Tournament Theory question

[David Sklansky]

It is the first day of a long 10K no limit tournament. All the players are about equal except for me. I estimate my chances of getting to 20K by playing normally is 60%. On the first hand a player pushes all in in the SB and unwittingly exposes his cards to me in the big blind (no penalty). No other players

The following question assumes that I am not concerned about hourly rate or getting into a side game if I go broke. It also assumes that the size of the blinds is insignificant. In other words, assume I am getting exactly even money if I call.

I should call if I think my chances of winning are

A. Anything over 50%

B. A bit less than 60%

C. Anything over 60%. Exactly 60% is a push.

D. A bit more than 60%

E. Significantly more than 60%.

[Wake up CALL]

E. Significantly more than 60%

I'll let others elaborate.

[RiverMel]

Aren't options C and D the same? Isn't 'a bit more than 60%' just a different way of saying 'anything over' unless of course you are using the term 'a bit' in a non-standard way, to mean 'substantially', in which case options D and E are the same? Either way, you have one too many choices.

[tw texas]

A, unless you plan on quitting once you get to 20k. In that case it would be C, because that would improve your odds from 60% to...more than 60%.

[Piiop]

C.
Any chance of winning the hand above 60% would accomplish the same goal as if you were "playing normally". The only difference is that you will reach 20k faster because you're putting it all on one hand. Also, if the chances are above 60% then its clearly a better decision (than not calling and playing normally) solely based that percent to win 20k. In this situation, I dont think anything else can be a factor in making the decision.

This is my first post. So, hi everyone.

[RiverMel]

It depends how good you think you are compared to the other players. Your stat about what % chance you think you have to get to 20k doesn't tell us anything about your perceived edge over the field. Does 'playing normally' mean eschewing all bets where your edge is no more than 50%? 51%? The problem with this question is that to be able to answer it, you need to know what bets you would take under 'normal' conditions. And once you know that, the solution to your question is trivial.

For us to be able to answer the question, we need to know the perceived advantage. Your figure about reaching 20K doesn't tell us this, because how exactly are we supposed to compute your perceived edge over the field by your perceived chances of doubling your stack?

[TimTimSalabim]

I'll be willing to sacrifice a bit of equity to have my double-up come now instead of later, because in a no-limit tourney there is a certain advantage to wielding a big stack. But I'm not going to sacrifice a lot, so I'll take answer B.

[David Sklansky]

When I said you have a 60% chance of getting to $20,000 if you played "normally", I should have just said that you have a 60% chance of getting to $20,000 if you fold this first hand.

Also C and D are not the same. C is 60.000001%. D is meant to mean more like 62%.

[magic_man]

My initial instinct is to agree with E here, though I admit that part of my reason is the mere fact that you asked the question. If you call all-in with a 60% chance of winning, it's true that you'll have the same odds of doubling up as if you played normally. The problem here, I think, is that you have a 40% chance of going broke instantly (assuming you call with exactly a 60% chance of winning). If you play normally, I doubt your chances if busting out will be that high. You stated merely that your chances of reaching 20k were 60%, which is much, MUCH different than saying that your chances of busting out were 40%, as they would be if you called. This is one of those situations where you have a fairly high risk for a reward that you can get just as well by playing good, conservative poker (which means lower risk).

There may be something more important here though. You stated that your chances of doubling to 20k were 60% playing normally. If you win this hand, what will be your chances of doubling again to 40k? If you feel it is still 60%, I may change my opinion. I just started on a new level of tournament play, so I'm interested in other thoughts on this one.

~Magic Man

[baggins]

D seems intuitively to be the best answer. but i really don't know for sure.

[Piiop]

Magic Man makes good point that his chances of busting out are not 40%, rather he has a 60% chance of reaching 20k. That leaves the remaining 40% to mean that he can obtain anything from 0 to 19,999. And he could finish as high as 2nd place. Even if he does not make that 20k, he is still definitely alive in the tournament, though he may be at chip disadvantage, that doesn't really apply to the question.

