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How to calculate the mathematical profit limit
Why consider a "mathematical profit margin" at all
The "mathematical profit margin" is theoretically the maximum average return you can aim for over a long distance under given limits: initial bankroll, risk profile, game variance, betting limits, time and number of sessions. This is not a "how much do you win tomorrow" forecast, but an upper bound that cannot be steadily exceeded without raising the risk of ruin.
In fact, the limit is set by three layers of mathematics:1. Expected return (expected, EV).
2. Risk and spread (variance/volatility, risk of ruin).
3. Limitations (bank, limits, time horizon, rate/withdrawal cap, psychological and operational barriers).
1) Base Quantity - Expectation (EV)
For one bet/round:[
EV = \sum_i p_i \cdot x_i
]where (p_i) - probability of outcome, (x_i) - profit/loss in monetary terms.
If (EV <0) (as in most casino games due to the advantage of the establishment), the theoretical profit limit at a distance is negative: the larger the volume of the game, the closer the actual result is to the minus.
If (EV> 0) (less commonly: bonus arbitration, skewing of coefficients, pricing error), there is a positive limit - but it will be "cut off" by risk and constraints.
Average earnings per N rounds:[
\mathbb{E}[\Pi_N] = N \cdot EV
]However, simply "multiply by N" ignores volatility and the probability of being out of the game before reaching N.
2) Variance, volatility and risk of ruin
The variance determines how widely the results will fluctuate around the EV. For the same (EV), a more volatile strategy requires a smaller leverage (bank share) and gives a lower safe growth rate.
The key practical metric is Risk of Ruin (RoR): the likelihood that a bank will fall to a critical level (for example, to zero or a given "stop level") before your long advantage is realized.
Intuitively: The higher the variance and the more aggressive the bet size, the higher the RoR - and the lower the sustainable profit margin because you're more likely to "fall out."
3) Profit limit through the prism of capital growth (log-criterion)
If the goal is the maximum long-term rate of capital growth, logarithmic utility and Kelly's test are used. For one "small" rate with advantage (e) (expected yield in percentage per dollar) and volatility (\sigma), in independent tests, the marginal growth rate is approximated:[
g \approx \mathbb{E}[\ln(1+R)] \approx e - \frac{\sigma^2}{2}
]where (R) is the yield per round. The maximum is reached at the optimal rate share (f ^) (half-Kelly/Kelly - depending on the form of distribution and your risk).
Kelly criterion (intuitive)
For Bernoullian advantage (e.g. "bet with winning probability (p) and coefficient (b) to 1 "):[
f^=\frac{bp-(1-p)}{b}
]Game meaning: we put the share of the bank proportional to the advantage and inversely proportional to the price of the error.
The profit margin in the log sense is the maximum sustained growth rate achieved at (f ^). Any rate above (f ^) increases the risk of "deep drawdown" and reduces long-term growth (overbetting "eats up" the advantage).
In practice, half-Kelly (0.5 × (f ^)) is often used to reduce volatility and the risk of ruin with almost no loss of growth rate in real, final horizons.
4) Time horizon and "cap" of infrastructure restrictions
Even with (EV> 0) and competent (f ^), your "mathematical ceiling" is cut:- Rate and turnover limits (max rate, frequency, deposit/withdrawal limits).
- Time resource (how many rounds/events you really play out over the period).
- Diminishing advantage over time (market adapts, stocks/bonuses change).
- Psychological limitations (fatigue, erroneous decisions in drawdowns).
Bottom line: real limit = "ideal log limit" × "reachability coefficient," which is often below 1 due to the above.
5) Working methodology for estimating the "mathematical limit"
Suppose you are analyzing a strategy/game and want to get a landmark of the upper bound.
Step 1. Rate EV and variance of one round
Build a table of outcomes: probabilities, payments, costs.
Calculate (EV).
Estimate the variance (\mathrm {Var} (R)) and standard deviation (\sigma) of returns per round.
Step 2. Select a target limit metric
Capital growth rate (log-criterion) - for infinite/long distance and the main goal of "growing as fast as possible."
Expected profit at limited RoR - if it is more important to keep the risk of ruin below a given threshold (for example, <1%).
Step 3. Find the optimal rate share (f)
Use the Kelly formula (or its approximations).
