Two Plus Two Poker Forums Elaborating On My Multiplayer Toy Game Article
 Register FAQ Search Today's Posts Mark Forums Read Video Directory TwoPlusTwo.com

 Notices

 Poker Theory General poker theory

 12-04-2017, 06:25 PM #1 David Sklansky Administrator     Join Date: Aug 2002 Posts: 14,022 Elaborating On My Multiplayer Toy Game Article Anyone want to take a crack at it? (Assuming it hasn't already been done.) https://www.twoplustwo.com/magazine/...r-toy-game.php
 12-05-2017, 08:41 AM #2 Yadoula8 adept   Join Date: Aug 2014 Location: Royal Leamington Spa Posts: 1,045 Re: Elaborating On My Multiplayer Toy Game Article Aww bless, you really like these little puzzles don't you Sklansky Last edited by Yadoula8; 12-05-2017 at 08:48 AM.
 12-05-2017, 12:47 PM #3 robert_utk old hand     Join Date: Jan 2005 Location: ValueTown Posts: 1,296 Elaborating On My Multiplayer Toy Game Article Looks like a variant of Nash-Shapley poker. I shall endeavor to provide the solution, plus I was planning to do so anyway. Simply a fantastic article btw! Should be solvable for any n-person game. Last edited by robert_utk; 12-05-2017 at 12:56 PM.
12-05-2017, 01:03 PM   #4
just_grindin
Pooh-Bah

Join Date: Dec 2007
Posts: 4,397
Re: Elaborating On My Multiplayer Toy Game Article

Quote:
 Originally Posted by robert_utk Looks like a variant of Nash-Shapley poker. I shall endeavor to provide the solution, plus I was planning to do so anyway. Simply a fantastic article btw! Should be solvable for any n-person game.
Why only 9 players?

 12-05-2017, 01:04 PM #5 robert_utk old hand     Join Date: Jan 2005 Location: ValueTown Posts: 1,296 Re: Elaborating On My Multiplayer Toy Game Article Wait, you want THE SOLUTION to n-person pre-flop Hold’em poker? I will only provide the answers up to 9 players, with 4 significant digits. More precision may give away the solution, which I have not found published, and do not wish to publish if I am successful.
12-05-2017, 01:07 PM   #6
robert_utk
old hand

Join Date: Jan 2005
Location: ValueTown
Posts: 1,296
Re: Elaborating On My Multiplayer Toy Game Article

Quote:
 Originally Posted by just_grindin Why only 9 players?

Yeah, I ninja edited that, jg.

 12-05-2017, 01:17 PM #7 robert_utk old hand     Join Date: Jan 2005 Location: ValueTown Posts: 1,296 Re: Elaborating On My Multiplayer Toy Game Article Oh, no raises allowed.
 12-05-2017, 01:40 PM #8 robert_utk old hand     Join Date: Jan 2005 Location: ValueTown Posts: 1,296 Re: Elaborating On My Multiplayer Toy Game Article First guess, on a napkin.... Let r be the rank of the highest previous limper from the ranks [1,...n] Any player should limp with hand x: (0,1) where x >= r/(r+1) Oops, the blind... So now it should read where x >= r+1/(r+2) Test for player 9 after 7 limpers (player 1 was forced in) 7+1/(7+2)=8/9=pot odds offered to player 9 player if all had limped. Test for player 8 after 3 previous limpers 3+1/(3+2)=4/5=pot odds offered to player 8 versus 4 players. Last edited by robert_utk; 12-05-2017 at 02:06 PM. Reason: Using n* was misleading/confusing so “r” instead
 12-05-2017, 05:47 PM #9 robert_utk old hand     Join Date: Jan 2005 Location: ValueTown Posts: 1,296 Re: Elaborating On My Multiplayer Toy Game Article So much for napkins.... Lets try it this way. Let n be the number of previous limpers already in the pot... Player 1 is the blind, forced in and not counted as a limper. Hands are x: (0,1) where 1 is the nuts Any player requires a hand such that x >= 1/(2+n) to limp +EV vs folding (no raises allowed) for any limper should be (n+1)x
 12-06-2017, 11:49 AM #10 robert_utk old hand     Join Date: Jan 2005 Location: ValueTown Posts: 1,296 Re: Elaborating On My Multiplayer Toy Game Article No guarantee the above is correct. Can we get a “yay or nay” from anyone? I may have precisely missed the point of the gto solution, since there is no value in “limp bluffing”. -Rob
 12-06-2017, 11:52 AM #11 robert_utk old hand     Join Date: Jan 2005 Location: ValueTown Posts: 1,296 Re: Elaborating On My Multiplayer Toy Game Article Also, the nature of the infinite game played by infinite participants is quite informative, imho.
 12-06-2017, 01:35 PM #12 robert_utk old hand     Join Date: Jan 2005 Location: ValueTown Posts: 1,296 Re: Elaborating On My Multiplayer Toy Game Article I left out the antes. Probably belongs next to n in the above solution. n(amount of ante)
12-06-2017, 02:10 PM   #13
just_grindin
Pooh-Bah

Join Date: Dec 2007
Posts: 4,397
Re: Elaborating On My Multiplayer Toy Game Article

Quote:
 Originally Posted by robert_utk No guarantee the above is correct. Can we get a “yay or nay” from anyone? I may have precisely missed the point of the gto solution, since there is no value in “limp bluffing”. -Rob
If n is only dependent on previous limpers, then that cannot be correct. Take the 3 player example from the article.

