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Are two heads better than one?

eieio.games

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Re: Are two heads better than one?

#11

Post by zahlman »

I imagined it like:

    F  T (Alice)
  F xx ????????
    xx ????????

  T ?? vvvvvvvv
    ?? vvvvvvvv
  ^ ?? vvvvvvvv
  B ?? vvvvvvvv
  o ?? vvvvvvvv
  b ?? vvvvvvvv
  v ?? vvvvvvvv
    ?? vvvvvvvv
(where "F" describes cases where the specified person tells you a Falsehood, and "T" labels the cases of that person telling you the Truth)

In the check-mark (v) region, you get the right answer regardless; they are both being truthful, and of course you trust them when they agree. Similarly you get the wrong answer regardless in the x region.

In the ? region you are no better than a coin flip, regardless of your strategy. If you unconditionally trust Alice then you win on the right-hand side, and lose on the left-hand side; and whatever Bob says is irrelevant. The situation for unconditionally trusting Bob is symmetrical (of course it is; they both act according to the same rules, on the same information). If you choose any other strategy, you still have a 50-50 chance, since Alice and Bob disagree and there is no reason to choose one over the other.

Since your odds don't change with your strategy in any of those regions of the probability space, they don't change overall.

Re: Are two heads better than one?

#13

Post by stared »

In voting parity matters. In some cases, effects are interesting - like in some cases if the mafia/werewolf game, when adding a citizen/villager decreses their winning chance by sqrt(pi/2), vide https://arxiv.org/abs/1009.1031

When it comes to the wisdom of crowds, see https://egtheory.wordpress.com/2014/01/30/two-heads-are-bett...

Re: Are two heads better than one?

#14

Post by benlivengood »

nick238 wrote: post #10

This kinda reminds me of error correction, and where at some level you can have detectable but not correctable error conditions. Adding Bob is just like adding a parity bit: can give you a good indication someone lied, but won't fix anything. Adding Charlie gives you the crudest ECC form, a repetition code (though for storing one bit, I don't think you can do better?)

I guess if you only need to store one bit you could store either 0 or 11 and on average use less than two bits (for bit flips only), or 111 if you have to also worry about losing/duplicating bits.

Re: Are two heads better than one?

#17

Post by thomascountz »

Let's pretend Alice tells the truth 100% of the time, and Bob is still at 20%. It's more easy to intuit that Bob's contribution is only noise.

Slide Alice's accuracy down to 99% and, again, if you don't trust Alice, you're no better off trusting Bob.

Interestingly, this also happens as a feature of them being independent. If Bob told the truth 20% of the time that Alice told a lie, or if Bob simply copied Alice's response 20% of the time and otherwise told the truth, then the maths are different.

Re: Are two heads better than one?

#18

Post by cortesoft »

pavon wrote: post #8

It depends on what you are doing with the guess. If it is just a question of how frequently you are right or wrong the second person doesn't help. But if you are, for example, betting on your guess the second person improves your odds of coming out ahead significantly, since you can put down a higher wager when they agree than when they disagree.

The post discusses this exact point near the end.

Re: Are two heads better than one?

#20

Post by titanomachy »

gweinberg wrote: post #3

Bob isn't giving you any actionable information. If Alice and Bob agree, you're more confident than you were before, but you're still going to be trusting Alice. If they disagree you're down to 50% confidence, but you still might as well trust Alice.

Better than 50% confidence: they only lie 20% of the time, so when they disagree it's still 64% likely to be heads (.8 x .8)
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