Risk-of-Ruin Max Stake Calculator — Ceiling, Not Target

From a Stylised Trade to a Real Session

In the TMDA post we asked whether one repeated wager is profitable, sustainable, and survivable, and found the largest stake fraction whose probability of a 50% drawdown inside 2,300 bets stayed under 5%. Clean inputs, clean answer.

A real exchange session is not like that. There is no single win probability and no single price. What there is is a history: a spreadsheet of every bet, with its stake and the market's net result. This post turns the survivability question loose on such a history and asks: what is the most I can put at risk in one market without a realistic chance, over time, of halving the starting bank?

Conservative ceiling

0.52%

€52.13 per market on a €10,000 bank

Basis

330 markets

5,880 bets · racing Win markets only

Observed return

+5.77%

€913.27 on €15,824.53 liability

Actual mean liability

€47.95

median €43.00 · max €252.98

Uncomfortable finding: the trader's average liability over the period (€47.95) sat within 8% of the ceiling the history itself implies (€52.13). The largest single-market liability was 4.9× over it. The session was profitable, but it was not being run comfortably under its own survivability limit — it was being run at it, with excursions well above.

The Question TMDA Left Open

So the question becomes: given this history, and a bank I refuse to see halved more than one time in twenty, what is the maximum liability I should carry into any one market?

Key Insight: the answer is a ceiling, not a target. It says where survivability breaks down, not where growth is fastest. Kelly asks the growth question and accepts a one-in-two chance of ever halving the bank; this asks for one-in-twenty, so the ceiling sits well below the Kelly fraction (roughly m / v, about 1.6% on this session).

The Model: Fixed-Stake Ruin

Treat the bank as a random walk with drift. Each market adds a result drawn from the same distribution: mean m and variance v per unit of liability. Stake a fixed fraction s of the starting bank in every market. The classical continuous-time result (Feller, Ross) gives the probability that the walk ever falls by D before drifting away:

RoR = exp( −2 · m · D / (s · v) )
s_max = 2 · m · D / ( v · ln(1/r) )

Symbol	Meaning	Value
s_max	Maximum liability per market, as a fraction of the starting bank. Held fixed for a session; recalculated at session end.	answer
m	Mean market result per unit of liability	history
v	Variance of those results	history
D	Drawdown that counts as ruin, from the starting balance	50%
r	Accepted probability of ever hitting that drawdown	5% → ln(1/r) = 2.9957

Two properties matter for what follows. First, s_max is linear in m and inversely proportional to v: how you measure the mean and variance from history decides the number. Second, if m ≤ 0 the ceiling is zero — no positive stake is safe without an edge, and the formula says so bluntly.

Attribution: this expression is often called the “Mason Malmuth formula” in poker circles. That attribution has been publicly retracted by Malmuth himself. It is the standard gambler's-ruin result for Brownian motion with drift and has no single eponym; cite Feller (1971, Vol. 2, Ch. XIV) or Ross (1996). Chen & Ankenman (2006, Ch. 22) give the poker-facing derivation.

The Data: An Exchange Report

The input is the exchange's own market P&L report, one row per bet: Date, Market, Ref No, Selection, Odds, Stake, Bet P&L, Market P&L, Net P&L. The file covers 2 March to 30 April 2026. A market is one betting event; its liability is the sum of stakes placed into it, and its result is the net P&L the exchange reports for it after commission. The calculator keeps racing Win markets only — other market types are other strategies and are sized separately — and drops any market whose recorded liability cannot bound its loss.

Quantity	Win markets only
Bets (rows)	5,880
Markets	330
Total liability	€15,824.53
Total net P&L	€913.27
Overall return on liability	5.77%
Bets per market (mean)	17.8

The result per market is expressed as return on liability: net P&L divided by liability. A total loss is −1.00; scratching is 0.00; the best Win market in the period (€26 liability, €300.54 net) is +11.56. These ratios are the raw material for m and v.

Three Ways to Read the Same History

The formula wants a mean and a variance. There are at least three defensible ways to extract them from 330 market results, and they disagree by a factor of 3.3. Working through them is the point of the exercise.

