Hypothetical Heads Up Gambling Situation
 

I was playing live today and I was thinking of offering my buddy this deal.

For $200, I will play this game with you:

We both get dealt two preflop cards. I will call your all in at any point preflop. You either go all in or fold. If you fold, we do this again, for ten total times. On the tenth hand, if you havent gone all in preflop, you are forced to play whatever hand you get.

If you are my buddy, what hand are you looking for preflop and how long do you wait to get it?

Have fun.
Nichomacheo

Re: Hypothetical Heads Up Gambling Situation
 

I would not take this bet... and neither should anyone else.

*****SPECIAL NEWS BULLETIN!!*****
NICHOMACHEO IS THE LARGEST LUCK-BOX IN THE WORLD

~rob

Re: Hypothetical Heads Up Gambling Situation
 

AA-22, Ax, KQ, KJ, QJ, KTs, QTs, JTs
From hand 7, add in any two cards 9 or higher.

You?

Re: Hypothetical Heads Up Gambling Situation
 

AA
KK
QQ
JJ
AK s
AQ s
TT
AK
AJ s
KQ s
99
AT s
AQ
KJ s
88
QJ s
KT s
AJ
A9 s
QT s
KQ

EDIT: this was my first reaction, though after a few minutes thought, im thinking a MUCH wider range is appropriate

Re: Hypothetical Heads Up Gambling Situation
 

PS is there no blind structure? Otherwise I will play A8s

Re: Hypothetical Heads Up Gambling Situation
 

I would play the above range first up, with the range increasing on each failure. On the final attempt, I would play any hand, on the penultimate, I would play any 50% hand.

EDIT: this is not true, since you are 50% at the end anyway, you must play only 50% hands

Re: Hypothetical Heads Up Gambling Situation
 

[ QUOTE ]
AA-22, Ax, KQ, KJ, QJ, KTs, QTs, JTs
From hand 7, add in any two cards 9 or higher.

You?

[/ QUOTE ]

22 is a mistake, you will play against 2 overcards for a coin toss, imho

Re: Hypothetical Heads Up Gambling Situation
 

Some analysis.

The expected median on n trials on the beta distribution is the top hand minus 1/n. So in 10 goes you normally expect the maximum to peak around 90% therefore playing one of these hands is a good idea.

As you play more your bet range increases so hat you aim to play more hands. On the last trial, anything about 50% is playable. On the penultimate, a hand in the top third, one before that a hand in the top quarter etc.

I cant prove it, but does that sound logical?

Re: Hypothetical Heads Up Gambling Situation
 

chainsaw,

I'm not sure you 100% understood. On the tenth hand you must play whatever hand you are dealt. Therefore, if you fold nine times, you are forced into a "coin flip" situation -- whoever wins the hand after five cards are dealt on the table. But along the way, you can decide "Hey, I want to wait a few hands because even though Q8o might be favorite against a random hand, I will probably get a better hand soon." The dilema is that the longer you wait for a good hand, the wider your calling range has to be. You might fold K3 on the first hand waiting for an ace, but eventually you have to call with K3 because thats favorite over when you can expect to get on the next two or three hands. You might go on a bad run -- 27, 39, T4 or something. OR you could get your AT. Gambool.

I'll add a twist when I get more initial responses.

Re: Hypothetical Heads Up Gambling Situation
 

Wait for any top 49% hand and push. You'll be a more than 1-to-1 favorite on the pot, while the pot odds are always 1-to-1 (I don't consider blinds structure here).
This is enough to give you the edge.
The least favorable case is when you reach the 10th hand, but :
1) it is VERY VERY unlikely
2) it's a coin flip, you are not an underdog

So your buddy has a very favorable deal here...

Re: Hypothetical Heads Up Gambling Situation
 

I did understand but didnt really express myself well.

Lets change the 10 to a 1 and ask what hand you would play. Well any 50% hand is better than a coin flip, so the correct answer is any hand of 50% or better i.e Q5 and above. QED.

