How to Tune a Feature

Imagine having a feature that a casino plans to turn on a few times each week.  The feature is meant to keep players engaged and give them goals, but isn't necessarily a big money mover, so that the dynamics of the economy should be mostly stable.

One way to interpret this is by observing the frequent truth of constant wallets.  In terms of the economy equation derived in the previous post,

ft + et + pt = (Wt-Wt-1)/Bt + rake,

simplifies,

ft + et + pt = rake. (1)

Where the change of wallets (W-W') = 0. ft, et and pt are free, earned and purchased credits divided by total bets, respectively.  Finally rake is the expected (1- return to player) of the casino.

The casino is going to pay out et = 2%, in an equilibrium economy with ft-1 + pt-1 = rake.  This is enough to derive a useful equation, when we realize that ft=Ft/Bt , etc, and the casino is going to require an increase c in Bets before granting the reward.  Since Bt = (1+c)Bt-1, we get from equation (1),

ft-1/(1+c) + et + pt-1/(1+c) = rake,

simplifies,

et*(1+c) = c * rake. (2)

The structure of et = k*St-1/Bt, or in words, the earned credits are equal to k times some player statistic, St-1.  This allows us to rewrite (2) as,

k*St-1/Bt*(1+c)  = c * rake. (3)

As stated above the casino targets 2% feature return, i.e.,

k*St-1/Bt = 2%. (4).

Again substituting Bt = (1+c)Bt-1, and assuming the casino uses a bet based player statistic we take St-1=Bt-1 which gives,

k/(1+c) = 2%, (5)

and,

k = c * rake. (6)

(5) and (6) are two linear equations in two unknowns, c and k, which means we can solve them.  Taking rake = 8% we determine c = 33.3% and k = 2.67%.

The importance of this result is that you can choose rtp of the feature but then you must choose either how much you want players to increase bets or how much you want wallets to change.  There is only one value of c and k consistent with 2% feature return and no change in wallets.

More generally we may want to decrease wallets in the hopes that players purchase to maintain them, etc.  Following most of the above argument but with, (Wt-Wt-1)/Bt = dw, we would get

k - rake * c = dw, (7)

or

2%/(1+c) - rake * c = dw. (8)

The following is a plot of  dw vs c from (8).

These simple equations should supplement the tuning of regularly occurring features, so that overall changes in wallets can be predicted and tracked.  Too often the casino asks the players to increase bets by what "feels right" or doesn't test at all against the economy goals, with a resulting falling apart of the equilibrium.  What is necessary is to make explicit rake, feature return and desired change in wallets.  From there, a model that specifies "bet ask" can be identified.