A puzzle for causal decision theory
This is probably old, so pointers to the literature are welcome. Consider this game between Column and Row:
| C1 | C2 | |
|---|---|---|
| R1 | 0,0 | 2,2 |
| R2 | 2,2 | 1,1 |
What should Column and Row do if they know that they are equally rational and can't communicate with one another? The game doesn't have a Nash equilibrium has no unique Nash equilibrium, nor is there a dominant strategy (Thanks Marc!), so perhaps there is no determinate answer.
But now suppose the two players are perfect duplicates of one another situated in duplicate environments in a deterministic universe, and know with absolute certainty that this is so. Hence they can be certain that whatever they choose (option 1 or option 2), the other one will choose the same: they will inevitably end up at either 0,0 or 1,1. Isn't in this case option 2 the better strategy?
I would like to reach this conclusion even with causal decision theory, but I'm not sure how. Choosing option 1 would not cause the other player to choose option 1. Indeed, suppose they both choose option 2 and now we look at the relevantly closest world at which Row chooses R1. At this world, a small miracle occurs in Row's brain that makes him choose R1. There is no reason to suppose that a similar miracle would have to occur in Column's brain. Hence in the relevantly closest world at which Row chooses R1, Column still chooses C1. So if Row had chosen R1, they would have ended up with payoff 2,2. Wouldn't that have been the better choice then? -- On the other hand, as soon as Row actually chooses R1, of course no miracle will occur and thus Column is guaranteed to choose C1 as well.
Should we perhaps not look unrestrictedly at the closest worlds in which the subject makes a particular choice, but rather at the closest doxastically possible worlds where she does? Is it illegitimate to justify a move by saying (truly!) that it has a good chance of leading to X if at the same time you are 100% certain that X will never occur? Or is it important that our game does not have a stable solution? (My intuition that one should play option 2 here is stronger than my intuition that one should one-box in a Newcomb problem with a 100% reliable predictor.)
Hi Wo,
Just to make sure that I understand what you're after. When you say that the game "doesn't have a Nash equilibrium", do you mean that the game has not a *unique* NE in pure strategies? -- For, without further assumptions, <R2,C1> and <R1,C2> are (strong) NE. Consider the profile <R2,C1>: If Column plays C1, the best Row can do is to play R2. If Row plays R2, then the best Column can do is to play C1. So <R2,C1> is each other's best response and therefore a Nash equilibrium.
- Marc