Decision making

One important use that models of belief can be put to is as part of a theory of rational decision. IP is no different. Decision making with imprecise probabilities has some problems, however.
The problem for IP decision making, in short, is that your credal committee can disagree on what the best course of action is, and when they do, it is unclear how you should act (recall the definitions in section 1.1). Imagine betting on a coin of unknown bias. Consider the indicator gambles on heads and tails. Both bets have imprecise expectation [ltr][0,1][/ltr]. How are you supposed to compare these expectations? The bets are incomparable. (If the coin case appears to have too much exploitable symmetry, consider unit bets on Elga pulling toothpaste or jellyfish from his bag.) This incomparability, argues Williamson, leads to decision making paralysis, and this highlights a flaw in the epistemology (2010: 70). This argument seems to be missing the point, however, if one of our motivations for IP is precisely to be able to represent such incompatibility of prospects (see section 2.2)! The incommensurability of options entailed by IP is not a bug, it’s a feature. Decision making with imprecise probabilities is discussed by Seidenfeld (2004),Troffaes (2007), Seidenfeld, Schervish, and Kadane (2010), Bradley (2013), Williams (2014),Huntley, Hable, and Troffaes (2014).
A more serious worry confronts IP when you have to make sequences of decisions. There is a rich literature in economics on sequences of decisions for agents who fail to be orthodox expected utility maximisers (Seidenfeld 1988; 1994; Machina 1989; Al-Najjar and Weinstein 2009, and the references therein). This topic has recently been brought to the attention of philosophers again after the publication of Elga’s (2010) paper Subjective Probabilities Should Be Sharp which highlights the problem with a simple decision example, although a very similar example appears in Hammond (1988) in relation to Seidenfeld’s discussion of Levi’s decision rule “E-admissibility” (Seidenfeld 1988).
A version of the problem is as follows. You are about to be offered two bets on a coin of unknown bias, [ltr]A[/ltr] and [ltr]B[/ltr], one after the other. The bets pay out as follows:

[ltr]A[/ltr] loses 10 if the coin lands heads and wins 15 otherwise
[ltr]B[/ltr] wins 15 if the coin lands heads and loses 10 otherwise

If we assume you have beliefs represented by [ltr]P(H)=[0,1][/ltr], these bets have expectations of [ltr][−10,15][/ltr]. Refusing each bet has expectation of 0. So accepting and refusing [ltr]A[/ltr] are incomparable with respect to your beliefs. Likewise for [ltr]B[/ltr]. The problem is that refusing both bets seems to be irrational, since accepting both bets gets you a guaranteed payoff of 5. Elga argues that no decision rule for imprecise probabilities can rule out refusing both bets. He then argues that this shows that imprecise probabilities are bad epistemology. Neither argument works. Chandler (forthcoming) and Sahlin and Weirich (2014) both point out that a certain kind of imprecise decision rule does make refusing both bets impermissible and Elga has acknowledged this in an erratum to his paper. Bradley and Steele (2014a) argue that decision rules that make refusing both bets merely permissible are legitimate ways to make imprecise decisions. They also point out that the rule that Chandler, and Sahlin and Weirich advocate has counterintuitive consequences in other decision problems. Even if Elga’s argument worked and there were no good imprecise decision rules, that wouldn’t show that IP was a faulty model of belief. We want to be able to represent the suspension of judgement on various things, including on the relative goodness of a number of options. Such incommensurability inevitably brings with it some problems for sequential decisions (see, for example, Broome 2000), but this is not an argument against the epistemology. As Bradley and Steele note, Elga’s argument—if it were valid—could mutatis mutandis be used as an argument that there are no incommensurable goods and this seems too strong. Moss (forthcoming) relates Elga-style IP decision problems to moral dilemmas and uses the analogy to explain the conflicting intuitions in Elga’s problem.

Decision making

Decision making :: تعاليق

Decision making