Modal SAT: Self Cooperation

post by Scott Garrabrant · 2015-08-10T05:48:33.000Z · LW · GW · 4 comments

Contents

  Theorem: If there exists a modal agent M such that Ci cooperates with M for each Ci∈C and such that M defects against Di for each Di∈D, then there exists an M′ which satisfies the above properties and cooperates with itself.
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4 comments

Post 2 in in Modal SAT series. In this post, we show that SC Modal SAT is equivalent to Modal SAT.

For this, we need just need to prove the following theorem:

Theorem: If there exists a modal agent such that cooperates with for each and such that defects against for each , then there exists an which satisfies the above properties and cooperates with itself.

Proof: Let be greater than the total number of boxes in agents in and . Consider the agents and defined by .

We define by

  1. if and and

  2. if and and

  3. if and and

  4. otherwise

Rule 1 says that cooperates with , rule 2 says that defects against , and rule 3 says that cooperates with anyone who (provably assuming ) cooperates with and (provably assuming )defects against . Thus, cooperates with itself.

For any bot with fewer than boxes, the conditions of 1, 2, and 3 are all false. For 1 and 2, this is because the actions of such bots against CoopearteBot stabilize by the time you assume . For 3, this is because these bots cannot distinguish between and .

Therefore behaves the same as on all inputs with fewer than boxes, so cooperates with every bot in and defects against every bot in .

Note that I was lazy here, and took way more longer to cooperate with myself than I had to. In principle, if there are only bots that I need to consider (including bots I need to consider because they are referenced by bots I care about), then regardless of how many boxes are in each bot, It should be possible to achieve self cooperation within worlds of the Kripke frame. That is, worlds to identify a single bot that is distinguishable from all other bots, and another worlds to ensure that the actions of differ on that bot from all other bots, so that can identify itself without changing its behavior against any other bot.

EDIT: Actually, I think + a small constant should suffice, but it does not matter much.

4 comments

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comment by orthonormal · 2015-08-10T00:05:48.000Z · LW(p) · GW(p)

I'm confused; does your definition of imply that it evaluates those conditionals in order? If so, consider the example of and DefectBot. The you construct will cooperate in world 0, and thus it will not get the cooperation of , while does. What am I missing?

Replies from: Scott Garrabrant
comment by Scott Garrabrant · 2015-08-10T05:41:38.000Z · LW(p) · GW(p)

Your confusion was justified. It was wrong before. I think it is fixed now.

The conditionals are checked in order. As it is written now, none of the conditionals except the last one should trigger until world .

The first only triggers against , the second only triggers against and the third only triggers against .

Replies from: agilecaveman
comment by agilecaveman · 2015-08-11T06:01:46.000Z · LW(p) · GW(p)

I am also confused. How does this do against EABot, aka C1=□(Them(Them)=D) and M = DefectBot. Is the number of boxes not well defined in this case?

Replies from: Scott Garrabrant
comment by Scott Garrabrant · 2015-08-11T07:31:48.000Z · LW(p) · GW(p)

So according to the original Modal Combat framework, EABot is not a Modal Agent. The bots are not allowed to simulate Them(Them).