Robin asks the following question here: How does the distribution of truth compare to the distribution of opinion? That is, consider some spectrum of possible answers, like the point difference in a game, or the sea level rise in the next century. On each such spectrum we could get a distribution of (point-estimate) opinions, and in the end a truth. So in each such case we could ask for truth’s opinion-rank: what fraction of opinions were less than the truth? For example, if 30% of estimates were below the truth (and 70% above), the opinion-rank of truth was 30%.
Robin Hanson pergunta: How does the distribution of truth compare to the distribution of opinion? That is, consider some spectrum of possible answers, like the point difference in a game, or the sea level rise in the next century. On...
Ethan, imagine that A is worth 2/3 and B is worth 1/3 and that there are a hundred searchers. If everyone decides to search A, and one searcher realizes that this will happen, he can do better by searching B. And if everyone reasons that way, then everyone will search B, which is even sillier. So obviously the game-theoretic equilibrium is going to involve randomizing your moves or looking at what other players are doing, but it's certainly not going to involve all searchers going to the same node. There's a standard way to calculate the equilibrium randomization strategy where nobody can do better by tweaking their parameters, which I like totally forget, but it's a really elementary calculation, which is why Robin Hanson told you to go read up on basic game theory.
Robin, I didn't ask for one. And yes, your solution is an equilibrium. But there's no way for that equilibrium to emerge (other than by chance) in the situation you described in your post. That is, players cannot deduce the equilibrium prior to the first move and play accordingly.
Thanks Doug.
In a new field, where the first rounds of the game are played simultaneously, before the equilibrium emerges, we should see the distribution that Andrew describes - players clustering around the locations with highest expected value.
It is only later, after the equilibrium has been reached, (especially as more players enter the game) that the sequential-play dynamics of the easter egg hunt emerge, under which opinions more closely approximate truth.
You suggested that truth's opinion value might vary in different topics. And it might - I'd suggest it varies over time, relating to the 'maturity' of the topic, for the above reason.
Ethan, an equilibrium is reached when no player could have done better with a different strategy, assuming nobody else could have chosen a different strategy. Hence, the following are all equilibrium states:
1 and 2 always search A while 3 always searches B1 and 3 always search A while 2 always searches B2 and 3 always search A while 1 always searches B1, 2, and 3 all search A with probability 2/3 and B with probability 1/3
The choice of which player searches B by himself is arbitrary.
Strangely, I *did* take a course on Game Theory in college and quite enjoyed it. And upon checking with Wikipedia (admittedly, not a standard text), my use of "simultaneous" and "sequential" is perfectly appropriate.
In the simultaneous game, why does 3 choose B? Why doesn't 1 choose B? After all, there is no play order, so no player can infer what a 'previous' player would have done. If 1 chooses A, why don't they all choose A? Why is 3 special?
Or did you mean something else (more technical and counter-intuitive) by "simultaneous"?
I may have misunderstood your game. By "sequential", I meant that the players would play in a definite order, with complete knowledge of the preceeding players' moves. I meant by "simultaneous" that the players would have no knowledge of where the other players are looking prior to revealing their choices - and that there was no ORDER of play. As such, 3 has no knowledge that 1 and 2 have chosen A. Indeed, when 3 is choosing, 1 and 2 have not chosen yet at all! 3 knows that the egg has 2/3 chance to be at A. But 3 has no way to know if 1 and/or 2 will chose A. As the moves are simultaneous, 3 can only guess based on what he thinks 1 and 2 are likely to choose. Does he think 1 or 2 is especially clever? In this case, the egg hunt dynamics break down.
Perhaps my use of a distinction between simultaneous and sequential play is non-standard - it has been awhile since I took Game Theory.
Ethan, I'm not sure you understand the concept of a game theoretic equilibrium. Imagine there are two places to look, A and B, and three people, 1, 2, and 3. If the chance the egg is at A is 2/3, then one simultaneous choice equilibrium is 1 and 2 look at A, and 3 looks at B.
In a simultaneous Easter Egg game, no player has information about the other player's choices. So a player choosing a search location must weight their choice of location by their distribution of the probability of finding the egg, modified by their estimation of the likely strategies of their opponents. Doesn't the uncertainty of that second level consideration remove the Easter Egg dynamic of seeking to search not where the total probabilities are greatest, but where the probability-per-marginal-searcher are greatest?
Ethan, the simultaneous version is the easiest to analyze, and if the time to move is small compared to the time to search, I don't think it makes much difference.
Robin, in the Easter Egg game, are you positing simultaneous moves or sequential moves for choice of search location?
If sequential, then there is a clear difference between strategies for the first and any subsequent player. The first player searches in the most likely location - an expected value solution. But the search radius of any player is fairly discrete - much more so than the continuous nature of the probability distribution. As such, the second player should look for a location that has the highest expected value, after taking into account the search radius of the first player (and the consequent decrease in likelihood of success within that radius). So we quickly get into the Easter Egg dynamic you describe, where Bayesian estimations of overall probability are adjusted by the expected utility of searching in a particular location based on the number of others already searching that location.
But if play is simultaneous, then there is no information about previously-claimed search territory.
Further, I think it depends on whether the game is competitive or cooperative. The more intense the competition (because of payoff size or relative gains/loss), the more an Easter Egg dynamic would emerge as the expected value of losing falls sharply compared to the benefit of covering all search territory.
Yes of course point estimates of expected values, including Bayesian estimates, will fall in the middle of the distribution of truth. My point was that perhaps we can understand human point estimates as often being moves in an Easter egg like game, rather than expected values. In this case we need to look at actual human point estimates.
