Ok so I've had this offered to me at the poker table and was wondering what is the most +ev decision.

My opponent will offer to flip $1000 with me and will put in $1000 dark without looking at his cards
Hero gets to look at his cards but must put in $200 dark.
Hero then gets the option of going all in for $1000 total or folding.

Should we be folding parts of our range that don't have 40% equity vs a 100% range? or should we just continue going all in no matter what we have?


I've seen this guy make this offer to several players and most people who take this offer and fold the bottom parts of their range almost always inevitably lose in the long run.

I assumed in the next that you are not flipping for 200+200 total pot if you "fold" but that you lose the 200 right away in that case and move to the next hand or you put all in and risk 1000.

You can see this game as the same with putting down always 1000 and electing to take 800 back losing the 200 or leaving it in and taking the bet for all 1000.

If you have a fraction f you fold then the remaining you call with, that is say 1-f, will need to have an equity E vs the random range that is such that the following inequality is satisfied;

(1-f)*E*2000+f*800>1000

or 2*E*(1-f)+0.8*f>1 or E>1/2*(1-0.8*f)/(1-f)

So for example if you were folding 5% of the hands (the bottom 5%) making f=0.05 then
E(of 95% vs 100%) must be larger than (1-0.8*0.05)/2/(1-0.05)=0.5053

Now notice that E(94.6% vs 100%) (or dropping only is 82o-32o) has E=50.9%

It needed to break even 0.5057. So it clears it.



So you can pick any range that satisfies that inequality. We just established that one can fold the bottom 5% and see profit. But you can do better.

You can try to maximize the EV of the strategy per hand dealt that is

EV=(1-f)*E*2000+f*800-1000.

This appears to be happening at about 88.2% range or all hands except 93o-43o and 92o-32o.

The EV is then about +9.79 (it was around +6.5 at 95% range so not a huge difference but important to fine tune it)

Notice however that this is not the end of the story.

Since 1000 is a great deal of money and 9.79 per 1000 is only like playing a game that is roughly 50.5% to 49.5% in EV, if this guy has a lot of cash and you have little, he may be able to take you money very often simply because your ruin probability is significantly larger than his ruin probability (or the give up level). If you had 10k to lose you may drop once you are down 10k.

If the guy sees a tight player he will play him indefinitely. But if he sees one that is properly folding he may drop out of the game after losing 20-30 k say having realized he plays a negative EV game vs a properly aware opponent.

As an illustration if this guy has 100k to lose and you only 10k and you have 50.5% on avg vs him playing effective flips with 50.5% vs 49.5% edge then your risk of ruin is roughly 79.6%

If he had a practicality unlimited bankroll (say millionaire) your risk of ruin is basically

(0.495/0.505)^10=81.9%!!!

So if the guy is very loaded having found many victims to play him before and limiting losses vs very well adjusted opponents by ending the games at will, he can then play others that are not familiar with the game and its volatility and deplete their stack very often simply by pure luck because he can afford wider losses and the edge even at best range selection is not huge. So even good players will tend to lose very often because they dont have enough money. Additionally if there is rake involved (and rake back incentives) there may be other benefits for him to do that and then the above perfect range calculation needs to take into account that exact rake scheme.


That above risk of ruin is not exact here because the game is not exactly all in every time at 0.505 probability because you fold also some 12% but the sd of the real game is not particularly different if you do the math to the all in every time game equivalent EV version.

If you came to the game with 20k to lose then your risk of ruin is

(0.495/0.505)^20=67%

20k is no unimportant sum of money so vs someone deep enough you will hit it 67% of the time and you will fail to leave a winner even at 20k.


If you play a disciplined game of course vs a loaded player, say who reaches first 20k profit leaves (if your tolerance is 20k or 20 units) your risk of ruin becomes;


((q/p)^20-(q/p)^40)/(1-(q/p)^40)) with q=0.495 and p=0.505 or

a far less risk of 40%. You win 20k 60% of the time and lose 20k 40% of the time then.

As expected when the range of losses for both is the same the player with the edge will win more often.


Given that usually most people will not be aware coming into this game what is the right fold probability/range or the effect of risk of ruin they have, they will likely either deplete their bankroll fast and drop out if the opponent is very loaded or they will even give him a profit or he will drop out and end it at will to move to a next victim.

