Bounty Builder tournament

Question

Is it profitable to call two players with a smaller stack with any 2 cards in hand if the total gain per knocking out is more than a buy-in?

For example:

Buy-in = $33

Total received Knockouts = $37

Opponent 1 = 29845 chips (Bounty = $10.50)

Opponent 2 = 34112 chips (Bounty = $10.50)

Me = 52200 chips

Opponent 1 on UTG1 all-in, all players fold, Opponent 2 on SB raise to all-in. I have any two: call or fold?

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UPD: I speak English very bad, sorry.

Answer 1

I think the best way to answer this, as it's applicable to any and all should you call situations as it provides the base maths on equity, is to look at your equity. Sorry if the figures are a little off, just doing this quickly so rounded some figures up or down on percentages, but this will show you a basic framework.

There are four parts to this:

1. Calculate your required equity
2. Calculate your opponents range
3. Calculate your equity against your opponent's range
4. Compare your equity to required equity

Required Equity

So the calculation on required equity is very simple and something you can do at the table.

Required equity = price of calling / (pot size + price of calling)

In your example this would be the following:

Price of calling = 34,112 - 1,200 (your big blind) = 32,912

Pot Size = 1,200 + 34,112 + 29,845 = 65,157

So when we substitute the above into our equation we get the following:

32,912 / (65,157 + 32,912) = 32,912 / 98,069 = 0.3356 or 0.34 = 34%.

To make this call and it be a mathematically correct play you need at least 34% equity.

Opponents Range

Simple one here, but think of what your opponents could possible have here to do what they have done. I won't do this part for you as I didn't play against the people, but when working out a range it's an assumption of how you think they would play, and what you think they would, in this case, go all-in with.

I'll give you an example, and I am just going to assign a type to each player:

Opponent 1, is very aggressive, but we haven't seen him/her go all-in without 99+ or AQ. So their range in this case would be {AQs,AQo,AKs,AKo,99,10,JJ,QQ,KK,AA}.

Opponent 2, let's say is very tight, you don't think they would go all in without AKs or QQ+, so their range would be {AKs,QQ,KK,AA}.

Equity against your opponent's range

So know you have a guesstimate on their range, and now you can compare how many hands in your range (i.e. what you will call with), and how your range does against their range. So if you have say JJ, and you compare what that beats from the above range you can work that out by doing the following. Note I'm giving rounded percentages to make the calculation easier.

Against Opponent 2: JJ vs AKs = 56% JJ vs QQ = 18% JJ vs KK = 18% JJ vs AA = 18%

You will also need to know how many combinations of each hand can exist.

AA = 6 combinations (Any pocket pair is always 6 combinations) KK = 6 QQ = 6 AKs = 4

Total combinations = 22

So the equation your equity against a range is the following:

(equitycombinations + equitycombinations(repeat for number of hands in range)) / Total combinations

(56 * 4 + 18 * 6 + 18 * 6 + 18 * 6) / 34 = (224 + 108 + 108 + 108) / 34 = 548 / 34 = 16% equity

Compare Equities

Very simple part, is your required equity to make this call profitable less than or greater than your equity verse your opponent range.

If your equity vs their range is greater than your required equity = profitable call

If your equity vs their range is less than your required equity = not profitable call

So your example on is the call profitable here

(I just grabbed these equities from an online calculator, check here)

1. Hero: Q3o - 17.4%
2. Opponent 1: AKs - 60.5%
3. Opponent 2: QKs - 20.3%

So we know from above, you required equity to call be profitable is 34%. We also know from the above your exact equity is 17.4% (note this is using the calculator, but you could easily use the opponents range here to work out this as a more rough calculation)

17.4% is less than 34% therefore not a profitable call.

Closing things to consider

This is a very specific isolated pure maths on a single situation, other things to consider are ICM when it comes to tournaments, which I won't go into, however even when you consider ICM it is not profitable to always call with any two, even if you're in profit for the tournament. Always try to consider your required equity vs what you your equity against your opponents range.

You don't need to work out this stuff yourself when playing online, they're online calculators use them. Check here too.

Answer 2

Since you have already made a profit from bounties in a tournament, you could call with any two cards and eventually get knocked out of the tournament yourself and make a profit.

However, this is not the correct way to think about the tournament that you are playing. If you have the resources to go further in the tournament and get more profit, you should go for it. You may get lucky and knock two players out with any two cards but this is highly unlikely and will most likely bring your tournament life to an end very quickly.

I would loosen up my calling range in that spot (facing 2 all in players that have a sizable bounty), because the bounties are very valuable. However, do not loosen that range up so wide that you are just calling with any two cards. If your hand is really bad and your stack is not too short, it would be much more profitable to wait for a better spot where you are more likely to win a bounty and increase your stack size.