So, back to the question, I think that I'm changing my answer to E. Unless the chance of me winning is very, very high, I'm folding. When I do fold where does it leave me? The same amount of chips as everyone else and I still have a good chance of continuing and possibly winning the tournament.

Just for fun I might call at 60% anyway /images/graemlins/smile.gif

[slavic]

In a side game A would be the obvious answer, the one and done aspect of a NL Tourney makes A anything but correct in this aspect. Since you have a 60% chance of doubling up anyway, simply folding gives you a better shot of doubling up with a much lower variance than A, B, and C.

Now we add in that if you double up now and you are significantly better than you competition you can marshall yourself into the money a higher percentage of the time. So even though you add to your risk, you may very well out distance your average opponent so far as to minimize the luck factor of a large tournament. So the added gain pushes us back towards B or C. More to the point B.

I really don't like this answer and probably wouldn't push it in with anything less than 4.5 TO 1 but that's my tight early tourney play comming out. This would also be why your slider strategy seems to work so well.

[youtalkfunny]

[rayrns]

"player pushes all in in the SB and <b>unwittingly exposes</b> his cards to me in the big blind (no penalty)"
"And no other players."

At this point knowing what the player has, you should be 100% sure of a win or loss. Therefore why would you need more than a 50% chance of winning. I'll be different and vote for A.

[7stud]

The answer would be anything over 60% if when you folded you had a 40% chance of going bust. However, that's not necessarily the situation as magic man pointed out. If you play on, you also have chances of getting to 19k, 18k, 17k, .....4k, 3k, or 1k as well as 0 the other 40% of the time. If the probability of all those outcomes were equally likely, then the other 40% of the time, you would average a net gain or loss equal to 0 relative to your 10k now. Setting up some equations with that assumption in an attempt to quantify the difference in the expected value of calling versus folding:

x = probability of winning the current hand
y = probablilty of losing the current hand

EV(calling) = x(+10,000) + y(-10,000)
EV(of folding) = .6(+10,000) + .4(0)<---net change of 0
x + y = 1.0

Setting EV(calling) equal to EV(of folding):

10,000x - 10,000y = 6,000

Substituting in:

x = 1 -y

for x in the equation gives:

10,000(1-y) - 10,000y = 6,000
10,000 - 10,000y - 10,000y = 6,000
20,000y = 4,000
y = .20
=>
x= .80

How valid is the assumption that all the other outcomes below 20k are equally likely? It seems fairly reasonable, and in fact, just because you have a 60% chance of getting to 20k doesn't mean you have 0 chances of getting to 40k or 60k or even 100k the other 40% of the time, so the expected value of folding could be even higher.

I'll choose E.

[risen]

[ QUOTE ]
At this point knowing what the player has, you should be 100% sure of a win or loss. Therefore why would you need more than a 50% chance of winning. I'll be different and vote for A.

[/ QUOTE ]

Wow I can't think of any situation where you are a 100% favorite preflop. On the first day of a big money and what I hope to be a long tournament for me, I see no reason to ever be all in preflop. I'd muck AA vs a flashed 73o right here rather than go home with nothing to show for my entry fee but a bad beat story to post on RiveredAgain.com or somethin /images/graemlins/grin.gif Doubling up this early is overrated anyway, you're not going to make it to the final table by default off of this win, and as you said you will more than likely get to 20K with your normal play. Put another way, would you put up the deed to your house and title to your car on a wager with a 40% chance of leaving you homeless?

