Slots Aren't Rigged - Math Beats Us

Additive and multiplicative games. What’s the difference?

Not all gambling games are equally dangerous in a mathematical sense. The difference is determined by the payout mechanics.

An additive game – you bet a fixed amount each time regardless of your current balance. You win $10 – you lose $10. The distribution of outcomes is relatively symmetric, tails are thin, the mean works as a guide.

A multiplicative game – the result of each round is multiplied by the current capital. That’s how roulette behaves when playing a Martingale system and any strategies where bet size is tied to the bankroll. Slots with a fixed stake are formally additive – but more on that below. In multiplicative processes tails are fat, dispersion is huge, and ruin over the long run is mathematically inevitable.

How the game type affects risk

Casinos often promote strategies like “double your bet after a loss” not because they are profitable for the player, but because they turn the game from additive into multiplicative, accelerating the inevitable.

Why players “irrationally” keep playing

Psychologists have for decades explained the propensity to gamble by cognitive biases: illusion of control, availability bias (“I saw my neighbor win a jackpot”), the near-miss effect. All of that is real – but it’s not the whole picture.

Researcher Ole Peters, within so-called ergodic economics, offered another explanation: the player behaves completely rationally – he maximizes what he perceives as his time average. The problem is that the casino methodically substitutes his reference point.

Advertising shows the “ensemble average”: someone won a million, someone hit the jackpot, average RTP 96%. The player unconsciously transfers this to himself – and decides as if he were immediately 10,000 people, not one specific person with a finite bankroll and finite time.

Practical takeaway – how to think about bets and bankroll

Classical expected utility theory (von Neumann-Morgenstern) implicitly assumes ergodicity – it evaluates bets by the ensemble mean. This is the root of the “paradox” of gambling: a game that seems profitable by expected value can in practice lead to ruin. The work of Ole Peters and Murray Gell-Mann showed that people intuitively adjust their behavior to the time average, not the ensemble average.

Professional poker players and sports bettors know: a profitable game kills you with the wrong bet size. That’s why they use the Kelly criterion – a formula that defines the optimal fraction of the bankroll to wager.

If the probability of winning is 60% and the odds are 2.0 (i.e. on a win you get back double your stake, net profit equals 1x), the Kelly criterion recommends betting no more than 20% of your bankroll at a time. Bet more – even with positive expectation you’ll eventually go bust. This is not caution, it’s the math of multiplicative processes.

For casino games, where expectation is negative by definition (house edge from 1% in blackjack to 25% in keno), the Kelly criterion gives an even harsher answer: the optimal bet tends to zero. The only way to “beat” a non-ergodic game with negative expectation is not to play long.

Inequality of winnings is math. Not luck!

The distribution of winnings in casinos always follows the same pattern: the vast majority of players lose money, a few scoop large prizes. This looks like unfairness or a marketing trick – but in reality it’s an inevitable mathematical consequence of multiplicative dynamics with fat tails.

According to gambling math data, the house edge for slots reaches 15%, for keno up to 25%, while for blackjack with correct strategy it drops below 1%. The difference is colossal – but even 1% over a long enough horizon and with multiplicative mechanics leads to inevitable depletion of the bankroll.

“Lucky” players who regularly win large sums generally do not stay ahead over a span of several years. Their stories are used in advertising precisely because they are rare – just as plane crashes make the news, not the millions of safe flights. The ensemble average lies to you again.

Play responsibly!

Gambling is fundamentally a non-ergodic system. Looking at the “average payout” of a group of players and extrapolating it to yourself is a fundamental mistake that casinos and gambling platforms exploit. So accept as a given that for your money there’s only one road in a casino – out of your wallet – and play smart.