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Author Topic: Math Games  (Read 59734 times)

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faust

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Re: Math Games
« Reply #375 on: February 21, 2017, 11:19:56 am »

Why is it stupid? Choosing B is also an equilibrium strategy, and in fact in a sense it is way more stable than A. In practice, with the number of people we have, I calculated that if each other person has even a 2% chance of choosing B, then I should choose B. There's no assuming others are stupid at all.
I don't know by which term you define "more stable"; both A and B are ESS, but A's output is higher than B's.
Nevermind, A may not be ESS; I don't have the exact definition in mind. Still, if a single player deviates from A, they can only lose.
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liopoil

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Re: Math Games
« Reply #376 on: February 21, 2017, 02:31:15 pm »

Why is it stupid? Choosing B is also an equilibrium strategy, and in fact in a sense it is way more stable than A. In practice, with the number of people we have, I calculated that if each other person has even a 2% chance of choosing B, then I should choose B. There's no assuming others are stupid at all.
I don't know by which term you define "more stable"; both A and B are ESS, but A's output is higher than B's.
Nevermind, A may not be ESS; I don't have the exact definition in mind. Still, if a single player deviates from A, they can only lose.
Yeah, looks like A is not ESS, but I don't really know anything about that. By stable I meant that if anyone deviates from the equilibrium A then B becomes better, while lots of people can deviate from the equilibrium B and have B stay best until literally everyone has deviated. The fact that a single person deviating from A can only lose just means that it is an equilibrium, which is why an argument can be made for choosing A at all.
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schadd

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Re: Math Games
« Reply #377 on: February 21, 2017, 05:36:34 pm »

B is ESS (Extremely Shitty and Stupid)
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Axxle

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Re: Math Games
« Reply #378 on: February 21, 2017, 06:44:33 pm »

You're playing this game against people who aren't perfect logicians. If you don't assume some people will play suboptimally, then that's playing suboptimally.
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scott_pilgrim

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Re: Math Games
« Reply #379 on: February 21, 2017, 07:38:14 pm »

Thread Locked
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scott_pilgrim

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Re: Math Games
« Reply #380 on: February 21, 2017, 07:46:32 pm »

5. Guess an Unpopular Outcome of Coin Flips (by Tables)   
   
Rules      

Thank you to Tables for suggesting the main ideas of this game!      
This game will be played in three rounds.  THERE IS TO BE NO COMMUNICATION AFTER RULES ARE POSTED.      
In each round, n coins will be flipped, where n varies from round to round.  Each player may submit r integers*, each from 0 to n (inclusive), where r is the current round number.      
Each submission represents the number of coins that the player thinks will land heads.      
Each player who correctly guesses the number of heads flipped shares in a point prize of p points, where p varies from round to round (rounding up to avoid decimals).      
Clarification: When I say that they "share" in the prize, I mean that the p points are divided up evenly among the winners.  Therefore, it is better to win by yourself than with many players.      

*A player may "repeat" guesses, to give themselves extra "shares" in that guess.  For example, if Alice guesses 2, 2, 3, and Bob guesses 2, 3, 4, and there end up being 2 heads (and no one else guessed 2), then Alice will win two thirds of the points and Bob will win one third of the points.      

Round Number/Number of Guesses   n (Number of Coins Flipped)   p (Points for Correct Guesses)
1                                5                             12
2                                8                             16
3                                (number of players)           18


In Round 3, instead of flipping coins, each player will be given control over the outcome of a single coin.  When you PM me for round 3, please also say whether you would like your coin to be heads or tails.      
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scott_pilgrim

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Re: Math Games
« Reply #381 on: February 21, 2017, 07:47:17 pm »

Deadline for Round 1 will be 8:00pm tomorrow (Wednesday).
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scott_pilgrim

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Re: Math Games
« Reply #382 on: February 22, 2017, 08:04:41 pm »

Round 1 Results:

Calamitas   1
UmbrageOfSnow   (no submission)
gkrieg13   3
sudgy   2
liopoil   1
2.71828…   3
Dylan32   3
faust   0
ghostofmars   2
schadd   4
GendoIkari   3
LaLight   1
   
Outcome: HTHHT (3 heads)   
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scott_pilgrim

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Re: Math Games
« Reply #383 on: February 22, 2017, 08:10:13 pm »

Round 2 deadline is 8:00pm tomorrow (Thursday).
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scott_pilgrim

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Re: Math Games
« Reply #384 on: February 23, 2017, 08:05:39 pm »

Round 2 Results:

Calamitas   2,3
UmbrageOfSnow   3,6
gkrieg13   4,5
sudgy   4,5
liopoil   4,6
2.71828…   4,5
Dylan32   3,3
faust   (no submission)
ghostofmars   3,4
schadd   2,5
GendoIkari   4,4
LaLight   2,6
   