For example the guy wins if people fold more often than 35%. If he finds not very deep people that fold anywhere near 30% say he will take their money over 80-90% of the time. He can keep playing them indefinitely if he sees they are near 30% anyway and win in other ways with rake or luck.


Do you have rake in that game?


Also notice that the risk of ruin calculations are done to get an idea in a similar EV similar volatility game but it is not exactly the same as here and this may be important if you want to be very accurate. What do i mean by that? Well for example 25% of the time here you will have a top 50% hand and he will have a bottom 50% and you will then have 64.77% chance to win the all in (and ~20% the reverse only a bit better). So your win probability is not fixed at some 50.5%, it is moving all over the place typically from 80% to 20% and most of the time its 40-60% but sometimes it can be 5% or 95% eg kk vs k2o. I do not expect substantial deviations from the general picture though in the risk of ruin sense moreover this wider selection of equities. It will tend to average out to near 50% soon. It would be interesting for someone that has poker code in their simulations to run it and see how far the picture is from a 50.5% to 49.5% game if one say has 10k to risk.

Just to clarify in case you play another game than the one i solved;

Are you always seeing the 200 lost when you do not push all in or is it simply played as 200 vs his 200 and whoever wins gets 400 back (or 200 back in ties)?

If this is what is going on and not what i solved above then you EV equation is ;(we fold f and go all in 1-f)

EV=(1-f)*(E(top 1-f of range vs 100% range)*2000-1000)+f*(E(bottom f of range vs 100% range)*400-200)



In that case the solution is different.

But i didnt consider this version because then the opponent will always lose in EV nomatter what we did as only if we played 100% its even. The other times its always positive for us and it becomes only a question of what is the best push all 1000 instead of just risk 200 range.

In any case if this is what you are after then a numerical search shows that;

The best is to push the top 46% and keep it at 200 the other 54% for an EV of +62.5.

But i doubt its this game that is offered because he needs to find some victims that lead to him winning and it looks like the 200 is lost when you do not push all 1000.

It is interesting that he may be offering this game in a rake environment that makes it close to impossible to play profitably but he has a special deal for rake that leads to advantages for him if he is very rich already and can sustain the volatility and meets often enough people that cannot fold just 12% and tend to 30% or worse.

This is amazing, the level of depth and accuracy for one post. I tip my hat off to you sir.

I agree, an excellent detailed answer.

yeah that was interesting

yeah u r getting 1.5:1 so if u have 40% or more equity vs 100% call, if not fold.

Wow great explaination do you mind me asking you what your major was in college if you went.

Thanks. I studied Physics both in undergrad and grad school.

Whilst I'm sure it's a very good post, I got confused reading it and it doesn't appear to have a clear definitive answer regarding what hands we should be folding and how big our ev is in this game overall?

So I decided to make a simpler solution that more people may be able to understand. So for each hand we need 40% equity against a random hand for going all in to be more ev than folding. The hands that don't satisfy this criteria are as follows: 92o-32o, 83o-43o, 74o-54o, 65o, 72s-32s, 63s-43s.

With the remaining hands that we can call, we have 52.74% equity. The range of hands which we call is 83.11%. Which means when we call, we will win on average 2000x0.5274 = 1054.8. Thus we win 54.8bb*. This will happen 83.11% of the time, so 54.8x0.8311 = 45.54bb won on average when we call.

When we fold, we lose 200bb - this will be the case 16.89% of the time. So we simply have to do 200x0.1689 = 33.78bb that we will lose on aveage when we fold.

Then it's a case of subtracting the 2 numbers together: 45.54-33.78 = 11.76bb profit.

Our edge in the game can be found out by adding 11.76 to 1000 and dividing by 2000 - which gets us 50.59% (rounded to 2dp)

Which as pointed out by masque de Z is a very small edge in a game where we are risking $1000 each time. As such, we would need a large bankroll to make the game worthwhile playing with as little risk of ruin as possible.

*For some reason I worked it out in bb even though it's not relevant here, but 1bb = $1.

NB. This is all assuming no rake too. If the game is raked then it's unlikely to be +ev at all to play unless it's very low rake (don't have time to work out min rake for game still to be +ev)

You know thanks for doing this hand by hand clearance F_Ivanovic that gives more accuracy and illuminates a problem not with the method i used but the way it appears pokerstove is ranking hands at the bottom of the range when you open up range from 100% towards AA hand by hand.

