(By the way, the numbers I gave are the same as the ones you gave, only I cancelled a common factor of (C+D+X))
And you don't want to combine my method with Zack's. You don't want a probabilistic strategy - you want to figure out what your beliefs are as far as "how many people do I expect to be in categories C, X, and D". Given your beliefs about how your choices affect others, there's exactly one right choice.
I think that your "one right choice" might sometimes be a probabilistic one. To make this more obvious, consider a game where the value of the prize is maximal when exactly half of the participants choose C, and the value goes down as the proportion gets further from a half (and any of the participants is equally likely to win the prize). Then I think it's obvious that the correct strategy is to estimate C, D, and X as before, and then cooperate with probability p so that C+pX=D+(1-p)X. Then because everyone else in X acts as you do you'll end up with exactly half the people choosing C, which is what you want.
Note that even some of the people in X who you are "acausally controlling" still end up choosing a different option from you (assuming that your random number generators are independent). This allows you to exactly optimise the proportion of people who choose C, which is what makes the strategy work.
I think the same thing applies in Yvain's game. In particular, if we thought that C=D=0 then I think that Zack's analysis is exactly correct (although I wouldn't have used exactly the same words as he does).
EDIT: I retract the last sentence. Zack's calculation isn't what you want to do even in the C=D=0 case. In that case I endorse cooperating with p=1. But I still think that mixed strategies are best in some of the cases with C or D non-zero. In particular what about the case with D=0 but C=X? Then I reckon you should pick C with p=0.724.
I think this is it. Suppose there are C CooperateBots, D DefectBots, and X players who Cooperate with probability p. The expected utility of the probabilistic strategy is (proportional to) (p(C + pX) + 4(1-p)(C + pX))/(C + 4D + pX + 4(1-p)X). Then (he said, consulting his computer algebra system) if C/X < 1/3 then p = 1 (Cooperate), if C/X > 3 then p = 0 (Defect), and p assumes intermediate values if 1/3 < C/X < 3 (including 0.7239 if C/X = 1, as you mention).
It's that time of year again.
If you are reading this post, and have not been sent here by some sort of conspiracy trying to throw off the survey results, then you are the target population for the Less Wrong Census/Survey. Please take it. Doesn't matter if you don't post much. Doesn't matter if you're a lurker. Take the survey.
This year's census contains a "main survey" that should take about ten or fifteen minutes, as well as a bunch of "extra credit questions". You may do the extra credit questions if you want. You may skip all the extra credit questions if you want. They're pretty long and not all of them are very interesting. But it is very important that you not put off doing the survey or not do the survey at all because you're intimidated by the extra credit questions.
It also contains a chance at winning a MONETARY REWARD at the bottom. You do not need to fill in all the extra credit questions to get the MONETARY REWARD, just make an honest stab at as much of the survey as you can.
Please make things easier for my computer and by extension me by reading all the instructions and by answering any text questions in the simplest and most obvious possible way. For example, if it asks you "What language do you speak?" please answer "English" instead of "I speak English" or "It's English" or "English since I live in Canada" or "English (US)" or anything else. This will help me sort responses quickly and easily. Likewise, if a question asks for a number, please answer with a number such as "4", rather than "four".
Last year there was some concern that the survey period was too short, or too uncertain. This year the survey will remain open until 23:59 PST December 31st 2013, so as long as you make time to take it sometime this year, you should be fine. Many people put it off last year and then forgot about it, so why not take it right now while you are reading this post?
Okay! Enough preliminaries! Time to take the...
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2013 Less Wrong Census/Survey
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Thanks to everyone who suggested questions and ideas for the 2013 Less Wrong Census/Survey. I regret I was unable to take all of your suggestions into account, because of some limitations in Google Docs, concern about survey length, and contradictions/duplications among suggestions. I think I got most of them in, and others can wait until next year.
By ancient tradition, if you take the survey you may comment saying you have done so here, and people will upvote you and you will get karma.