Method 1 — Wins and Losses

The spreadsheet-trader's instinct: count profitable markets, compute a strike rate p, take the average odds of the winning selections O, and treat every market as a coin that pays O − 1 or loses 1.

p = 108 / 330 = 32.73%    O = 25.00 (mean odds of winning selections)
m = p · (O − 1) + (1 − p) · (−1) = 0.3273 × 24.00 − 0.6727 = 7.182
v = p · (24 − m)² + (1 − p) · (−1 − m)² = 137.60
s_max = 2 × 7.182 × 0.50 / (137.60 × 2.9957) = 1.74% = €174.22

Read m = 7.182 again. It claims the average market returns seven times its liability. The session's actual return was 5.77%. The method has silently assumed each win puts the whole liability on one selection at 25.0, when in fact the trader spreads eighteen bets across the field and a winning market returns about 2.6× liability, not 24×. Both m and v are inflated by orders of magnitude; the ratio happens to net out to a stake 2.5× too generous.

Why it flatters: Wins-and-Losses discards outcome size. It cannot distinguish a session of small wins and rare catastrophic losses from a session of large wins and modest losses, so long as the counts match. Prefer the size-weighted figures.

Method 2 — Profit and Loss (banded)

The spreadsheet approach: sort each market's return-on-liability into five bands, weight each band by how often it occurs and by its mean result. Sizes now survive, but each band is collapsed onto its mean.

Why these five bands, and not some other five: the cut points are not tuned to the data; they are the three values of return-on-liability that mean something to a trader. −1 is a total loss, the floor for anyone backing — you cannot lose more than you staked. 0 is a scratch: liability committed, nothing won or lost. +1 is the point where a market returns more than the liability that was risked in it. Those three points partition the number line into exactly five regions — total loss, partial loss (reds), scratch, modest win (greens), and outsized win — which is also how a trader describes a day. The regions do not overlap and, between them, cover every result a backer can record, so each market lands in exactly one band and the probabilities must sum to 100% by construction. When the checksum reads anything else, the bands are not what is wrong; the data is.

Band	Meaning	Count	Probability	Mean result	Contribution to m
> 1	returned more than the liability	80	24.24%	+258.72%	+0.62720
0 .. 1	modest wins	28	8.48%	+62.80%	+0.05329
= 0	scratched	9	2.73%	0.00%	0.00000
−1 .. 0	partial losses	13	3.94%	−66.86%	−0.02634
= −1	total losses	200	60.61%	−100.00%	−0.60606
Total		330	100.00%		m = +0.04809

m = Σ p_i · x_i = 0.04809    v = Σ p_i · (x_i − m)² = 2.2775
s_max = 2 × 0.04809 × 0.50 / (2.2775 × 2.9957) = 0.70% = €70.49

Now m is a believable 4.8% per unit of liability, in line with the observed 5.77% return. Three in five markets are total losses; one in four returns more than 2.5×. This is what a dutching session at long prices looks like, and the stake falls to 0.70%.

Method 3 — Raw Outcomes

Drop the bands. Weight all 330 market results equally at their own return on liability and compute the moments directly.

m = mean(x_i) = 0.04809    v = var(x_i) = 3.0796
s_max = 2 × 0.04809 × 0.50 / (3.0796 × 2.9957) = 0.52% = €52.13

The mean is identical to Method 2 — a probability-weighted mean does not care how finely you partition the outcomes. The variance is 35% higher. Banding replaced eighty individual wins ranging from +1.0 to +11.6 with a single value of +2.59, and two hundred losses with exactly −1; it threw away the spread within each band. The true per-market variance includes that spread, so the Raw stake is never higher than the banded one, and here it is a quarter lower.

Raw Outcomes is the honest method. It makes no distributional assumption beyond “the future draws from the same urn as the past”, it loses no information, and it is the one to act on.

Comparison and the Conservative Ceiling

Method	m	v	s_max	Cash	Verdict
Wins and Losses	7.18179	137.602	1.74%	€174.22	discards size
Profit and Loss (banded)	0.04809	2.278	0.70%	€70.49	understates v
Raw Outcomes	0.04809	3.080	0.52%	€52.13	ceiling

Wins and Losses

1.74% · €174.22

Profit and Loss

0.70% · €70.49

Raw Outcomes

0.52% · €52.13

Wins-and-Losses is 2.5× the banded figure; banded is 1.4× Raw. The calculator reports the minimum of the three as the conservative ceiling. In practice that is always Raw, and the other two are there so you can see how much the shortcut methods would have over-staked you.

Ceiling Versus What Was Actually Staked

The per-market detail the script writes alongside the report makes one more comparison possible: the ceiling against the liabilities the trader actually carried.

