Too Many Donks – Game Theory And Stupidity At The SNG Bubble

Here I Look At The Common Belief That It Is Impossible To Beat Too Many Donks,
Looking At Morton’s Theorem And The Sit N Go Bubble

This article is inspired by a game theory article I read over at Science 2.0, this looked at the interesting question as to why stupidity has not been removed from the gene pool via natural selection… a great read!

They use a tennis example to demonstrate that a population of stupid individuals making irrational decisions can actually benefit each other at the expense of a ‘smart’ individual who always makes a rational choice – especially when the supposedly smart individual makes assumptions that others will choose rationally too.

In poker there is a theorem put forward by Andrew Morton which covers implicit collusion in multi-way pots. This provides scenarios where several individuals making bad decisions (for example to draw) actually benefit (the schooling effect) in the long run against a player behaving ‘rationally’.

What I wanted to do is look at the bubble of Sit N Go tournaments – and find out whether having 2 or more ‘stupid’ opponents can actually work against the rational player over time. Since mistakes in terms of prize pool equity are large and obvious, it should be simple enough too see any effect. I have already worked through these in my mind, and the conclusion is an interesting one – though maybe not for the reason you think.

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Game Theory And Schooling At The Sit N Go Bubble

Here is our starting hypothesis – one good player knows his ICM and calls / pushes all-in correctly based on what he thinks his opponents should push and call with themselves. The other 3 players are perfectly nice people, however they have no idea about prize pool equity and will push and call based on what they think is a 'good hand' or to ‘stop someone pushing them around’.

The question is whether all those 3 donks playing badly together actually give our rational player a negative expectation when playing the correct way?

I’ll simplify things into 2 main mistakes and then look at the effect of these with match-ups with a good player vs a bad one and 2 bad ones together, assuming equal stacks and a $100 prize pool to keep things easy.

  • Mistake 1: Calling too light, way too light – light enough to spew half of their own prize pool equity and half of their opponent’s equity over the table each time they do it!
  • Mistake 2: Pushing All-in too tight, missing positive expectation pushes when nobody else has entered the pot is a common and often expensive mistake, say $2 to $5 in equity per ‘miss’ on average - depending on the blind levels and situation.

Here is where it gets interesting.

Player A (the ‘smart’ one) pushes all in against player B who calls horribly, spewing $25 of his own equity and $25 of player A's equity over the table – players B and C benefit to the tune of $7 each while players A and B both lose an average of $7. This is based on the assumption that one of these players wins the big pot and doubles up, with the remaining equity ($13 or so) going to the players not in the hand. For more on how this math is calculated see my Introduction To ICM.

If this also happens with players C and D then each of the 3 'bad' players has actually benefited twice and paid once by making these bad calls… if they were only made against the single rational player. This appears to be a strong demonstration of the Morton effect, their bad calls collectively benefit them at the expense of the player who does not make these calls!

Hang on a moment though - we also need to account for the fights between B, C and D – some of which should benefit player A when he is not in the hand.

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Well, lets think about this, ‘bad players’ do not know that they can shove practically any 2 on the bubble against a single opponent with high blinds – they do not know that their opponent can not call them often enough to ensure a profit… which is actually a good thing for them! Since their opponents do not know this and so call horribly wide. The relatively tighter pushing ranges a ‘non ICM’ player would push with actually balance these wider calling ranges to an extent. The damage in equity spew is smaller and our ‘rational’ player could still lose out. Because of their tighter pushing ranges the donks are not spilling equity as often enough to repay the $7 (well, just under) which they are taking away.

Tight Folds.

We can go through the same type of arguments with the tight open-folding. Sure, the smart player benefits when the player to his right gives him a walk. Here the ICm savvy player should benefit, all else being equal he has 1 walk for every 2 shared between the 3 donks. When a bad player folds a hand that would have been a positive expectation all-in, that is real money he is giving up and the primary beneficiary is the player who gets the blinds and antes.

All-in all, when we combine the effects of bad calls and tight folds we get a noticable, though not dramatic effect on profits. Even 3 complete novices at the table should only put a small dent in the smart players equity – this is noticeable, however not a disaster. The size of this effect is primarily shaped by how wide they push all-in against each other. Remember also that this is based on the idea that the ‘smart’ player is playing with the assumption that the other players will be making ‘rational’ decisions.

Sit n Go Bubbles Full Of Donks – Can We Construct A Worse Scenario?

Almost certainly, for example a 5-handed table with another ICM savvy player to our right, a ‘super donk’ who will call with almost anything to our left and a propensity of 1 or more other players to raise 3x the big blind for 1/3rd of their stack would be pretty bad.

Here the savvy player to our right would (correctly) steal when folded to, negating the occasional ‘walk’, and the minefield of the others would be in tact… ugly.

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Too Many Donks At The Bubble – Here Is Where The Logic Falls Apart!

So, we have shown an effect! Too many donks all playing horrible poker does have a negative effect on your profit (and they gain!).

To get to that point we had to make an assumption about our ‘smart’ ICM-savvy player. That he would make decisions based on assuming that his opponents were rational – that they would fold hands when it was -$ev to do so and shove with hands which were +$ev against rational calling ranges.

Since opponents are not rational, no truly smart player would play as if they were.

Instead of pushing all-in hands which figure to be +$ev against a range of hands they should call with, the ICM expert can start with a wide range of hands which a typical donk would call with, and work backwards from there.

The real beauty of Nash equilibrium is that every calling range has a pushing all-in range based on the stacks, prize distribution and blind levels… every single one.

A smart player can adjust so that only hands with a positive expectation against those horror ranges are used to push all-in… and can likewise call based on the somewhat tighter range a bad opponent might choose to push with.

Without the equity spew caused by ‘bad calls’, the smart player gains.

A truly smart player should welcome as many donks as possible – they are extremely profitable.

Now - Remind Me Why You Play So Many SNGs With 'Regulars' Instead Of Seeking Out The Donks??

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