Abstract
Benchmark theory (BT), introduced by Ralph Wedgwood, departs from decision theories of pure expectation maximization like evidential decision theory (EDT) and causal decision theory (CDT) and instead ranks actions according to the desirability of an outcome they produce in some state of affairs compared to a standard—a benchmark—for that state of affairs. Wedgwood motivates BT through what he terms Gandalf’s principle, that the merits of an action in a given state should be evaluated relative only to the performances of other actions in that state, and not to their performances in other states. Although BT succeeds in selecting intuitively rational actions in a number of cases—including some in which EDT or CDT seem to go wrong—it places constraints on rational decision-making that either lack motivation or are untenable. Specifically, I argue that as it stands BT is committed both to endorsing and rejecting the independence of irrelevant alternatives. Furthermore its requirement that weakly dominated actions be excluded from consideration of rational action lacks motivation and threatens to collide with traditional game theory. In the final section of the paper, I construct a counterexample to BT.
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Notes
Specifically, this is typically taken to be a von Neumann–Morgenstern utility function, i.e. it is unique up to positive affine transformations: neither utility 0 nor utility 1 has special significance, but same-sized differences in utility are significant. Nothing in what follows turns on the details of these functions. Indeed, strictly speaking, utility is a measure of subjective preference and Wedgwood is keen to emphasize that he does not assume in his theory that such preferences are measurable by utility functions. In this paper, I use utility in a looser sense to denote the payoffs an agent associates with outcomes; these need not be interpreted as measures of subjective preference, but whatever values Wedgwood needs to underwrite the details of his theory.
As sketched in the last footnote, more structure is usually built into these utility functions than mere monotonic preservation of preference orderings, but the details are not important here since this is not relevant for Wedgwood’s theory.
Some theorists advocate modifications to this basic model in order to account for e.g. the Egan cases discussed below. In this paper, I will focus only on Wedgwood’s response to these kinds of cases.
See Jeffrey (1983) for a classic treatment.
This term is taken from Gibbard and Harper (1978).
(Rational) conditional credences conform to Bayes’ Rule, i.e. \(Cr(P|Q)=Cr(P\cap Q)/Cr(Q)\), for \(Cr(Q)\not = 0\).
For an example of such research, see Marcus et al. (1997).
For an exposition of this constraint, see Lewis (1981).
This does not generalize to cases with more options: BT, as I have presented it thus far, sometimes will choose a dominated action. See below for discussion.
This is Wedgwood’s term.
In a decision problem with infinitely many actions to choose from (and only such problems), every action can be strictly (or weakly) dominated. It does not seem in general therefore that it is unreasonable to rule that every action in a problem is irrational to choose. In an infinite problem, there might always be a better action available than whatever one chooses, and this is why one’s choice might always be irrational. This plainly cannot be the case in a finite problem. In particular in a two-action problem in which both actions are nearly dominated, they cannot both be better than each other. An anonymous reviewer comments that it is implausible that every action in a decision problem be irrational to choose. While I’m not sure that I agree with this point, if it is true then it serves to emphasize that it could be rational to choose a nearly dominated action.
I have, for clarity of exposition, relabelled actions to accord with my description of IIA above. However, the structure of the problem and the values used for credences and utilities are true to Wedgwood’s example.
Wedgwood claims that wherever the benchmark is set, BT requires choosing \(A\). This is false. There is a relatively small region of values (where the weighting given to the top-ranked action is small in comparison to the weighting assigned to the second-ranked action) where the ECV of \(B\) exceeds that of \(A\). In particular \(63 < 72w - 16v\), where \(v\) is the weighting given to the top-ranked action and \(w\) that given to the second, is a sufficient and necessary condition for \(B\) to receive a higher ECV than \(A\). As an aside, \(C\) is never preferred to \(A\) in this problem but there is a small region of values in which \(C\) receives a higher ECV than \(B\).
He cannot use probabilistic choice dependence as any grounds for rejecting IIA because his only reason for thinking that IIA fails when there is choice dependence is that BT is inconsistent with IIA under these circumstances. It would be begging the question to use this as a defence of BT. As far as I am aware, he has no independent reason for thinking that IIA fails in these dependency situations.
Note that Briggs’ proof that BT agrees with EDT and CDT in cases where the state of the world is probabilistically independent of one’s choice of action assumes that weakly dominated actions are not ruled out beforehand.
Proof sketch: First note that since weak domination is irreflexive and transitive, every player with only finitely many strategies must have some undominated strategy. Nash showed that every finite game has a Nash equilibrium, and hence if we reduce a (finite) game \(\varGamma \) by eliminating a weakly dominated strategy, the resulting game will have one, \(\pi \). But every strategy did at least as well in the non-reduced game \(\varGamma \), since we only eliminated a weakly dominated strategy. Therefore \(\pi \) must contain only best responses in \(\varGamma \) and is therefore Nash in \(\varGamma \). It follows by induction that equilibria of \(\varGamma \) will remain regardless of how many weakly dominated strategies are eliminated (or indeed if they are eliminated iteratively).
Or at least if there is, this seems to cast a long shadow over the prospects of unifying decision theory and game theory.
Of course, in the benchmark theorist’s defence, independent motivation for the strict dominance principle will be more easily found than that for the weak dominance principle.
This, of course, should not be given a causal or counterfactual reading. In the lottery ticket example, if it turns out that the ticket is a winner then I must’ve chosen it. In Newcomb’s problem, if it turns out that box \(B\) has the million then I must’ve chosen to one-box.
An anonymous reviewer suggests that perhaps Wedgwood does not conceive of ratifiability as a necessary condition on rational decision-making, but rather a feature that counts in favour of rationally choosing an action. If this is so, then Wedgwood can maintain that the ratifiability of \(A\) may explain why it is rationally permissible to choose, while denying that \(B\) and \(C\) are rationally impermissible simply because they are not ratifiable choices. However, if Wedgwood does think this, it still seems that we are owed some explanation of why \(B\) and \(C\) are rationally permissible when they are to be thought of as using one’s powers badly. Wedgwood might say here that, while this fact should count against them as rational choices, there remain other considerations in their favour that it does not outweigh.
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Acknowledgments
I am very grateful to Dorothy Edgington, Corine Besson, Dan Adams, Jonathan Nassim, Chris Sykes and to two anonymous referees for this journal for helpful suggestions and comments on this paper.
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Bassett, R. A critique of benchmark theory. Synthese 192, 241–267 (2015). https://doi.org/10.1007/s11229-014-0566-3
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DOI: https://doi.org/10.1007/s11229-014-0566-3