Outcome: THTTHHTH (4 heads)

16 points split among 7 shares, so each player who guessed correctly gets 3 points.  I realize now that it is ambiguous in the rules whether GendoIkari should get 5 or 6 points (because it's not stated whether rounding happens before or after counting the multiple shares), but I think rounding first is the more natural interpretation, so he gets 6 points.  Let me know if you have a problem with this ruling.
« Last Edit: February 23, 2017, 08:06:53 pm by scott_pilgrim »
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scott_pilgrim

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Re: Math Games
« Reply #385 on: February 23, 2017, 08:06:44 pm »

Round 3 deadline is 8:00pm tomorrow (Friday).  There are 12 players currently, so there will be 12 coins total.  If someone does not submit anything, I will determine their flip randomly to ensure that there are 12 coins.
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scott_pilgrim

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Re: Math Games
« Reply #386 on: February 24, 2017, 08:00:33 pm »

Thread Unlocked

Results coming shortly.
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scott_pilgrim

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Re: Math Games
« Reply #387 on: February 24, 2017, 08:02:05 pm »

Round 3 Results:

Calamitas   4,5,6   T
UmbrageOfSnow   9,10,11   H
gkrieg13   7,8,9   H
sudgy   7,8,9   H
liopoil   6,7,9   H
2.71828…   3,4,3   T
Dylan32   8,8,9   H
faust   8,9,10   H
ghostofmars   7,8,9   H
schadd   2,4,5   T
GendoIkari   5,6,7   H
LaLight   3,6,9   T

8 heads total; each share is worth 18/6 = 3 points.
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schadd

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Re: Math Games
« Reply #388 on: February 24, 2017, 08:06:50 pm »

well, futz
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schadd

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Re: Math Games
« Reply #389 on: February 24, 2017, 08:07:41 pm »

to be fair i was right for round 1
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I thought you thought it was a slip because I said 'Jake's partners' instead of 'Roadrunner7671.'

gkrieg13

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Re: Math Games
« Reply #390 on: February 24, 2017, 08:16:01 pm »

That was definitely the best game I've had so far.
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liopoil

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Re: Math Games
« Reply #391 on: February 24, 2017, 08:21:10 pm »

Okay, so I used a random number generator and it told me to guess 6,7,8. Then I decided that was boring and said:

Round 3 Submission: My coin is heads, I guess 6,7, and 9 heads

Take that, random number generator! Just watch there be 8 heads though.

Random number generator wins this time...
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liopoil

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Re: Math Games
« Reply #392 on: February 24, 2017, 08:27:16 pm »

I think that this is a pretty neat game. This is what I wrote for round 1:

There are only 6 options, and 32 possible ways the flips can go. The relative likelihoods of the outcomes are:

1 5 10 10 5 1

The relative expected value of a choice is the relative likelihood on the outcome divided by the (1 + the expected number of other players who choose that outcome). There are eleven other players. The equilibrium distribution would have all six outcomes equally valuable, and so in the limit with lots and lots of players, would mirror the binomial distribution. With just eleven other players it doesn't quite work like that though. Since it's symmetric, I can arbitrarily decide which half to choose from right now, and I think that fewer people will guess <3 heads, so that's what I'll do. If the probabilities of guessing 0, 1, or 2 heads are x, y, and z, then in the equilibrium we would have:

x + y + z = 1/2
1/(1 + 11x) = 5/(1 + 11y) = 10/(1 + 11z)

Re-arranging, 10 + 110x = 2 + 22y = 1 + 11z = q. Then 16q = 110(x + y + z) + 30 = 85, so q = 85/16. That means that:

z = 69/176
y = 53/352
x = -15/352

Okay, I think that means that it's actually never a good idea to choose 0 heads with this few people. In fact I think this makes sense; there's no way the eleven people can make the 5s and 10s bad enough. Okay, well let's redo it with just y and z. Now we have:

y + z  = 1/2
2 + 22y = 1 + 11z = q

Then 3q = 22(y + z) + 4 = 15, so q = 5. Then z = 4/11 and y = 3/22; I can definitely believe that. Then since I've decided to choose from the lower half, I'm going to guess 2 heads 8 times out of 11 and guess 1 heads the other 3 times out of 11. If everyone did this, then the average person would expect to face a distribution like:

0 1.5 4 4 1.5 0

This person still would not want to guess 0 or 5, and they wouldn't care otherwise. Unfortunately this ignores the fact that 12 is not divisible by 5, which is quite ugly and so I will choose to ignore it even though it makes the center choices slightly more appealing. In fact, I'm willing to bet that more people will be more inclined to guess 1 or 4 heads than they should be, and that somebody is going to guess 0 or 5 heads even though that's a bad guess. So maybe I should be choosing 2 heads more than 8/11 = ~73% of the time, but I'm just going to play the equilibrium. Time to go to a random number generator.