Ceiling (Raw, s_max)

€52.13

Median liability

€43.00

Mean liability

€47.95

Largest liability

€252.98

The median market was staked at 82% of the ceiling; the mean market at 92% of it; the largest at 485%. Caveat 1 says to stay comfortably underneath. This session was sitting close to the line and periodically jumping over it. It was profitable in the period — but with a 5% lifetime chance of halving at s_max, and materially worse odds on the €250 markets, “profitable in the period” is exactly what the 95% branch is supposed to look like.

Actionable: cap per-market liability at the ceiling and treat anything above two-thirds of it as needing a reason. On this bank that is roughly €35–€52 per market, recalculated at the end of each session (week or month) from the updated history.

Interactive Dashboard

To explore the ceiling across different parameters, we have created an interactive dashboard where you can:

• Load the three measured (m, v) pairs from this session, or type your own from your report
• Move the bank, the drawdown threshold and the risk tolerance
• See s_max, the cash liability, and the round-trip RoR check update live
• Read the ruin probability at multiples of the ceiling, with an Accept / Reduce / Avoid verdict

(Opens in a new window; allow popups if prompted)

How This Relates to TMDA

	TMDA (Feb 2026)	This calculator
Input	One wager: P, O	A history: 330 market results
Staking	Fixed fraction of current bank (compounding)	Fixed fraction of starting bank per market (fixed stakes)
Space	Log-returns: μ, σ	Arithmetic P&L per unit liability: m, v
Horizon	Finite: n bets	Infinite: “ever”
Question	Is this stake survivable over n bets?	What is the largest stake that is survivable at all?
Decision	Accept / Reduce / Avoid	A ceiling; m ≤ 0 ⇒ ceiling is zero (Avoid)

They are the same diffusion, asked two different questions. TMDA's finite horizon makes it less pessimistic than the “ever” version here; its compounding makes ruin in the strict sense impossible and so it measures drawdown instead, which is the same thing this calculator does with D.

The TMDA post's closing section on asymmetric payoffs is directly relevant. Whelan (2025) showed the diffusion approximation understates ruin when wins are rare and large. The Win session here has a 33% strike rate and winning selections at average odds of 25 — an asymmetry ratio of order K ≈ 24, far beyond the K = 9 example where diffusion missed by ten percentage points. So the 5% in “5% chance of halving at s_max” is best read as a lower bound; the discrete-chain figure will be materially higher. One more reason the number is a ceiling to stay under, not a target to reach.

Caveats

These travel with every result the script prints. Numbered so they can be cited.

1. Treat s_max as a ceiling to stay comfortably underneath, not a target to reach.
2. Asks whether you will ever hit the drawdown, not whether you will hit it in the next N markets. Pessimistic on that axis.
3. Measures drawdown from the starting balance, not the running peak, which flatters the result — on balance the number errs generous.
4. Assumes the reported history represents the session you are about to trade.
5. Assumes fixed stakes: no resizing or moving down as the bank moves.
6. Treats an observed edge as the true edge; optimistic on small samples. 330 markets is not many for a 33% strike rate at long prices.
7. Continuous approximation of discrete outcomes. With K ≈ 24 the approximation understates ruin (Whelan 2025).
8. Wins-and-Losses discards outcome size, so it flatters a session of small wins and rare large losses. Prefer the size-weighted figures.
9. Profit-and-Loss collapses each band onto its mean, understating variance. Raw Outcomes is the honest one; prefer that.
10. If m ≤ 0, s_max is 0%: no positive stake is safe without an edge.

Bottom Line: TMDA told you whether a bet is survivable. This tells you, from your own record, how big a bet could be before it stops being survivable. On this session the answer is under half a percent of the bank per market — and the session was already there.

References

• Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. 2, Ch. XIV. Wiley.
• Ross, S. M. (1996). Stochastic Processes, 2nd ed. Wiley.
• Chen, B. & Ankenman, J. (2006). The Mathematics of Poker, Ch. 22. ConJelCo.
• Whelan, K. (2025). “Ruin Probabilities for Strategies with Asymmetric Risk.” University College Dublin. [PDF]
• matekus (2026). TMDA — Profitable, Sustainable, Survivable.

Claude used for the final sanity check of the analysis and for the complete generation of the visual theme.

EV-LP Review And Update

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Expected Value tells us whether an edge exists. Likely Profit tells us whether our bankroll can sustain the way we choose to bet it.

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Sports bettors often ask the question: Is this bet profitable (+EV) on average? This is the right place to start, but it is not the right place to finish.

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In three earlier posts, we developed our approach incrementally:

• Pairing Expected Value (EV) with Likely Profit (LP).
• Separating the Bookmaker (Additive) and Bettor (Multiplicative) Playbooks.
• Converting both EV and LP into practical probability thresholds.

