<p>The previous answers suggesting B has an advantage are correct <em>under several tacit assumptions</em>. However, in the general case it is impossible to say who has an advantage. This depends on the question and on the precision of the information the players possess.</p>

<p>Example: Consider the question: <em>How many legs does a beetle have?</em> Obviuosly, player A has an advantage here.</p>

CC BY-SA 4.0
2020-03-20T18:47:38.193Z
I think in most cases it is fair to assume that the probability of A getting the number exactly right is 0.
Caspar
CC BY-SA 4.0
2020-03-19T15:07:36.900Z
2020-03-19T15:07:36.900Z
VARulle
<p>This is an interesting question!</p>

<p>It would seem that the game is not fair: Player A is at a pretty serious disadvantage.</p>

<p>Let's imagine each player has their own probability distribution of beliefs that is a sampling distribution of some population distribution centered at the true value. Both players are aware of this, and know their own distribution but have no further information on the other's belief distribution.</p>

<p>The optimal strategies are for player A to guess at the center of their own distribution and for B to guess "More" if the center of B's distribution is above A's guess and "Less" if it's below.</p>

<p>Player A will win if their guess is between the true value and Player B's guess.</p>

<p>-Since there is a 50/50 shot for either guess to be above or below the true value, 50% of the time the guesses will lie on either side of the true value and player B will win, regardless of whose guess is closer.</p>

<p>Via symmetry:</p>

<p>-25% of the time their guesses will be on the same side of the true value and player B's will be closer. Player B will also win here (deservedly).</p>

<p>-25% of the time their guesses will be on the same side of the true value and player A's will be closer. Here player A will win (deservedly).</p>

<p>Hence, player A will win only 1/4 of the time.</p>

<p>However the game also becomes increasingly complicated once you allow each player to have first order beliefs about the other player's beliefs (ie. specific information about the shape and center of the other's belief distribution), because then there's the opportunity for say, A to exploit their belief that B's guess is below the true value by guessing slightly lower than they would otherwise. And with second-order beliefs things get even more complex, and so on and so on. I'm not sure whether player A's disadvantage is robust to these nuances, but it's still quite striking to me that a game that at first glance seems fair is actually under normal conditions pretty lopsided!</p>

CC BY-SA 4.0
2020-03-19T09:02:42.450Z
"*each player has a guess that they are 100% confident in*" What exactly does this mean? Are they irrational? If not, how are they 100% confident in a guess?
Giskard
CC BY-SA 4.0
2020-03-19T09:02:57.397Z
"*We also assume each player has no information about the other player's beliefs. The optimal strategies are*..." How do players know what is optimal? This seems to depend on the other's decision, which depends on their beliefs. Otherwise interesting answer.
Giskard
CC BY-SA 4.0
2020-03-19T10:52:41.663Z
Fair. I guess the only model that formally makes sense is the one where they each have their own probability distribution and are guessing ar its center, and they know the other player is in the same situation but have no idea what the others distribution looks like
H Rogers
CC BY-SA 4.0
2020-03-19T10:53:03.520Z
This will yield the same result. I'll edit my answer in a bit.
H Rogers
CC BY-SA 4.0
2020-03-19T11:08:45.967Z
"*have no idea what the others distribution looks like*" If you assume this you will keep having similar problems. This game is not solvable without a common prior.
Giskard
CC BY-SA 4.0
2020-03-19T11:14:52.027Z
I disagree. In the absence of information on B's probability distribution, A's rational move is to approximate it with their own, since both distributions are drawn from the same distribution.
H Rogers
CC BY-SA 4.0
2020-03-19T11:57:27.627Z
Haha yep! Provided that by "not sure" you mean you have literally no clue whether he has more or less money than you (ie. to you, 'Putin' is just some random name), and you have the median amount of money out of everyone you know, which are the assumptions you'd have to make for that analogy to apply.
H Rogers
CC BY-SA 4.0
2020-03-19T12:07:35.723Z
I have edited my original answer to account for your feedback. Thanks!
H Rogers
CC BY-SA 4.0
2020-03-19T01:52:11.783Z
2020-03-19T12:05:49.273Z
2020-03-19T12:05:49.273Z
H Rogers
<p>The game is not fair. B has an advantage.</p>

<p>The definition of the word "information" in the below analysis means knowledge of the subject area of the game.</p>

<p>Consider the following scenarios.</p>

<p>Scenario one: The players have equal information.</p>

<p>If the game were fair in this scenario, both players should have an equal likelihood of winning. We can see this is the case. Assume both players have zero information. Then A has a 50/50 chance of being too high or too low. And B has a 50/50 chance of correctly calling the result. Both players have even odds of winning.</p>

