4 Causal
It shouldn’t be controversial to claim that game theory textbooks are committed a broadly causal version of decision theory.1 For one thing, they always recommend defecting in Prisoners’ Dilemma, even when playing with a twin. As David Lewis showed, this is equivalent to recommending two-boxing in Newcomb’s Problem (Lewis, 1979). They endorse the causal decision theorist’s signature argument form: the deletion of strongly dominated strategies. Indeed, the typical book introduces this before it introduces anything about probability. When they do get around to probabilities, they tend to define the expected value of a choice in a way only a causal decision theorist could endorse. In particular, they define expected values using unconditional, rather than conditional, probabilities.2 And the probabilities are simply probabilities of states, not probabilities of any kind of counterfactual. Indeed, you can go entire textbooks without even getting a symbol for a counterfactual conditional.
1 This point is made by Harper (1988), and many (though not all) of the conclusions I draw in this chapter will be similar to ones he drew.
2 See, for instance, the introduction of them on page 136 of Bonanno (2018). And note that we get 135 pages before the notion of an expectation is introduced; that’s how much is done simply with dominance reasoning
3 Recall that I’m using ‘CDT’ here as the name for a family of theories. The point of this chapter will be to argue that the member of the family I prefer is immune to several challenges that have been raised against other members of the family.
What’s more controversial is that they are right to adopt a kind of causal decision theory (CDT).3 In the recent literature, I think there are three main kinds of problem that have been raised for CDT. First, it does badly in cases where there are no pure ratifiable strategies. Second, it gives strange results in cases involving betting on grand propositions about the past history of the world, or about the laws of nature. Third, it leaves its proponents will less money in Newcomb’s Problem than EDT does. In this chapter I’ll respond to each of these in turn.
4.1 No Ratifiable Choices
In some simple cases, neither of the options on the table look very good by the lights of GDT. Table 4.1 is an almost maximally simple case.
| PA | PB | |
|---|---|---|
| A | 0 | 100 |
| B | 100 | 0 |
Everyone agrees that there is nothing to choose between the options here.4 Things get complicated when A gets somewhat sweetened, say by adding 99 to the payouts if Chooser selects A, resulting in Table 4.2.
4 Almost everyone agrees that Chooser should be indifferent between A and B here. I don’t, for reasons that will become important presently, and will be discussed much more in Chapter 8. I think A and B should be treated symmetrically, of course, but they are incomparable not equally good.
| PA | PB | |
|---|---|---|
| A | 99 | 199 |
| B | 100 | 0 |
Many versions of CDT, including GDT, do not unconditionally recommend A in this case. Yet there are many problems with the same structure as Table 4.2 where people have insisted that intuition says A is the only rational choice, and it is a problem for a decision theory that doesn’t agree.5
5 This objection goes back to Richter (1984). The most memorable version of an asymmetric problem with no ratifiable choice is the Psychopath Button case presented by Andy Egan (2007).
A related objection is raised by Arif Ahmed (2014) and by Jack Spencer and Ian Wells (2019). I’ll focus on the latter version, but the points are essentially the same. Start with Table 4.1 and add a third option X. X gets a guaranteed return of 40, and if Demon predicts X, then A and B return 50. They call this problem The Frustrator.
| PA | PB | PX | |
|---|---|---|---|
| A | 0 | 100 | 50 |
| B | 100 | 0 | 50 |
| X | 40 | 40 | 40 |
They say, and many seem to agree, that intuition says that taking X is the only rational option. This is a bit surprising since there are two things wrong with X. Or, better, there is one thing wrong with X that has two manifestations which turn out to be equivalent.
For one thing X is strongly dominated by the mixed strategy of playing A and B with probability 0.5 each. Whatever Demon does, that mixed strategy has an expected return of 50, while X has a guaranteed return of 40.
For another thing, there is no probability distribution over the three states, PA, PB, and PX, such that X maximises expected utility. If Pr(PA) ≥ Pr(PB), then A will have a higher expected return than X, and if Pr(PB) ≥ Pr(PA), then B will have a greater return than X. In game-theoretic terms, X is not a best response (Bonanno, 2018: 41); no matter what Chooser thinks Demon will do, indeed no matter what probability Chooser has over Demon’s choices, X is sub-optimal.
The last two things turn out to be equivalent. As Pearce (1984) shows, an option is sometimes a best response iff it is not strictly dominated by any other pure or mixed strategy.6 That is, for any option O, there is some probability distribution over the states such that no alternative to O has a higher expected utility iff there is no mixture of alternatives to O that has a higher expected return than O in all states.
6 Bonanno (2018: 207) attributes this to Pearce, who has a rather elegant proof of the result that involves turning the decision problem into a zero-sum game, and applying a famous result of Nash’s.
