How buying "back in" in poker tournament changes odds

Question

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?

Answer 1

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.

Answer 2

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)

Answer 3

(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.