Volatility, Spins and RTP

As is often the case, the right answer is found by intelligent workers in a field before theoretical analysis catches up.   For example, many financial models are used by investors before they are formally analyzed by financial and economic researchers.  I'm thinking historically of CAPM or even the factors of Fama-French portfolios.

In the case of casino mathematics, we have a similar situation when measuring volatility.  Somehow, many mathematicians at different companies independently came to use median spins on a fixed bankroll to rank the volatility of machines.  This article will explore properties of the median spins statistic.

For example, a player with $20.00 making a 40 credit bet with 1 credit = 1 cent begins with 50 initial bets.  But if they play until they are out of money, the median player will get to play for many more spins, say 150 spins.  If playing with the same initial bankroll and making the same size bets on a different game the median player only gets 100 spins, we say that this game is more volatile than the other.

In the previous article we explored the graph of spins players make when they make a certain return from the casino.  We pointed out that the median player's rtp was therefore related to this overall market change in spins.  Interestingly, spins is also related to rtp at the individual level.

Consider a player with initial Wallet, W, who will make a sequence of bets {B1, B2, ...} until they can no longer afford to bet.  We get a sequence of budget constraints:

0 <= W - B1,

0 <= W - B1 - B2 + Win1,

0 <= W - B1 - B2 - B3 + Win1 + Win2,

... 

0 <= W - B1 - B2 - B3 - ... - BN+ Win1 + Win2 + ... + WinN-1, (1)

In words, the player is only allowed to bet B1 up to the size of wallet W.  Then adjusting wallet by the amount bet and what was won on that first spin, Win1, B2 must be no greater than that new wallet amount.  We continue in this way until the game stops on the Nth spin.

Since any win is greater than or equal to 0, we can bound (1) with

0 <= W - B1 ... - BN + Win1 + ... + WinN-1 <=  W - B1 ... - BN + Win1 + ... + WinN. (2)

Next divide everything through by Total Bets = B1 + ... + BN,

0 <= W / Total Bets - 1 + RTP, (3)

or by rearranging,

Rake <= W / Total Bets. (4)

On the other hand in order to have stopped at spin N the difference of bets from wins and initial wallet must be less than the minimum bet B.

B > W - B1 ... - BN + Win1 + ... + WinN. (5)

Since Average Bet >= B we get,

Average Bet >= B > W - B1 - ... - BN + Win1 + ... + WinN, (6)

and dividing by Total Bets,

Average Bet / Total Bet > W / Total Bet - 1 + RTP. (7)

Since Total Bet = N * Average Bet, we can reorganize (7) as

Rake > (W / Average Bet) / N - 1/N. (8)

Combining (4) and (8) we get

(W / Average Bet) / N - 1/N < Rake <= W / Total Bets = (W / Average Bet) / N. (9)

So, when players bet until they run out of coins, we know within 1/N coins that the casino raked (W / Average Bet) / N.  Furthermore, games mathematicians usually take B=B1=...=BN for an "apples to apples" comparison across games, by comparing Median(RTP) or the statistic Median(N) across the games starting with an equal initial wallet in terms of multiples of bet B.

One point to make is that in practice different companies like to use the median spins stat differently.  Some prefer a 50 bet initial wallet, while others use a different number.  Some even prefer thinking about it in terms of fixed $ amounts like $20.  Deep knowledge of stopping games from stochastic processes or the seemingly unrelated signal processing field reveals that it doesn't matter too much.

The above graph compares the log(Median(N)) for different slot games in order to empirically derive the relationships of their Median(N) if you change initial bet size for a fixed $ amount.  It is clear that it isn't hard to do.  Fundamentally, volatility is a function of starting wallet, bet size and some property of the game that can be measured by statistics like Median(N) or Median(RTP) and such statistics are all very closely related to each other.

Bringing this full circle to the economy accounting identity in equilibrium,

f + e + p = rake, (10)

and recall the denominator is total bets,

( (F + E + P) / Average Bet ) / Total Spins = rake. (11)

What does that mean?  In equilibrium, where wallets are constant, the sum of free, earned and purchased coins plays the role of W in (9).  This is how the volatility of games influences total spins in the economy.  What is even more interesting about (11) though, is that players are free to choose the size of bets strategically, and of course the casino influences F, E and P.  So, volatility is determined in a strategic game between the player and casino even after the slot machine volatilities are set.