You are playing a game against ROB, the robber. In this game, ROB will choose two arbitrary (real/computable) numbers, A and B. ROB will send both numbers to TABI the arbiter, who will secretly flip a fair coin and then hand you either A or B depending on the outcome of the flip. Your task is to come up with an algorithm which does better than 50% accuracy to determine if you have the larger number.
I will now insert line breaks, then some hints, then more line breaks, then the solution. Edit: This sentence is meant to remind you that you can only discover the answer once, and it is sweet victory to solve puzzles.
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Note that ROB is obviously not uniformly choosing any two arbitrary real numbers—that is incoherent! Much worse, ROB is stealing your algorithm and choosing his numbers from a probability distribution which minimizes your changes of winning.
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You can probably solve this if you restrict ROB to only picking two of {0,1,2} with different probability distributions, and then extrapolate.
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The solution is to choose any strictly increasing function P whose range is (a subset of) (0,1), and guess that your number X is the larger number with probability P(X). For any given pair A<B selected by ROB, you will guess “larger” correctly with probability P(B)2, and you will guess “larger” incorrectly with probability P(A)2; since P is increasing, it is clear you will guess “larger” correctly more often than you guess “larger” incorrectly. I hope the remaining details are clear.
Shout-out to jacob.wood who posted a solution first, and dxu who posted the entire solution first.
This is naively surprising to those who know that strategies should often not be probabilistic, even when the games you are playing are. Yudkowsky’s newest novel demonstrates other areas where probabilistic strategies dominate. I will cover an example in my next post, as well as illustrate real world use cases where I have immediately applied the lesson.
Deterministic Strategies Can Be Suboptimal
Yesterday I posted the following problem
I will now insert line breaks, then some hints, then more line breaks, then the solution. Edit: This sentence is meant to remind you that you can only discover the answer once, and it is sweet victory to solve puzzles.
.
.
.
.
.
.
.
.
.
Note that ROB is obviously not uniformly choosing any two arbitrary real numbers—that is incoherent! Much worse, ROB is stealing your algorithm and choosing his numbers from a probability distribution which minimizes your changes of winning.
.
.
.
.
.
.
.
.
.
You can probably solve this if you restrict ROB to only picking two of {0,1,2} with different probability distributions, and then extrapolate.
.
.
.
.
.
.
.
.
.
The solution is to choose any strictly increasing function P whose range is (a subset of) (0,1), and guess that your number X is the larger number with probability P(X). For any given pair A<B selected by ROB, you will guess “larger” correctly with probability P(B)2, and you will guess “larger” incorrectly with probability P(A)2; since P is increasing, it is clear you will guess “larger” correctly more often than you guess “larger” incorrectly. I hope the remaining details are clear.
Shout-out to jacob.wood who posted a solution first, and dxu who posted the entire solution first.
This is naively surprising to those who know that strategies should often not be probabilistic, even when the games you are playing are. Yudkowsky’s newest novel demonstrates other areas where probabilistic strategies dominate. I will cover an example in my next post, as well as illustrate real world use cases where I have immediately applied the lesson.
The link to Yudkowsky’s newest novel is here: https://www.glowfic.com/posts/4582