Motivations

Let’s consider, in general terms, what sort of motivations one might have for adopting models that fall under the umbrella of IP. The focus will be on models of rational belief, since these are the models that philosophers typically focus on, although it is worth noting that statistical work using IP isn’t restricted to this interpretation. Note that no one author endorses all of these arguments, and indeed, some authors who are sympathetic to IP have explicitly stated that they don’t consider certain of these arguments to be good (for example Mark Kaplan does not endorse the claim that concerns about descriptive realism suggest allowing incompleteness).

2.1 Ellsberg decisions

There are a number of examples of decision problems where we are intuitively drawn to go against the prescriptions of precise probabilism. And indeed, many experimental subjects do seem to express preferences that violate the axioms. IP offers a way of representing these intuitively plausible and experimentally observed choices as rational. One classic example of this is the Ellsberg problem (Ellsberg 1961).

اقتباس :: I have an urn that contains ninety marbles. Thirty marbles are red. The remainder are blue or yellow in some unknown proportion.

Consider the indicator gambles for various events in this scenario. Consider a choice between a bet that wins if the marble drawn is red (I), versus a bet that wins if the marble drawn is blue (II). You might prefer I to II since I involves risk while II involves ambiguity. A prospect is risky if its outcome is uncertain but its outcomes occur with known probability. A prospect is ambiguous if the outcomes occur with unknown or only partially known probabilities. Now consider a choice between a bet that wins if the marble drawn is not blue (III) versus a bet that wins if the marble drawn is not red (IV). Now it is III that is ambiguous, while IV is unambiguous but risky, and thus IV might seem better to you if you preferred risky to ambiguous prospects. Such a pattern of preferences (I preferred to II but IV preferred to III) cannot be rationalised as the choices of a precise expected utility maximiser. The gambles are summarised in the table.
Table 1: The Ellsberg bets. The urn contains 30 red marbles and 60 blue/yellow marbles

	R	B	Y
I	1	0	0
II	0	1	0
III	1	0	1
IV	0	1	1

Let the probabilities for red, blue and yellow marbles be [ltr]r[/ltr], [ltr]b[/ltr] and [ltr]y[/ltr] respectively. If you were an expected utility maximiser and preferred I to II, then [ltr]r>b[/ltr] and a preference for IV over III entails that [ltr]r+y[/ltr]. No numbers can jointly satisfy these two constraints. Therefore, no probability function is such that an expected utility maximiser with that probability would choose in the way described above. While by no means universal, these preferences are a robust feature of many experimental subjects’ response to this sort of example (Camerer and Weber 1992; Fox and Tversky 1995).
The imprecise probabilist can model the situation as follows: [ltr]P(R)=1/3,P(B)=P(Y)=[0,2/3][/ltr]. Note that this expression of the belief state misses out some important details. For example, for all [ltr]p∈P[/ltr], we have [ltr]p(B)=2/3−p(Y)[/ltr]. For the point being made here, this detail is not important. Modelling the ambiguity allows us to rationalise real agents’ preferences for bets on red. To flesh this story out would require a lot more to be said about decision making, (see section 3.3) but the intuition is that aversion to ambiguity explains the preference for I over II and IV over III.
As Steele (2007) points out, the above analysis rationalises the Ellsberg choices only if we are dealing with genuinely indeterminate or unspecific beliefs. If we were dealing with a case of imperfectly introspected belief then there would exist some [ltr]p[/ltr] in the representor such that rational choices maximise [ltr]E[size=13]p[/ltr]. For the Ellsberg choices, there is no such [ltr]p[/ltr].[/size]
This view on the lessons of the Ellsberg game is not uncontroversial. Al-Najjar and Weinstein (2009) offer an alternative view on the interpretation of the Ellsberg preferences. Their view is that the distinctive pattern of Ellsberg choices is due to agents applying certain heuristics to solve the decisions that assume that the odds are manipulable. In real-life situations, if someone offers you a bet, you might think that they must have some advantage over you in order for it to be worth their while offering you the bet. Such scepticism, appropriately modelled, can yield the Ellsberg choices within a simple game theoretic precise probabilistic model.

Motivations

2.1 Ellsberg decisions

Motivations :: تعاليق

Motivations