Expected Value

Expected Value (EV) is the outcome multiplied by the probability of that outcome. For example if a bet pays off 10 dollars 30% of the time, the EV of that bet is $10\ast .03=3$. Expected Value is often used to evaluate actions in poker. Often this is expressed in terms of winning or losing:

$$EV=(Xw\ast Rw)+(1-Rw)\ast Xl$$

Where:

• $Xw$ equals win value
• $Rw$ is win rate
• Lose rate $R_l=(1-Rw)$
• $Xl$ = Lose value.

In a scenario where we win $200 30% of the time and lose 95 the rest of the time:

$$\begin{array}{c}EV=(200\ast .3)+(-95\ast .7)\\ EV=60+-66.5\\ EV=-6.5\end{array}$$

We expect to lose $6.5 every time this situation occurs, this is a bad play.

For a more complicated example suppose I am last to act on the river in Texas hold ’em, I am heads up against an opponent that I think has me beat 60% of the time. If I raise 30 dollars into a $100 pot they fold 30% of the time, let’s consider:

Assume there are two actions for the opponent and we are ignoring that they could raise. This means that EV is equal to the value of each of these actions at their equivalent rate: 30% folding and %70 calling. The total EV is the sum of these two outcomes:

$$EV=FV+CV$$

Where FV is the expected value when the opponent folds and CV is the expected value when they call. Fold value is straight forward, it happens 30% of the time and we get the pot ($100) .

$$\begin{array}{cc}FV=& R_f\ast V_f\\ FV=& .3\ast 100\\ FV=& 30\end{array}$$

So we expect to get $30 from the opponent folding. For the expected value of a call by the opponent we have to consider the scenarios of them winning and losing

$$\begin{array}{cc}CV=& R_c\ast (V_w+V_l)\\ CV=& R_c\ast ((R_w\ast V_w)+((1-R_w)\ast V_l))\end{array}$$

As stated above we expect to win 40% of the time, and we will win the pot and the opponent’s call $100+30=130$ . 60% during a call, we lose our bet of $30. So:

$$\begin{array}{cc}CV=& R_c\ast ((R_w\ast V_w)+((1-R_w)\ast V_l))\\ CV=& R_c\ast ((.4\ast 130)+(.6\ast -30)))\\ CV=& .7\ast (52-18)\end{array}$$

$30+23.8=53.8$

I expect this raise to have an EV of $53.8 dollars. Lets consider the same situation with a $100 dollar raise where the opponent folds 60 percent of the time:

$$\begin{array}{cc}CV=& R_c\ast ((R_w\ast V_w)+((1-R_w)\ast V_l))\\ CV=& R_c\ast ((.4\ast 200)+(.6\ast -100)))\\ CV=& .4\ast (80-60)\end{array}$$

$60+.4(20)=68$

The $100 raise is EV $68 dollars. But what if the opponent was equally as likely to fold for $30 as $100 (i.e. 30 %).

$EV=30+.7(20)=44$

The $100 bet is now a much worse option, but still expected to win money.

Weak points of Expected Value

Expected value is useful for decision making when you will encounter a situation many times and so variance will be smoothed out. This is why it is useful for poker where you will encounter many similar spots. However when you will only encounter a spot or decision once, for example whether to take a job or start a company or making a one-time investment, EV can be a poor measure because your sample size is so small. Jason Cohen poses a scenario that illustrates this well¹:

Investment A

Invest 50k, with a 99% guaranteed profit of million dollars in 1 year.

$$\begin{array}{cc}EV=& (.99\ast 1,000,000)+(.01\ast -50,000)\\ EV=& 989,500\end{array}$$

Investment B Invest 50k, with a 10.3432% pay out of 10 million dollars in 1 year.

$$\begin{array}{cc}EV=& 0.103433\ast 10,000,000+.0.896567\ast -50000)\\ EV=& 989,502\end{array}$$

These two investments are roughly equivalent in EV value (in fact investment B is slightly favorable). However, you don’t get to run this investment scenario 10 or a 100 or 1000 times, you get to run it once and choosing option A is far preferable because 99% of the time you are walking with 1M whereas 90% of the time in option B you are losing 50k.

1. Cohen, J. Deciding whether an investment is worthwhile. https://longform.asmartbear.com/investment/ (2024).