Well, the random number generator spit out 9 when I asked for a number between 1 and 11, so I should guess 1 heads. Round 1 Guess: 1 coin will land heads

For round 2 the computations were way nastier (the final denominator of the probabilities was 340) and less solidly grounded but I did them anyway and went with the random numbers again. For round 3 my analysis was so meaningless that I decided to ignore what the random number generator said, oops.
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Dylan32

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Re: Math Games
« Reply #393 on: February 24, 2017, 09:15:58 pm »

Darn it. I initially had typed 8 three times in the 3rd round, but then decided to change one to 9 just to give me another chance at points... That would have been beautiful.
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GendoIkari

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Re: Math Games
« Reply #394 on: February 24, 2017, 10:19:17 pm »

I don't understand anyone who said anything other than 2 or 3 for round 1. They were simply mathematically the best options; and there was no human component to consider. Similar for round 2, except you could argue 3,4 or 4,5 are better than 4,4.

For round 3, I failed to account for the human component and still expected about half heads.

*edit* Ok, after reading liopoil's post, I see th human component now; the hope that you split the win less. But I still think it just makes the most sense to go with the most likely win.
« Last Edit: February 24, 2017, 10:24:19 pm by GendoIkari »
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liopoil

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Re: Math Games
« Reply #395 on: February 24, 2017, 11:31:51 pm »

I don't understand anyone who said anything other than 2 or 3 for round 1. They were simply mathematically the best options; and there was no human component to consider. Similar for round 2, except you could argue 3,4 or 4,5 are better than 4,4.

For round 3, I failed to account for the human component and still expected about half heads.

*edit* Ok, after reading liopoil's post, I see th human component now; the hope that you split the win less. But I still think it just makes the most sense to go with the most likely win.
If everyone chooses 2 or 3, then it would increase your expected point yield to choose 1 or 4 instead. The equilibrium state where you don't benefit from switching has to be some ratio between choosing 1/4 and 2/3, with 2/3 more likely.
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Axxle

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Re: Math Games
« Reply #396 on: February 25, 2017, 02:35:52 am »

I don't understand anyone who said anything other than 2 or 3 for round 1. They were simply mathematically the best options; and there was no human component to consider. Similar for round 2, except you could argue 3,4 or 4,5 are better than 4,4.

For round 3, I failed to account for the human component and still expected about half heads.

*edit* Ok, after reading liopoil's post, I see th human component now; the hope that you split the win less. But I still think it just makes the most sense to go with the most likely win.
This is actually similar to something that daily fantasy players have to take into consideration: ownership percentage. So you sometimes have to play the guy who is projected to get fewer points, but fewer other managers are playing. You might hit less often than the other players, but when you do hit it matters a lot more since you beat a larger percentage of the opposition if the underdog does go off.
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Calamitas

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Re: Math Games
« Reply #397 on: February 25, 2017, 06:41:38 am »

I don't understand anyone who said anything other than 2 or 3 for round 1. They were simply mathematically the best options; and there was no human component to consider. Similar for round 2, except you could argue 3,4 or 4,5 are better than 4,4.

For round 3, I failed to account for the human component and still expected about half heads.

*edit* Ok, after reading liopoil's post, I see th human component now; the hope that you split the win less. But I still think it just makes the most sense to go with the most likely win.
You need to maximize your expectancy value not your winning probability. If more than twice as many people choose 2/3 than 1/4, 1/4 becomes best. So if you expect that to happen, 1/4 is the right call.
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Tables

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Re: Math Games
« Reply #398 on: February 25, 2017, 06:43:48 pm »

For those interested, this game is actually quite heavily modified from what I originally suggested - definitely in a good way for this kind of thing - and I originally made it for a probability game in the classroom. The original suggestion was much simpler: Everyone writes down their guess on a whiteboard, roll 2d6, everyone who guessed correctly shares the points in the same way as here, repeat. It would work well with a group of about 20 or so, but less well with smaller numbers (although I did suggest using coins or similar). Scott then took the idea further by having multiple guesses and more complex numbers in each round, which I think is better with this kind of player count and ability level than just repeating the basic game.
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...spin-offs are still better for all of the previously cited reasons.
But not strictly better, because the spinoff can have a different cost than the expansion.

scott_pilgrim

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Re: Math Games
« Reply #399 on: February 25, 2017, 07:46:08 pm »

Random note: GendoIkari's and/or faust's scores may be wrong in the OP.  At some point I will try to double-check everything and maybe make a Google doc to track where points are coming from.
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