Verdade?
Robin Hanson pergunta: How does the distribution of truth compare to the distribution of opinion? That is, consider some spectrum of possible answers, like the point difference in a game, or the sea level rise in the next century. On...
Thanks Eliezer.
Ethan, imagine that A is worth 2/3 and B is worth 1/3 and that there are a hundred searchers. If everyone decides to search A, and one searcher realizes that this will happen, he can do better by searching B. And if everyone reasons that way, then everyone will search B, which is even sillier. So obviously the game-theoretic equilibrium is going to involve randomizing your moves or looking at what other players are doing, but it's certainly not going to involve all searchers going to the same node. There's a standard way to calculate the equilibrium randomization strategy where nobody can do better by tweaking their parameters, which I like totally forget, but it's a really elementary calculation, which is why Robin Hanson told you to go read up on basic game theory.
Robin, I didn't ask for one. And yes, your solution is an equilibrium. But there's no way for that equilibrium to emerge (other than by chance) in the situation you described in your post. That is, players cannot deduce the equilibrium prior to the first move and play accordingly.
Thanks Doug.
In a new field, where the first rounds of the game are played simultaneously, before the equilibrium emerges, we should see the distribution that Andrew describes - players clustering around the locations with highest expected value.
It is only later, after the equilibrium has been reached, (especially as more players enter the game) that the sequential-play dynamics of the easter egg hunt emerge, under which opinions more closely approximate truth.
You suggested that truth's opinion value might vary in different topics. And it might - I'd suggest it varies over time, relating to the 'maturity' of the topic, for the above reason.
Ethan, an equilibrium is reached when no player could have done better with a different strategy, assuming nobody else could have chosen a different strategy. Hence, the following are all equilibrium states:
1 and 2 always search A while 3 always searches B1 and 3 always search A while 2 always searches B2 and 3 always search A while 1 always searches B1, 2, and 3 all search A with probability 2/3 and B with probability 1/3
The choice of which player searches B by himself is arbitrary.
Ethan, I'm not going to write you a whole tutorial on game theory here. End of discussion.
Strangely, I *did* take a course on Game Theory in college and quite enjoyed it. And upon checking with Wikipedia (admittedly, not a standard text), my use of "simultaneous" and "sequential" is perfectly appropriate.
In the simultaneous game, why does 3 choose B? Why doesn't 1 choose B? After all, there is no play order, so no player can infer what a 'previous' player would have done. If 1 chooses A, why don't they all choose A? Why is 3 special?
Or did you mean something else (more technical and counter-intuitive) by "simultaneous"?
Ethan, you need to go learn game theory.
In short - why in your simultaneous equilibrium does 3 choose B and not A?
I may have misunderstood your game. By "sequential", I meant that the players would play in a definite order, with complete knowledge of the preceeding players' moves. I meant by "simultaneous" that the players would have no knowledge of where the other players are looking prior to revealing their choices - and that there was no ORDER of play. As such, 3 has no knowledge that 1 and 2 have chosen A. Indeed, when 3 is choosing, 1 and 2 have not chosen yet at all! 3 knows that the egg has 2/3 chance to be at A. But 3 has no way to know if 1 and/or 2 will chose A. As the moves are simultaneous, 3 can only guess based on what he thinks 1 and 2 are likely to choose. Does he think 1 or 2 is especially clever? In this case, the egg hunt dynamics break down.
Perhaps my use of a distinction between simultaneous and sequential play is non-standard - it has been awhile since I took Game Theory.
Ethan, I'm not sure you understand the concept of a game theoretic equilibrium. Imagine there are two places to look, A and B, and three people, 1, 2, and 3. If the chance the egg is at A is 2/3, then one simultaneous choice equilibrium is 1 and 2 look at A, and 3 looks at B.
ethanj above makes an interesting point.
In a simultaneous Easter Egg game, no player has information about the other player's choices. So a player choosing a search location must weight their choice of location by their distribution of the probability of finding the egg, modified by their estimation of the likely strategies of their opponents. Doesn't the uncertainty of that second level consideration remove the Easter Egg dynamic of seeking to search not where the total probabilities are greatest, but where the probability-per-marginal-searcher are greatest?
Ethan, the simultaneous version is the easiest to analyze, and if the time to move is small compared to the time to search, I don't think it makes much difference.
Robin, in the Easter Egg game, are you positing simultaneous moves or sequential moves for choice of search location?
If sequential, then there is a clear difference between strategies for the first and any subsequent player. The first player searches in the most likely location - an expected value solution. But the search radius of any player is fairly discrete - much more so than the continuous nature of the probability distribution. As such, the second player should look for a location that has the highest expected value, after taking into account the search radius of the first player (and the consequent decrease in likelihood of success within that radius). So we quickly get into the Easter Egg dynamic you describe, where Bayesian estimations of overall probability are adjusted by the expected utility of searching in a particular location based on the number of others already searching that location.
But if play is simultaneous, then there is no information about previously-claimed search territory.
Further, I think it depends on whether the game is competitive or cooperative. The more intense the competition (because of payoff size or relative gains/loss), the more an Easter Egg dynamic would emerge as the expected value of losing falls sharply compared to the benefit of covering all search territory.
Yes of course point estimates of expected values, including Bayesian estimates, will fall in the middle of the distribution of truth. My point was that perhaps we can understand human point estimates as often being moves in an Easter egg like game, rather than expected values. In this case we need to look at actual human point estimates.