<p>Scenario two: A has more information than B; but A's information is not perfect.</p>

<p>In this case, "fair" should mean that A has a better chance of winning than B. But we can see this is not true. Assume B has zero information and their answer comes by flipping a coin. B's odds of winning would still be 50/50. B could essentially flip a random coin for their answer and expect to win the game 50% of the time.</p>

<p>Scenario three: B has more information than A; but B's information is not perfect.</p>

<p>In this case, we would expect fair to mean that B's odds of winning would be greater than A's odds. We can see this is the case. Assume, for example, A has zero information and, therefore, their guess is made by choosing a random number. If B has any information at all, their odds of winning are significantly greater than 50%.</p>

<p>Since the game yields a fair outcome in scenarios one and three but the outcome is weighted in favor of B in scenario two, we must conclude the game is not fair and B has an advantage.</p>

CC BY-SA 4.0
2020-03-19T10:51:08.990Z
If you see my answer, even in the case where the players have the same level of information, B will still have an advantage.
H Rogers
CC BY-SA 4.0
2020-03-19T11:08:56.940Z
Actually sorry, I guess this depends on how you choose to formally model "information"
H Rogers
CC BY-SA 4.0
2020-03-19T09:45:13.690Z
2020-03-19T10:59:00.990Z
2020-03-19T10:59:00.990Z
FreeMarketUnicorn
<p>I think this holds up in the case where we assume each player draws from a normal distribution where the actual answer is the mean (on average they guess correctly). If Player A makes some guess G1 then the probability that the guess of the second person is on the same side as the mean (and therefore encompasses the mean) is greater the the probability it is not.<a href="https://i.sstatic.net/Pm5RN.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/Pm5RN.jpg" alt=" "></a></p>

CC BY-SA 4.0
2020-03-19T04:41:48.420Z
2020-03-19T04:41:48.420Z
Pingboy
<p>I sometimes play a guessing game with a friend. So fx we might try to guess how many people live in Russia. So person A says 100 milion and then person B says either 'more' or 'less'. If B says 'more' and in fact more than 100 milion live in Russia, then B wins and conversely. Is this game fair? I have an intuition that B probably has an advantage, but I'm not sure how to formally analyse the situation.</p>

CC BY-SA 4.0
2020-03-18T23:26:27.220Z
What do you mean by fair?
Herr K.
CC BY-SA 4.0
2020-03-18T23:34:20.533Z
@Herr K . I'm pretty sure that the OP means: "Does each person have an equal chance of winning ? I'm not sure how to formalize but I don't think so because, for it to be fair, each person would have to know the expectations of other person. The chances of that being true are slim.
mark leeds
CC BY-SA 4.0
2020-03-18T23:50:05.550Z
Well I'm not really sure what I mean. Specifying what is fair probably would do a lot of the work of formalizing the situation. I guess what I am going for is something like this: The person who is epistemically superior should always win the game. If the correct answer is 101 milion and B thinks that the correct answer is 200 milion, then A is in some sense more right but B will win. But I would like to see what analytical tools exists to analyse situations like these.
Caspar
<game-theory>
How to analyze this guessing game?
Caspar
# How to analyze this guessing game?

(Asked by: username_0 on 2020-03-18)

I sometimes play a guessing game with a friend. So fx we might try to guess how many people live in Russia. So person A says 100 milion and then person B says either 'more' or 'less'. If B says 'more' and in fact more than 100 milion live in Russia, then B wins and conversely. Is this game fair? I have an intuition that B probably has an advantage, but I'm not sure how to formally analyse the situation.

**Tags:** game-theory
> **Comments:**
> **username_1**: What do you mean by fair?  
> **username_2**: @Herr K . I'm pretty sure that the OP means: "Does each person have an equal chance of winning ? I'm not sure how to formalize but I don't think so because, for it to be fair, each person would have to know the expectations of other person. The chances of that being true are slim.  
> **username_0**: Well I'm not really sure what I mean. Specifying what is fair probably would do a lot of the work of formalizing the situation. I guess what I am going for is something like this: The person who is epistemically superior should always win the game. If the correct answer is 101 milion and B thinks that the correct answer is 200 milion, then A is in some sense more right but B will win. But I would like to see what analytical tools exists to analyse situations like these.  

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## 4 answers

### Answer by username_3 on 2020-03-19. Score: 2

The previous answers suggesting B has an advantage are correct *under several tacit assumptions*. However, in the general case it is impossible to say who has an advantage. This depends on the question and on the precision of the information the players possess.

Example: Consider the question: *How many legs does a beetle have?* Obviuosly, player A has an advantage here.

> **Comments:**
> **username_0**: I think in most cases it is fair to assume that the probability of A getting the number exactly right is 0.  