B's calling situation is not solely based on whether no callers have entered the pot - he has to assume that C will enter the pot some of the time and thus cannot call as often as if A had been the only other player.

Some how I want to say Bernoulli trials and by proxy the binomial dostribution would be helpful with this. You could model each player as calling success and folding failure. Not sure though would have to put more thought into it.

 12-06-2017, 02:12 PM #14 robert_utk old hand     Join Date: Jan 2005 Location: ValueTown Posts: 1,296 Re: Elaborating On My Multiplayer Toy Game Article Ahh, yes. Are we to assume how many players are seated at the table? This will be part of the solution, and if all have antes it gets quite interesting!
12-06-2017, 03:02 PM   #15
just_grindin
Pooh-Bah

Join Date: Dec 2007
Posts: 4,397
Re: Elaborating On My Multiplayer Toy Game Article

Quote:
 Originally Posted by robert_utk Ahh, yes. Are we to assume how many players are seated at the table? This will be part of the solution, and if all have antes it gets quite interesting!
You can use the notation you were using for an in determinant amount of callers.

Each caller is dependent on how many callers in front of them and how many potential callers behind them because they all contribute to how often they have to win.

Basically player n can have any number of calling scenarios where the number of callers varies from 0-(n-1).

Luckily if player n doesn't have n-1 callers in front of her, her calling decision reduces to the same decision as a player in the spot (n-(n-(k+1))) where k is the number of callers in front of player n.

So for example say 9 players total with 6 callers in front. Player 9's calling decision is the same as player (9-(9-(6+1))) = 9-2 = 7 when all players have called before. Yeah I think that should be right.

Almost seems like there is a series construction in there, but not sure how you account for callers behind.

 12-06-2017, 03:07 PM #16 robert_utk old hand     Join Date: Jan 2005 Location: ValueTown Posts: 1,296 Re: Elaborating On My Multiplayer Toy Game Article Potential callers and their antes must be predetermined, so the final solution will be a function of the number of players seated.
12-06-2017, 03:10 PM   #17
just_grindin
Pooh-Bah

Join Date: Dec 2007
Posts: 4,397
Re: Elaborating On My Multiplayer Toy Game Article

Quote:
 Originally Posted by robert_utk Potential callers and their antes must be predetermined, so the final solution will be a function of the number of players seated.
True. Sorry all over the place today.

 12-06-2017, 04:23 PM #18 robert_utk old hand     Join Date: Jan 2005 Location: ValueTown Posts: 1,296 Re: Elaborating On My Multiplayer Toy Game Article The only real dislike I have is that antes are a reason to raise, but the toy game forbids this, but will serve to illustrate the value of raising I suppose.
 12-06-2017, 04:44 PM #19 robert_utk old hand     Join Date: Jan 2005 Location: ValueTown Posts: 1,296 Re: Elaborating On My Multiplayer Toy Game Article As a matter of fact, we should only solve it with the addendum that illustrates how badly the EV of a limper is affected by a potential raise from a player ahead.
 12-06-2017, 10:16 PM #20 David Sklansky Administrator     Join Date: Aug 2002 Posts: 14,022 Re: Elaborating On My Multiplayer Toy Game Article Why not start with the simplest case? One blind. No ante. Three players. If the hands goes from zero to one then player B's minimum hand is x and player C's minimum hand (when B plays) is y, then: xy =1/2 and y[(y-x)(1-x)] = 1/3
 12-07-2017, 04:31 AM #21 browni3141 old hand   Join Date: Aug 2015 Location: South Florida Posts: 1,607 Re: Elaborating On My Multiplayer Toy Game Article I have used fi to denote the minimum calling hand of the ith player, A is sum of the antes. Skipping some steps because typing this out is tedious. Basically I just setup the system of equations and solved by substitution. f2f1 = 1/(A+2) f1 = 1/(f2(A+2)) 0 = f2(f2-f1)/(1-f1)(A+3)-1 0 = f23(A+2)(A+3)-f2(2A+5)+1 There are not always nice solutions when A != 0. To get exact solutions when the equation doesn't factor nicely we'd need to use the cubic formula. With no antes there are nice solutions. f2 = (3+sqrt(3))/6 f1 = (3-sqrt(3))/2 For more players it quickly gets messy, as the calling frequency of each player will depend on which players have already called. For example a fourth player has different calling frequencies depending on whether two alone has called, three alone has called or both two and three have called. The equations get too complicated to be worth attempting to solve by hand. I'd just use a computer if approximate solutions are wanted for more players. EDIT: BTW, I have not checked my partial solution for anything other than A = 0. Feel free to check and point out if it's in error. Last edited by browni3141; 12-07-2017 at 04:38 AM.
 12-07-2017, 02:57 PM #22 robert_utk old hand     Join Date: Jan 2005 Location: ValueTown Posts: 1,296 Re: Elaborating On My Multiplayer Toy Game Article I think it will suffice if we arrive at an expression of the calling hand, which accounts for a known number of players behind yet to act, but also takes into account the limps that have already happened, most importantly the most recent limp and how many players were behind that player when that player limped. So, the pot size is total limpers +1 The hand to beat, is the last limp hand, taking into account the number of players behind THAT player at that moment. Finally, we need to increase on that previous best hand by just enough to beat it, and still break even versus every player left behind us. The closer we are to the end of the line, then the closer we can be to the previous last best hand that called.
12-07-2017, 10:10 PM   #23
whosnext
Pooh-Bah

