Wolfgang Schwarz

Representation theorems and the indeterminacy of mental content

To what extent are the beliefs and desires of rational agents determined by their actual and counterfactual choices? More precisely, suppose we are given a preference order that obtains between a possible act A and a possible act B iff the relevant agent is disposed to choose A over B. Say that a pair (C,V) of a credence function C and a utility (desirability) function V fits the preference order iff, whenever A is preferred over B, then A has higher expected utility than B by the lights of (C,V). Now, to what extent does a rational preference order constrain fitting credence-utility pairs?

Various "representation theorems" in decision theory appear to show that an agent's preference order, if it meets certain qualitative constraints, completely determines their beliefs and desires: if (C,V) and (C',V') both fit the preference order, then C=C' and V' is a positive linear transformation of V. (Since utilities have no fixed zero and unit, V and V' thereby represent the very same distribution of desire.)

On the other hand, philosophers often say that an agent's choice dispositions leave their attitudes radically underdetermined. The claim is usually backed by the vague suggestion that one can always find deviant assignments of beliefs compensated by correspondingly deviant assignments of desires that rationalise the very same (actual and counterfactual) choices. But is this really possible? If it is, what's wrong with the representation theorems?

In "New Work for a Theory of Universals", Lewis offers what looks like a proof for the underdetermination claim. The argument is very compressed, so here is a reconstruction. (I leave out Lewis's input constraints and the diachronic constraints, since the representation theorems don't consider them either.)

Following Jeffrey, we identify states, acts and outcomes with sets of possible worlds. For simplicity, let's pretend that the space of worlds W is finite. Define the expected utility of an act A by the lights of (C,V) as follows (again following Jeffrey):

EU(A) = \sum_{w\in W} C(w/A) V(w).

Observe that if A is any act and S a partition of A, then instead of calculating EU(A) by summing over all A-worlds at once, we can do the sum piecemeal for each member S of S (boldface S is the partition):

EU(A) = \sum_{S\in S} C(S/A) \sum_{w} C(w/S)V(w)

So the expected utility of A is determined by the distribution of credence C over the cells in S and the C-expectation of V within each cell.

Now consider the space of all possible acts over which the agent's preference order is defined. We do not assume that the acts are mutually exclusive. However, we can define a partition S by the equivalence relation that holds between worlds w and w' iff for every possible act A, w and w' are either both inside A or both outside A. So the cells in S are sets of worlds that are perfectly alike with respect to the agent's acts.

By the above observation, it now follows that the expected utility of every possible act is determined by the distribution of credence C over the cells in S and the C-expectation of V within each cell. Hence any permutation of a fitting assignment (C,V) that holds these aspects fixed is also a fitting assignments.

To illustrate, suppose (as seems plausible) that any complete specification of a possible way the agent might act is compatible with both the hypothesis that unobserved emeralds are green and the hypothesis that unobserved emeralds are blue. (Let's abbreviate these hypotheses with Green and Grue, respectively.) So each member of S contains some Green worlds and some Grue worlds. Now take any fitting assignment (C,V) and make the following change for every cell S: if x is the C-probability of S and v the C-expected utility of S, choose an arbitrary Grue world w in S and stipulate that C'(w)=x, V'(w)=v, and C'(w')=0 for all other members w' of S. The result is an equally fitting pair (C',V') on which the agent is certain that unobserved emeralds are blue. By the same technique, we can find another fitting assignment on which she is certain that they are green, or red, or purple.

This makes a real difference: assigning high probability to Green worlds is a genuinely different attitude than assigning high probability to Grue worlds. By comparison, consider the argument whereby Stalnaker supports underdetermination in chapter 1 of Inquiry. Stalnaker argues that choice dispositions do not settle at which individuals the agent's beliefs and desires are directed. From what we've just learned, we can fill in the details as follows. Let E be a relation that holds between worlds w and w' iff they differ only in the haecceitistic respect that Alice and Bob have traded places. Suppose the agent's possible acts are individuated so that they do not involve the identity of Alice and Bob. (For instance, there is no such act as greeting Alice). Then we can always move around credence and utilities between E-equivalent worlds while preserving fit. The problem with this argument is that it's controversial whether our attitudes distinguish between merely haecceitistic differences. I think there's a useful notion of belief and desire on which they don't. The same is true for merely quidditistic differences. It is then best to exclude such differences from the space W over which C and V are defined. If the differences are nevertheless included, underdeterminaction is to be expected, and unproblematic.

Back to the representation theorems. If Lewis's argument works, then where do the theorems go wrong? Here are some considerations which, I think, do not get at the heart of the matter.

