Expected Value and Likely Profit II: Different Playbooks

'EV' vs. 'EV+LP'

For the Bookmaker, EV adds up. For the Bettor, EV+LP goes forth and multiplies!

In our original post, we introduced Likely Profit (LP) as a complementary metric to Expected Value (EV). To recap clearly:

• Expected Value (EV) measures the average profit per bet. It is an additive metric, suited to the bookmaker's scenario of multiple parallel bets.
• Likely Profit (LP) measures the expected geometric (logarithmic) growth rate of our bankroll. It is a multiplicative metric, reflecting the bettor's sequential reality and limited bankroll.

Mathematically, these metrics are defined as follows:

$$\begin{align} \tag{a} EV = (WB \times P) + (LB \times (1 - P)) - 1 \end{align}$$

$$\begin{align} \tag{b} LP = (WB^{P} \times LB^{(1 - P)}) - 1 \end{align}$$

Where:

• WB (Win-Balance multiplier) is the bankroll multiplier if the bet wins (e.g., $WB = (1 + (F * (O - 1)))$).
• LB (Loss-Balance multiplier) is the bankroll multiplier if the bet loses (e.g., $LB = (1 - F)$).
• F is the stake.
• O is the decimal odds.
• P is the probability of winning.

Bookmaker's Additive World: Why EV is King

Take a small edge from every bet and let volume do the rest.

Bookmakers handle hundreds or thousands of bets simultaneously, each priced with a built-in margin ("vig"), ensuring each bet has a positive expected value from the bookmaker's perspective. With effectively unlimited bankroll and diversified action, bookmakers invoke the law of large numbers, ensuring actual profits converge closely to the expected profits.

In the bookmaker's additive world, EV is the definitive metric because:

• Volume smooths out variance, making EV directly translate into predictable profits.
• Bankroll size is enormous, meaning the bookmaker does not worry about short-term fluctuations from individual outcomes.
• Profitability is guaranteed over time, given a positive EV across many bets.

Thus, bookmakers focus exclusively on EV, setting odds to guarantee their additive advantage. They care about aggregate profit, not about individual outcomes.

Bettor's Sequential Reality: Why EV Needs LP

Maximize bankroll growth, not just average payout.

Now, switch seats to a bettor's perspective. Unlike bookmakers, bettors cannot make thousands of simultaneous bets. Instead, bettors sequentially place bets over time with limited bankrolls. Each bet outcome directly influences future betting capacity, making their reality multiplicative rather than additive. The multiplicative effect means that the order and size of wins and losses matter a lot.

Here is where Likely Profit (LP) becomes essential:

• LP measures the expected geometric (logarithmic) growth rate of our bankroll.
• LP is closely related to the Kelly criterion, well-known in finance and betting for maximizing long-run bankroll growth.
• LP effectively considers that bankroll growth is multiplicative, and thus, variance and bet sizing matter greatly for survivor bias and long-term wealth accumulation.

While EV alone indicates average profit per bet, LP indicates how our bankroll is expected to compound over time. A bet with positive EV but negative LP could lead to significant bankroll volatility, potentially resulting in ruin before the theoretical EV manifests.

Practical Example: EV vs EV+LP in Action

Scenario: You have a $1,000 bankroll and have decided to stake $100 (10% of our bankroll) on one of two bets. Both bets have the same EV (+$10), but differ significantly in risk and profile:

• Bet X (High-risk, high-reward bet):

  • Odds: +1000 (decimal 11.0)
  • Our estimated true probability: 10% (implied odds = 9.1%)
  
  Crucially, LP is negative, indicating expected bankroll shrinkage over repeated bets of this type at this stake fraction.

• Bet Y (Lower-risk, moderate-reward bet):

  • Odds: +100 (decimal 2.0)
  • Our estimated true probability: 55% (implied odds = -122)
  
  LP is positive, indicating expected bankroll growth over repeated sequential bets of this type.

Both bets have identical EV, yet Bet Y clearly outshines Bet X when evaluated holistically using EV+LP. Bet Y's positive LP means it's better suited for sustainable bankroll growth, lower variance, and greater certainty of survival.

This example starkly highlights that as a bettor, we must consider LP as well as EV to intelligently balance profit potential, risk, and bankroll longevity.

Different Goals, Different Metrics

Bookmakers and bettors have fundamentally different goals and constraints:

• Bookmakers operate in an additive world with massive diversity and bankroll, making EV sufficient.
• Bettors face multiplicative outcomes with limited bankrolls, making EV necessary but insufficient. They must also consider LP to ensure survival through inevitable volatility and to maximize bankroll growth.

Thus, the metrics they optimize diverge:

Perspective	Metric	Goal
Bookmaker	EV (additive)	Maximize total profit
Bettor	EV + LP (multiplicative)	Maximize long-term bankroll growth

Playing Correct Perspective

In sum, to succeed as a bettor, we must combine the mathematical rigor of EV identification with the strategic prudence of LP-based bankroll management. Selecting only positive-EV bets is necessary but not sufficient. We must also size our bets to ensure positive LP, aligning our strategy with geometric bankroll growth and survival.

In short:

• EV answers: "How much do I win on average per bet?"
• LP answers: "How will my bankroll realistically grow over many sequential bets?"

A savvy bettor understands both and uses them in tandem, ensuring a profitable journey not just theoretically, but practically.

By clearly understanding these dual metrics, we can meaningfully improve our betting strategy, aligning our actions with our real-world constraints and maximizing our probability of success in the long run.

Enjoy!

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Note: The final draft of this post was sanity checked by ChatGPT.

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 multiplier (WB) and the Loss-Balance multiplier (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.