Some counterexamples to the Benchmark Theory
In "Gandalf's solution to the Newcomb problem" (2013), Ralph Wedgwood proposes a new form of decision theory, Benchmark Theory, that is supposed to combine the good parts of Causal and Evidential Decision Theory.
Like many formulations of Causal Decision Theory, Benchmark Theory (BT) assumes a privileged partition of states that are outside the agent's causal control. Like Evidential Decision Theory, BT only considers the probability of these states conditional on a given act A. However, what is weighted by the conditional probabilities P(S_i/A) is not the absolute utility of S_i & A, but the comparative utility of S_i & A, which is determined by the difference between the absolute utility U(S_i & A) and the average absolute utility U(S_i & A') for all options A'. (This average is the benchmark B(S_i).) So the degree of choiceworthiness of an act A is given by
ECU(A) = \sum_i P(S_i/A) (U(S_i & A) - B(S_i)).
The choice of conditional probabilities P(S_i/A) is motivated by the assumption that we shouldn't ignore (as CDT allegedly does) the evidential impact a given choice has on our beliefs about the world. The motivation for looking at comparative utilities is that what matters for the evaluation of an option A under a state S_i is only how much better or worse we could have fared in that same state; it doesn't matter how much better or worse we could have fared in other states, since we have no control over the states.
To illustrate, consider Newcomb's problem.
| predicted 1 | predicted 2 | |
|---|---|---|
| take 1 | $1000000 | $0 |
| take 2 | $1001000 | $1000 |
| Absolute utilities | ||
| predicted 1 | predicted 2 | |
|---|---|---|
| take 1 | -500 | -500 |
| take 2 | +500 | +500 |
| Comparative utilities | ||
Since most of the probability lies on the upper-left and the lower-right cells, both EDT and BT focus on these cells when evaluating the options. However, EDT looks at the absolute utilities in the left table, while BT looks at the relative utilities in the right table. Consequently, EDT recommends one-boxing and BT two-boxing.
The deviation from standard CDT is motivated by Andy Egan's Psychopath example (from Egan 2007).
| psycho | not psycho | |
|---|---|---|
| press | -90 | +10 |
| not press | 0 | 0 |
| Absolute utilities | ||
| psycho | not psycho | |
|---|---|---|
| press | -45 | +5 |
| not press | +45 | -5 |
| Comparative utilities | ||
This time, BT agrees with EDT: in either table, comparing the upper-left and lower-right cell favours the second option, not pressing. By contrast, CDT says that you may press the button iff you're rationally confident that you won't press it. (As I've argued in my paper on the absentminded driver, I actually think CDT gets these cases right.)
Wedgwood discusses some problems for BT. In particular, BT's recommendations can be affected by adding intuitively irrelevant further options to a decision problem. For example, suppose there's a third option (call it X) in the psychopath case whose choice is probabilistically independent of the two states:
| psycho | not psycho | |
|---|---|---|
| press | -90 | +10 |
| not press | 0 | 0 |
| X | -210 | -10 |
| Absolute utilities | ||
| psycho | not psycho | |
|---|---|---|
| press | +10 | +10 |
| not press | +100 | 0 |
| X | -110 | -10 |
| Comparative utilities | ||
(Both 'press' and 'not press' should be understood to entail 'not X'.) Now pressing the button suddenly comes out better than not pressing the button!
Wedgwood offers two lines of response to this kind of problem. One is that we shouldn't expect that adding new options never makes a difference to the ranking between other options. I agree. The problem for BT is not that it violates some general principle on the "independence of irrelevant alternatives". The problem is that according to BT, adding new options can make a difference in many particular cases where it clearly shouldn't make a difference.
Wedgwood's second response is that "insane" options such as X should be excluded before computing the benchmark. What is an "insance" option? Wedgwood gives two sufficient conditions: an option is insane if (a) it is dominated, or (b) if the agent is certain from the outset that she won't carry it out. Condition (b) doesn't help much, since presumably we can just stipulate that our agent isn't certain she won't choose X. (We can't assume that every irrational choice is one of which the agent is certain she won't make it -- otherwise condition (b) makes the rest of decision theory redundant.)