The method i used is correct and is identical to the 40% clearance argument but it has an advantage hence why i used it, provided the numerical tool (poker enumerator etc) uses hands ranking according to equity vs a random hand at the bottom of the range at least. But it appears it doesnt.

For example when i put it at 88.2% ie 22+,A2s+,K2s+,Q2s+,J2s+,T2s+,92s+,82s+,72s+,62s+,5 2s+,42s+,32s,A2o+,K2o+,Q2o+,J2o+,T2o+,94o+,84o+,74 o+,64o+,54o

and then go one hand less it removes next 94o while still keeping inside hands like 64o or 65o that have worse equity vs 100% range and are in fact below 40% by a bit. Yet 94o has over 40% vs 100%.

As a result if i use the Equity of range vs 100% approach, like i did, this requires that the function E(range) is moving range from 100% to eg AA by reducing one by one the worse equity hands vs random hand. It appears Pokerstove is using another ranking that may relate either to tighter range opponent or some form of playability of the hand at flop (ie 65o or even 63s is more playable than 94o) i am not sure how they have ranked it and need to search the original program FAQ if it explains.

So when i solved it i used a spreadsheet and ran the range from 100 to lower levels until it maximized the EV as i described. This is identical to stopping at 40% for each hand that clears it but it has the advantage that it finishes fast the process of search provided the pokerstove range opens according to the equity each hand has vs the 100% range. If they use a different ranking it will fail to be accurate.

And this is why it gives a different (if not very big difference) result that affects a bit the EV too.

If i do it using the 40% criterion and go over the suspect hands and clear them one by one to find the island that is below 40% i get the same results you did.

Thanks for posting so that i can know this about pokerstove from now on and try to search how they rank their range opening at the lower levels and whether they use the same for the top ones too. (if you see i had also posted which hands you fold but it was according to this pokerstove methodology hence a 4-5% different hands )

How you rank hands in poker is in general a problem because for example when you play with some all ins or 3bets etc you are not against a random 100% hand, so the ranking has to depend on what is usually your opponent. Maybe they use a 50% range or something.

I see for example that 94o vs 50% gives 94o 31.97% but for 65o is 35.17% vs 50% range ie much greater. So the rationale to remove first 94o before eg 65o, that is actually removed much later by pokerstove, is probably because they are ordering them with respect to some tighter than 100% range as criterion.

No problem - I was using equilab to do hand for hand but also noticed that for hand ranking at 83.11% it includes some hands that I had as folds and excludes some hands I had as calls. eg. T3-T2o and 94o are excluded where-as some of the low suited combos are included. Low suited hands perform better as jams against a calling range that is <100% eg. 32s has more equity than T3o when you jam and are called by a 67% range.

I'm curious what bankroll you would need to have to make this game "sensible" to play?

F_Ivanovic, using your numbers:
- 0.1689 lose $200
- 0.4383 win $1000
- 0.3928 lose $1000

Let T be our bankroll. We want expected utility (play) > expected utility (not play), i.e.

0.1689*LN(T-200) + 0.4383LN(T+1000) + 0.3928 LN (T-1000) > LN(T)

Which stops being true for bankrolls below ~$35.6K. So if your bankroll was $36K it would be worth having a go, though if you lost the first showdown (or folded 3 times before a showdown) you should stop.

Hi, thanks for that. Can I ask what formula is that you are using and what does LN mean?

Using the Kelly Criterion we should only bet 1.18% of our BR but since we're sometimes just losing $200 outright and sometimes gambling $1000 I got confused how to work it out.

By LN he means "natural log". He's looking to see what value of T you can have where taking the met maximizes the logarithm of your bankroll.

Just applying the Kelly Criterion isn't enough when you do not have an option about how much to bet. The Kelly value is the optimal amount to bet, not that maximum you should take the bet. E.g. It is generally going to be better to bet 1.5 Kelly than 0, if they are your only two options

Interesting. From Wizard of Odds:

"While betting more than Kelly will produce greater expected gains on a per-bet basis, the greater volatility causes long-term bankroll growth to decline compared to exact Kelly bet sizing. Betting double Kelly results in zero expected growth. Anything greater than double Kelly results in expected bankroll decline."

Didn't realize 2x was the other break even point.