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This review follows that same progression, from finding value to protecting growth.

Starting Point: Expected Value and Likely Profit

Suppose a bettor risks a fraction \(F\) of their current bankroll at decimal odds \(O\), with an estimated win probability \(P\).

The bankroll has two possible one-bet multipliers:

$$ WB = 1 + F(O-1) $$

$$ LB = 1-F $$

Here, \(WB\) is the Win-Balance multiplier and \(LB\) is the Loss-Balance multiplier. Decimal odds include the returned stake, so the winning net profit is \(F(O-1)\), not \(FO\).

The original equations deliberately use the same outcomes and probability weights:

$$ EV = (WB*P) + (LB*(1-P)) - 1 $$

$$ LP = (WB^P * LB^{1-P}) - 1 $$

This parallel structure is central to the framework. EV takes the probability-weighted arithmetic mean of the two bankroll outcomes; LP takes their probability-weighted geometric mean. The inputs are symmetrical in form, while the different averaging operation is precisely what makes LP more sensitive to losses and stake size.

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Expected Value: Is There An Edge?

The EV equation simplifies to:

$$ EV = F(PO-1) $$

Therefore, \(EV>0\) means that the bettor's estimated probability is high enough to make the offered odds profitable on average.

EV is the arithmetic view. It answers:

How much should this bet return on average?

Likely Profit: Does The Stake Imply Positive Geometric Growth?

The companion LP equation can equivalently be written as an expected log-growth rate:

$$ g = P*\ln(WB) + (1-P)*\ln(LB) $$

Because \(LP=e^g-1\), \(LP\) and \(g\) always have the same sign. A positive LP indicates positive long-run geometric growth under repeated bets with the same probability, odds, and bankroll fraction.

LP is not the probability of finishing ahead over a fixed number of bets, and it does not describe the path, drawdowns, or risk of ruin. It is a transformed expected log-growth measure whose long-run interpretation assumes that the estimated probabilities remain valid and that repeated outcomes behave consistently with them.

LP is the compounding view. It answers:

Does this stake imply positive geometric growth under the model?

That distinction produced the original decision rule:

EV	LP	Interpretation
Positive	Positive	The edge and stake size support bankroll growth
Positive	Zero or negative	The bet may have value, but the stake is too aggressive
Zero or negative	Any	No estimated edge at the offered odds

Positive EV is necessary. Positive LP asks whether the bettor is using that edge to generate bankroll growth.

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Two Playbooks

The second stage of the framework explained why EV can be enough for one side of the market but incomplete for the other.

Bookmaker's Additive World

At the level of a broad, well-capitalized, and actively risk-managed book, bookmakers seek to collect small pricing advantages across many bets. Diversification and volume can make aggregate results more predictable, although correlated liabilities, model error, and capital constraints still require risk management.

Their playbook is therefore additive:

Take a small edge repeatedly and let volume accumulate the profit.

For pricing and aggregate margin, EV is a central metric. Individual outcomes matter less than the average margin across the full book, but they do not cease to matter for liability management.

Bettor's Multiplicative World

A bettor faces a different constraint: finite capital. A loss reduces the base available for the next stake, while a win increases it. When stakes are fractions of the current bankroll, returns compound multiplicatively.

That makes sizing inseparable from selection. Two bets can offer the same +EV and still have very different consequences for long-run growth (-LP).

Consider a $1,000 bankroll with a $100 stake:

	Bet X: High Risk	Bet Y: Moderate Risk
Decimal odds	11.0	2.0
Estimated win probability	10%	55%
Expected Value (EV)	+$10	+$10
Win-Balance multiplier	1.9	1.1
Loss-Balance multiplier	0.9	0.9
Likely Profit (LP)	-3.02%	+0.50%

Correction to the earlier post: Recalculation gives \(LP_X=-3.0174\%\) and \(LP_Y=+0.5021\%\). These replace the previously published values of \(-1.32\%\) and \(+0.27\%\).

Both bets have identical positive EV. Bet X, however, combines a low hit rate with a 10% bankroll stake. Its negative LP says that repeatedly taking this risk at that size would shrink the typical compounded bankroll. Bet Y clears both tests.

This is the bettor's playbook:

Find positive EV, then size the bet so that expected bankroll growth remains positive.

LP does not replace EV, and it does not turn a poor probability estimate into a good one. It simply exposes the interaction between edge, odds, and stake size.

