Comments on: The Coin Guessing Game http://www.overcomingbias.com/2007/01/the_coin_guessi.html Overcoming Bias is economist Robin Hanson’s blog, on honesty, signaling, disagreement, forecasting, and the far future. Wed, 17 Mar 2010 21:44:15 -0400 http://wordpress.org/?v=2.8.5 hourly 1 By: Calca http://www.overcomingbias.com/2007/01/the_coin_guessi.html#comment-423202 Calca Thu, 04 Jan 2007 20:28:39 +0000 http://prod.ob.trike.com.au/2007/01/the-coin-guessing-game.html#comment-423202 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. 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.

]]> By: conchis http://www.overcomingbias.com/2007/01/the_coin_guessi.html#comment-423201 conchis Wed, 03 Jan 2007 09:26:03 +0000 http://prod.ob.trike.com.au/2007/01/the-coin-guessing-game.html#comment-423201 Thanks Hal. You're right of course - a fact which I realised shortly after posting. Silly me. :) Thanks Hal. You’re right of course – a fact which I realised shortly after posting. Silly me. :)

]]> By: Hal Finney http://www.overcomingbias.com/2007/01/the_coin_guessi.html#comment-423200 Hal Finney Wed, 03 Jan 2007 04:14:26 +0000 http://prod.ob.trike.com.au/2007/01/the-coin-guessing-game.html#comment-423200 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. 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.

]]> By: Roy Yates http://www.overcomingbias.com/2007/01/the_coin_guessi.html#comment-423199 Roy Yates Tue, 02 Jan 2007 16:09:04 +0000 http://prod.ob.trike.com.au/2007/01/the-coin-guessing-game.html#comment-423199 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. 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.

]]> By: conchis http://www.overcomingbias.com/2007/01/the_coin_guessi.html#comment-423198 conchis Tue, 02 Jan 2007 12:12:45 +0000 http://prod.ob.trike.com.au/2007/01/the-coin-guessing-game.html#comment-423198 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 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?

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