### FAQ: Expected Value (EV) Graphs

Posted:

**Tue Nov 30, 2010 10:57 am**What is All-in EV?

EV stands for "Expected Value". In situations where you or your opponents are all-in, we can calculate what the result would be if no more cards were to be dealt and instead each player was returned their equity in the pot. Your All-in EV is calculated by subtracting the total amount you bet from your equity in the pot.

What is equity in the pot?

Equity in the pot is the percentage you expect to win the pot multiplied by the size of the pot. This will tell you how much of the pot you would get if the cards were run an infinite number of times. Note that if players are all-in on the river or one player is guaranteed to win, the players' equity and actual results are the same. Let us consider the following example. Hero has AA and Villain has KK, and Hero has Villain's suits dominated. Hero and Villain are both all-in before the flop for $100 each, so the total pot is $200 (we will assume the blinds account for the rake in this example to keep the math simple). In this case, Hero will win 82.6% of the time. Thus Hero's equity in the pot is $200 times 82.6% or $165.20 and Villain's equity in the pot is $34.80 ($200 * 17.4%).

What does the graph show?

In the above example, Hero and Villain both put $100 into the pot. Thus Hero's All-in EV is +$65.20 ($165.20 - $100), and Villain's All-in EV is -$65.20 ($34.80 - $100). If Hero wins the hand (Figure 1), his Amount Won for the hand will increase by $100, but his Net Expected Won line will only increase by $65.20. If Hero loses the hand (Figure 2), his Amount Won will decrease by $100, but his Net Expected Won line will not change.

When in the hand are these calculations performed?

These calculations are performed at the point of the all-in. If, in our above example, the all-in did not occur until the flop and the flop was K72, Hero would then expect to win 8.6% of the time instead of 82.6%, changing the results drastically.

What happens in multi-way all-in pots?

If all players have the same stack size, it is calculated as above for each player. If the players have differing stack sizes, and that creates multiple all-in pots, each pot's equity is calculated separately. First we calculate for the main pot for all players who are all-in for that pot, then we calculate the first side pot for those who are all-in for that pot, and so on. Imagine we have an all-in pot with three players, A, B and C. The pot is all-in preflop. A and B have 100 big blinds and C has 20 big blinds. A holds AK, B holds QQ and C holds JJ. In this case we have two pots, one pot with 60 big blinds in it that all players can win, and one pot with 160 big blinds in it that only A or B can win. In the small pot, A is 36.6% to win, B is 46.2% to win, and C is 17.2% to win. So we can calculate C's All-in EV directly: 17.2% times 60 big blinds is 10.32 big blinds, subtracted from 20 big blinds is -9.68 big blinds (17.2% * 60 - 20). For A and B we must look at the other pot. In the side pot A is 44.2% to win and B is 55.8% to win. So A's equity in the first pot is 60 * 36.6% (21.96), and his equity in the second pot is 160 * 44.2% (70.72), so his EV is 21.96 + 70.72 - 100, or -7.32 big blinds. B's All-in EV is 60 * 46.2% + 160 * 55.8% - 100, or 17.

So how do I read the All-in EV graph?

The All-in EV graph shows your normal results for hands that were not all-in but for all-in hands it shows your All-in EV instead of your actual results. The all-in line will differ from your actual results. If you won the example hand, your Amount Won (or green) line will be 34.8 big blinds higher than your Net Expected Won (or gold) line, since you won 100 bbs but expected to win only 65.2. If you lost the example hand your green line would be 165.2 big blinds lower than your gold line because you lost 100bbs but expected to win 65.2.

What happens in the long run?

In the long run, the Amount Won and Net Expected Won lines should end up being fairly close together* since your skill will eventually dominate the luck portion of the game. However, the long run can take an extremely long time – millions of hands or more, depending on your style of play, especially if you play with multiple stack sizes.