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From Metrics to Probability Thresholds

The third stage made the process directly actionable. Instead of calculating EV and LP after choosing \(P\), we can solve for the minimum probability needed to clear each test.

EV Threshold

Setting \(EV=0\) gives:

$$ P_{EV} = \frac{1-LB}{WB-LB} = \frac{1}{O} $$

The result is the market's implied probability. Ignoring commission and other costs, positive EV requires:

$$ P > \frac{1}{O} $$

This threshold depends on the odds, but not on the stake fraction. Changing the stake changes the amount won or lost; it does not create an edge.

LP Threshold

Setting \(LP=0\), or equivalently \(g=0\), gives:

$$ P_{LP} = \frac{\ln(1/LB)}{\ln(WB/LB)} $$

Unlike \(P_{EV}\), this threshold depends on both the odds and the stake fraction through \(WB\) and \(LB\). For any non-zero stake below the full bankroll, the LP threshold is stricter than the EV threshold. The larger the stake, the more certainty is required to overcome volatility drag.

Canonical Example

Take decimal odds of 1.9091 and a stake equal to 1% of bankroll:

$$ F=0.01, \qquad WB=1.009091, \qquad LB=0.99 $$

The thresholds are:

$$ P_{EV}=\frac{1}{1.9091}\approx52.38\% $$

$$ P_{LP}=\frac{\ln(1/0.99)}{\ln(1.009091/0.99)}\approx52.62\% $$

An estimated probability of 55% clears both. An estimate between 52.38% and 52.62% identifies a positive-EV bet that is nevertheless oversized at a 1% bankroll stake.

This leads to a practical sequence:

1. Estimate the outcome probability conservatively.
2. Compare it with \(P_{EV}\) to test whether an edge exists.
3. Compare it with \(P_{LP}\) to test whether the proposed stake supports geometric growth.
4. Reduce the stake or pass when the LP test fails.

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Closing Line Value

Closing Line Value (CLV) asks whether the price taken by the bettor was better than the market's closing price. It is usually interpreted as a process metric: consistently beating a liquid closing market suggests that the bettor is identifying information or mispricing before it is fully absorbed.

There is no single universal CLV equation. Let:

• \(O_B\) be the decimal odds when the bet was placed;
• \(O_C\) be the closing decimal odds; and
• \(q_B=1/O_B\) and \(q_C=1/O_C\) be their raw implied probabilities.

For a back bet, common definitions include:

CLV definition	Equation	Interpretation
Odds difference	\(O_B-O_C\)	Simple price movement, but difficult to compare across odds ranges
Odds-ratio CLV	\(\frac{O_B}{O_C}-1\)	Percentage improvement in gross payout relative to the close
Probability CLV	\(q_C-q_B\)	Change in implied probability, measured in percentage points
Relative probability CLV	\(\frac{q_C}{q_B}-1\)	Relative repricing; algebraically equal to odds-ratio CLV
Log CLV	\(\ln(O_B/O_C)\)	A symmetric, additive measure that is convenient for aggregation

Positive back-bet CLV means that the taken odds were higher than the closing odds: \(O_B>O_C\). For a lay bet, the direction reverses because laying at lower odds than the eventual close is favorable.

Closing-Price EV Proxy

The connection to DMDA is direct if we treat the de-vigged closing probability \(p_C\) as the market's best available estimate of the true probability. The expected return per unit staked at the original price is then:

$$ EV_{C,\text{stake}}=p_CO_B-1 $$

If the closing market is fair, so that \(p_C=1/O_C\), this becomes:

$$ EV_{C,\text{stake}}=\frac{O_B}{O_C}-1=CLV_{\text{odds}} $$

Under those assumptions, odds-ratio CLV is not merely correlated with value: it is the bet's closing-price EV estimate per unit staked. In the bankroll-scaled DMDA notation:

$$ EV_C=F\left(\frac{O_B}{O_C}-1\right) $$

This also links CLV to the first probability threshold:

$$ CLV_{\text{odds}}>0 \iff p_C>\frac{1}{O_B} \iff p_C>P_{EV} $$

Positive CLV therefore says that, under the closing-price proxy, the closing market supports the conclusion that the entry price cleared the EV threshold.