GDT’s response to these kinds of problems is in two parts. One part is to argue, as I will in Chapter 8, that the intuitions behind cases like this are unstable. The other is to argue that in fact the right choice here is to play the mixed strategy. That requires arguing that mixed strategies are available, as I will in Chapter 5, and indeed saying more about what it is to play a mixed strategy. And then it requires appealing to Pearce’s result to say that any option ruled out in cases like The Frustrator is in fact strictly dominated, and so not choice-worthy.
There are a lot of promissory notes in the last paragraph, but hopefully I’ve said enough to say how GDT will, eventually, respond to cases like these.
4.2 Laws, and other Non-Events
- Cite Ahmed book for the general problem
- Cite Hedden’s recent version
I’m not going to reply to this problem as much as dismiss it. There are three reasons to think that it isn’t the kind of thing we should worry about given the purposes for which CDT is used.
One immediate reason for dismissing it is that this is not a case that satisfies the idealising constraints, and CDT is a theory of ideal decision. If what Chooser can do is determined by the past and the laws, and Chooser doesn’t know the past or the laws, then Chooser doesn’t know what they can do. And in ideal decision theory, Chooser always knows what they can do. Just what decision theory should look like in situations where the range of possible choices is potentially unknown is quite tricky; I suspect it should not be expectationist. But it’s not the problem CDT, or for that matter EDT, is addressing.
Second, I’m not sure we should go along with the claim that the laws are causally independent of our actions. Distinguish the following two claims:
- A is causally independent of B iff it is not the case that B causes, in whole or in part, A.
- A is causally independent of B iff A and B are events such that it is not the case that B causes, in whole or in part, A.
When A is a proposition about the laws, and B is a proposition about a particular action, the first is somewhat plausible. (Though I’ll dispute it presently.) But the second is rather implausible. There was no event of the laws of nature being imposed, perhaps in some divine law-giving ceremony.
The claim I’m making here is roughly that talk of causal independence presupposes that the allegedly independent things are the right kinds of things to stand in causal relationships. That is, that they are events. When I say that the states of the world must be causally independent of Chooser’s action, I mean that they are the kind of thing that could have been caused by some action, but in fact they are not.
The main concern with this reply is that it overgenerates. We want to apply decision theory to bets on laws of nature, as when a chemist is betting on which research program will be most worthwhile. And some applications of causal decision theory, like the Spence model of education [insert citation here!] also seem to rely on states, like the underlying cognitive skill of the decider, that aren’t obviously events. So we’d have to see if imposing this constraint would undermine desired applications of decision theory.7
7 I rather doubt EDT is compatible with the use of signaling games to model real-world phenomena, so this objection isn’t entirely good news for the critic of CDT.
This gets to the final objection: perhaps we do in fact cause the past. It’s really tricky in the context of determinism to even say just what it is for a proposition to even be about a time, let alone motivate a causal independence claim.
Consider a toy example. I start eating my lunchtime burrito precisely four hours after my cat, Squid, started eating her breakfast. Call this proposition p. What time is the proposition p about? Is it when Squid ate her breakfast, or when I ate my lunch? It seems at least in part to be about the earlier time. Add to the story that my burrito was delayed because the person making it messed up a step. Did that mess-up cause p, and thereby cause a proposition in part about the past to be true? I think it sort of did; the bar on backward causation only applies to very special kinds of propositions, and p is not one of them.
Now change the example a bit. Squid sleeps in and doesn’t have breakfast. But we’re now in a deterministic world. At nine o’clock, the world is such that it’s determined I will start eating a burrito at one o’clock. Did the person who messed up the order cause me to start eating at one o’clock? Sure - I would have eating earlier without them. Did they cause the world at nine o’clock to be such that I’d start eating at one o’clock? Well, that’s a harder question. On the one hand, it would seem like a weird kind of backwards causation if they did so. On the other hand, if they did not, we have to distinguish the causal role of claims that are nomologically equivalent. That sounds bad too.
I think the best thing to say here is, echoing the previous point, that claims like Messing up the order caused the world at nine o’clock to be such that I started eating at one o’clock have a presupposition failure, and are neither true nor false. And the causal independence constraint is that it is false that the action Chooser takes causes which state of the world is actual.
Things are even clearer when it comes to propositions about the laws. Like David Lewis, I’m a Humean about laws of nature. So I think laws are constituted by all the actions in the world, past, present, and future. So I think we all totally cause, and constitute, the laws in everything we do. Now I don’t expect everyone to agree with this kind of Humeanism. But I do find it very odd that something which is set up as an objection to Lewis on decision theory8 simply starts with the assumption that Lewis is wrong about laws.