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### Answer by username_4 on 2020-03-19. Score: 2

This is an interesting question!

It would seem that the game is not fair: Player A is at a pretty serious disadvantage.

Let's imagine each player has their own probability distribution of beliefs that is a sampling distribution of some population distribution centered at the true value. Both players are aware of this, and know their own distribution but have no further information on the other's belief distribution.

The optimal strategies are for player A to guess at the center of their own distribution and for B to guess "More" if the center of B's distribution is above A's guess and "Less" if it's below.

Player A will win if their guess is between the true value and Player B's guess.

-Since there is a 50/50 shot for either guess to be above or below the true value, 50% of the time the guesses will lie on either side of the true value and player B will win, regardless of whose guess is closer.

Via symmetry:

-25% of the time their guesses will be on the same side of the true value and player B's will be closer. Player B will also win here (deservedly).

-25% of the time their guesses will be on the same side of the true value and player A's will be closer. Here player A will win (deservedly).

Hence, player A will win only 1/4 of the time.

However the game also becomes increasingly complicated once you allow each player to have first order beliefs about the other player's beliefs (ie. specific information about the shape and center of the other's belief distribution), because then there's the opportunity for say, A to exploit their belief that B's guess is below the true value by guessing slightly lower than they would otherwise. And with second-order beliefs things get even more complex, and so on and so on. I'm not sure whether player A's disadvantage is robust to these nuances, but it's still quite striking to me that a game that at first glance seems fair is actually under normal conditions pretty lopsided!

> **Comments:**
> **username_5**: "*each player has a guess that they are 100% confident in*" What exactly does this mean? Are they irrational? If not, how are they 100% confident in a guess?  
> **username_5**: "*We also assume each player has no information about the other player's beliefs. The optimal strategies are*..." How do players know what is optimal? This seems to depend on the other's decision, which depends on their beliefs. Otherwise interesting answer.  
> **username_4**: Fair. I guess the only model that formally makes sense is the one where they each have their own probability distribution and are guessing ar its center, and they know the other player is in the same situation but have no idea what the others distribution looks like  
> **username_4**: This will yield the same result. I'll edit my answer in a bit.  
> **username_5**: "*have no idea what the others distribution looks like*" If you assume this you will keep having similar problems. This game is not solvable without a common prior.  
> **username_4**: I disagree. In the absence of information on B's probability distribution, A's rational move is to approximate it with their own, since both distributions are drawn from the same distribution.  
> **username_4**: Haha yep! Provided that by "not sure" you mean you have literally no clue whether he has more or less money than you (ie. to you, 'Putin' is just some random name), and you have the median amount of money out of everyone you know, which are the assumptions you'd have to make for that analogy to apply.  
> **username_4**: I have edited my original answer to account for your feedback. Thanks!  

---

### Answer by username_6 on 2020-03-19. Score: 1

The game is not fair. B has an advantage.

The definition of the word "information" in the below analysis means knowledge of the subject area of the game.

Consider the following scenarios.

Scenario one: The players have equal information.

If the game were fair in this scenario, both players should have an equal likelihood of winning. We can see this is the case. Assume both players have zero information. Then A has a 50/50 chance of being too high or too low. And B has a 50/50 chance of correctly calling the result. Both players have even odds of winning.

Scenario two: A has more information than B; but A's information is not perfect.

In this case, "fair" should mean that A has a better chance of winning than B. But we can see this is not true. Assume B has zero information and their answer comes by flipping a coin. B's odds of winning would still be 50/50. B could essentially flip a random coin for their answer and expect to win the game 50% of the time.

Scenario three: B has more information than A; but B's information is not perfect.

In this case, we would expect fair to mean that B's odds of winning would be greater than A's odds. We can see this is the case. Assume, for example, A has zero information and, therefore, their guess is made by choosing a random number. If B has any information at all, their odds of winning are significantly greater than 50%.

Since the game yields a fair outcome in scenarios one and three but the outcome is weighted in favor of B in scenario two, we must conclude the game is not fair and B has an advantage.

> **Comments:**
> **username_4**: If you see my answer, even in the case where the players have the same level of information, B will still have an advantage.  
> **username_4**: Actually sorry, I guess this depends on how you choose to formally model "information"  

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### Answer by username_7 on 2020-03-19. Score: 0

I think this holds up in the case where we assume each player draws from a normal distribution where the actual answer is the mean (on average they guess correctly). If Player A makes some guess G1 then the probability that the guess of the second person is on the same side as the mean (and therefore encompasses the mean) is greater the the probability it is not.[![ ](https://i.sstatic.net/Pm5RN.jpg)](https://i.sstatic.net/Pm5RN.jpg)


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