[Leonardo]

I think its a very simple question. The answer is E, and I am 100% certain. If you have a 0.6 chance of getting to 20k, you are saying you have a 0.6 chance of doubling though. I will take that as a given that you always have a 0.6 chance of doubling up. So we are no longer playing cards, rather flipping coins(biased coins). You cant take a 0.6 chance for your whole stack. It would be much more prudent to take tonnes of 0.6 chances with small amounts. If the amounts were small enough, you are effectively guaranteed to win. The larger the amounts get, and in this case, 100% of your stack, the more you lose. I figure that taking this bet on the first hand with 60% chance of victory is taking a HUGE LOSS, way more than 10k, could be more than $1 Million. If you always have a 0.6 chance of doubling up, you would be mad to push your whole stack in (unless of course the blinds are huge, which you say they arent), with anything. Even AA may be better in the muck for the whole stack (am i taking this too far here?? /images/graemlins/smile.gif )
regards
Leonardo

[ohnonotagain]

"I estimate my chances of getting to 20K by playing normally is 60%."

I understand this to mean that 60% is the chance of getting to 20K *before getting knocked out*. So it is meaningless to say that the statement implies a 40% chance of getting to some lesser amount: if you get a lesser amount you are still just on the way to trying to get 20K (or more, or to the money).

I think this is true because it is also fair to assume, in the context of a long 10K tournament, that the chances of having 20K (or less) being worth anything in itself is very small. The chances of getting into the money while *never* having had as much as 20K are very small indeed, although it does happen (note that many of the people who limp into the money with e.g. one chip left previously had a good deal more, and certainly twice the buy in). So you almost certainly have to get to 20K on the way to winning any money.

So I think DS's question really relates to the added value, if any, of having 20K (much) earlier than you "normally" would. if you have 20K after the first hand, I would say that is going to give you a significant period at the start of the tournament, when many other players are typically playing very tight, when you cannot go broke in a single hand, and everyone else (or at least many people) at your table can. In the terminology of DS's tournament book, "they're broke, they're done" holds good, but "you're broke, you're done" does not. This should allow you to steal a lot of chips.

If you have the skills to exploit this situation, then I would say that the value of having 20K earlier than "normal" is substantial. This leaves options "A" and "B" as possible answers. I do not know how to quantify the above advantage, but very much doubt if it is worth a 10% difference in your chance of reaching 20K. So I would answer "B".

Oh no!! Not again!

[Al_Capone_Junior]

I think that this situation should go on the principle that it's better to pass up a merely good bet now if losing that bet will prevent you from making an even better bet later. So if you have a 50.00001-59.99999% chance of doubling now, but losing would prevent you from getting a 60% chance of doubling later, then you should definitely fold.

Therefore the answer must be C. You should not pass up a better bet now if you doing so only allows you to make a worse bet later.

al

[Al_Capone_Junior]

I did not consider the possibility of NOT doubling, but also NOT going broke by passing up this bet. Since going broke is of course bad, I should have figured in the possibility of not going broke, but not doubling either, thus still having a chance to win anyway. Assuming this, I would accept that E) significantly more than 60% could be the best answer.

al

[storm_madden]

Without looking at the other answers I say E.

The other 40% of the time you dont make 20K by playing normally you could have nonzero amounts

Some of the sixty % of the times you make 20K you could have much more than 20K.

[storm_madden]

After reading the clarification I interpret it to mean 40% of the time you will go broke since David did not include a time frame. It's different I think if he were to say 60% chance of doubling up on the first day or a given time frame.

A hand like AcQh vs KcJd is a 63% to 37% matchup (Wins 62.8 ties 0.2). Would I call with it?

62.8% of the time I double up, which is better than the 60% if I just play normally. There is no reason why a big stack will hurt me in the future, it will probably help me. It's better to double up now than later at the same odds because the blinds rise.

This hand allows you to fast forward the tournament at better than expected odds. It's even better than a fast forward, more like a time warp where you go forward and back in time with your chips.

It's almost like asking would you like 20K in chips at the beginning of the first day or at the end. Do you want to be chip leader after first hand or another chump with a par stack of 20K at the end of day one?