8 Ahmed says that this case is an objection to CDT more generally, but it’s clear from the introduction to his book that he’s really just objecting to Lewis. He gestures in the direction of an argument that Lewis’s version of CDT is sufficiently general that objections to it will apply to all other versions. But that claim is false, despite it’s Ludovician pedigree. O“ne really shouldn’t take objections to Lewis’s version of CDT to apply to the versions defended by Skyrms, or Joyce, or Fusco, or me.
- Note I’m really dismissing this problem not replying to it
- Objection 1: This violates the idealising constraints; we don’t know what we can do.
- Objection 2: These are not events, and causal independence requires independent events. (Caveat: All that matters is that it can be treated as an event, as in Spence or ChoKreps. Can it? Eh, maybe.)
- Objection 3: I think we do maybe cause the laws (or the past). One reason - Humeanism implies future events are partially constitutive of the laws. (Caveat: My response to Hawthorne.) And if determinism is true, then counterfactuals go all funny; maybe the past is counterfactually sensitive to current actions (for yes, see Lewis; for no, see Dorr).
- General objection: The point is to explain, predict, and maybe evaluate, ordinary people making ordinary decisions. And ordinary people come with ordinary pictures of how the world works. If it turned out the theory didn’t even apply to people with non-standard notions of causation, or who applied the notion of causation beyond its intended scope of understanding current day interactions between medium sized dry goods, I wouldn’t be too worried.
4.3 War on WAR
That leaves the point that CDT leaves one poorly off in Newcomb’s Problem, while other theories, like evidential decision theory (EDT) leave one well off. This isn’t a particular mark against CDT, since other theories, like EDT, leave one poorly off in some situations. Here is one such case.
There are two demons, who will predict what Chooser will do. Both of them are arbitrarily good, though not quite perfect, and their errors are independent. Both demons will predict what Chooser does before anything else happens. Chooser will play either the left or right game in Table 4.4.
| PA | PB | |
|---|---|---|
| A | 2 | 0 |
| B | 3 | 1 |
| PA | PB | |
|---|---|---|
| A | 1002 | 1000 |
| B | 1003 | 1001 |
If Demon-1 predicts that Chooser will play A, Demon-1 will offer Chooser Table 4.4 (a); if Demon-1 predicts that Chooser will play B, Demon-1 will offer Chooser Table 4.4 (b). And Chooser knows what they are playing, that’s part of what it is to play a game, so Demon-1’s prediction will be announced, though Demon-2’s prediction will be secret. After Chooser makes a decision, Demon-2’s prediction will be used for determining whether the payout is from column PA or PB. In almost all cases, if Chooser uses CDT, they will get 1001, while if they use EDT, they will get 2. So in this case, CDT will get more than EDT.
This case is not meant as an objection to EDT. It is perfectly fair for the evidential decision theorist to complain that they have simply been the victim of a Demon who intends to punish users of EDT, and reward users of CDT. That seems a perfectly fair complaint. But if the evidential decision theorist makes it, they cannot object when causal decision theorists, such as Lewis (1981), use the same language to describe Newcomb’s Problem. The ‘objection’ that CDT leaves one poorly off in one particular case is equally an objection to everyone, and so it is an objection to no one.
One might object that this is unfair because at the time they make decisions, the EDTer and the CDTer have different evidence. After all, they will know what Demon-1 predicted and it will (almost certainly) be different in each case. Can we get rid of that step? We can, but it’s a bit complicated, and I’ve put that case in Appendix D. The short version is that there is an example where some versions of CDT predictably do better than EDT, even though at every point the followers of EDT and (those versions of) CDT have the same evidence when they are making choices.
While followers of EDT end up with more money than followers of causal theories when playing Newcomb’s Problem, this is not because of the distinctive money-making powers of EDT. It’s because Newcomb’s Problem is designed to leave causally based decision theories badly off. Design a case to leave evidential theories badly off, and they’ll be badly off. The “Why Ain’Cha Rich?” consideration tells against everyone, so it overgenerates, so it should be rejected.
Well, not quite everyone. It doesn’t tell against some kind of ‘resolute’ decision theories which recommend one-box in Newcomb’s Problem and Up in Table 4.4.9. Those theories leave their proponents well off in all the cases. The so-called ‘foundational decision theory’ that Levinstein & Soares (2020) endorse also endorse the same three choices. But those theories are vulnerable to much more serious objections, that I’ll come to in Section 7.4.
9 These theories recommend always playing Up in Figure D.1, the ‘complicated’ example I mentioned above
So I conclude that there is no good objection to adopting a broadly causal decision theory, much as the game theorists do. But which version of CDT do they adopt, and are they right to do so? That will take us much more time.