Notice however that for Logical Counterfactual Mugging to be well defined, you need to define what Omega is doing when it is making its prediction. In Counterfactuals for Perfect Predictors, I explained that when dealing with perfect predictors, often the counterfactual would be undefined. For example, in Parfit’s Hitchhicker a perfect predictor would never give a lift to someone who never pays in town, so it isn’t immediately clear that predicting what such a person would do in town involves predicting something coherent.
Another approach is to change the example to remove the objection.
The poker-like game at the end of Decision Theory (I really titled that post simply “decision theory”? rather vague, past-me...) is isomorphic to counterfactual mugging, but removes some distractions, such as “how does Omega take the counterfactual”.
Alice receives a High or Low card. Alice can reveal the card to Bob. Bob then states a probability p for Alice’s card being Low. Bob’s incentives just encourage him to report honest beliefs. Alice loses p2.
When Alice gets a Low card, she can just reveal it to Bob, and get the best possible outcome. But this strategy means Bob will know if she has a high card, giving her the worst possible outcome in that case. In order to successfully bluff, Alice has to sometimes act like she has different cards than she has. And indeed, the optimal strategy for Alice in this case is to never show her cards.
This example will get less objections from people, because it is grounded in a very realistic game. Playing poker well requires this kind of reasoning. The powerful predictor is replaced with another player. We can still technically ask “how is the other player deal with undefined counterfactuals?”, but we can skip over that by just reasoning about strategies in the usual game-theoretic way—if Alice’s strategy were to reveal low cards, then Bob could always call high cards.
We can then insert logical uncertainty by stipulating that Alice gets her card pseudorandomly, but neither Alice nor Bob can predict the random number generator.
Not sure yet whether you can pull a similar trick with Counterfactual Prisoner’s Dilemma.
== nitpicks ==
Applying the Counterfactual Prisoner’s Dilemma to Logical Uncertainty
Why isn’t the title “applying logical uncertainty to the counterfactual prisoner’s dilemma”? Or “A Logically Uncertain Version of Counterfactual Prisoner’s Dilemma”? I don’t see how you’re applying CPD to LU.
Symmetric? The original is already symmetric. But “symmetric” is a concept which applies to multi-player games. Counterfactual PD makes PD into a one-player game. Presumably you meant “a one-player version”?
where regardless of whether the coin comes up heads or tails you are asked to pay $100 and you are then paid $10,000 if Omega predicts that you would have paid if the coin had come up the other way. If you decide updatelesly you will always received $9900, while if you decide updatefully, then you will receive $0.
This is only true if you use classical CDT, yeah? Whereas EDT can get $9900 in both cases, provided it believes in a sufficient correlation between what it does upon seeing heads vs tails.
So unlike Counterfactual Mugging, pre-committing to pay ensures a better outcome regardless of how the coin flip turns out, suggesting that focusing only on your particular probability branch is mistaken.
I don’t get what you meant by the last part of this sentence. Counterfactual Mugging already suggests that focusing only on your particular branch is mistaken. If someone bought that you should pay up in this problem but not in counterfactual mugging, I expect that person to say something like “because in this case that strategy is guaranteed better even in this branch”—hence, they’re not necessarily convinced to look at other branches. So I don’t think this example necessarily argues for looking at other branches.
I’m curious, do you find this argument for paying in Logical Counterfactual Mugging persuasive? What about the Counterfactual Prisoner’s Dilemma argument for the basic Counterfactual Mugging?
Another approach is to change the example to remove the objection
Interesting point about the poker game version. It’s still a one shot game, so there’s no real reason to hide a 0 unless you think they’re a pretty powerful predictor, but it is always predicting something coherent.
I don’t see how you’re applying CPD to LU
The claim is that you should pay in the Logical Counterfactual Prisoner’s Dilemma and hence pay in Logical Counterfactual Mugging which is the logically uncertain version of Counterfactual Mugging.
Symmetric? The original is already symmetric. But “symmetric” is a concept which applies to multi-player games. Counterfactual PD makes PD into a one-player game. Presumably you meant “a one-player version”?
Edited now. I meant it’s a symmetric version of counterfactual mugging. So not in the game theory sense, but just that there is now no difference between heads and tails.
This is only true if you use classical CDT, yeah? Whereas EDT can get $9900 in both cases, provided it believes in a sufficient correlation between what it does upon seeing heads vs tails.
Point noted. Maybe I should have been more careful about specifying what I was comparing
Another approach is to change the example to remove the objection.
The poker-like game at the end of Decision Theory (I really titled that post simply “decision theory”? rather vague, past-me...) is isomorphic to counterfactual mugging, but removes some distractions, such as “how does Omega take the counterfactual”.
Alice receives a High or Low card. Alice can reveal the card to Bob. Bob then states a probability p for Alice’s card being Low. Bob’s incentives just encourage him to report honest beliefs. Alice loses p2.
When Alice gets a Low card, she can just reveal it to Bob, and get the best possible outcome. But this strategy means Bob will know if she has a high card, giving her the worst possible outcome in that case. In order to successfully bluff, Alice has to sometimes act like she has different cards than she has. And indeed, the optimal strategy for Alice in this case is to never show her cards.
This example will get less objections from people, because it is grounded in a very realistic game. Playing poker well requires this kind of reasoning. The powerful predictor is replaced with another player. We can still technically ask “how is the other player deal with undefined counterfactuals?”, but we can skip over that by just reasoning about strategies in the usual game-theoretic way—if Alice’s strategy were to reveal low cards, then Bob could always call high cards.
We can then insert logical uncertainty by stipulating that Alice gets her card pseudorandomly, but neither Alice nor Bob can predict the random number generator.
Not sure yet whether you can pull a similar trick with Counterfactual Prisoner’s Dilemma.
== nitpicks ==
Why isn’t the title “applying logical uncertainty to the counterfactual prisoner’s dilemma”? Or “A Logically Uncertain Version of Counterfactual Prisoner’s Dilemma”? I don’t see how you’re applying CPD to LU.
Symmetric? The original is already symmetric. But “symmetric” is a concept which applies to multi-player games. Counterfactual PD makes PD into a one-player game. Presumably you meant “a one-player version”?
This is only true if you use classical CDT, yeah? Whereas EDT can get $9900 in both cases, provided it believes in a sufficient correlation between what it does upon seeing heads vs tails.
I don’t get what you meant by the last part of this sentence. Counterfactual Mugging already suggests that focusing only on your particular branch is mistaken. If someone bought that you should pay up in this problem but not in counterfactual mugging, I expect that person to say something like “because in this case that strategy is guaranteed better even in this branch”—hence, they’re not necessarily convinced to look at other branches. So I don’t think this example necessarily argues for looking at other branches.
Also, why is this posted as a question?
I’m curious, do you find this argument for paying in Logical Counterfactual Mugging persuasive? What about the Counterfactual Prisoner’s Dilemma argument for the basic Counterfactual Mugging?
Interesting point about the poker game version. It’s still a one shot game, so there’s no real reason to hide a 0 unless you think they’re a pretty powerful predictor, but it is always predicting something coherent.
The claim is that you should pay in the Logical Counterfactual Prisoner’s Dilemma and hence pay in Logical Counterfactual Mugging which is the logically uncertain version of Counterfactual Mugging.
Edited now. I meant it’s a symmetric version of counterfactual mugging. So not in the game theory sense, but just that there is now no difference between heads and tails.
Point noted. Maybe I should have been more careful about specifying what I was comparing
Accident. It’s fixed now