Lets now say you have 2 chances, what hands do you play? Well on the 2nd trial, you would take the strategy above. On the first trial you can shorten your calling range, but to what I dont know - perhaps you would want a hand in the top third, my gut tells me. Maybe not.

You can continue the extrapolation until you get 10 goes, and I'm again guessing that on the first go you will need a hand in the top 1/10th, etc.

Does that make more sense, i'll try and work on a proof.

Re: Hypothetical Heads Up Gambling Situation
 

thechainsaw : on the last hand, you can't choose your hand, you are forced to push

Re: Hypothetical Heads Up Gambling Situation
 

Proving the correct ranges is near enough impossible as a hand that is of a rank X wins with a % time Y, X & Y are not linearly related and are also not evenly distributed (X is, Y isnt)

It would be a much simpler game if you just both got dealt 1 card and the highest card wins, I'm sure a correct answer could be found for that as you know your % chance of winning, getting a bettter card, and that the chances of a better card are evenly distributed.

Re: Hypothetical Heads Up Gambling Situation
 

Before we answer the problem with 10 hands , we should find the solution for 2 hands and then proceed from there .

Since you are forced to shove on the second hand , you must take the gamble if your first hand is better than 50 % .

Here is my conjecture . If you play this game with n trials , then you should wait to shove with a hand in the top 1/(n-1)*100% of all hands . For n=10 , you should expect to see a hand in the top 11.11% of hands on your 9th trial .

top 11.1% of hands includes 66+ ,a7S,k10s+ a10+ kq

However , the hands you decide to shove all in with changes with each hand dealt . You have to loosen your shoving range .

Re: Hypothetical Heads Up Gambling Situation
 

Now this changes once you don't hit on the second hand .

For instance , if my second hand isn't in the top 11.1% of hands , then I should shove if my hand is in the top 1/8 or 12.5 % of all hands .

Re: Hypothetical Heads Up Gambling Situation
 

Here, knowing the % winning chance of a given hand against any other RANDOM hand is enough.

Re: Hypothetical Heads Up Gambling Situation
 

Final answer :

On first deal you shove with the top 11.1%(1/9) of hands
On second deal " " " " " 12.5 %(1/8) of hands
On third deal " " " " " 14.2%(1/7) of hands
On fourth deal " " " " " 16.6% (1/6) of hands
.
.
.
On eighth deal you shove with the top 50 % (1/2) of hands

Re: Hypothetical Heads Up Gambling Situation
 

[ QUOTE ]
Before we answer the problem with 10 hands , we should find the solution for 2 hands and then proceed from there .

Since you are forced to shove on the second hand , you must take the gamble if your first hand is better than 50 % .

Here is my conjecture . If you play this game with n trials , then you should wait to shove with a hand in the top 1/(n-1)*100% of all hands . For n=10 , you should expect to see a hand in the top 11.11% of hands on your 9th trial .

top 11.1% of hands includes 66+ ,a7S,k10s+ a10+ kq

However , the hands you decide to shove all in with changes with each hand dealt . You have to loosen your shoving range .

[/ QUOTE ]

I like the conjecture.

I think it can be proved inductively.

let n be the number of trials before a shove and P(n) your probability of winning.

if n = 0, P(0) = 50%

P(1) = .50*.50 + .50*.75 = .50(1.25) = 62.5%

and so on - i think it's possible to find a formula - basically if P(n-1) is favourable to P(n), fold it, otherwise shove

Re: Hypothetical Heads Up Gambling Situation
 

I'm liking jay's logic...

I know nicho's "twist" (I am his roommate) and I'm really excited to see how your answers change... But he has forbade me from twisting it myself... so for now I'm patient.

I like to work backwards myself...
on the 10th hand- I'll play 100% (OBV)
on the 9th- top 50%
on the 8th- top 1/3

etc... this is different than jay's... i say the first hand should be top 10%... because before you are dealt the first hand, in 10 hands you can expect 1 hand in the top 10%...

