The true limit would be 0 if everyone was rational, and aware of the rationality of everyone else, but rational people in the real world should be taking into account...
the fact that most people only seem to be able cope with two or three recursions before they get bored and pick an option
So what you should do, based on that, is try to figure out how many iterations “most people” will do, and then estimate the smaller percentage of the “rank one pragmatic rationalists” who will realize this and aim for 2⁄3 of “most people” and so on until you have accounted for 100% of the population. The trick is knowing that some people aren’t logical means logical strategy (that will actually win) requires population modeling rather than game theory. Hearing your average of 30 makes me hypothesize that the distribution of people looks something like this:
0.1% 100 Blind number repeaters
4.9% 66 Blind multipliers
17% 44 Beating the blind!
28% 30 "most people"
20% 20 Read a study or use a "recurse three times" heuristic
15% 13 Contrarian rationalists
10% 9 Meta-contrarian rationalists
4.9% 6 Economists (I kid!)
0.1% 0 Obstinate game theorists
Suppose you run the experiment again (say, on a mixture of people from LW who have read this combined with others) where you expect that a lot of people might have actually seen this hypothesis, but a lot of people will naively play normally. I think the trick might be to figure out what the percentage of LWers you’re dealing with is and then figure out what to do based on that.
I’m tempted (because it would be amusing) to try to estimate the percentage of LWers relative to naive, and then model the LWers as people who execute some variant of timeless decision theory in the presence of the baseline people. If I understand timeless decision theory correctly, the trick would be for everyone to independently derive the number all the LWers would have to pick in order for us all to tie at winning given the presence of baselines. It seems kind of far fetched, but it would be ticklesome if it ever happened :-D
OK, now I’m curious! I think I might try this at the next SoCal meetup.
ETA: …or does utilitarian consequentialism mean we need to calculate the value we’d all have to vote to maximize the number of winners, whether or not they were expected to participate in the calculation? That sounds even harder.
This is a follow up to my comment just above. At the meetup this game was played with six people: me, three other people deeply read in the sequences, and two people who hadn’t read the sequences.
The results were: four people guessed 0, a deeply read person guessed 3, and one of the people not familiar with the sequences guessed 23. The target was (26/6)*(2/3) = 2 and 8/9ths, so the guess of 3 won with an error of only 1/9th.
I’ll leave the other people to explain what their thinking was if they want, but I directly estimated “the obvious thing” for each of the other players and got an estimated sum of 36. Then I tried to imagine what would happen if some of us chose to do what I was doing (either with TDT or without). It became clear that even with TDT as an intention, differences in estimates could lead to slightly different fixed point calculations and someone winning over someone else “by accident”.
I didn’t personally have the mental arithmetic skills to even calculate the fixed point fast enough but I noticed that I expected at least one other 0 guess anyway. And people’s guesses would be revealed so I thought maybe it would be an opportunity to signal, and I wanted to signal “non-defection” and figured other people who had this insight would also want to signal that way. So zero started to look like “the moral schelling point” that might be everyone’s guess and would at least give me a good signaling outcome in compensation for not “winning along with all the other 0′s”, so I guessed 0.
In the immediate aftermath I felt pretty good about it when the numbers were revealed because I got the signaling outcome I wanted and my guesstimates for other people were off by enough that if I had defected I would have probably been at like 5 or 6 and lost anyway to the 3. So it appears that I was too close to the median to out-clever this group, and though I wasn’t honest enough to realize this at the time my desire for a good signaling outcome left me with exactly the compensation for losing that I was hedging for, and I wasn’t left high and dry :-)
Writing this, I think the “small group with guesses revealed” aspect of the situation definitely influenced me, may have been a major influence on other players as well.
If I were doing a study using a one or two shot version of this game as the paradigm for political cooperation I think I might try to experimentally vary small versus large groups, the mixture of players with game naivety and/or pre-existing relationships, and player’s expectations of post-hoc analysis to see if they affect group outcomes.
My hypothesis would be that with at least one game-naive player and a significant mixture of pre-existing relationships, groups smaller than ~8 would usually converge towards signaling cooperation and game theoretic “optimal guesses”, groups larger than ~20 would usually converge on vying for recognition as “the winner” with precisely calibrated “very small answers”, and that post-hoc analysis expectations would positively correlate with convergence effects. My hypothesis is probably wrong in some details, but I think an experiment designed to test it would be fascinating :-)
The true limit would be 0 if everyone was rational, and aware of the rationality of everyone else, but rational people in the real world should be taking into account...