DMDA-Qualified CLV

Positive CLV does not automatically mean that the stake cleared the stricter LP threshold. DMDA can extend the usual CLV test by evaluating the original bet at the closing probability:

$$ LP_C=WB_B^{p_C}LB^{1-p_C}-1 $$

where:

$$ WB_B=1+F(O_B-1), \qquad LB=1-F $$

Within this framework, we call the result DMDA-qualified CLV when the closing-price proxy supports both positive EV and positive LP. This is a framework-specific classification rather than a universal industry term. The closing-price proxy supports positive geometric growth only when:

$$ p_C>P_{LP}(O_B,F) $$

This creates three useful post-bet classifications:

Closing assessment	Interpretation
\(p_C\le P_{EV}\)	Negative or zero CLV; the close does not support the entry edge
\(P_{EV}<p_C\le P_{LP}\)	Positive CLV, but not enough to support the chosen stake
\(p_C>P_{LP}\)	DMDA-qualified CLV; the proxy supports both the edge and positive geometric growth at the chosen stake

For the canonical bet at \(O_B=1.9091\) and \(F=1\%\), consider a closing price of \(O_C=1.905\):

Measure	Result
EV probability threshold, \(P_{EV}\)	52.380703%
LP probability threshold, \(P_{LP}\)	52.618839%
Closing implied probability, \(p_C=1/O_C\)	52.493438%
Odds-ratio CLV, \(O_B/O_C-1\)	+0.215223%
Closing-price EV per unit staked	+0.215223%
Closing-price LP, \(LP_C\)	-0.002395%

The close moved the implied probability by approximately 0.112735 percentage points and produced positive odds CLV. Under the closing-price proxy, it therefore supports the entry price against the EV threshold. However, \(52.493438\%<52.618839\%\), producing \(LP_C=-0.002395\%\). The proxy therefore does not support positive geometric growth at the chosen 1% stake.

Converting the LP probability threshold back into closing odds gives:

$$ O_{C,LP}=\frac{1}{P_{LP}}=1.900460 $$

At \(O_C=1.900460\), closing-price LP is approximately zero. Strictly positive LP requires the back selection to close at odds below this boundary, subject to de-vigging and commission adjustments.

Measurement Cautions

CLV is only as reliable as the closing benchmark:

• In multi-outcome bookmaker markets, use de-vigged closing probabilities rather than raw \(1/O_C\) values.
• On betting exchanges, the final liquid price or Betfair Starting Price may provide a cleaner benchmark, but thin markets and late shocks can still make it noisy.
• Commission should be included in DMDA calculations. For a simplified back-bet calculation with commission rate \(c\) on net winnings, use \(O_{\text{eff}}=1+(1-c)(O-1)\); actual exchange commission is settled on net market winnings.
• Compare like with like: same market, selection, rules, and settlement terms.
• One bet's CLV proves little. The signal becomes useful through repeated, preferably liquidity-weighted observations and calibration against outcomes.

CLV does not replace the bettor's own probability estimate. It supplies a second, later estimate from the market. That makes it a valuable audit trail for DMDA:

Pre-bet DMDA asks whether our probability clears the EV and LP thresholds. Post-bet CLV asks whether the closing-price proxy supports that assessment.

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Full DMDA Framework

The three stages fit together as a single discipline:

First, identify positive expected value bets (+EV). EV compares the bettor's probability estimate with the price offered by the market.

Second, respect your bankroll constraints. A finite bankroll compounds, so variance and stake size cannot be ignored.

Third, demand enough growth. The EV threshold establishes the minimum for an edge; the higher LP threshold establishes the minimum for growth at the chosen stake.

In compact form:

$$ \text{Bet only when } P>P_{EV} \text{ and } P>P_{LP} $$

For real betting-exchange decisions, these thresholds should be calculated after allowing for commission on net market winnings. Commission reduces the effective winning return, raises the break-even probability, and narrows small apparent edges.

No formula can rescue a badly calibrated probability estimate. DMDA is a decision framework, not a prediction engine. Its value is that it forces the bettor to state three things explicitly: the estimated probability, the offered price, and the fraction of bankroll at risk.

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Conclusion

Expected Value remains the foundation of rational betting, but a bettor cannot live on arithmetic averages alone. Bankrolls grow geometrically, losses reduce future capacity, the sequence of wins and losses exposes us to volatility drag, and excessive sizing can turn a genuine edge into negative long-run growth.

The completed DMDA framework therefore asks two questions in order:

1. Is the price favorable (+EV)? That is the EV test.
2. Is the stake sustainable (+LP)? That is the LP test.

The objective is not merely to find bets that look profitable on paper. It is to manage risk and stake sensibly enough to give a real edge the opportunity to compound.

EV finds the opportunity. LP determines whether the bankroll can sustain it.

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Note: Sanity Check of final draft using ChatGPT.

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