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Examining tournament EV

 In the bubble article of the MTT course, we encountered the following example hands.
 
 
Let's have a closer look at the mathematics behind the first interactive example. It is not necessary for everyone to read this, but it gives an indication of how advanced players will approach the decision and calculate the merits of each play.
 
To recap: you are on the bubble of the Sunday Million and the aggressive big stack moves all in from the button with 100,000 chips. Blinds are 1,500/3,000 and you have to decide if you want to call or to fold in the big blind with pocket nines and a 35,000 stack. 
 
We will try to calculate how much each of your decisions is worth in EV terms. EV stands for "expected value" and it is the attempt to determine how many chips/how much money you would expect to gain on average if the same situation occurred over and over again.
 
FOLD 
 
If you fold you will have 32,000 chips left. It is hard to say how much this stack would be worth exactly, but there are some clues. 
 
Firstly, you will almost certainly get into the money if you play the next couple hands carefully. The smallest prize is worth about $300, so that is the minimum your stack is worth. 
 
You also have a chance to reach one of the higher spots to win more money. At the beginning of the tournament, 10,000 chips are worth $200 (the buy-in). After the bubble, about 30 per cent of the prize money is distributed already (15 per cent of the players at least double their money). The other 70 per cent will be determined in the end-game of the tournament. 
 
A decent guess is that this extra chance is worth about $500. That means 10,000 chips will be worth about $140 on average after the bubble - not counting the $300 every player will get after the bubble no matter what. 
 
You have 32,000 left which is 3.2 times 10,000. So your chips are worth about 3.2*$140 = $448.
 
Therefore:
 
Your total EV = [guaranteed prize for making it to the money] + [worth of your stack]
= $300 + $448 = $748
 
That means you will win about $750 on average if you fold.
 
CALL
 
There are two possible results if you call: you either win or lose. (Let's not think about split pots. It is very unlikely.) So we need to examine your EV for either of these possibilities.
 
EV if you call and lose - That's easy. If you lose you are out and won't win anything. Therefore your EV = 0
 
EV if you call and win - Now you would have about 70,000 chips. This is easy to calculate because you will have almost exactly an average stack. In real money terms, the average stack is worth the price pool divided by the number of players left in the tournament. 
 
So your EV if you call and win = $1,400.
 
The total EV of a call (factoring in the potential of winning and losing the pot) is the EV if you win multiplied with the probability to win plus the EV if you lose multiplied with the probability you lose. 
 
Here it is written as a formula:
 
EV call = equity * [EV call and win] + (1-equity) * [EV call and lose]
= (65% * $1400) + (35% * 0)
= (65% * $1400)
= $910
 
So the total EV of calling equals $910.
 
SUMMARY:
 
The EV of calling is about $910 and the EV of folding is about $750. That means your EV is about $160 larger if you call. That is a huge difference. Therefore you should call and a fold would be a costly mistake.
 
If you make slightly different assumptions about the hands your opponent will go all in with, and about the value of your stack if you fold, your results will be slightly different. 
 
But in the end you should reach the same conclusion, which is that a call is far superior to a fold.
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