So what you should do, based on that, is try to figure out how many iterations “most people” will do, and then estimate the smaller percentage of the “rank one pragmatic rationalists” who will realize this and aim for 2⁄3 of “most people” and so on until you have accounted for 100% of the population. The trick is knowing that some people aren’t logical means logical strategy (that will actually win) requires population modeling rather than game theory. Hearing your average of 30 makes me hypothesize that the distribution of people looks something like this:
0.1% 100 Blind number repeaters4.9% 66 Blind multipliers17% 44 Beating the blind!28% 30 "most people"20% 20 Read a study or use a "recurse three times" heuristic15% 13 Contrarian rationalists10% 9 Meta-contrarian rationalists4.9% 6 Economists (I kid!)0.1% 0 Obstinate game theoristsSuppose you run the experiment again (say, on a mixture of people from LW who have read this combined with others) where you expect that a lot of people might have actually seen this hypothesis, but a lot of people will naively play normally. I think the trick might be to figure out what the percentage of LWers you’re dealing with is and then figure out what to do based on that.
I’m tempted (because it would be amusing) to try to estimate the percentage of LWers relative to naive, and then model the LWers as people who execute some variant of timeless decision theory in the presence of the baseline people. If I understand timeless decision theory correctly, the trick would be for everyone to independently derive the number all the LWers would have to pick in order for us all to tie at winning given the presence of baselines. It seems kind of far fetched, but it would be ticklesome if it ever happened :-D
OK, now I’m curious! I think I might try this at the next SoCal meetup.
ETA: …or does utilitarian consequentialism mean we need to calculate the value we’d all have to vote to maximize the number of winners, whether or not they were expected to participate in the calculation? That sounds even harder.
This is a follow up to my comment just above. At the meetup this game was played with six people: me, three other people deeply read in the sequences, and two people who hadn’t read the sequences.
The results were: four people guessed 0, a deeply read person guessed 3, and one of the people not familiar with the sequences guessed 23. The target was (26/6)*(2/3) = 2 and 8/9ths, so the guess of 3 won with an error of only 1/9th.
I’ll leave the other people to explain what their thinking was if they want, but I directly estimated “the obvious thing” for each of the other players and got an estimated sum of 36. Then I tried to imagine what would happen if some of us chose to do what I was doing (either with TDT or without). It became clear that even with TDT as an intention, differences in estimates could lead to slightly different fixed point calculations and someone winning over someone else “by accident”.
I didn’t personally have the mental arithmetic skills to even calculate the fixed point fast enough but I noticed that I expected at least one other 0 guess anyway. And people’s guesses would be revealed so I thought maybe it would be an opportunity to signal, and I wanted to signal “non-defection” and figured other people who had this insight would also want to signal that way. So zero started to look like “the moral schelling point” that might be everyone’s guess and would at least give me a good signaling outcome in compensation for not “winning along with all the other 0′s”, so I guessed 0.
In the immediate aftermath I felt pretty good about it when the numbers were revealed because I got the signaling outcome I wanted and my guesstimates for other people were off by enough that if I had defected I would have probably been at like 5 or 6 and lost anyway to the 3. So it appears that I was too close to the median to out-clever this group, and though I wasn’t honest enough to realize this at the time my desire for a good signaling outcome left me with exactly the compensation for losing that I was hedging for, and I wasn’t left high and dry :-)
Writing this, I think the “small group with guesses revealed” aspect of the situation definitely influenced me, may have been a major influence on other players as well.
If I were doing a study using a one or two shot version of this game as the paradigm for political cooperation I think I might try to experimentally vary small versus large groups, the mixture of players with game naivety and/or pre-existing relationships, and player’s expectations of post-hoc analysis to see if they affect group outcomes.
My hypothesis would be that with at least one game-naive player and a significant mixture of pre-existing relationships, groups smaller than ~8 would usually converge towards signaling cooperation and game theoretic “optimal guesses”, groups larger than ~20 would usually converge on vying for recognition as “the winner” with precisely calibrated “very small answers”, and that post-hoc analysis expectations would positively correlate with convergence effects. My hypothesis is probably wrong in some details, but I think an experiment designed to test it would be fascinating :-)