I want to try to understand the nature of logical correlations between agents a bit better.
Consider two agents who are both TDT-like but not perfectly correlated. They play a one-shot PD but in turn. First one player moves, then the other sees the move and makes its move.
In normal Bayesian reasoning, once the second player sees the first player’s move, all correlation between them disappears. (Does this happen in your TDT?) But in UDT, the second player doesn’t update, so the correlation is preserved. So far so good.
Now consider what happens if the second player has more computing power than the first, so that it can perfectly simulate the first player and compute its move. After it finishes that computation and knows the first player’s move, the logical correlation between them disappears, because no uncertainty implies no correlation. So, given there’s no logical correlation, it ought to play D. The first player would have expected that, and also played D.
Looking at my formulation of UDT, this may or may not happen, depending on what the “mathematical intuition subroutine” does when computing the logical consequences of a choice. If it tries to be maximally correct, then it would do a full simulation of the opponent when it can, which removes logical correlation, which causes the above outcome. Maybe the second player could get a better outcome if it doesn’t try to be maximally correct, but the way my theory is formulated, what strategy the “mathematical intuition subroutine” uses is not part of what’s being optimized.
So, I’m not sure what to do about this, except to add it to the pile of unsolved problems.
Coming to this a bit late :), but I’ve got a basic question (which I think is similar to Nesov’s, but I’m still confused after reading the ensuing exchange). Why would it be that,
The first player would have expected that, and also played D.
If the second player has more computer power (so that the first player cannot simulate it), how can the first player predict what the second player will do? Can the first player reason that since the second player could simulate it, the second player will decide that they’re uncorrelated and play D no matter what?
That dependence on computing power seems very odd, though maybe I’m sneaking in expectations from my (very rough) understanding of UDT.
Now consider what happens if the second player has more computing power than the first, so that it can perfectly simulate the first player and compute its move. After it finishes that computation and knows the first player’s move, the logical correlation between them disappears, because no uncertainty implies no correlation. So, given there’s no logical correlation, it ought to play D. The first player would have expected that, and also played D.
The first player’s move could depend on the second player’s, in which case the second player won’t get the answer is a closed form, the answer must be a function of the second player’s move...
But if the second player has more computational power, it can just keep simulating the first player until the first player runs out of clock cycles and has to output something.
I don’t understand your reply: exact simulation is brute force that isn’t a good idea. You can prove general statements about the behavior of programs on runs of unlimited or infinite length in finite time. But anyway, why would the second player provoke mutual defection?
But anyway, why would the second player provoke mutual defection?
In my formulation, it doesn’t have a choice. Whether or not it does exact simulation of the first player is determined by its “mathematical intuition subroutine”, which I treated as a black box. If that module does an exact simulation, then mutual defection is the result. So this also ties in with my lack of understanding regarding logical uncertainty. If we don’t treat the thing that reasons about logical uncertainty as a black box, what should we do?
ETA: Sometimes exact simulation clearly is appropriate, for example in rock-paper-scissors.
I want to try to understand the nature of logical correlations between agents a bit better.
Consider two agents who are both TDT-like but not perfectly correlated. They play a one-shot PD but in turn. First one player moves, then the other sees the move and makes its move.
In normal Bayesian reasoning, once the second player sees the first player’s move, all correlation between them disappears. (Does this happen in your TDT?) But in UDT, the second player doesn’t update, so the correlation is preserved. So far so good.
Now consider what happens if the second player has more computing power than the first, so that it can perfectly simulate the first player and compute its move. After it finishes that computation and knows the first player’s move, the logical correlation between them disappears, because no uncertainty implies no correlation. So, given there’s no logical correlation, it ought to play D. The first player would have expected that, and also played D.
Looking at my formulation of UDT, this may or may not happen, depending on what the “mathematical intuition subroutine” does when computing the logical consequences of a choice. If it tries to be maximally correct, then it would do a full simulation of the opponent when it can, which removes logical correlation, which causes the above outcome. Maybe the second player could get a better outcome if it doesn’t try to be maximally correct, but the way my theory is formulated, what strategy the “mathematical intuition subroutine” uses is not part of what’s being optimized.
So, I’m not sure what to do about this, except to add it to the pile of unsolved problems.
Coming to this a bit late :), but I’ve got a basic question (which I think is similar to Nesov’s, but I’m still confused after reading the ensuing exchange). Why would it be that,
If the second player has more computer power (so that the first player cannot simulate it), how can the first player predict what the second player will do? Can the first player reason that since the second player could simulate it, the second player will decide that they’re uncorrelated and play D no matter what?
That dependence on computing power seems very odd, though maybe I’m sneaking in expectations from my (very rough) understanding of UDT.
The first player’s move could depend on the second player’s, in which case the second player won’t get the answer is a closed form, the answer must be a function of the second player’s move...
But if the second player has more computational power, it can just keep simulating the first player until the first player runs out of clock cycles and has to output something.
I don’t understand your reply: exact simulation is brute force that isn’t a good idea. You can prove general statements about the behavior of programs on runs of unlimited or infinite length in finite time. But anyway, why would the second player provoke mutual defection?
In my formulation, it doesn’t have a choice. Whether or not it does exact simulation of the first player is determined by its “mathematical intuition subroutine”, which I treated as a black box. If that module does an exact simulation, then mutual defection is the result. So this also ties in with my lack of understanding regarding logical uncertainty. If we don’t treat the thing that reasons about logical uncertainty as a black box, what should we do?
ETA: Sometimes exact simulation clearly is appropriate, for example in rock-paper-scissors.