Inflation in Token Economies

Headline inflation is fundamental to most economies and your slot economy is no exception.  Sometimes it is a slow and steady inflation as players are rewarded for moving up the experience curve.  Other times it is a surprise event wherein the casino increases the rewards for all the inflows.  This article will use the accounting identity for slot economies and the basic theories of feature tuning and sales to construct a larger analysis of a few simple strategies for inflating an economy.  The core empirical result to capture is that economic activity in total scales by exactly the rate of inflation.

To start we will consider free coins Ft, earned coins Et(Bt,Bt-1) as a function of today's and yesterday's bets, Bt and Bt-1, and purchased coins Pt = Dt*Xt where Dt is $ demand and Xt is the exchange rate.  RAKEt is the total amount of coins the casino rakes and Wt indicates wallets.  To inflate the economy we will multiply free inflows, initial wallets and the exchange rate by 10.

So,

Ft + E(Bt,Bt-1) + Pt = Wt - Wt-1 + RAKEt (1)

becomes

10*Ft-1 + Et(Bt,Bt-1) + Dt*10*Xt-1 = Wt - 10*Wt-1 + RAKEt. (2)

It's clear the casino manager's job isn't finished as they also need to think about earned credits.  Imagine first that they leave the same features running with the same rules as before the inflation,

meaning,

Et(Bt,Bt-1) = Et-1(Bt,Bt-1) = k*Bt-1*I(Bt>=(1+c)*Bt-1). (3)

(Today I've introduced the notation I(Bt>=(1+c)*Bt-1), which equals one if todays bets exceed (1+c) times yesterdays bets and is zero otherwise.  It can be added into the previous feature tuning post for greater generality.  But just ignore it, if it is confusing.)

The empirical result we see when inflating an economy is Bt  = 10*Bt-1 and Wt =10*Wt-1. So substituting (3) into (2) and dividing through by Bt gives

ft-1 + et-1/10 + Dt*Xt-1 = rake. (4)

we see that to reach equilibrium $ demand Dt must increase a lot compared to Dt-1 or the casino must drop rake, because earned credits have been divided by 10.  Maybe that will be too much of a change to ask of players.  So, instead we can modify (3) by demanding that players spend 10 times (1+c)*Bt-1 and will reward them correspondingly with 10 times as much coins.

So,

Et(Bt,Bt-1) = 10*Et-1(Bt,10*Bt-1) =10*k*Bt-1*I(Bt>=10*(1+c)*Bt-1). (5)

Now, substituting (5) into (2) and dividing through by Bt =10*Bt-1 gives us the usual equilibrium condition:

ft-1 + et-1 + Dt*Xt-1/Bt-1  = rake. (6)

Equilibrium is reached with Dt = Dt-1 and we should expect no fundamental changes to the variables Ft, Et, Bt or Wt.

The force of the casino making a profit through inflating requires balancing between these two extremes.  For example by only rewarding earned credits with a factor of 8x and requiring the players to purchase some small difference to reach equilibrium wallets.  Any irrational exuberance from feeling richer that would make the players buy more coins doesn't seem to happen as a matter of logic or in practice without some prompting.

That being said there is a hidden part to this story.  These variables like Bt are aggregates and this variable in particular, which is total bets made in coins, can be expressed as the product of total spins St and average bets,

or,

Bt = avg(Bt) * St. (7)

Understanding this means that even though Bt jumps up to 10*Bt, avg(Bt) does not have to.  In fact I have seen St increase to make up the difference.  Managers having grown used to an economy where median wallets tracked mean wallets but with less noise were surprised to see median wallets growing.  In fact this had to happen because players were spinning more.

To understand this look at spins vs players bucketed by rtp before and after the inflation.  The central limit theorem gives shape to two beautiful curves, but the one after the inflation will be taller and narrower.  The curve with higher spins has a higher median player rtp.

Applying the accounting identity for slot economies to just those players around the median rtp gave very precise estimates of how much median wallets should have been growing.  This leads us to the more interesting questions.  Why did players' desire for volatility decrease? and what if anything should the casino do about it?

Eventually the average bets grew in size, spins dropped down to normal levels, and median and mean wallets moved in synchronization again.  I conjecture this is a long run equilibrium condition deeply rooted in the players' tastes for volatility.  In the next post we will see more precisely how spins St relates to volatility.