Condition (a) is also problematic. For one thing, it is not at all obvious that dominated options are always insane. EDT recommends choosing the dominated option in Newcomb's problem, and a significant number of philosophers think that's correct. It may still be true that it is always wrong to choose dominated options, but this needs to be shown as a consequence of a general decision theory. (Wedgwood himself ignores his condition (a) when he discusses Newcomb's problem. On his official account, there is only one genuine option in Newcomb's problem, since one-boxing is dominated.)
Now, it might be OK to remove dominated options from the outset if we could show that according to BT, it is never rational to choose a dominated option. (Just as one-boxing is irrational according to BT.) But that's not true. Consider the following scenario. You can press a red button, a green button or neither. In another room, your duplicate twin is facing the same options, and you are confident that (s)he will make the same choice. The outcomes for you are as follows, depending on the choice made by your twin.
| red | green | neither | |
|---|---|---|---|
| red | $5 | $30 | $200 |
| green | $55 | $0 | $40 |
| neither | $60 | $60 | $60 |
| Absolute utilities | |||
| red | green | neither | |
|---|---|---|---|
| red | -35 | 0 | +100 |
| green | +15 | -30 | -60 |
| neither | +20 | +30 | -40 |
| Comparative utilities | |||
Here BT recommends pressing the green button, although this option is strongly dominated by pressing neither. In fact, in this scenario it would really be insane to prefer the second option over the third: you know for certain that the third option will get you $60 and that the second will give you less! Yet BT says that you should choose the second.
--- Of course, with the additional rule that dominated options must be removed in advanced, BT no longer recommends choosing the second option. But that's precisely the point: why is it OK to exclude the dominated option if the very principles of BT entail that if we include it, it comes out as best?
Another problem with Wedgwood's response is discussed by Rachael Briggs (Briggs 2010): the basic issue illustrated by the addition of option X to the psychopath example doesn't really hang on the fact that X is dominated. Suppose there's a further state of affairs with negligible probability, and in that state X is better than the other two options. It still seems that adding X shouldn't change the ranking of these options.
Here are some more problems.
Consider the following variation of the psychopath case, in which you give somewhat higher value to killing all the psychopaths in a situation where you're not one of them.
| psycho | not psycho | |
|---|---|---|
| press | -90 | +100 |
| not press | 0 | 0 |
| Absolute utilities | ||
| psycho | not psycho | |
|---|---|---|
| press | -45 | +50 |
| not press | +45 | -50 |
| Comparative utilities | ||
Now BT recommends pressing the button. Does that make sense? As I said, I don't have the intuition that it's definitely wrong to press the button in the original scenario, but to the extent that I can put myself into a state of mind in which that seems right -- "pressing the button is virtually guaranteed to kill myself!" -- it doesn't make a difference whether I value press & not psycho at +10 or +100.
Next, consider this variation of the two buttons scenario.
| red | green | neither | |
|---|---|---|---|
| red | $10 | $0 | $10 |
| green | $0 | $10 | $-40 |
| neither | $5 | $5 | $0 |
| Absolute utilities | |||
| red | green | neither | |
|---|---|---|---|
| red | +5 | -5 | +20 |
| green | -5 | +5 | -30 |
| neither | 0 | 0 | +10 |
| Comparative utilities | |||
Here EDT supports choosing either red or green. In CDT, there are two (non-trivial) deliberation equilibria, red and green, with red having a much larger base of attraction. BT, by contrast, says you should press neither button. To me, that seems clearly wrong. (Note that the third option here isn't dominated, nor even ``almost dominated''.)
One more variation.
| red | green | neither | |
|---|---|---|---|
| red | $10 | $0 | $10 |
| green | $5 | $5 | $-40 |
| neither | $0 | $10 | $0 |
| Absolute utilities | |||
| red | green | neither | |
|---|---|---|---|
| red | +5 | -5 | +20 |
| green | 0 | 0 | -30 |
| neither | -5 | +5 | +10 |
| Comparative utilities | |||
In this case, both EDT and sensible forms of CDT say that red is the only rational choice: it is the only deliberation equilibrium, and the only ratifiable option. Yet BT wrongly says you should press neither button.