumably not all of it. Some people borrow money at what they take to be a positive real interest rate. And that can be, in some circumstances, rational.

Perhaps there is a simple explanation here, but it seems there is a very large argumentative gap in Diachronic Dutch Book arguments that isn’t there in Synchronic Dutch Book arguments.

6 Replies to “Diachronic Dutch Books”

  1. Isn’t the usual economic story something to do with ‘smoothing’ lifetime consumption (since we tend to have more resources in the future)?

    It seems a quick fix for the diachronic dutch book arguments would be to set all the money transfers to occur simultaneously at some future time t, no matter when the bet itself was placed. (This assumes the agent has no temporal bias, but rather a constant concern for their wealth at time t. Otherwise, we would have to find some other timeless preferences they have, and gamble on their satisfaction instead.)

  2. I think there are multiple reasons for it being rational to borrow at positive interest rates. One might be smoothing. But I think future discounting is generally (if not quite universally) thought to be rationally permissible.

    Having the payouts occur in the future is a good way to resolve some of these problems though. (Maybe all of them.) I should have thought of that.

  3. Here is my thought: Brian supposes that “the marginal utility of money is constant enough,” but he considers a situation in which “having money to use is worth something.” By the latter sentence, I suspect that he means “having money to use now is worth something.”

    Now, consider the case of “borrowing money at a positive interest rate.” If my suspicion is correct, then the utility of money is not constant in this case (which contradicts Brian’s assumption). Typically, if one agrees to borrowing $X at t and returning $X+i at t’>t, she does because having $X at t has the same or greater utility than having $X+i at t’.

    For instance, the borrower may invest the $X to improve the equipments of her restaurant, expecting her income to be $j>=i more than when she does not improve them. In such a case, borrowing $X at a positive interest rate is not irrational (in a way that does not contradict the core idea of the diachronic dutch book argument).

    In contrast, when we talk about the diachronic dutch book argument, we (implicitly or explicitly) assume that the same amount of money has the same value at different times, and assume that the money had at an earlier time does not produce additional profit over time. In such a case, being exploitable by a bookie will perhaps show the irrationality of the potential victim’s belief state.

    My last point: When Brian says “Part of the answer has to do with expected inflation,” he seems to be aware of the possibility that in a typical case of borrowing money at a positive interest rate, the value of money may not be constant due to the inflation. In my above suggestion, I pointed out that there may be another way in which money fails to have a constant value over time, for example, by producing profit by investment. I suspect that such situations explain the rationality of borrowing money paying interest, not undermining the core idea of the diachronic dutch book.

  4. But you also have the use of 60 cents for a day. Now 17% per day is probably a high price to pay for the use of that money. But we think having money to use is worth something. A non-zero liquidity preference is not irrational.

    One problem is that you’d sell at t and purchase at t’, even if t’ were a few moments away from t. Worse, I think, you’d more or less give it away at t, if your credence for p then approximated 0, and buy it back at t’ for more or less the value of p (whatever the value of p!), if your credence then approximated 1. That can’t be rational.

  5. I’ve never thought that dutch book arguments hinged on facts about the value of money. As I understood them (maybe because I learned this stuff from Brian Skyrms), given a certain set of degrees of belief at a time, or in the diachronic case, a set of degrees of belief at a time + a strategy for revising those beliefs given new information, if one ADDS to that a coherent set of things (utils) which are valued in a linear, time-invariant, way, then one is willing to accept a set of books on which one is guaranteed a loss of those utils, whatever they are. On the assumption that it is perfectly all right to have something of which one values in a linear, time-invariant way, the problem must be one’s set of degrees of belief+one’s belief-revision strategy, namely, non-conditionalization. The point of the dutch book is that it shows the incoherence of the belief-revision strategy; starting with a perfectly all right sets of beliefs, and a perfectly all right set of utils, you necessarily end up in a predicament you deem to be not perfectly all right. Conditionalizers, so the argument goes, do not end ever necessarily end up in such a predicament, given the same starting points.

    If this is the right way to understand diachronic dutch books, then facts about the value of money seem irrelevant.

  6. Barry (I guess “Lam” is your last name), I suspect that using utils (or hedons?) instead of dollars (or euros, yens, wons, whatever actual currencies) to formulate the Diachronic Dutch Book argument (hereafter: DDB) will cure the potential problem of potential inflation or deflation (or in my opinion, the potentially higher profitability of money available at an earlier time) if the DDB is carefully constructed.

    However, we are talking about the traditional formulations of the DDB argument, in which usually dollars or pounds have been used. In those traditional formulations, the fluctuations of the values of the given currency certainly matter. Plus, I am not sure that the utilities (represented by utils or hedons) that we assign to various items can be always given a sort of linear ordering, as seemingly required by the DDB argument.

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