Since I’m used to hearing Dutch Book arguments as the primary way of defending expected utility maximization, I was intrigued to read this passage (from here):
The Dutch book argument concerns the long-term consistency of accepting bets. If probabilities are assigned to bets in a way that goes against the principles of CP [Classical Probability] theory, then this guarantees a net loss (or gain) across time. In other words, probabilistic assignment inconsistent with CP theory leads to unfair bets (de Finetti et al. 1993). [...]
These justifications are not without problems. Avoiding a Dutch book requires expected value maximization, rather than expected utility maximization, that is, the decision maker is constrained to use objective values rather than personal utilities, when choosing between bets. However, decision theorists generally reject the assumption of objective value maximization and instead allow for subjective utility functions (Savage 1954). This is essential, for example, in order to take into account the observed risk aversion in human decisions (Kahneman & Tversky 1979). When maximizing subjective expected utility, CP reasoning can fall prey to Dutch book problems (Wakker 2010).
The Wakker 2010 reference is to a book; searching it for “dutch book” gets me the footnote
In a later chapter on expected utility we will show that a Dutch book and arbitrage are possible as soon as there is risk aversion (Assignment 3.3.6).
And looking up assignment 3.3.6 gets
Assignment 3.3.6. This assignment demonstrates that risk aversion implies
arbitrage. More generally, it shows that every deviation from risk neutrality implies
arbitrage.
You may assume that risk neutrality for all 50–50 prospects implies complete risk
neutrality. Show that arbitrage is possible whenever there is no risk neutrality, for
instance as soon as there is strict risk aversion. A difficulty in this assignment is
that we have defined risk neutrality for decision under risk, and arbitrage has been
defined for decision under uncertainty. You, therefore, have to understand §2.1–2.3
to be able to do this assignment.
Since I don’t really have the time or energy to work my way through a textbook, I thought that I’d ask people who understood decision theory better: exactly what is the issue, and how serious of a problem is this for somebody using the Dutch Book argument to argue for EU maximization?
Dutch Book arguments as the primary way of defending expected utility maximization
The von Neumann-Morgenstern theorem isn’t a Dutch book argument, and the primary purpose of Dutch book arguments is to defend classical probability, not expected utility maximization. von Neumann-Morgenstern also assumes classical probability. Jaynes uses Cox’s theorem to defend classical probability rather than a Dutch book argument (he says something like using gambling to defend probability is uncouth).
I don’t really understand what issue the first reference you cite claims exists. It doesn’t seem to be what the second reference you cite is claiming.
I don’t really understand what issue the first reference you cite claims exists. It doesn’t seem to be what the second reference you cite is claiming.
I’m not really sure whether the parts of Wakker that I quoted are the parts that the first cite is referring, either—it could be that the first cite is talking about something completely different. That was the only part in Wakker that I could find that seemed possibly relevant, but then my search was extremely cursory, since I don’t really have the time to read through a 500-page book with dense technical material.
You can elicit probabilities from risk averse agents by taking the limit of arbitrarily small bets. There is an analogy with electro-magnetism, where people who want to give a positivist account of the electro-magnetic field say that it is defined by its effect on charged particles; but since the charges affect the field, one talks of an infinitesimal “test charge.”
Since I’m used to hearing Dutch Book arguments as the primary way of defending expected utility maximization, I was intrigued to read this passage (from here):
The Wakker 2010 reference is to a book; searching it for “dutch book” gets me the footnote
And looking up assignment 3.3.6 gets
Since I don’t really have the time or energy to work my way through a textbook, I thought that I’d ask people who understood decision theory better: exactly what is the issue, and how serious of a problem is this for somebody using the Dutch Book argument to argue for EU maximization?
The von Neumann-Morgenstern theorem isn’t a Dutch book argument, and the primary purpose of Dutch book arguments is to defend classical probability, not expected utility maximization. von Neumann-Morgenstern also assumes classical probability. Jaynes uses Cox’s theorem to defend classical probability rather than a Dutch book argument (he says something like using gambling to defend probability is uncouth).
I don’t really understand what issue the first reference you cite claims exists. It doesn’t seem to be what the second reference you cite is claiming.
I’m not really sure whether the parts of Wakker that I quoted are the parts that the first cite is referring, either—it could be that the first cite is talking about something completely different. That was the only part in Wakker that I could find that seemed possibly relevant, but then my search was extremely cursory, since I don’t really have the time to read through a 500-page book with dense technical material.
Wouldn’t this trivially go away by redenominating outcomes in utilons instead of dollars with diminishing marginal returns?
Then the bookie doesn’t always profit from your loss. I don’t know if that matters to you, though.
You can elicit probabilities from risk averse agents by taking the limit of arbitrarily small bets. There is an analogy with electro-magnetism, where people who want to give a positivist account of the electro-magnetic field say that it is defined by its effect on charged particles; but since the charges affect the field, one talks of an infinitesimal “test charge.”