Theory post....ICM,Bubble factor (part 2)

Bubble factor
Bubble factor quantifies the degree to which ICM considerations involved in a stepped payout structure skews the pot odds needed to make a call correct.
In the previous example, Player B needs only 50% equity (odds of evens) to make the call correct in terms of chip pot odds (slightly less in actual fact taking blinds into account), but needs odds of 3 to 1 (75% equity) for it to be correct in tournament equity terms. This is because he loses 3 times as much equity if he calls and loses than he gains if he calls and wins. Bubble factor in this case is 3.
Once you know the bubble factor in a given situation, you can determine whether it is correct to call or not based on bubble factor and pot odds.
Pot odds required = odds of winning multiplied by bubble factor

In the example, if player B believes he is a 6 to 4 favourite (60% equity) against the shover’s range and is getting evens on the call
Pot odds required = 4/6 by 3
Here he would need pot odds of 2 to 1, so getting odds of only evens, folding is correct.

For the call to be correct, player B must be a 3 to 1 favourite (have 75% equity):
Pot odds required = 1/3 by 3
So calling getting odds of 1 (evens) is correct.

This illustrates the first key point of ICM:
In a tournament, it is never correct to call getting the bare pot odds required in chip terms. The true odds required are odds of winning multipled by bubble factor

A slightly more complicated example
Two package $6K satellite with 3 players left, but player A has twice as many chips (8000) as the other 2 (4000 each).
Similar logic as in the previous example tells us that if the three players were evely stacked, their tournament equity would be $4k (one third of the prize pool), and bubble factor is 2.
Does the fact that one player has half the chips mean he has half the equity?
Player A:
50% chance of winning. Equity $3k
If player A doesn’t win, either player B or C must. Since player A has twice as many chips as the other player in this case, it follows that he will be second twice as often as the other player in this scenario. So 50% of the time, player A doesn’t win, but in two thirds of these cases, he comes second and wins $6K. So one third of the time, he wins $6k, representing tournament equity of $2K.
The rest of the time (one sixth), he finishes third and wins nothing.
Total expectation: $5K
Player B:
25% chance of winning. Equity $1500
50% of the time, player A wins, and half of these times, player B comes second (25%)
25% of the time, player C wins, and only one third of the time does player B then come second (8.33%)
So overall player B comes second one third of the time. Equity $2K
Total expectation: $3500
Player C:
Same as for player B

This illustrates the second key point of ICM:
Chips in smaller stacks are worth proportionally more than chips in a bigger stack (in a stepped payout structure). Doubling your chips late in a tournament does not double your equity

In the above example, player A has 8K in chips and $5K in equity. So each chip in his stack is worth 62.5 cents.
Players B and C have 4K in chips and $3500 in equity. Each chip in their stack is currently worth 87.5 cents.

Now let’s look at bubble factor in this case.
If player B (or C) moves all in and the other shorter stack calls:
The winning player’s equity increases to $6K (gain of $2500)
The loser’s drops to $0 (loss of $3500)
Bubble factor in this case is 3500/2500 (1.4)

If Player A gets all in with either B or C:
If he wins, his equity increases to $6K (gain of $1K)
If he loses, his equity drops to $3500 (loss of $1500)
So player’s A bubble factor in this case is 1500/1000 (1.5)

When player B (or C) gets all in with A:
If he wins, his equity increases to $5K (gain of $1500)
If he loses, his equity drops to $0 (loss of $3500)
His bubble factor in this case is 3500/1500 (2.3333)

This illustrates a third key point of ICM:
When stacks are uneven, players have different bubble factors relative to each other


The following table summarises the situation:

Current chips 8000 4000 4000
8000 X 1.5 1.5
4000 2.3333 X 1.4
4000 2.3333 1.4 X

This illustrates a fourth key point:
Bubble factors of smaller stacks to bigger stacks > bigger stacks to smaller stacks > smaller stacks to other smaller stacks

This example illustrates a number of key strategic concepts:
In these cases, nobody wants to get all in. In this specific case, the shorter stacks need to be a 1.4 to 1 favourite to justify getting it all in against each other, and a 2.3333 to 1 favourite to get all in with the big stack
The big stack doesn’t want to get all in either. Sometimes in these spots bad players who don’t understand ICM make loose calls on the misguided idea that “I can afford to lose the chips and it’s a chance to knock someone out”. To call an all in correctly, the big stack needs to be a 1.5 to 1 favourite
Given a choice, the small stack would prefer to get all in with the other small stack. This suggests that correct short stack strategy is to attack the other short stack and stay out of the way of the big stack
Since players need to believe themselves to be a big favourite before they can correctly call an allin, and since hands that are big favourites are rare three handed, correct strategy is to shove light and call tight. These situations have been compared to “a game of chicken”.
In particular, the short stacks need to be a massive favourite to call a big stack shove, so the big stack can shove pretty liberally

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