Some time back I thought of an example which shed light for me on some of the fail-to-disagree results. Imagine that two players, A and B, are going to play a coin-guessing game. A coin is flipped out of sight of the two of them and they have to guess what it is. Each is privately given a hint about what the coin is, either heads or tails; and they are also told the hint "quality", a number from 0 to 1. A hint of quality 1 is perfect, and always matches the real coin value. A hint of quality 0 is useless, and is completely random and uncorrelated with the coin value. Further, each knows that the hint qualities are drawn from a uniform distribution from 0 to 1 – on the average, the hint quality is 0.5.
I think the original argument is not completely incorrect, though it is phrased in a confusing manner. When A says "B's hint quality is uniformly distributed in [1/2,1]", he means that the distribution of B's hint quality conditional on the fact that he stuck with his original answer, but not conditional on A's hint quality, is uniformly distributed in [1/2,1]. This is a claim of the same form as the claim made in the first round, in which the assertion that B's hint quality is uniformly distributed in [0,1] is also not conditioned on A's hint or hint quality. The point is that comparing A's hint quality versus B's expected hint quality not conditional on A's hint quality tells you which hint is more likely to be correct.
The analysis is incomplete. Though it gets the right results, it does so using a flawed method.
The problem is that you did not consider the distributions each player has of the other's quality using bayesian reasoning.
Specifically, after observing a disagreement on the first round, the quality of the other player is no longer uniformly distributed. Higher qualities are less likely, but a higher quality also makes it better off to switch, resulting in a cutoff of 1/2. The same value as with naive reasoning, but the reasoning behind it is completely different.
"just a thought. if hints are opposite but hint qualities are exactly the same"
In that case the true probability would be exactly for 50% heads and for 50% tails. It is therefore completely expected that they would never be able to logically produce an agreement asserting "heads is more probable" or "tails is more probable".
Two hints pointing in the same direction increases the probability of a correct answer above quality of either hint alone, so agreeing on the first round maximizes the certainty of getting the true answer. If two hints point in opposite directions the qualities effectively work to cancel each other out. Each time the game advances to an additional round it strictly represents diminishing remaining information quality. The longer the game goes on the more your remaining information quality cancels towards zero. The longer the game goes on the closer your answer gets to "I have no information the probability is 50%-50%".
If the qualities are exactly equal there is zero information to pick heads or tails. It is impossible to agree on any answer unless you allow the players to eventually quit on an agreement of "I don't know" to indicate that the chance is very close to 50-50.
just a thought. if hints are opposite but hint qualities are exactly the same, then the process as you have defined it continues till both players figure out that they had opposite hints with the same hint quality. in this case, what happens: do they agree to disagree? (i do understand that the probability of such a situation is vanishingly small, which leads me to ask if aumann's result is almost always true, or just always true.)
Your game describes two players who compare each other's knowledge and step-by-step reach an agreement. However, a more realistic situation is one where A compares his knowledge with C and B compares her knowledge with D, etc...because that's what we usually do in debates, we try to find more and more people to quote in trying to win an argument. I'm new to game theory, but wouldn't all kind of paradoxes arise when moving from games with only a finite number of players to games with an infinite number of players?
Conchis - It's true that with similar uncertainty levels in this game, convergence could take a while. But in most cases it would involve both players changing sides a few times. In real life people tend not to adopt and then argue for their opponent's side, so I don't think this explains disagreement.
Roy, my idea was for a different kind of game. One example is where two coins are flipped out of sight, and two players are asked to estimate the probability that both are heads. A priori the probability is 1/4. Then each player is privately informed about the value of one of the coins. Suppose both coins are heads. Then each player updates his probability to 1/2. Upon informing each other of their views, which are the same, both players will decide that the probability is 1 (since hearing the other player say 1/2 means that his coin is heads; otherwise he would have judged the probability as 0).
Another example is the well-known "three hats" puzzle where you have to guess your hat color. It works out that players agree about probabilities for two rounds and then suddenly they change. With N hats they can stay the same for N-1 rounds and then suddenly change their opinions.
As far as "hint quality", that is just something I made up for this.
"I have constructed different games in which people can agree for two consecutive rounds and then disagree."
Do you mean instances of this particular game with particular values for the hint quality or do you mean similar but different in some fundamental way such as in the a priori assumpions or the iterative agreement procedure?
Also, I'm curious to know if "hint quality" is an established term. I took it to mean that a hint of quality p means that the actual coin outcome is passed as a hint with probability (1+p)/2 and the reverse coin outcome is hinted with probability (1-p)/2. That is, in a communication theory context, the binary coin outcome signal is passed through a binary symmetric channel with crossover probability e=(1-p)/2.