First, I mentioned that the representation theorems rely on certain qualitative constraints on preference orders. These constraints, e.g. in the form of Savage's axioms, are unrealistic and demanding. But I don't think that's relevant. Where would Lewis's argument rely on, say, non-transitive preferences or failures of the "Sure Thing Principle"?

Second, Lewis's argument uses Evidential Decision Theory. This is interesting because in Evidential Decision Theory, preferences (over acts, or even over the whole space of propositions P(W)) do not suffice to determine a uniquely fitting (C,V) pair, as Bolker (and Goedel) showed: whenever (C,V) fits, then there are constants k for which (C',V'), defined as follows, fits as well: C'(X) = C(X)(1+k V(X)), V'(X) = V(X)(1+k)/(1+kV(X)). In effect, a preference order can at most determine the products P(X)V(X), but not the exact contribution of P(X) and V(X). This is not so in Causal Decision Theory. I don't fully see how this relates to Lewis's argument, but I think the indeterminacy Lewis has in mind is different.

Can Lewis's argument be applied to Causal Decision Theory? Not directly. In Causal Decision Theory, the choiceworthiness of an act does not only depend on the probability distribution over act partitions and the expectation of V within each cell of the partition. Let A be a maximally specific act whose outcome depends on the colour of unobserved emeralds in a way that matters to the agent. In Evidential Decision Theory, the choiceworthiness of A is given by the C-expectation of V within A, which leaves C(A & Green) and C(A & Grue) completely open. In Causal Decision Theory a la Savage and Lewis, the choiceworthiness of A instead depends on the C-expectation of the evidential expected utility across a fixed set of states, and the states distinguish between Green and Grue. Since the very same probability distribution over states matters to the choiceworthiness of all acts, we can't simply redistribute probabilities and utilities within each act while preserving choiceworthiness. Unlike in Evidential Decision Theory, C(A & Green) also matters for the choiceworthiness of acts other than A.

How does this affect the underdetermination? I'm not sure. Lewis's argument no longer works, but I guess it can probably be adjusted to the Causal framework. In any case, it would be odd if there was radical indeterminacy in Evidential Decision Theory, but full determinacy in Causal Decision Theory.

The really crucial point, I think, is not the calculation of choiceworthiness, but the individuation of acts and outcomes.

Consider how a Savage-style decision theory would let us determine an agent's beliefs about unobserved emeralds. Let [Good] and [Bad] be two acts which are guaranteed to produce certain outcomes, which I'll call Good and Bad, respectively. Suppose the agent prefers [Good] over [Bad]. This reveals that V(Good) > V(Bad). Let [Green ? Good : Bad] be an act (a "gamble") that leads to outcome Good if unobserved emeralds are green, otherwise to Bad. Let [Grue ? Good : Bad] be an act that leads to Good in case of Grue, otherwise to Bad. Now we can ask whether the agent would choose [Green ? Good : Bad] over [Grue ? Good : Bad]. If yes, their credence in Green must be greater than their credence in Grue.

Savage assumes that the space of states and outcomes are externally fixed, and identifies acts with arbitrary functions from states to outcomes. This guarantees that the preference order is defined for "constant acts" like [Good] as well as gambles like [Green ? Good : Bad]. On a more ordinary understanding of acts, the existence of all these acts is problematic. Is there really a possible act, for every possible outcome, which guarantees the outcome no matter the state of the world? A version of this problem also affects Savage's conception of outcomes. If we identify outcomes with sets of worlds, then two alternative acts can never bring about the very same outcome, for the resulting worlds will always differ in their history. If you choose [Good] rather than a gamble, you end up in a world in which you chose [Good] and no gamble. But the gamble [Green ? Good : Bad] is supposed to yield the very same outcome as [Good] if unobserved emeralds are green. This cannot be right if outcomes are sets of worlds. (Although the utility of the two outcomes can still be the same.)

The two possible, and compatible, acts [Green ? Good : Bad] and [Good] together entail Green. So contrary to the "plausible" assumption we made above, there are complete specifications of acts which settle the colour of unobserved emeralds. If, in general, the act partition S partitions W into singleton sets, there is no room for redistributing probability and utility within the cells.

I am not sure what to make of all this. Savage's conception of acts and outcomes is certainly unrealistic. On the other hand, it doesn't look like we would need a lot of further constraints, in Causal Decision Theory, to vindicate the idea that an agent's beliefs about unobserved emeralds are revealed by the bets or gambles she is disposed to accept. In particular, what happens if we add some constraint to the effect that the utility of states is generally independent of the choices in their history?