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Crash games are online gambling games where players place bets on an increasing multiplier that can ‘crash’ at any moment. Players must cash out before the crash occurs; successful cash-outs yield the bet amount multiplied by the current multiplier, while failure results in total loss of the wager.

Crash game showing an airplane flying with increasing multiplier until it crashes

The game I came across uses an aircraft flight metaphor. Let’s call it Plane Game. What intrigued me wasn’t the game itself but that it said “provably fair” on the startup screen, which I assumed to be a typo at first. I stand corrected:

A provably fair gambling system uses cryptography to let players verify that each outcome was generated from fixed inputs, rather than chosen or altered by the operator after a bet is placed. The casino commits to a hidden “server seed” via a public hash, combines it with a player-controlled “client seed” and a per-bet nonce, and later reveals the server seed so anyone can recompute and confirm the result.

The stated Return-to-Player (RTP) of that specific game is 97%, implying a 3% house edge. After watching a few rounds, the perceived probability felt off. And if there’s something that gets my attention, it’s the combination of games and statistics. So I did what any reasonable person would do: I watched another 20,000 rounds over six days (112 hours total) and wrote a paper about it.

Script recording 20000 rounds over six days (112 hours total)

The distribution below shows the classic heavy tail: most rounds crash quickly at low multipliers, while rare events produce 100x or even 1000x payouts. The maximum I observed was 10,000x. This extreme variance creates the illusion of big wins just around the corner while the house edge operates relentlessly over time.

Heavy-tailed distribution of crash multipliers on log-log scale showing most rounds end at low multipliers while rare events exceed 100x or 1000x, with maximum observed at 10,000x

For a crash game with RTP = r (where 0 < r < 1), the crash multiplier M follows a specific probability distribution. The survival function is particularly relevant:

$$P(M \geq m) = \frac{r}{m}$$

This means the probability of reaching at least multiplier m before crashing equals r/m. For any cash-out target, the expected value of a unit bet works out to:

$$E[\text{Profit}] = P(M \geq m) \times m - 1 = \frac{r}{m} \times m - 1 = r - 1 = -0.03$$

This mathematical property makes crash games theoretically “strategy-proof” in expectation. No cash-out timing strategy should yield better long-term results than another.

Survival probability curve on log-log scale showing probability of reaching target multiplier: 2x succeeds 48.5% of the time, 5x at 19.6%, 10x at 9.7%, 50x at 2.0%, and 100x at just 1.1%

The empirical data matches theory almost perfectly. A 2x target succeeds about 48.5% of the time. A 10x target works only 9.7% of rounds. The close fit between our observations and the theoretical line is consistent with the stated 97% RTP.

My analysis supports the fairness claim. Using three different statistical methods (log-log regression, maximum likelihood, and the Hill estimator), I estimated the probability density function exponent at α ≈ 1.98, within 2.2% of the theoretical value of 2.0. This contrasts with Wang and Pleimling’s 2019 research that found exponents of 1.4 to 1.9 for player cashout distributions. Their deviations reflect player behavioral biases (probability weighting), rather than game manipulation. These results support the stated payout model over this sample.

A close fit cannot rule out every deviation. Statistical tests also cannot establish cryptographic security. Under the 97% RTP model, tests on 20,038 rounds lose power as the cash-out threshold rises and hits become rarer. The graph compares their chance of detecting 5% and 10% relative shortfalls in hits.

Detection power for one-sided 5% tests with 20,038 rounds under a 97% return-to-player null model. Power declines at higher cash-out thresholds for alternatives with 5% or 10% fewer hits; a dotted line marks 80% power.

I then ran Monte Carlo simulations of 10,000 betting sessions under four different strategies: conservative 1.5x cashouts, moderate 2.0x, aggressive 3.0x, and high-risk 5.0x targets.

Modelled net profit over 100 equal fixed-stake bets at cash-out targets of 1.5x, 2x, 3x and 5x under a 97% return-to-player model. Higher targets widen the percentile ranges; expected profit remains minus 3% of total wagers.

Every strategy has the same negative expected return per unit wagered. Lower cash-out targets reduce variance; higher targets widen the range of possible results.

Expected net loss under a 97% return-to-player model equals 3% of total wagers. After 179 completed bets of one fixed stake each, expected loss is 5.37 stake units.

The consumer protection angle is what concerns me most. My data revealed 179 rounds per hour with 16-second median intervals. If a player completes 179 equal fixed-stake bets, a 3% house edge implies an expected loss of 5.37 stake units. That is 3% of the total amount wagered. The manual cashout mechanic creates an illusion of control over a game with a persistent expected cost.

Cryptographic fairness lets players check whether a game follows its published rules. A game can follow those rules and still expose players to substantial losses.

The only winning strategy is not to play

The full paper preprint with methodology and statistical details is available on SSRN. Code and data are on GitHub.