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Does stopping a slot machine help?

The random numbers that determine your outcome for that spin have already been set before you stop the reels. Game designers call it “illusion of skill.” You feel like you have control of the game. Your timing in stopping the reels doesn't really make any difference.

tunicatravel.com - Does Stopping the Slot Reels Early Change Your Result? - Tunica
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Does Stopping the Slot Reels Early Change Your Result?

By John Grochowski on Monday March 5, 2012

It’s a pretty common feature on modern video slots that once you push the button to start the reels spinning, you can push it again to stop them. You don’t have to wait for them to stop by themselves.

That prompted a question from a reader who usually plays table games.

“I dabble in slots, usually when my wife and I play together,” he wrote. “Does stopping the reels yourself also stop the random number generator?” I told him that it really makes no difference. Nearly all games remain random when you stop the reels yourself. The random numbers that determine your outcome for that spin have already been set before you stop the reels. Game designers call it “illusion of skill.” You feel like you have control of the game. Your timing in stopping the reels doesn’t really make any difference. There are a couple of games that involve actual skill in IGT’s Reel Edge series, including Blood Life and Triple Red White and Blue. In those, you touch each reel to stop it individually, and you can affect the outcome. In theory, a player who was really, really good at making the reels stop on the right symbols could get an edge. In practice, the reels spin extremely fast and the symbols are mostly a blur. In a Blood Life test, I decided the green 7s stood out enough that I might be able to pick them off. And I did make all three 7s stop in the slot window — but not on the same payline. I have my doubts that anyone could play well enough to get a edge. On other video slots, but you’re going to get whatever the random number generator says you’re going to get, with or without an early stop.

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Does a martingale have drift?

That martingales have no drift: As LHS has no quadratic variation, so must the RHS and thus c(t,X)=b(t,X), leaving a(t,X) to be identically 0. Hence, nothing that is a martingale can have drift.

quant.stackexchange.com - brownian motion - Drift Term in Black-Scholes Model Martingale

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That no drift is a martingale:

That ito integrals are martingales requires a simple but algebraically cubersome proof. You can refer to Shreve (continuous time) for the proof. You can also intuitively observe it as Brownian increments that are multiplied with their respective integrands are allocated independently of the integrand value. Thus, when all terms are added, the sum isn't biased upward or downward. This is exactly the martingale property.

That martingales have no drift:

The martingale representation theorem (MRT) asserts that (loosely speaking) a martingale can be represented as a stochastic integral. This necessitates that any other representation of the martingale as an Ito process:

$dX/X = a(t,X)dt+b(t,X)dW(t)$

must have $a(t,X):=0$. If not, we have:

$a(t,X)dt+b(t,X)dW(t)=c(t,X)dW(t)$ for all $t$, for some function $c$ due to MRT.

$a(t,X)dt=c(t,X)dW(t)-b(t,X)dW(t)$ for all $t$

As LHS has no quadratic variation, so must the RHS and thus $c(t,X)=b(t,X)$, leaving $a(t,X)$ to be identically 0.

Hence, nothing that is a martingale can have drift.

Thus martingales are equivalent to no drift.

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