But, I guess once you are delivered the first hand you must choose between it and 9 more chances (which you are likely to get a top 1/9 hand)... so I guess taking a 10.5% hand is plus EV after the first hand is dealt... kinda tricky

so i change my answer to 1/9 (1st) 1/8 (2nd) etc... down to 1/2 (8th)... but what do you do on the 9th hand? folding ensures a "coinflip" on hand 10... so i guess you push with anything better than average... top 1/2...

good question...

it is still -EV to play this game against the luck-box

~rob

Re: Hypothetical Heads Up Gambling Situation
 

What does "top 10%" of hands mean to you all?

Re: Hypothetical Heads Up Gambling Situation
 

55+,ATo+,A8s+,KQs is top 10%, according to SNGPT

Re: Hypothetical Heads Up Gambling Situation
 

[ QUOTE ]
What does "top 10%" of hands mean to you all?

[/ QUOTE ]

22+, A2+ K2+ and 69s, T2o and of course Q5.

More or less

~rob

Re: Hypothetical Heads Up Gambling Situation
 

Great. Still, what does that mean?

Re: Hypothetical Heads Up Gambling Situation
 

The most important thing is that your shoving range changes each time you don't hit your desired shoving range .

Any hand included in the top x % of hands should be played .

http://home.earthlink.net/~forrestmo...l#Wild%20Cards

This ranks the order of all the hands which can be helpful .

Re: Hypothetical Heads Up Gambling Situation
 

[ QUOTE ]
Great. Still, what does that mean?

[/ QUOTE ]

see harrington's book, he lists the top 10%

A8s and above

Re: Hypothetical Heads Up Gambling Situation
 

[ QUOTE ]
I'm liking jay's logic...

I know nicho's "twist" (I am his roommate) and I'm really excited to see how your answers change... But he has forbade me from twisting it myself... so for now I'm patient.

I like to work backwards myself...
on the 10th hand- I'll play 100% (OBV)
on the 9th- top 50%
on the 8th- top 1/3

~rob

[/ QUOTE ]

i can see why the top half, but why the top 1/3rd? i would say the top 37.5%

Re: Hypothetical Heads Up Gambling Situation
 

Rob changed his answer after . It was a little tricky for him .


As for the proper definition of the top 10 % of all hands .
Since there are 1326 different hand combinations , then you should select the top 1326*0.1 = 132.6 hands but you must choose the top 132 hands . The 133rd hand is not included in the top 10 % and so you can do slightly better by folding it .

Re: Hypothetical Heads Up Gambling Situation
 

Here are my figures - make it that you should get a hand rougly in the top 14% in 10 goes

Trials Fold Shove Fold EV Shove ev %median
1 0 1 0 0.5 0.5
2 0.5 0.5 0.25 0.38 0.63
3 0.63 0.38 0.39 0.3 0.7
4 0.7 0.3 0.48 0.26 0.74
5 0.74 0.26 0.55 0.22 0.78
6 0.78 0.22 0.6 0.2 0.8
7 0.8 0.2 0.64 0.18 0.82
8 0.82 0.18 0.67 0.16 0.84
9 0.84 0.16 0.7 0.15 0.85
10 0.85 0.15 0.72 0.14 0.86

Re: Hypothetical Heads Up Gambling Situation
 

[ QUOTE ]
Rob changed his answer after . It was a little tricky for him .

[/ QUOTE ]

Jay... I was just showing my thought process for those viewers who might not know how to approach this problem... this was a problem solving lesson...

MORE IMPORTANTLY...

TWIST TWIST TWIST TWIST TWIST TWIST TWIST TWIST TWIST TWIST TWIST TWIST TWIST TWIST TWIST TWIST TWIST TWIST TWIST TWIST TWIST TWIST TWIST TWIST TWIST TWIST TWIST TWIST TWIST TWIST

Re: Hypothetical Heads Up Gambling Situation
 

My answer --

This was doing using the numbers from the page that Jay posted. I arranged these in Excel by win percentage, then grouped them by what hand you should call on.