There are no real hands in Holdem that are exactly 60% propositions. But is it a push? Well, I think I'm better off taking a 60% proposition immediately rather than spend all day slugging at it. Again, do you want your 20K now or at the end of day one? So I'll take a bit less than 60%.

I take B.

[Josh W]

But most (all) tourny pro's accept E as the correct answer, and its not close.

[TimTimSalabim]

Actually, most top pros would probably assess their chances of getting to 20k at much greater than 60%. But if it is indeed 60% as stated in the problem, why not take the chance now instead of later? You lose nothing, and in fact you gain a slight edge having the chips now. I'm sticking with "B".

[Bozeman]

This is what I hate about the questions posed by Sklansky here. He poses a question in a tone that the answer is obvious, but often incompletely specifies the conditions such that multiple answers are possible. Maybe he just likes to have multiple answers so that more back and forth goes on.

If he means that there is a 60% chance you will have 20K at the end of the first day (or even some time during the first day), E is the obvious answer.

If instead he means that you have a 60% chance that you largest stack achieved in the tournament is 20K or greater, then C seems to be the answer. B is also a possible answer since having a large stack now may give you more opportunities. Some would, however, say D because they overestimate the difference in chip values between small and large stacks.

Since the second alternative has more gray area, he must have meant the first, though the wording seems closer to the second.

Arghhhhhh,
Craig

[elysium]

hi mr. sklansky
i think you would need significantly greater than the 60%. the reason is that by playing normally in many, many non-all-ins, and spreading the risk over a wide area, you can reserve your all-in calls for times when you have significantly greater than the 60%.

if you have a 60% chance when playing normally in the non-all-ins, but then are forced to play 60% all-ins also, al-ins in which you have a 60% chance of winning, the odds of surviving go down to about 40%. and playing normally cannot include going all-in with a hand that you normally wouldn't go all-in with.

i think you would need to be about a 70% favorite, excluding the effect of blinds, in order to go all-in hot and cold.

[BB King's]

... B !
Allow me to ellaborate a little !
C is of course wrong. Call now - earn you 2k and use the rest of the day for futher improvent.

[Cosimo]

My first guess would have been E. Al Capone Jr makes a good point here. Mathematically, C is correct.

A timeframe for getting to 20k IS specified, implicitly: the whole tournament. I am choosing to ignore the chance of getting into the money without ever having broken 20k, because I think that is so low as to not affect the decision. It's a long tournament; I give it a 0% chance.

The correct conclusion is that you have a 40% chance of going broke before making it to 20k if you fold. Likewise, on this hand, you have a 40% chance of going broke before making it to 20k. The two options are a push, answer C.

Saying "if you make an infinite number of small bets then you are guaranteed to get to 20k because you have a 60% chance of winning" (or whatever) is incorrect. If your play after this hand gave you a greater than 60% chance of getting to 20k, then the problem would be "I estimate my chances of getting to 20k to be 100%." Whether you double up now or fold and go for it later does not matter; both ways you still have a 60% chance of getting to 20k before going broke.

It doesn't matter what "most pros" do--that's arguement from authority.

-Cosimo

[7stud]

Leonardo,

I like your analysis and your confidience, however as you must know, if the player folds, it is most likely that not a single one of the ensuing hands will ever be equal to a 60% coin flip. The player will have to play the future hands according to whether they are +EV or not--not wait and play only those hands where he has a 60% chance of winning the hand. Hands he plays will sometimes have a 30% chance of winning if the pot odds are right, and some other hands may have a 65% chance of winning. The good player will be able to extract some extra value for each hand he does play. If the player waited and only played those hands where he had a 60% chance of winning, his stack would get blinded away, and the tourney would end with him maybe never even having played a single hand.

In the end, having a 60% chance of getting to 20k is not the same thing as having a 60% chance of winning every hand, so the player cannot fold and then get many 60% coin flips instead.

[Deadbuddha]

Id go for d on this one, no point on risking what i can most likely get by playing normally.

