**Note: **Not at the old Poker1 site. A version of this entry was originally published (2008) in Casino Player*. *

I’m about to give you some poker tournament advice based on a strange mathematical truth. But first I’d like to talk about the whole notion of mathematics in poker.

Too many players worry that they aren’t good at calculating poker odds. They believe that this failure will greatly diminish their profits or cause them not to win at all. The truth is that you don’t need to calculate poker odds.

I’ve never known a world-class player who did complex calculations during a poker hand. I don’t.

Why? It’s because most poker strategy is obvious to me. It’s been pre-calculated away from the table. About the only thing I do with math at the poker table is something like this: I reason that the pot is $10,000, it costs me $3,000 to call and see the river card, and if I do, there are nine cards left that will make my flush, out of 46 unknown cards.

So, that’s 37 cards that will miss for me against 9 that will hit. The 37-9 reduces to a bit worse than 4-to-1 against me.

**Other factors**

I’m only getting 3.33-to-1 on the money, and my odds against making the flush are longer than that. So, should I call?

Here’s where I take other factors into consideration to see if I can make up the statistical shortfall. How much extra, on average, can I earn on the final-round wagering if I make the winning flush? How much chance is there that I could successfully bluff on the river? What is the chance that I might make a winning pair? What chance is there that I can make my flush and still lose — perhaps to a bigger flush or to a full house or four-of-a-kind if the board pairs with a card of my suit?

The point is, I’m only going to be able to guess; the mathematics will deteriorate to an estimate. And that, despite what you’ve been told, is how most poker decisions are made — educated guesswork. So, if you’ve been wondering how non-mathematical players sometimes can fare well against statistical wizards, that’s why.

Reasonable choices are obvious for most situations. It’s the quality of your estimates that determines how often you do the right thing.

**Two hours late**

** **Okay, I said I had a mathematical secret to share about tournament poker. Here it is.

Although I seldom play poker tournaments, I’m here at the World Series of Poker to do a private seminar for Doyle’s Room players. Since I’m here, I entered a few events and, by golly, I stumbled upon a new policy. They’re letting players enter up to two hours after an event starts.

You get to take a seat in the event that’s already underway and start with your full amount of chips. Advantage or a disadvantage?

On the disadvantage side, you can point out that by sitting out the first couple hours, you’re missing a chance to gather chips. This is especially meaningful if you have an edge against the average field of players. And since weaker players tend to get eliminated early, you’re joining a slightly stronger field, on average, when you enter late. Even if only a few players have been eliminated, the stacks usually have shifted toward the stronger players, so not as many chips are in the hands of weak foes.

**Simple truth**

But those considerations may be overshadowed by a simple mathematical truth. In a proportional payout tournament — where first-place only wins a fraction of the prize pool, second-place a smaller fraction, and so forth — your chips are worth more and more as the tournament progresses.

In that light, let’s revisit the hand we talked about earlier. Although it might be profitable to play that flush draw in a regular non-tournament game, it isn’t profitable to play it in a proportional-payout tournament.

Now let’s return to our discussion of whether to exercise the option to buy in late. A sophisticated player once told me he wouldn’t waste his money rebuying late in a tournament for the original entry price.

**Not logical**

** **He reasoned that the average stacks opposing him would be almost insurmountable after so many other players had been eliminated and donated their chips to the few remaining players.

That may seem logical at first, but it isn’t. Let’s say it’s a standard proportional payoff tournament in which first place gets $1 million, second place gets $500,000, and third place gets $300,000. If you could buy-in for $10,000 when there were only two players left, would you? Of course! You’d take third place automatically — and maybe better.

Of course that’s a silly example, based on something that could never happen. You can buy in up to two hours late, but not at the final table.

Fine. The point is, the later you buy in, the more your chips are worth. So, other things being equal, it’s a slight mathematical advantage to enter those WSOP events late. **— MC**

Mr Caro…

After (admittedly) just a quick read of your article here, I do tend to see the logic of a late BI to an mtt, for the same cost-to-chip PRICE as a BI prior to the first hand.

However what about the math considerations behind a RE-BUY (or more specifically, multiple re-buys), as relates to your late article reference to the “sophisticated player”?

It strikes me that the “sophisticated player” you reference might have a point, because a re-buy and a late BI entail 2 different investment costs for the MTT.

Consider…

With a late BI (assuming no one can re-buy/re-enter for simplicity), your cost is the same as anyone else’s to enter. Therefore you may buy an equity stake in a prize pool and a chance to play against chip stacks that have decreased in individual chip value (because they’ve INCREASED in size). This sets up a situation wherein you tend to benefit (perhaps) from these now larger chip stacks being more accepting of chip loss to you (because they can better “afford it”), at a price to you EQUAL to the one you may have paid to start at a time that willingness may not have existed.

A re-buy, however, entails a DOUBLING of the cost of entry to retain that prize pool equity (assuming it is not some juiced up re-buy event where a stack to return costs far less than the initial BI, as seen in some charity type events).

Therefore, would not the benefit derived from re-buying need to be very much GREATER, from a pure tourney equity standpoint, than is likely to be derived from simple increase of the remaining stacks you would be playing against if you choose to re-buy?