Hand 1 - Top 9th - A9s+, AJo+, 66+, KQs
Hand 2 - Top 8th - A8s+, ATo+, 66+, KJs+
Hand 3 - Top 7th - A7s+, ATo+, 66+, KTs+, KQo
Hand 4 - Top 6th - A7s+, A9o+, 55+, KTs+, QJs, KJo+
Hand 5 - Top 5th - A5s+, A8o+, 55+, K9s+, QJs, KTo+
Hand 6 - Top 4th - A3s+, A5o+, 55+, K8s+, QTs+, K9o+, QJo
Hand 7 - Top 3rd - A2s+, A3o+, 44+, K5s+, Q8s+, K8o+, Q9o+, J9o+
Hand 8 - Top Half - A2s+, A2o+, 33+, K2s+, K3o+, Q3s+, Q6o+, J7s+, J8o+, T7s+, T8s+, T9, 98s+
Hand 9 - Same

PM me for Excel chart.

Thoughts?

Re: Hypothetical Heads Up Gambling Situation
 

Great question Nicho .

Here is a follow-up question . What range of hands should the opponent be calling the all in with ?

If he knows player A's optimal shoving range , then player B should know the optimal calling range . This is a classic game theory problem .

Re: Hypothetical Heads Up Gambling Situation
 

You mean if he had the option to call or fold? It would complete change the problem because then I wouldnt have to call with a random hand. Very hard problem without a doubt.

Re: Hypothetical Heads Up Gambling Situation
 

I verified that my conjecture is true for all n .

Pretty cool .

Re: Hypothetical Heads Up Gambling Situation
 

<font color="blue">Question 2. Now the person offering the bet is allowed to fold once, except for the tenth hand. Now hat is the optimal strategy for both players?</font>

Keep in mind, once the fold is used, the person offering cannot fold again and must call the opponent's next push.

Re: Hypothetical Heads Up Gambling Situation
 

Now this is poker... no longer is it just odds... both players can think on many levels...

the pushing player can push his first decent hand in hopes that the offering player will use his fold early... so that later on with a better hand the pusher can 'force' the deal.

and obv since the offering player knows that the pusher is thinking this way would be correct to loosen his standards against that first push

but if the pusher knows his opponent is a saavy multi level thinker he will be sure to push the first time not to force the fold but rather trap him into calling

of course it follows that the offering player would be better to fold anyways just in case the pushing player realizes that his pushing will be called loosely and pushes with the goods

but, if the pushing player knows that he will fold for the aforementioned reasons he is best off pushing the first hand no matter what (blind) so that the second player is forced to a decision for all of his chips (secret of poker according to mike) and he can later appropriately push with a better hand if the offering player folds.

~rob

Re: Hypothetical Heads Up Gambling Situation
 

The first player should still shove with the range of hands as before . The caller has to be aware of the optimal shoving range by player A . If player A shoves on the first hand , then B knows he's shoving with the top ninth of all hands . B will only call if his hand is better than 50 % against the shovers all in range .

ie , ace king wins 53 % of the time against the top 9th percentile of hands . Therefore ace king must be included in the calling range for the first shove . It turns out player B should call with AK and 10+ .

You keep on working this out for the 2nd, 3rd ,4th hand etc

I realized after that I answered a slightly different problem .

ok a random hand against Player A's first shove wins 30 % of the time . That means if player B's first hand does better than 30 % , then he should call the all in .

Re: Hypothetical Heads Up Gambling Situation
 

Disagree jay...
knowing that the offering player is unlikely to have AK and TT+ (or whatever) he should push to eliminate his fold.

~Rob

Re: Hypothetical Heads Up Gambling Situation
 

lol Rob

Re: Hypothetical Heads Up Gambling Situation
 

jay... i'm serious... your read on your opponent SHOULD affect your play...

bluffing is a possibility... and correct to consider

~rob

Re: Hypothetical Heads Up Gambling Situation
 

I think this is a very different problem Jay. Like Rob said, you now have to consider that the pusher is trying to make you lose your fold because its higher EV later when the caller doesnt have this option.

As Rob said, knowing that your opponent might push with air here will change the dynamics a lot.