I haven't worked this out, but this sort of game would seem to me to suggest that the closer are the quality of individual hints, the longer the process is going to take to converge (and, more speculatively, that introducing uncertainty about the quality of one's own hint could spin the process out even further). If that's true, it suggests two further questions: (a) whether this would also be true of Aumann's more general result; and (b) whether (or under what conditions) lack of convergence could then explain much of the existing disagreement in the world?
I think the original argument is not completely incorrect, though it is phrased in a confusing manner. When A says "B's hint quality is uniformly distributed in [1/2,1]", he means that the distribution of B's hint quality conditional on the fact that he stuck with his original answer, but not conditional on A's hint quality, is uniformly distributed in [1/2,1]. This is a claim of the same form as the claim made in the first round, in which the assertion that B's hint quality is uniformly distributed in [0,1] is also not conditioned on A's hint or hint quality. The point is that comparing A's hint quality versus B's expected hint quality not conditional on A's hint quality tells you which hint is more likely to be correct.
The analysis is incomplete. Though it gets the right results, it does so using a flawed method.
The problem is that you did not consider the distributions each player has of the other's quality using bayesian reasoning.
Specifically, after observing a disagreement on the first round, the quality of the other player is no longer uniformly distributed. Higher qualities are less likely, but a higher quality also makes it better off to switch, resulting in a cutoff of 1/2. The same value as with naive reasoning, but the reasoning behind it is completely different.
"just a thought. if hints are opposite but hint qualities are exactly the same"
In that case the true probability would be exactly for 50% heads and for 50% tails. It is therefore completely expected that they would never be able to logically produce an agreement asserting "heads is more probable" or "tails is more probable".
Two hints pointing in the same direction increases the probability of a correct answer above quality of either hint alone, so agreeing on the first round maximizes the certainty of getting the true answer. If two hints point in opposite directions the qualities effectively work to cancel each other out. Each time the game advances to an additional round it strictly represents diminishing remaining information quality. The longer the game goes on the more your remaining information quality cancels towards zero. The longer the game goes on the closer your answer gets to "I have no information the probability is 50%-50%".
If the qualities are exactly equal there is zero information to pick heads or tails. It is impossible to agree on any answer unless you allow the players to eventually quit on an agreement of "I don't know" to indicate that the chance is very close to 50-50.
just a thought. if hints are opposite but hint qualities are exactly the same, then the process as you have defined it continues till both players figure out that they had opposite hints with the same hint quality. in this case, what happens: do they agree to disagree? (i do understand that the probability of such a situation is vanishingly small, which leads me to ask if aumann's result is almost always true, or just always true.)
Hal,
Your game describes two players who compare each other's knowledge and step-by-step reach an agreement. However, a more realistic situation is one where A compares his knowledge with C and B compares her knowledge with D, etc...because that's what we usually do in debates, we try to find more and more people to quote in trying to win an argument. I'm new to game theory, but wouldn't all kind of paradoxes arise when moving from games with only a finite number of players to games with an infinite number of players?
Calca.
Thanks Hal. You're right of course - a fact which I realised shortly after posting. Silly me. :)
Conchis - It's true that with similar uncertainty levels in this game, convergence could take a while. But in most cases it would involve both players changing sides a few times. In real life people tend not to adopt and then argue for their opponent's side, so I don't think this explains disagreement.
Roy, my idea was for a different kind of game. One example is where two coins are flipped out of sight, and two players are asked to estimate the probability that both are heads. A priori the probability is 1/4. Then each player is privately informed about the value of one of the coins. Suppose both coins are heads. Then each player updates his probability to 1/2. Upon informing each other of their views, which are the same, both players will decide that the probability is 1 (since hearing the other player say 1/2 means that his coin is heads; otherwise he would have judged the probability as 0).
Another example is the well-known "three hats" puzzle where you have to guess your hat color. It works out that players agree about probabilities for two rounds and then suddenly they change. With N hats they can stay the same for N-1 rounds and then suddenly change their opinions.
As far as "hint quality", that is just something I made up for this.
When you write
"I have constructed different games in which people can agree for two consecutive rounds and then disagree."
Do you mean instances of this particular game with particular values for the hint quality or do you mean similar but different in some fundamental way such as in the a priori assumpions or the iterative agreement procedure?
Also, I'm curious to know if "hint quality" is an established term. I took it to mean that a hint of quality p means that the actual coin outcome is passed as a hint with probability (1+p)/2 and the reverse coin outcome is hinted with probability (1-p)/2. That is, in a communication theory context, the binary coin outcome signal is passed through a binary symmetric channel with crossover probability e=(1-p)/2.
I haven't worked this out, but this sort of game would seem to me to suggest that the closer are the quality of individual hints, the longer the process is going to take to converge (and, more speculatively, that introducing uncertainty about the quality of one's own hint could spin the process out even further). If that's true, it suggests two further questions: (a) whether this would also be true of Aumann's more general result; and (b) whether (or under what conditions) lack of convergence could then explain much of the existing disagreement in the world?