Session Boundaries & The Optional Stopping Theorem

1. Martingales, Supermartingales, and Slot Bankrolls

To analyze stopping rules mathematically, we must formalize the bankroll trajectory in the framework of filtered probability spaces $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \ge 0}, P)$.

Let $B_t$ denote the player's bankroll after spin $t$. Let $X_t$ be the payout multiplier of spin $t$, where $\mathbb{E}[X_t] = ext{RTP} < 1.00$. The net return of spin $t$ with constant unit stake $W = 1$ is $\Delta B_t = X_t - 1$, and its conditional expectation given the historical filtration $\mathcal{F}_{t-1}$ satisfies:

\mathbb{E}[B_t - B_{t-1} \mid \mathcal{F}_{t-1}] = \mathbb{E}[X_t - 1 \mid \mathcal{F}_{t-1}] = 	ext{RTP} - 1 = -	ext{HE} < 0

Because the conditional drift is strictly negative, the bankroll process $B_t$ is a strict supermartingale:

\mathbb{E}[B_t \mid \mathcal{F}_s] \le B_s, \quad \forall s \le t

In a fair game ($ ext{RTP} = 1.00$), $B_t$ would be a martingale ($\mathbb{E}[B_t \mid \mathcal{F}_s] = B_s$). In an advantageous game ($ ext{RTP} > 1.00$), it would be a submartingale. A slot machine is unconditionally a supermartingale.

2. Doob's Optional Stopping Theorem and Its Inviolable Bound

A stopping time $ au$ is a random variable taking values in $\{0, 1, 2, \dots\} \cup \{\infty\}$ such that for every finite $t$, the event $\{ au \le t\}$ belongs to $\mathcal{F}_t$. In plain English, a player's decision to stop playing at spin $t$ can depend on outcomes up to spin $t$, but cannot look into the future.

Doob's Optional Stopping Theorem states that if $B_t$ is a supermartingale and $ au$ is a stopping time satisfying any of the following standard regularity conditions:

• $ au$ is almost surely bounded (there exists an integer $K < \infty$ such that $P( au \le K) = 1$); or
• $\mathbb{E}[ au] < \infty$ and the increments $|B_t - B_{t-1}|$ are uniformly bounded; or
• $B_t$ is uniformly integrable;

Then the expected value of the bankroll at the stopping time $ au$ satisfies the fundamental inequality:

\mathbb{E}[B_	au] \le \mathbb{E}[B_0] = B_0

Furthermore, because the incremental drift is strictly negative ($- ext{HE} < 0$), if the expected session length is finite ($\mathbb{E}[ au] > 0$), the inequality is strict:

\mathbb{E}[B_	au] = B_0 - 	ext{HE} \cdot \mathbb{E}\left[\sum_{t=1}^{	au} W_t
ight] < B_0

This mathematical proof is absolute: no stopping rule $ au$, however complex, adaptive, non-linear, or psychologically inspired, can ever produce an expected bankroll greater than the starting capital. The house edge cannot be circumvented by choosing when to stop.

3. Asymmetric Double-Barrier Mechanics: Locking in Positive Variance

If stopping rules cannot produce positive expectation, why do professional quantitative analysts emphasize session boundaries?

The answer lies in the distinction between expectation (an infinite-sample average) and realized sample-path probability.

Without an upper stopping boundary, the player continues spinning indefinitely. Because the supermartingale has negative drift, the asymptotic probability of hitting the zero absorbing barrier is exactly 1:

\lim_{t 	o \infty} P(B_t = 0 \mid B_0) = 1.00

However, if the player introduces an Asymmetric Double-Barrier Stopping Rule:

	au = \inf\{ t \ge 0 : B_t \ge U 	ext{ (Take-Profit)} \quad 	ext{or} \quad B_t \le L 	ext{ (Stop-Loss)} \}

Where $L = B_0 - D_{ ext{loss}}$ and $U = B_0 + G_{ ext{target}}$.