Join Date: Mar 2009
Location: California
Posts: 4,036
Re: Elaborating On My Multiplayer Toy Game Article

Presuming browni3141's equations are correct (they look good to me) for the 3-person case, here are the numerical solutions from solving the cubic equation with various antes.

AnteMin Call for BMin Call for C if B Calls
0.00.6340.789
0.10.6140.775
0.20.5970.762
0.30.5800.750
0.40.5650.738
0.50.5500.727
0.60.5370.716
0.70.5240.706
0.80.5130.697
0.90.5020.687
1.00.4910.679

Not shown is the min call for C if player B folds. Clearly this is the standard simple pot odds calculation (which I think means the min call is given by 1/(3A+2)).

I hope these are correct but mistakes are always possible.

 Thread Tools Display Modes Linear Mode

 Posting Rules You may not post new threads You may not post replies You may not post attachments You may not edit your posts BB code is On Smilies are On [IMG] code is On HTML code is Off Forum Rules
 Forum Jump User Control Panel Private Messages Subscriptions Who's Online Search Forums Forums Home Links to Popular Forums     News, Views, and Gossip     Beginners Questions     Marketplace & Staking     Casino & Cardroom Poker     Internet Poker     NL Strategy Forums     Poker Goals & Challenges     Las Vegas Lifestyle     Sporting Events     Politics     Other Other Topics Two Plus Two     About the Forums     Two Plus Two Magazine Forum     The Two Plus Two Bonus Program     Two Plus Two Pokercast     The Best of Two Plus Two Marketplace & Staking     Commercial Marketplace     General Marketplace     Staking - Offering Stakes     Staking         Staking - Offering Stakes         Staking - Seeking Stakes         Staking - Selling Shares - Online         Staking - Selling Shares - Live         Staking Rails         Transaction Feedback & Disputes     Transaction Feedback & Disputes Coaching & Training     Coaching Advice     Cash Game Poker Coach Listings     Tournament/SNG Poker Coach Listings Poker News & Discussion     News, Views, and Gossip     Poker Goals & Challenges     Poker Beats, Brags, and Variance     That's What She Said!     Poker Legislation & PPA Discussion hosted by Rich Muny     Twitch - Watch and Discuss Live Online Poker     Televised Poker     Two Plus Two Videos General Poker Strategy     Beginners Questions     Books and Publications     Poker Tells/Behavior, hosted by: Zachary Elwood     Poker Theory     Psychology No Limit Hold'em Strategy     Medium-High Stakes PL/NL     Micro-Small Stakes PL/NL     Medium-High Stakes Full Ring     Micro-Small Stakes Full Ring     Heads Up NL     Live Low-stakes NL Limit Texas Hold'em Strategy     Mid-High Stakes Limit     Micro-Small Stakes Limit Tournament Poker Strategy     STT Strategy     Heads Up SNG and Spin and Gos     Mid-High Stakes MTT     Small Stakes MTT     MTT Community     Tournament Events Other Poker Strategy     High Stakes PL Omaha     Small Stakes PL Omaha     Omaha/8     Stud     Draw and Other Poker Live Poker     Casino & Cardroom Poker         Venues & Communities         Regional Communities     Venues & Communities     Tournament Events         WPT.com     Home Poker     Cash Strategy     Tournament Strategy Internet Poker     Internet Poker         Winning Poker Network         nj.partypoker.com         Global Poker     Commercial Software     Software         Commercial Software         Free Software General Gambling     Backgammon Forum hosted by Bill Robertie.     Probability     Sports Betting     Other Gambling Games 2+2 Communities     Other Other Topics         OOTV         Game of Thrones     The Lounge: Discussion+Review     EDF     Las Vegas Lifestyle     BBV4Life         omg omg omg     House of Blogs Sports and Games     Sporting Events         Single-Team Season Threads         Fantasy Sports     Fantasy Sports         Sporting Events     Wrestling     Golf     Chess and Other Board Games     Video Games         League of Legends         Hearthstone     Puzzles and Other Games Other Topics     Politics     History     Business, Finance, and Investing     Science, Math, and Philosophy     Religion, God, and Theology     Travel     Health and Fitness     Laughs or Links!     Computer Technical Help     Programming International Forums     Deutsch         BBV [German]     Français     Two Plus Two en Espańol

All times are GMT -4. The time now is 02:48 PM.

 Contact Us - Two Plus Two Publishing LLC - Privacy Statement - Top