*This assumes a constant winrate and standard deviation over time. The remaining distance between the lines will be dictated by the mathematical confines of the standard deviation.

EV stands for "Expected Value". In situations where you or your opponents are all-in, we can calculate what the result would be if no more cards were to be dealt and instead each player was returned their equity in the pot. Your All-in EV is calculated by subtracting the total amount you bet from your equity in the pot.

What is equity in the pot?

Equity in the pot is the percentage you expect to win the pot multiplied by the size of the pot. This will tell you how much of the pot you would get if the cards were run an infinite number of times. Note that if players are all-in on the river or one player is guaranteed to win, the players' equity and actual results are the same. Let us consider the following example. Hero has AA and Villain has KK, and Hero has Villain's suits dominated. Hero and Villain are both all-in before the flop for $100 each, so the total pot is $200 (we will assume the blinds account for the rake in this example to keep the math simple). In this case, Hero will win 82.6% of the time. Thus Hero's equity in the pot is $200 times 82.6% or $165.20 and Villain's equity in the pot is $34.80 ($200 * 17.4%).

What does the graph show?

In the above example, Hero and Villain both put $100 into the pot. Thus Hero's All-in EV is +$65.20 ($165.20 - $100), and Villain's All-in EV is -$65.20 ($34.80 - $100). If Hero wins the hand (Figure 1), his Amount Won for the hand will increase by $100, but his Net Expected Won line will only increase by $65.20. If Hero loses the hand (Figure 2), his Amount Won will decrease by $100, but his Net Expected Won line will not change.

When in the hand are these calculations performed?

These calculations are performed at the point of the all-in. If, in our above example, the all-in did not occur until the flop and the flop was K72, Hero would then expect to win 8.6% of the time instead of 82.6%, changing the results drastically.

What happens in multi-way all-in pots?

If all players have the same stack size, it is calculated as above for each player. If the players have differing stack sizes, and that creates multiple all-in pots, each pot's equity is calculated separately. First we calculate for the main pot for all players who are all-in for that pot, then we calculate the first side pot for those who are all-in for that pot, and so on. Imagine we have an all-in pot with three players, A, B and C. The pot is all-in preflop. A and B have 100 big blinds and C has 20 big blinds. A holds AK, B holds QQ and C holds JJ. In this case we have two pots, one pot with 60 big blinds in it that all players can win, and one pot with 160 big blinds in it that only A or B can win. In the small pot, A is 36.6% to win, B is 46.2% to win, and C is 17.2% to win. So we can calculate C's All-in EV directly: 17.2% times 60 big blinds is 10.32 big blinds, subtracted from 20 big blinds is -9.68 big blinds (17.2% * 60 - 20). For A and B we must look at the other pot. In the side pot A is 44.2% to win and B is 55.8% to win. So A's equity in the first pot is 60 * 36.6% (21.96), and his equity in the second pot is 160 * 44.2% (70.72), so his EV is 21.96 + 70.72 - 100, or -7.32 big blinds. B's All-in EV is 60 * 46.2% + 160 * 55.8% - 100, or 17.

So how do I read the All-in EV graph?

The All-in EV graph shows your normal results for hands that were not all-in but for all-in hands it shows your All-in EV instead of your actual results. The all-in line will differ from your actual results. If you won the example hand, your Amount Won (or green) line will be 34.8 big blinds higher than your Net Expected Won (or gold) line, since you won 100 bbs but expected to win only 65.2. If you lost the example hand your green line would be 165.2 big blinds lower than your gold line because you lost 100bbs but expected to win 65.2.

What happens in the long run?

In the long run, the Amount Won and Net Expected Won lines should end up being fairly close together* since your skill will eventually dominate the luck portion of the game. However, the long run can take an extremely long time – millions of hands or more, depending on your style of play, especially if you play with multiple stack sizes.

*This assumes a constant winrate and standard deviation over time. The remaining distance between the lines will be dictated by the mathematical confines of the standard deviation.