So, I’ve been obsessed lately with a couple of really good posts over on Thoughts, Arguments, and Rants about decision theory. In particular, there’s this post about a game Brian Weatherson calls (for reasons that are clear if you read the post) “Asymmetric Death in Damascus.”
Weatherson’s Original Example
Here’s a quick summary of the game. Predictor has placed money in two boxes, and gives you a choice of picking Box A or picking Box B. (He hates mixed strategies, so won’t put any money in the boxes if he thinks you’re going to use one). He allocates the money as follows:
- If he predicts you’re going to choose Box A, he puts $100 in A and $1400 in B.
- If he predicts you’re going to choose Box B, he puts $800 in A and $700 in B.
Now, I think there’s an obvious right choice here–box B. Note that the standard “two-boxing” argument doesn’t argue for either A or B, and the standard “one- boxing” argument argues for B. But as Weatherson points out, there’s a problem with following the “one-boxing” argument: one-boxing arguments don’t work.
Still, though, the intuitive pull of Box B is pretty strong here. And note that, while Weatherson demolishes the argument that chosing Box B is rational (at least for us two-boxers out there), he doesn’t actually present an argument that it’s not rational. It’s just that the standard techniques (such as ratifiability) available to two-boxers don’t endorse it.
What would be really cool is if we had a plausible general account of rational decisions that mandated two-boxing in Newcomb’s paradox, but also mandated picking Box B in Asymmetric Death in Damascus (and, dare we dream, ensured that any game with a finite set of choices had at least one rational choice?). And I think I do.
My Account
This is an account of the rational choice (or choices, if multiple choices are equally rational) for single move-games with finite possible moves. That last part is important, because there are steps here that are undefined for infinite-choice games. But I think infinite-choice games have problems up the wazoo; I’m not really convinced that the notion of an infinite-choice game is even coherent if examined too closely (that’s a topic for another post, though). It relies on a rather weaker notion than ratifiability, which I’ll call cyclic ratifiability, and then adds a totally different onto cyclic reliability.
To make the setup easier, let’s start with a game where every possible outcome is distinct. We’ll make the necessary adaption for games where multiple choice/circumstance combinations can lead to the same outcome afterwards.
For any choice in this game, c, I’m going to define the decision chain from c as an infinite sequence defined as follows:
- c[0] = c.
- For any n>=0, c[n+1] is the action with the highest payoff, given that you’ve chosen action c[n].
c is cyclically ratifiable if c[n]=c for some n>0. It should be fairly obvious that among a finite set of choices, there must always be at least one cyclically ratifiable choice (because every decision chain will eventually become cyclic–at or before n=number of choices, by the pigeon-hole principle, if you count being constant as being cyclic with period 1).
My claim is that, in this case, the uniquely rational choice is the cyclically ratifiable choice such that, given that you’ve chosen it, it has the highest payoff. That is, it’s the ideal one-boxer solution among the cyclically ratifiable choices.
What’s the intuitive justification for this? Well, just as the idea behind ratifiability is that you don’t want to pick an option that will instantly seem like a bad idea, the idea here is that, if you do pick an option that will instantly seem like a bad idea, you at least want to ensure that, at some point in the future, it will again seem like a good idea. It differs from ratifiability in that being “led away” from an option through failure of ratifiability isn’t necessarily bad, as long as you’ll periodically be led back to it. And given choices that are identical in that way, you might as well pick the one with the highest expected value.
And this account does have the big advantage of dealing with both Newcomb’s Paradox and Asymmetric Death in Damascus appropriately: It endorses two-boxing in Newcomb’s Paradox (as one-boxing isn’t cyclically ratifiable), and endorses Box B in Asymmetric Death (as both options are cyclically ratifiable, and at that point maximizing expected value takes over).
Now to adapt this to cases where the outcomes of multiple choice/circumstance combinations might be identical. The problem with the above is that there’s no unique decision chain from a given choice–the decision chain can branch when there are multiple actions with the maximal payoff when you’ve decided on choice c[n]. Instead, there’s an infinite “decision tree” (a very unfortunate name, since it means something else in other contexts, but I can’t think of a better one for now).
I think, though, that we can redefine cyclic ratifiability to work with decision trees too. A choice c here is cyclically ratifiable iff, for any choice d that’s a descendent of c in the ratifiability tree, c is also a descendent of d. The proof that there’s always at least one cyclically ratifiable choice is a bit more complex than in the chain case, but it’s not too hard. The proof’s by induction.
In a game with only one choice, of course that choice is cyclically ratifiable; the base case is easy.
Now, let’s suppose that, for any game with fewer than n choices, there is at least one choice that is cyclically ratifiable. Let’s introduce a game G with n choices, and suppose it’s has no cyclically ratifiable choices. Let’s let c be any choice in that game. Then, c has a descendent, d, such that c isn’t a descendent of d. That is, c does not occur anywhere in the decision tree for d. Let’s modify the game to include only the choices that do occur in d’s decision tree, and call this game G’. Now, note two things:
- The decision tree for any choice in G’ is the same as it would be for that choice in G, since no non-G’ choices will ever crop up in its G decision tree.
- Since G’ doesn’t contain c, it has fewer than n choices, so it has at least one choice (call it e) that is cyclically ratifiable.
Since e is cyclically ratifiable in G’, it’s a decendent of any node in its G’ decision tree. But since it’s G’ subtree and its G subtree are identical, it’s also a decendent of any node in its G decision tree. So it’s cyclically ratifiable in G, a contradiction, meaning that any game with n choices must have at least one cyclically ratifiable choice.
So, by induction, any game with a finite number of choices must have at least one cyclically ratifiable choice.
Since there’s no guarantee that all cyclically ratifiable choices in such a game will have distinct expected payoffs, we have to make a (fairly obvious) modification there too: Instead of the unique rational choice being the cyclically ratifiable choice with the highest payoff, any cyclically ratifiable choices with maximal expected payoffs are equally rational.
I’m very curious about what people think of this. Are there any cases where this recommends a decision which pulls away from intuition (at least, away from two-boxer intuition)?
Actually, I’m not sure applying this to infinite-choice games is a bad idea. If we *do* apply it to infinite games, it suggests that in some infinite-choice games there is *no* rational choice (because there aren’t always cyclically ratifiable options in infinite games). In particular, it suggests that the infinite game Weatherson presents here has no rational options, which (at least) strikes me as a much better claim than what ratifiability would suggest, which is that 0 (the worst possible move) is the unique rational option.
(Actually, it occurs to me that, as this shows, cyclic ratifiability isn’t a strictly weaker notion than ratifiability. It’s weaker for games where every outcome has a different payoff, but not necessarily for games where multiple outcomes have the same payoff–because some branches of the decision tree will never lead you back to the original choice, even if some will.)
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