Using the diffusion approximation for Brownian motion with drift $\mu = - ext{HE} \cdot W$ and variance $\sigma^2 = W^2 \cdot \sigma_{ ext{slot}}^2$, the exact probability of exiting at the upper profit boundary $U$ before collapsing to the loss boundary $L$ is:

P(	ext{Hit } U 	ext{ before } L) = \frac{1 - \exp\left( - \frac{2 \mu (B_0 - L)}{\sigma^2} 
ight)}{1 - \exp\left( - \frac{2 \mu (U - L)}{\sigma^2} 
ight)} = \frac{1 - \exp\left( \frac{2 \cdot 	ext{HE} \cdot D_{	ext{loss}}}{W \cdot \sigma^2} 
ight)}{1 - \exp\left( \frac{2 \cdot 	ext{HE} \cdot (D_{	ext{loss}} + G_{	ext{target}})}{W \cdot \sigma^2} 
ight)}

Consider an empirical example with $B_0 = \$1,000.00$, unit wager $W = \$2.00$, slot parameters $ ext{HE} = 0.04$ and $\sigma = 8$, setting a stop-loss at $D_{ ext{loss}} = \$300.00$ ($L = \$700$) and a modest take-profit target at $G_{ ext{target}} = \$150.00$ ($U = \$1,150$):

\frac{2 \cdot 	ext{HE}}{W \cdot \sigma^2} = \frac{2 	imes 0.04}{2 	imes 64} = \frac{0.08}{128} = 0.000625

P(	ext{Hit } +150 	ext{ before } -300) = \frac{1 - e^{0.000625 	imes 300}}{1 - e^{0.000625 	imes 450}} = \frac{1 - 1.2062}{1 - 1.3248} = \frac{-0.2062}{-0.3248} \approx 63.48\%

In 63.48% of sessions, the player will successfully touch the $+150.00 profit boundary and terminate with cash in hand. If they lack the discipline to stop at $+150.00$ and continue spinning, that 63.48% positive realization decays inexorably toward zero.

4. Expected Session Duration Under Double Barriers ($\mathbb{E}[ au]$)

By applying Dynkin's Formula or Wald's Identity to the supermartingale $B_t$, the expected duration of the session in spins until hitting either barrier is:

\mathbb{E}[	au] = \frac{\mathbb{E}[B_	au] - B_0}{-\mu} = \frac{B_0 - \left( P_{	ext{win}} \cdot U + (1 - P_{	ext{win}}) \cdot L 
ight)}{	ext{HE} \cdot W}

In our numerical example ($P_{ ext{win}} = 0.6348$, $U = 1150$, $L = 700$):

\mathbb{E}[B_	au] = 0.6348 	imes 1150 + 0.3652 	imes 700 = 730.02 + 255.64 = \$985.66

\mathbb{E}[	au] = \frac{1000 - 985.66}{0.04 	imes 2.00} = \frac{14.34}{0.08} \approx 179.25 	ext{ spins}

The expected duration to resolution is under 180 spins (approximately 15 minutes of play). The double barrier enforces a compact, tightly controlled stochastic exposure window where variance can be captured before house drift extracts its actuarial toll.

5. Psychological Pitfalls: The House Money Illusion

Behavioral finance studies identify two catastrophic cognitive biases that undermine stopping boundaries:

• The House Money Effect (Thaler & Johnson, 1990): When players accumulate early winnings, they mentally categorize profits as "the casino's money" rather than their own. This causes risk tolerance to expand dramatically, leading players to increase wager sizes or dismantle take-profit barriers, accelerating turnover bleed.
• Escalation of Commitment / Loss Chasing: When approaching a stop-loss boundary ($L$), players experience acute loss aversion. Instead of terminating, they lower or remove the stop-loss barrier ("just 50 more spins to break even"), ensuring full absorption.

6. Quantitative Governance Rules for Session Boundaries