Expected Value (EV) and Likely Profit (LP)

In betting analysis, there is a strong emphasis on only selecting Value Bets. To this end, we are advised to calculate the Expected Value (EV) of a proposed bet and, if the result is positive, then we have a potential value bet. However, this approach is very short-sighted as outlined below. This will lead us to add an additional metric - Likely Profit (LP) - and the Dual-Metric Decision Algorithm (DMDA).

We will focus on this canonical betting example:

Parameter	Value
Initial Bankroll (B)	$10000
Markets (M)	1
Decimal Odds (O)	1.9091
Win Probability (P)	55.00%
Stake Fraction (F)	1.00%

First, let us calculate the Win Balance (WB) and the Loss Balance (LB) in percentage terms. WB is the state of the bankroll after a winning bet and LB is the state of the bankroll after a losing bet. WB and LB are calculated as follows:

$$\begin{align} \tag{1} \mathit{WB} = (1 + (F * (O - 1))) \\ \tag{2} LB = (1 - F) \end{align}$$

The $EV_{per-unit-staked}$ is equal to:

$$\begin{align} \tag{3} EV = (\mathit{WB} * P) + (LB * (1 - P)) - 1 \end{align}$$

EV represents the average profit or loss per unit staked over a large number of bets, assuming the same odds and probability hold true.

But, to evaluate the bet in terms of our specific circumstances, we need an additional metric - Likely Profit (LP).

The $LP_{per-unit-staked}$ is equal to:

$$\begin{align} \tag{4} LP = (\mathit{WB}^{P} * LB^{(1 - p)}) - 1 \end{align}$$

LP represents the expected growth rate of the bankroll over a series of bets, assuming the same odds and probability hold true. It takes into account the compounding effect of wins and losses.

Also, the $Bankroll_{EV}$ and the $Bankroll_{LP}$ are equivalently:

$$\begin{align} \tag{5} B_{EV} = (1 + EV)^{M} * B \\ \tag{6} B_{LP} = (1 + LP)^{M} * B \end{align}$$

Note: Likely Profit (LP) is equivalent to expected bankroll growth!

This leads naturally to our Dual Metric Decision Algorithm (DMDA), which is best exemplified with the following Python snippet:

if ev_per_unit > 0 and lp_per_unit > 0:
    decision = 'Favorable bet; consider proceeding.'
elif ev_per_unit > 0 and lp_per_unit <= 0:
    decision = 'Positive EV but negative LP; reconsider stake size.'
else:
    decision = 'Negative EV; generally avoid this bet.'

Returning to our example above, we can calculate the various metrics as follows:

• WB

$$\begin{align} \tag{7a} \mathit{WB} = (1 + (F * (O - 1))) \\ \tag{7b} \mathit{WB} = (1.00 + (0.01 * (1.9091 - 1.00))) \\ \tag{7c} \mathit{WB} = 1.009091 \\ \end{align}$$

• LB

$$\begin{align} \tag{8a} LB = (1 - F) \\ \tag{8b} LB = (1.00 - 0.01) \\ \tag{8c} LB = 0.99 \\ \end{align}$$

• EV

$$\begin{align} \tag{9a} EV = (\mathit{WB} * P) + (LB * (1 - P)) - 1 \\ \tag{9b} EV = (1.009091 * 0.55) + (0.99 * (1.00 - 0.55)) - 1.00 \\ \tag{9c} EV = 0.0004995 \\ \end{align}$$

• LP

$$\begin{align} \tag{10a} LP = (\mathit{WB}^{P} * LB^{(1 - p)}) - 1 \\ \tag{10b} LP = (1.009091^{0.55} * 0.99^{(1.00 - 0.55)}) - 1.00 \\ \tag{10c} LP = 0.0004524 \\ \end{align}$$

• B_EV

$$\begin{align} \tag{11a} B_{EV} = (1 + EV)^{M} * B \\ \tag{11b} B_{EV} = (1.00 + 0.0004995)^{1} * 10000 \\ \tag{11c} B_{EV} = 10004.995 \\ \end{align}$$

• B_LP

$$\begin{align} \tag{12a} B_{LP} = (1 + LP)^{M} * B \\ \tag{12b} B_{LP} = (1.00 + 0.0004524)^{1} * 10000 \\ \tag{12c} B_{LP} = 10004.524 \\ \end{align}$$

Decision: Since both EV and LP per unit are positive, this is a favorable bet.

While the traditional EV approach can be useful for identifying potential value bets, it fails to capture the full picture when considering the long-term impact of betting decisions. By introducing Likely Profit (LP) as a complementary metric, we can gain a more comprehensive understanding of the expected bankroll growth over a series of bets. This dual-metric approach allows for a more nuanced decision-making process, enabling bettors to make informed choices that align with their individual risk profiles and long-term goals. However, the DMDA is a simplified approach and should be further refined. What is presented here is a starting point for this approach, and further research and development are needed to refine and optimize this framework for practical application in real-world betting scenarios.