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My buddy and I are arguing over something that cropped up in this past weekend's Texas Hold'em tournament.

A player got "knocked out" (lost all their chips) early on in the game. The person hosting the tournament told them they could buy back in, and my buddy got upset. He argued that it gave the player buying back-in an unfair advantage, and that it changed the odds for the other players.

I argued that as long as everyone has the ability to buy back in, then whatever the mathematical ripple effects are of buying back in, they get spread over all players equally.

But I have no way to prove this mathematically and was hoping a few math gurus could lend me some help. I'm a Java developer with a math minor, so don't hold back (I can't prove it myself but I can at least follow someone else's math)!

Thanks in advance for any help here.

Edit - An example:

  • Poker tournament contains 5 players
  • Each player puts in $20 and gets 80 chips (25 cents/chip)
  • Player 3 gets knocked out (0 chips) while Players 1, 2, 4 and 5 have 20, 50, 75 and 15 chips left respectively
  • Player 3 buys back in ($20; 80 chips)
  • Every other player may buy back in once if knocked out
  • Does this give players unfair advantages/disadvantages? If so, who, why and how?
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This is completely without any proof, but my thoughts are that (making certain simplifying assumptions about the probability distribution that are likely false) if you buy back into a tournament for the same chip amount that you began with, your chances of winning the second time around are considerably less than the first time around, since on average your opponents will have a greater chip counts than you to begin with, putting you at a material disadvantage. –  Arthur Fischer Mar 12 '12 at 22:00
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If the host said the rules before the tournament started, then your friend should have not been bothered by it. However, if your friend was playing the tournament and the rule was added, then he has the right of being upset because of the following: You only play buy back tourneys if you are willing to buy back, otherwise you do not play them. I will try to look for the article written by (I think if my memory is correct) Daniel Negreanu. So if your friend was not willing to pitch in an extra twenty if he was knocked out, then yeah it is unfair. –  Daniel Montealegre Mar 12 '12 at 22:21
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3 Answers

One of the biggest pitfalls of using math to prove a point is accidentally proving something that isn't quite the issue at hand. (or even unrelated)

While the specific point you have made is true, I suspect you have fallen into this trap: that you are completely neglecting two very important possibilities:

  • Buy-ins weren't an option for everybody, but was instead only given to this guy because the host felt bad for him.

  • If buy-ins weren't part of the tournament rules but the host changed them to allow everyone to buy-in, then even though the new rules are fair, it still disadvantages the people who were ahead and advantages the people who were behind (especially the guy who had already lost)

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(1) If it's a game among friends, then I don't think that worrying about who has a theoretical advantage is more important than having a good time and letting a buddy buy back into the game. But the buy-in structures should be designed so that all players can comfortably rebuy if it is a friendly game.

(2) Your example doesn't make sense because there are less chips in play than at the beginning of the game.

(3) That said, there are significant strategic differences when playing with rebuys. The basic theory of tournament poker is that the value of your chips is not the same as their nominal value. For example, suppose you have a chance to risk all of your chips on the first hand versus another player where you are 60% to win if you do so. In a regular game, this is a great opportunity to risk even money on a bet which you win 60% of the time. But in a tournament, the end result is 40% of the time you have no chance to win the tournament, while the other 60% of the time, your chances of winning the tournament have increased, but generally not by twice as much. So the correct strategy may be to pass on the seemingly favorable bet, especially if you are good player and expect to be able to use your skill to your advantage as long as you still have chips. This is especially true in larger fields which tend to last longer and allow for more skill differentiation, while not as relevant in a home game with 5 or 6 friends. (Just how much of an edge you should pass up early on depends on a lot of factors, including how the prize money is distributed.)

However, if you have a chance to rebuy, then your incentive changes to take more (favorable!) risks early on because losing your initial buy-in does not eliminate your chances of winning the tournament. In that sense, it is certainly unfair for rebuys to be allowed if only some of the players know this is the policy going in.

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I've sometimes wondered about what would happen if tournaments required that players start with half their chips on their table, and allow players to bring chips to the table between hands, subject to the constraint that a player bringing chips to the table must, if possible, raise his table stake back up to the initial minimum (i.e. half the starting total). In most cases, that wouldn't skew things much, but it would lessen the effect you describe in #3. As it is, a bad player could have a better than 5% chance of beating an expert in a no-limit heads-up contest... –  supercat May 15 '12 at 22:52
    
...by simply going all in with any two cards. If the bad player didn't bother looking at his cards, it wouldn't matter how good a "reader" the expert was. If the good player waits for pocket rockets, the bad player might pocket enough blinds before the good player gets them that a single loss wouldn't knock him out. Meanwhile, even with pocket rockets, the expert would face a greater-than-5% chance of being eliminated if he calls. –  supercat May 15 '12 at 22:55
    
In fact, heads up, no matter how deep the stacks are, an amateur who decides to go all-in every hand will win just under 15% of the time. Say there is a single blind of 1 unit and stacks are $N$ units deep for arbitrarily large $N$. For sufficiently large $N$, the optimal strategy is to wait for a pair of aces and then call your opponent's all-in bet. Against a random hand, AA wins 85.2% of the time after the community cards are dealt out. If $N$ is on the order of a more typical size of 50-100 times the big blind, then going all-in every hand fares even better. –  Michael Joyce May 15 '12 at 23:12
    
I thought AA had better hand equity than that heads-up. Still, what do you think of the notion of having players start with up to half the chips off the table? I would think it probably wouldn't affect things much, except to reduce the ability of wild players to skew the results by getting lucky. Incidentally, speaking of odds, a couple more thoughts: (1) What would be the worst pre-flop hand equity AA could have, pre-flop, against nine known opponents' hands? (2) Against e.g. nine unknown opponents, what would be the best hand equity one could have on the flop... –  supercat May 15 '12 at 23:32
    
...that could be totally destroyed (go to zero) on the river, against still-unknown opponents? My guess for the latter would be hand(33) flop(223) [the only better hand would have both remaining twos]. If the turn and river come up (22), the pocket 33 would lose to any other possible holding (any opponent would have to have at least one card higher than 3). –  supercat May 15 '12 at 23:36
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You can just argue that all players start with the same options and therefore (assuming equal skill) have the same chance of winning. What has changed is the value to each player of having somebody knocked out. If there are $n$ players originally, if there is no buy back in, your probability of winning if you are not the first knocked out is $\frac 1{n-1}$, but as the person who buys back in might win, you don't have as big an edge. This is balanced by the fact that you might be the first out, buy back in, and win.

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Thanks for responding - but if everyone can buy back in, then the scenario you mention (someone is first to be knocked out, buys back in and wins) doesn't hold. Am I missing something? –  Adam Tannon Mar 12 '12 at 21:59
    
@zharvey: Yes, the first player can buy back in and then knock each other player out twice. True, they also can buy back in, but presumably the once only applies to them, as well. –  Ross Millikan Mar 12 '12 at 22:01
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