For complex distributions (slots, multi-source rates), a numerical search (f) that maximizes (\mathbb {E} [\ln (1 + f\cdot R)]).
In a practical game, use half Kelly or part of Kelly (⅓- ½) as a compromise between growth and drawdowns.
Step 4. Sustainable growth forecast
With "small" (f): (g\approx f\cdot e -\frac {(f\sigma) ^ 2} {2}).
Maximum (g) at (f = f ^). This is the mathematical limit of sustainable growth without overstated risk.
Step 5. Take into account the limitations and "cap" of the volume
Determine the available amount of rounds per period (time × game speed × limits).
Consider profit cap from rate/payout limits.
Make a degradation benefit (expected rule/stock/pool changes).
Result: annual limit = (g_{\text{ustoychivyy}}) × effective number of growth cycles × reachability ratio (0. 5–0. 9 depending on realities).
6) Earnings cap on negative EV
If (EV <0), no rate progression will create a positive cap. The log-criterion will give a negative growth rate, and the optimal fraction (f ^) tends to zero (that is, not to play).
The only math that raises the "limit" in a minus game is a decrease in turnover (you play less → lose less) or a search for a positive sub-EV within the ecosystem (bonuses, cashback, rakeback, VIP statuses), which turn the general (EV) into non-negative.
7) Practical mini-calculator (paper version)
1. Rate (EV) per 100 bet units: for example, (+ 1. 5%) → (e=0. 015).
2. Rate (\sigma) per round (by session log or from the outcome table). Let (\sigma = 0. 2) (20%).
3. Approximation of the optimal fraction (f ^\approx\frac {e} {\sigma ^ 2} =\frac {0. 015}{0. 04}=0. 375) (37. 5%) - rough, but gives order. Really take ⅓ - ½ from this (12-20%).
4. Rate your annual growth rate: (g\approach f e -\frac {(f\sigma) ^ 2} {2}). At (f = 0. 2):[
g \approx 0. 2\cdot0. 015 - \frac{(0. 2\cdot0. 2)^2}{2} = 0. 003 - \frac{0. 0016}{2} = 0. 003 - 0. 0008 = 0. 0022,(0. 22% )\text {per round}
]Multiply by the number of "independent" rounds per year (taking into account limits and realism) to get a benchmark. If there are 5,000 rounds, the expected log growth ~ (1 - e ^ {-0. 022}\approx 2. 2%) (log-c complex percentage interpretation; actual monetary dynamics will be wider due to variance).
Important: this is a simplification. In slots, the distribution of heavy tails is made real (f ^) lower and requires simulations.
8) Common errors in limit estimation
Ignore variance: read only EV and scale linearly.
Overbetting: putting more Kelly → explosive growth of drawdowns, falling long-term profitability.
Reassessment of outcome independence: Correlated events reduce the effective number of attempts.
Ignoring restrictions: limits of rates/payments, time, cap promo - all this cuts off the "ideal" ceiling.
Survivor bias: Count on "like in the best episode," not the average scenario.
9) The final wording of the "mathematical limit of profit"
The mathematical profit margin for a long-range strategy is the maximum of a sustained rate of capital growth at an acceptable risk of ruin and given constraints. It is defined by:1. sign and value (EV);
2. variance/volatility of results;
3. optimal rate share (Kelly/Kelly share);
4. real limits for the volume of the game and infrastructure.
If (EV\le 0) - there is no "above zero" limit. If (EV> 0), marginal steady-state growth is achieved with a conservative fraction from Kelly, taking into account constraints and correlations.
10) Checklist for practice
Confirm that your total EV ≥ 0 (includes bonuses/cashback/rackback/promotions).
Evaluate (\sigma) and distribution tails (heavy tails → reduce proportion).
Calculate (f ^) and apply Kelly's fraction (⅓ - ½) at the start.
Tightly control RoR and maximum drawdown (DD).
Update the model when rules/limits/market change.
Capture sessions, update scores (EV), (\sigma), (f), and "reachability ratio."
This discipline will make it possible to turn the abstract idea of a "mathematical ceiling" into a working planning tool, keep the risk under control and aim not at one-time good luck, but at a stable, reproducible result.