[David Sklansky]

is B. The four players who got it right explained it well. HOWEVER if the tournament was a short one the answer would be D. Why?

[Al_Capone_Junior]

"Whether you double up now or fold and go for it later does not matter; both ways you still have a 60% chance of getting to 20k before going broke."

If you definitely will go broke before reaching the money 40% of the time (after skipping this bet), then C must be the correct answer.

We COULD consider that he MIGHT NOT reach 20,000 but ALSO might not go broke, or that he might reach 20,000 but still not make the money... and consider lots of other "what-ifs" but I doubt that's what DS was looking for.

So it seems you and I are in agreement.

al

[7stud]

/images/graemlins/frown.gif

[crockpot]

remind me to bribe the tournament director to get a good seat at your table, and push all in whenever i want to steal any money you've put into the pot.

i'm only kidding, of course, and i understand your tournament goal is to win pots without showing down hands, but i still don't see how you can turn down a free 70% increase in your tournament EV at this point.

[Rushmore]

Although I have already seen the fact that the answer is B, I think Tim makes a good point. Wouldn't almost anyone here at 2+2 consider himself better than 60% to double up eventually playing "normally?"

If so, wouldn't the answer have to be (as I had originally thought) to go with E, as opposed to B?

Or do I overestimate the edge the better players have when factoring in the dead money (of which there is obviously less than in most tourneys, as it's a "main event," yet still significant, as with all tournaments these days)?

[emanon]

Selecting B greatly increases your standard deviation.

In a long tournament, this is offset by the increased liklihood of making it into the money due to the advantage conferred by starting out with a large stack. ie, you are increasing your risk for a higher EV.

However, in a short tournament, there will be fewer opportunities to take advantage of your stack. Thus, you will not be appreciably increasing your chances of finishing in the money. Therefore, you will be increasing risk without increasing return.

Thus, in a short tournament D is the correct choice.

[TimTimSalabim]

If the tournament was short enough that you could conceivably make it into the money without reaching 20k, that would offset the advantage of having the chips now, because if you get knocked out now you have zero chance of making the money. So, the answer shifts to D.

[TimTimSalabim]

Actually, I said the "top pros" would gauge their chance to reach the first double-up at much greater than 60% in a typical tournament they enter. Of course, it's all relative, no matter who you are. I'm sure most 2+2'ers have entered tournaments where they had greater than 60% chance to reach that first doubleup, and other, tougher tourneys where it might have been less than 60%. This problem is a good illustration of how your strategy changes depending on how good you are relative to the rest of the field. The less skill advantage you have, the more you roll the dice. But given the parameters of the problem, the answer is always B.

[slavic]

In a short tourney having extra chip equity is not as valuable because the blinds increase too fast to allow you to properly pumel the small stacks. Thus your additional equity risk does not gain you as much making it improper to do.

[Copernicus]

On the surface it seems straightforward that anything above 60% on this hand gets you to 20k with greater probability than you would have gotten there. The "gotcha" in the question may be the suggestion that this is a "long" tournament, ie there are many players in it, and the ommission of any probabilties of advancing beyond 20k.

Doubling up instantly without significantly affecting the number of opponents (many minus one is still many) is equivalent to starting out with twice the TCs without investing anything further, and doubles your tournament equity instantly. That would imply that 33% is sufficient. However that is contradicted by "don't risk it all on one bet if you have an edge on future bets" so more than 33% but less than 60% is called for. Since getting to 20k "normally" is the sum of many wagers which have some small edge pushes my answer closer to 60% (would a casino rather risk all of its bankroll on one roll of the dice with a small edge, or many smaller wagers), but not all the way to 60% because of the tournament equity considerations.

[i]The above answer assumes I have intepreted the question correctly, and differently than some of the first few responses indicate. I assume the alternative to getting to 20k with 60% probability is 40% of going broke. If you dont get to 20k anything less is obviously not in the money because there are many players * 20k in TCs. [\q]