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Stroggoz   New Zealand. Oct 30 2019 11:27. Posts 4494
I am trying to learn game theory atm and came across this cool problem in my textbook:

You play a game where a fair coin is flipped until it comes up tails the first time. At that point the player wins $2^n, where n is the number of times the coin was flipped. How much should one be willing to pay for each game to still be +EV

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supposed to have greenstar not braceletLast edit: 30/10/2019 11:37

Oly   United Kingdom. Oct 30 2019 19:31. Posts 3574

$1

Is there some trick here? This seems simply ev = payoff*probability = (2^n)*(1/2^n) = 1

Researchers used brain scans to show that when straight men looked at pictures of women in bikinis, areas of the brain that normally light up in anticipation of using tools, like spanners and screwdrivers, were activated. 

Nitewin   United States. Oct 30 2019 22:34. Posts 1163

Raul Oly ...?

EV = 1 yes


Stroggoz   New Zealand. Oct 31 2019 02:53. Posts 4494

:O. how do u get a number from an equation with n in it and n isn't defined. It isn't 1.

there is no trick here but most people find the answer to be quite unexpected.

supposed to have greenstar not braceletLast edit: 31/10/2019 02:55

Oly   United Kingdom. Oct 31 2019 12:03. Posts 3574

Oh yes 1 is a silly answer when the player wins a minimum of 2!

So if you sum the ev over all possible outcomes (n=1 to infinity) you get the same cancelling and end up with the divergent series 1+1+1+... which implies you should pay any amount asked! If that’s right this time that is indeed a very cool problem!

Researchers used brain scans to show that when straight men looked at pictures of women in bikinis, areas of the brain that normally light up in anticipation of using tools, like spanners and screwdrivers, were activated. 

Oly   United Kingdom. Oct 31 2019 12:11. Posts 3574


  On October 30 2019 21:34 Nitewin wrote:
Raul Oly ...?

EV = 1 yes



Not him. I used to post here a fair bit a long time ago but I stopped playing professionally years ago but still occasionally surf liquipoker as the habit seems deeply ingrained. I’m so old school I played the original Party Poker Monster days. 3betting JJ+ was cutting edge.

Researchers used brain scans to show that when straight men looked at pictures of women in bikinis, areas of the brain that normally light up in anticipation of using tools, like spanners and screwdrivers, were activated. 

Stroggoz   New Zealand. Oct 31 2019 13:01. Posts 4494


  On October 31 2019 11:03 Oly wrote:
Oh yes 1 is a silly answer when the player wins a minimum of 2!

So if you sum the ev over all possible outcomes (n=1 to infinity) you get the same cancelling and end up with the divergent series 1+1+1+... which implies you should pay any amount asked! If that’s right this time that is indeed a very cool problem!



yup, exactly right. It's called the St petersburg paradox

supposed to have greenstar not bracelet 

Baalim   Mexico. Nov 01 2019 02:22. Posts 33082

pay me 100 bucks and you flip the coin and ill pay the $2 for every non tails


Ex-PokerStars Team Pro Online 

Stroggoz   New Zealand. Nov 01 2019 02:39. Posts 4494

$2 for every non tails? Then the summation converges on $2, lol. You would be +$98 EV.

supposed to have greenstar not bracelet 

Daut    United States. Nov 01 2019 03:18. Posts 8897

Change the problem to the following and it becomes something fun to solve (this way is trivial as the answer is 2*.5+4*.25+8*.125+...=1+1+1+1...= infinity):

You play a game where a fair coin is flipped until it comes up tails the first time. After each heads flip, you win back $n, where n is the number of times the coin was flipped thus far. How much should one be willing to pay for each game to still be +EV?

ex: H +1, H +2, H +3, H +4, T end and receive $10

NewbSaibot: 18 TIMES THE SPEED OF LIGHT. Because FUCK YOU, DautLast edit: 01/11/2019 03:24

Stroggoz   New Zealand. Nov 01 2019 04:50. Posts 4494

nvm, read it wrong... edit.

response to Daut:

1/2(2) +1/4(3) +1/8(6) +1/16(10)+...

=sum 0 to n: 1/2^n* n(n+1)/2=4

the EV is $4? is this correct?

supposed to have greenstar not braceletLast edit: 01/11/2019 05:00

Daut    United States. Nov 01 2019 05:09. Posts 8897

Yes, but prove it, no mathematica! I had some trouble, but found an interesting proof that it's 4

NewbSaibot: 18 TIMES THE SPEED OF LIGHT. Because FUCK YOU, DautLast edit: 01/11/2019 05:10

Daut    United States. Nov 01 2019 05:17. Posts 8897

+ Show Spoiler +

NewbSaibot: 18 TIMES THE SPEED OF LIGHT. Because FUCK YOU, Daut 

Stroggoz   New Zealand. Nov 01 2019 05:43. Posts 4494

nice proof, can use that same kind of argument to prove that 1+2+3+4+5...=-1/12

supposed to have greenstar not bracelet 

Baalim   Mexico. Nov 01 2019 05:45. Posts 33082

isn't the paradox caused because infinity becomes ergodic?

I mean the paradox as why it seems non-intuitive since its time probability and not ensamble probability

edit: nevermind its just a math glitch

Ex-PokerStars Team Pro OnlineLast edit: 01/11/2019 07:53

Stroggoz   New Zealand. Nov 01 2019 05:45. Posts 4494



This centrifuge problem applies to seating for positional neutrality in poker. (no god seating)

supposed to have greenstar not bracelet 

Daut    United States. Nov 01 2019 06:01. Posts 8897


  On November 01 2019 04:43 Stroggoz wrote:
nice proof, can use that same kind of argument to prove that 1+2+3+4+5...=-1/12



Been about 15years since I tutored Calc 2, but I believe you can only use that method for series that converge (otherwise S doesn't have a value that we can multiply). And I want to say the smallest same sign series that diverges is 1/n, so any monotone series that shrinks faster than 1/n converges. Gets much trickier with series that alternate signs.

NewbSaibot: 18 TIMES THE SPEED OF LIGHT. Because FUCK YOU, DautLast edit: 01/11/2019 06:35

Daut    United States. Nov 06 2019 05:49. Posts 8897

Strogg, where are you taking game theory classes?

NewbSaibot: 18 TIMES THE SPEED OF LIGHT. Because FUCK YOU, Daut 

Stroggoz   New Zealand. Nov 06 2019 07:15. Posts 4494

im not. i'm just reading through this book. https://www.amazon.com/Game-Theory-Introduction-Steven-Tadelis/dp/0691129088

supposed to have greenstar not bracelet 

 



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