So no offense to your “sophisticated player” friend, but I think his standard for re-buying late may be CORRECT, but for the wrong reasons.

(Side note: it is through reading your lessons for years, Mr Caro, that I have largely come to see the importance in poker of not only doing the correct thing, but doing them for the correct reasons. Thank you.)

My personal standard for re-buying late (or multiple re-buys) tends to be based largely upon the playability of my stack as relates to the depth of money I’d return at, whether or not I’d return to a table of known opponents and stack sizes, the distance left to various pay out plateaux, the relative value of a min cash as relates to my increased investment, and a rational assessment of my skill relationship to the remaining field.

In my thinking, Mr Caro, a decision to re-buying tends to often be far more complex, than a decision to BI late or early.

Am I off base in thinking there is more to one decision, than to another?

Hi, David —

Thank you for your well-reasoned and well-worded reply. You’ll find other articles at Poker1 that address some of your points. Most importantly…

* The calibre of competition tends to increase as players are eliminated.

* While large stacks can be used to intimidate, it’s only because their values are less than perceived. It’s possible that one large-stack player is betting $10,000 in prize pool value while a short-stack player would have to call $30,000, if translated into actual value.

* Nonetheless, buying in for an amount that is now less than the average stack size isn’t a disadvantage. It’s an advantage. In fact, if everyone else had to begin with 1,000 chips and pay $1,000 dollars and you were allowed to buy just 500 chips and pay just $500 dollars, you would have an immediate long-range expectation of profit — even discounting any disparity in skill.

You’re right — rebuy considerations are more complex. But they almost all favor rebuying.

Rebuying for a relatively small amount late in a tournament, if allowed, would clearly be profitable. Average opposing chip sizes mean very little, except that you’d rather they be unequal than evenly distributed.

Your thoughts are appreciated.

— Mike Caro

one key advantage to registering late is your “freshness”. If you start to fade or get sloppy after 6 or 8 or 10 hours, then starting late pushes back your fade time.

He said it ” Although it might be profitable to play that flush draw in a regular non-tournament game, it isn’t profitable to play it in a proportional-payout tournament.”

what ive been saying, advice is good for cash games, not Ts.

Accordind this late survive theory,is it appropiate to take long seconds deciding the moves instead to play rapidly?

problem I find is late entry means the big stacks play looser and skew the odds.

wow- an actual mention of non tournament vs proportional-payout tournament.

Mike, if the 2 hour late buy in is in a tournament where your stack has been losing blinds and possibly antes, assuming a 9 handed event, will your stack still be worth more than if you bought in at the beginning?

So, in a tournament, adjusting your strategy to your stack size might be a good idea, but for that we need to weight the progression of the tournament in our math…

I now understand that 3K near the bubble is worth more than 10K after 1 hand, but can we know the multiplier? For example, when do 1 chip is worth 2 starting chips?

By the way, you said “your chips are worth more and more as the tournament progresses”. That’s like an poker illumination for me. Thanks!

Vincent Martin

Mike, it seems to me to be more about blind size when rebuying than the stacks you might be facing. For instance. If, at the beginning of a tournament I have 500 big blinds and at the last rebuy round, my rebuy would only net me 20 big blinds, I'm out. If, on the other hand, I can rebuy in that same tournament for 300 big blinds, it makes sense. Reasonable?

If you think being card dead is a myth then you must not be living in the real World.

To Jackie Wesley – My way of playing is to always fold. And even when the other player is not aggressive. However, with a 3 or 4 to 1 chip lead, and I am leading out , then many times I will bet with anything – and most of the time no less than 3x the BB. Nelson B.

I was just discussing the math with my husband last night, how in the world to the Pros keep up with all the math involved EV, implied odds, pot odds, fold equity, after reading your math column again for the 2nd time I realized you mentioned they do the math away from the table b4 the games, now that makes sense to me i could never understand how someone could possibly factor all that in without the clock being called I usually play tournaments. I will remember this great advice.

MIKE-how about this big Poker game coming on 9 November. Is it proper for me to suggest that I think that young fella PHIL Collins or the other young-un Holden will win. I think Ben Lamb is too much of a gambler. And he could bluff right into opo’s better hand. It seems to me that the tightest players are PHIL and MR. Holden. The best- Nelson.

Hi, Nelson —

I haven’t been following it closely enough to have an opinion. Thanks for contribution your thoughts, though.

Straight Flushes,

Mike Caro

I have a reply to my comment made last Feb. there is no such thing as being card dead it is a myth, you must be able to take action with the cards you are dealt.

Also take your re-buys if want or not, the re-buys will only be helpful if you don’t need them, if you get on a hand and want to risk it all remember if you did not get your re-buy b4 the tournament you may have less effective chip stack.

I have a question:So what do you do when you cannot catch any hold cards that are playable or worth bluffing and you are at a very aggressive table? Also do take your rebuys at a tournament or wait untl you need them?

“so what do you do when you cannot catch cards….and you are at an aggressive table?” i think you lose your chips and leave! But seriously, any cards can be worth bluffing if you have not caught anything for hours.