Maths in Poker: An Exact Science?

With the boom of online training aids, such as forums, poker videos and dedicated training sites there is a lot greater emphasis on the maths side of the game than several years ago. Clearly poker involves a lot of maths, but there is a danger in that people often believe maths is an exact science, and while it usually is sometimes it is based on assumptions and if these assumptions are not reasonable then the theories based on them can be inaccurate. This applies more to poker than other fields of application as poker also has a psychological side to it. For the record, I come from a heavy maths background including a maths degree, so this is not a case of me bashing something because I don't like or understand it.

Let's take a simple scenario, I am not going to calculate any of the maths as that is not the point of the blog, you can find examples easily....Google is your friend!

OK so in this example we have 66 and open and get 3 bet shoved on by a 15bb stack. We need to decide whether to call, and for simplicity we are nowhere near the bubble and are only interested in calculating our cEV. To make this calculation we work out our opponents 3 bet shoving range and from that can calculate our equity. The result of this calculation will give us an exact figure and if its positive then we call, but if it's negative we fold. Maths has helped make exactly the correct decision, or has it? Well in making this calculation we had to make one assumption, what was that?

The assumption was working out the 3 bet shoving range of the villain. The accuracy of our resulting calculation is therefore based solely on the accuracy of our assumption of the range. What this means is closer the resulting calculation is to zero the more accurate our assumption needs to be otherwise we may make the incorrect decision. This highlights the dangers of blindly using maths and assuming the answer is always correct!

That leads me on nicely to the concept I really want to talk about, ICM. ICM stands for Independent Chip Model. This model is used to calculate your current equity (in terms of cash) based on your current stack (in tournament chips) taking into consideration all the other player's stacks and the payout structure etc. Again I am not going to go into detail, there is plenty written on the web about ICM in a much better way than I could explain. Feel free to Google and have a read before resuming the blog if you so wish.

ICM essentially assumes that all players have equal ability and that each players chances of winning is directly related to their current chip stack. This also means that ICM assumes that you have the same ability regardless of your stack size. So we have identified two assumptions required by ICM, so in order to determine how reliable the calculations are we need to establish how realistic the assumptions are.

The first assumption is that all players are of the same ability, how reasonable is that assumption? Well consider every single table you have ever played on, and then think how many times you have felt that all players on that table were of the same ability. Unless you are deep in a super strong event then it is extremely unlikely that everyone is at the same skill level.

The second assumption is one that is often missed by a lot of players who swear by ICM. It actually has equal importance to the first assumption. This assumption is that our own ability is a constant regardless of our stack size, and the same principle should hold for our opponents. Well ask yourself the question, do you feel you play equally as well with a 10bb stack as a 100bb stack? What about a 25bb or 40bb stack? The fact is most players don't play as well around 25bb as they do for other stack sizes, 25bb is a very awkward stack to play correctly so this also likely represents our opponents skill set as well. What this means is that our chances of winning with a 26bb stack are not twice that of a 13bb stack. It also may mean that having a 50bb stack is more than twice as good as a 25bb stack. All of this is further clouded by the other effective stacks of course, doubling to 50bb when everyone else only has 25bb isn't as good as we are still playing 25bb poker. Clearly this second assumption is also on shaky ground and cannot be relied on.

To further highlight the weaknesses in these assumptions I will give two quick examples in which ICM states our tournament equity is the same. Take a look and see if you think it actually is.

Example 1:

Seat 1: Hero has 25bb

Seat 2: Nigel the Nit has 100bb

Seat 3: Larry the Good LAG has 10bb

Seat 4: Player 4 has 30bb

Seat 5: Player 5 has 22bb

Seat 6: Player 6 has 17bb

Example 2:

Seat 1: Hero has 25bb

Seat 2: Larry the Good LAG has 100bb

Seat 3: Nigel the Nit has 10bb

Seat 4: Player 4 has 30bb

Seat 5: Player 5 has 22bb

Seat 6: Player 6 has 17bb

The only difference in the examples is in seats 2 and 3, it is a subtle difference that makes a big difference to our chances in the tournament. Being out of position to a good LAG player is horrible when you have a 25bb stack and clearly our chances of winning in example 2 are far less than in example 1 as Larry will make our life very difficult whereas in example 1 our life is easier against the big stack because they are a tight player. Despite this, iIf we calculated our ICM in these two examples they would be exactly the same. Hopefully this shows how flawed the assumptions in ICM are if taking the results to be an exact science

There are also other failings with ICM, it treats each hand in isolation and it doesn’t account for the structure, ironically in a turbo ICM is going to be more accurate as the luck element is increased. Nor does it look at the average stack, again if the average stack is lower than ICM will actually be more accurate due to the higher luck factor. ICM doesn't account for players moods which may affect their ability. Lastly ICM fails to account for future games.

ICM generally means you should fold a massive range to opponents shoves on bubbles, what this means is your opponents can shove any two profitably purely because you can hardly ever call according to ICM. However they can only do this if they know you will fold most of the time, all of a sudden if they know you will call lighter, then they are also putting themselves at risk which means that if you call them lighter in one or two tournaments then they should start shoving a bit tighter in future games. This is especially true in small player pool SNGs / MTTs where you play the same players a lot and have a lot of history on the bubble against ICM players. If they adjust correctly to your lighter calls then you should find yourself getting a few more walks, this can be worth making a slightly marginal call and giving up some small equity in earlier SNGs in order to realise extra equity in future ones.

Don't worry about making precise overly complex calculations based on unrealistic assumptions

So bearing all this in mind is it worth bothering with ICM at all? Well despite all I have said I believe it is. Despite its flaws, ICM is the most accurate mathematical model currently available for poker tournaments. It gives a good approximation and can help players achieve a better understanding of calling ranges when near the money. It is clearly better than just using chip EV.

That said I would never actually go to the trouble of working an ICM calculation out myself, as some players do when looking back at a hand and deciding whether to they made the correct decision on a marginal call. If the decision isn't obvious without calculating it then the calculation itself is likely to be marginal enough that given the assumptions made the results themselves won't be reliable. Instead I will look at the hand and the context to decide whether it is worth the risk. For example if I have 25bb and have a good loose player with a big stack who has position on me than I am usually keen to get away from a 25bb stack so I am more likely to make a marginal call etc. Also in turbos or if the stacks are shallower I am more likely to make a marginal call. In good structured events I am more risk averse when I feel I have an edge over the field and have a playable stack.

It is more the principle behind ICM that is important than the actual calculations in my opinion. You need to be aware that you usually should be calling with a smaller range than cEV would suggest when near the bubble due to the loss of equity when losing the hand being more than the gain in equity when winning the hand. Providing you bear this in mind when making decisions you shouldn't need to do heavy ICM calculations unless you play in super tough games. People who calculate ICM and treat it as an exact science may be making the incorrect decisions on occasions due to incorrect assumptions.

The same is true whenever doing any mathematical calculation, make sure you are aware of the assumptions and put the results into context based on how realistic the assumptions are, otherwise you may draw the wrong conclusion from those calculations. Sometimes you don't even need to calculate precise results if you understand the principles behind a theory. Hopefully that is a relief to all you non maths people!