Opinion: Dubious Casino Math Strikes Again in New Lawsuit Against Drake and Stake
An interesting new lawsuit has appeared against Drake, Stake, and a number of other defendants in New Jersey state court, which goes well beyond the now-familiar illegal gambling accusations. It portrays Stake as being at the heart of a larger conspiracy involving the manipulation of music rankings on streaming services, using the casino site’s tipping feature as a way to move money without leaving a trail.
A Virginia lawsuit from January made similar claims. However, this new suit also accuses Stake of rigging its games to allow celebrity streamers like Drake to win large jackpots at a higher rate than a regular user would. That claim seems to be based entirely on a single article that Bloomberg Businessweek published in February.
That leaves me in a bit of a predicament when it comes to writing about the case, because I already wrote a takedown of Businessweek’s methodology when that article first came out. (I was writing for PokerScout at the time).
To their credit, the editorial team at Businessweek made some corrections and clarifications to the article after our team reached out. I understand why they don’t want to retract it entirely, as it was clearly a lot of work, and their approach to gathering the data was clever. Unfortunately, the approach is predicated on a single bad assumption that, in my opinion, makes it dangerous to rely on the analysis for anything important. Like, say, in the courtroom.
Let me be clear: This is a mathematical opinion, not a legal one. I am not expressing an opinion about the case as a whole, or even about whether Stake rigs its games. All I am saying is that, based on my own understanding of slot machine math, I did not and still do not believe there are any conclusions that can be drawn from the numbers Businessweek collected.
Here’s why: Businessweek counted extremely small numbers of “bonus buy” plays as if they were large numbers of regular spins.
Bonus Buys Change Slot Math
Most slots feature bonus rounds, and much of the total prize money gets paid out through those bonuses. Normally, a player would only trigger that bonus every few hundred or thousand spins, but many modern slots allow the player to buy the bonus immediately, by paying a multiple of the base bet.
For instance, if you’re playing for $1 per spin, you might be allowed to pay $80 in order to guarantee that the bonus triggers. This is what most or all of the streamers tracked by Businessweek were doing.
That’s a problem for two reasons.
Unreasonably Small Sample Size
For one thing, it means the sample size is massively overstated. In one session, Adin Ross (who is a co-defendant in the lawsuit) was tracked over “2,000 spins” on a slot by the Stake-affiliated studio EasyGo. However, based on Businessweek’s update to its article, that turns out to be exactly three plays — one for 1,000x the bet and two for 500x.
That is not a meaningful sample size for any casino game. Or any statistical thing, really.
Different Bonus Buy Options Have Different Volatility
For another, it ignores that players can control their volatility by choosing which bonuses to buy. A slot doesn’t need to be rigged for one player to have more big wins than another if that player is buying bonuses more aggressively. It just means that the player would also experience bigger downswings.
It would be essentially impossible to play 2,000 regular spins and be down 2,000 bets — no one loses that consistently. It would be almost as hard to have that happen playing 25 bonus buys at 80x apiece. However, playing only three bonus buys for huge multiples, Ross could easily have lost most or all of his wagers. The unlucky streak required for that to happen is much shorter.
But volatility is always a two-edged sword. Ross would experience the same statistical return-to-player as a less aggressive bonus-buyer, because he’s still playing the same slot. The fact that he’s more likely to burn through 2,000 bets in his session means a correspondingly higher chance that he wins big, because those wins and losses have to balance out to the same average.
And in fact, Businessweek’s definition of a “big win” was 1,000x the bet. For Ross’s biggest bonus buy, just breaking even would have counted as a big win.
Statistics Are Tantalizingly Easy to Absue
Mark Twain is often credited with the phrase “lies, damn lies, and statistics.”
Ironically, that’s a lie. Twain claimed to have been quoting former British prime minister Benjamin Disraeli, but that may be a lie, too. Truth is, no one’s really sure who came up with it, and you can’t believe everything you read.
And yet, that phrase has endured as a reminder that it is easier to mislead people with statistics than with words alone. In school, we’re indoctrinated with the idea that math is rigorous, that it proves things. Numbers feel like hard evidence.
But statistics, on their own, never actually prove whether something is true. Used properly, they tell you how confident you should be that something is true.
Unfortunately, statistics are taught only in a superficial way until college, and they’re rarely used for their intended purpose outside of scientific publications. Politicians, journalists, and marketers all use them as rhetorical devices.
Quite often, the first person they fool is themselves.
I don’t think the Businessweek team had any nefarious intent with their analysis. I think they saw something that seemed intuitively suspicious and grabbed the first thing they could find that looked like evidence. Now that evidence appears headed for the courtroom, where it’ll be subject to much closer scrutiny.
Personally, I would not want to defend it in that environment.
(I asked Bloomberg and the legal team for the plaintiffs to comment for this story, but have not received a response from either.)
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