Think about a simple gacha model. Each pull costs 100g (gold) and there is an uniform chance of 1% to win a prize. Assume free trade of gold and items in an efficient market. What is the fair price to that prize?
This is very simple: 10000g, this is the expected cost for you to pull one from the gacha. If the market price is below 10000, every producer (gacha puller) would stop pulling so supply would halt, market makers would sweep items below 10000g, forcing trades to be conducted at 10000g. If the price is above 10000g, there will be extra producers pulling the gacha producing that item at the cost of 10000g per piece.
In reality however, situations can be well different. There are cases where the prize is being traded at prices widely off from the expected value, and similar phenomenons are consistent across all games in all platforms.
Why?
Well, note that we have already made too many assumptions, many of them are almost impossible in applications:
"Efficient market" is often the biggest lie among all. Are there actually enough buyers, sellers and market makers to keep the price competitive? Are all stakeholders smart (efficient) enough? Is it an environment with symmetric information? None of the above hold in our case.
In MMORPG, the free trade condition is also most likely obscured. The imposed tax, associated items being locked from trading, cost of marketing and trading, opportunity cost of trading slot occupation, they are all hidden cost. Ah...and time. Time cost is often the most expensive. When I say gacha in the first place the same question may well be asked in scenarios of game events or dungeons, which is equivalently gachas but with time cost involved. It is not easy to quantity such into (in-game) monetary cost, but a premium is certainly necessary as well as a widened ask-bid gap.
The most complicated but also most important reason though, is that you won't often find a gacha with a single return. There are multiple prizes in a pool, and accounting each and every single of them would be a nightmare. It is not only those items but all associated items with supply or demand disrupted by the existence of items in our gacha pool.
Still though, it is possible for us to take another step on gacha evaluation and to explain why the equilibrium could shift away from the expected value. This is our main focus today.
To derive such theory we would like to differentiate between wealth and utility.
Let us take the gacha games as an example again. There is a point where no matter how strong your units are, your in-game performance probably won't improve much -- all PVE contents cleared effortlessly, similar PVP ranking and so on. That is, the utility you gain when you are extremely wealthy would be exponentially small with a cap. This is called risk-aversion. On the other side, some seek for thrills and takes values extra risk positively, these are called risk-seeking. Between risk seekers and haters there are also risk-neutral players.
In common sense we should expect the fair price to be higher than the expected value for the risk-aversion case because you want others to take the risk for you, and you are paying a premium for it. For risk-seekers, they are taking extra risk for unproportional return, so the fair value would be lower than the expected value.
With the above intuition in mind, we can set up models and prove our claims mathematically.
There are many models to utility, but here is a simple one. Introducing the exponential utility $u(w) = -e^{-\alpha w}$ where $\alpha$ is the degree of risk preference, positive for risk-aversion and vice versa. This is derived from the constant relative risk aversion (CARA) assumption and the differential equation $-u''/u' = \alpha$, although the reason behind such differential equation would be out of scope here.
Let us call the cost of a single pull to be $x$ and the underlying value of the prize to be $V$. Call the original wealth of a player $W$. Ignoring the risk of going bankrupt while pulling the gacha (we will come back to this later), we know that the new wealth by obtaining the item precisely on the $k^{th}$ pull to be $W+V - kx$.
Assume the chance of winning the prize to be $p$, then the expected utility of pulling till you win the prize to be $EU(W,V,x,p) = \sum _{k=1}^{\infty} p(1-p)^{k-1} u(W+V-kx)$. Now we can convert that into a price $P$ that you are willing to pay instead of pulling the gacha by yourself. That gives the equation
$u(W-P) = EU(W,V,x,p)$.
First we can simplify $EU(W,V,x,p)$:
$EU(W,V,x,p) = \sum _{k=1}^{\infty} p(1-p)^{k-1} u(W+V-kx)$
$= \sum _{k=1}^{\infty} p(1-p)^{k-1} (-e^{-\alpha(W+V-kx)})$
$= -p e^{-\alpha (W+V)} e^{\alpha x} \sum _{k=1}^{\infty} ((1-p)e^{\alpha x})^{k-1}$.
Assuming the geometric factor $|(1-p)e^{\alpha x}|<1$ we obtain the form below:
$EU(W,V,x,p) = -p\frac{e^{-\alpha (W+V-x)}}{1-(1-p)e^{\alpha x}}$.
Such assumption is true for sufficiently small (aka valid) $\alpha$. It can be interpreted as that $\alpha$ must be small relative to the cost of a single pull $x$ such that each failed attempt does not inflict a large enough penalty to the point where pulling gacha itself becomes nonsense.
Plugging in $u(W-P) = -e^{-\alpha (W-P)}$ into the main equation, we obtain
$e^{-\alpha (V+P)} = p^{-1} e^{-\alpha x}(1-(1-p)e^{\alpha x})$
Taking log and some rearrangement gives
$V+P = x + \frac{1}{\alpha}(\ln p - \ln (1-(1-p)e^{\alpha x})) = f_{x,p}(\alpha)$.
The rest is slightly nasty calculus but I will leave as exercise here:
Proposition. Given $p\in (0,1)$ and $x > 0$, there exists $\alpha _0>0$ such that $f_{x,p}: (-\alpha _0,0)\cup (0,\alpha _0) \to \mathbb{R}$ is a valid function that satisfies the following:
1) $f_{x,p}$ can be continuously extended at $\alpha = 0$, denoted as $f^*_{x,p}: (-\alpha _0,\alpha _0) \to \mathbb{R}$.
2) $f^*_{x,p}$ is strictly increasing.
3) $f^*_{x,p}(0) = \frac{x}{p}$.
What does it say?
First we need to understand the term $V+P$.
$V$ is the underlying value of the item.
$P$ is the value you are willing to pay to buy the item instead of pulling.
Summing together, $V+P$ represents the value you are willing to buy the item -- or the market value of the item.
The above Proposition says that $V+P\to \frac{x}{p}$ as $\alpha \to 0$. That is, the market value of the item is equal to the value based on probability in the risk-neutral scenario, which is precisely what we claimed!!!
Since $f^*_{x,p}$ is strictly increasing, we know that $V+P > \frac{x}{p}$ when $\alpha > 0$ and vice versa. Hence there is a premium in a risk-aversion market and vice versa in a risk-seeking market.
*
It has long been something I would like to investigate since I came back to a RPG that allow trades. Some may have already noticed I wrote something about
in-game economies under tax a year ago, and one may treat the above kind of a sequel to that.
But it helps more than just on the pricing side. It is also about game design.
Gacha mobile games have been around long enough and is surely a mature industry. Many designs are pretty much fixed, and the gacha system is one of them. 90% of the games provide two options, a single pull or a package of $N$ pulls with some discount/promises. When such model can be analyzed under the above utility framework, one may as well ask if such gacha system is optimized to players or to developers. Is there an alternative that is at least as attractive (in terms of utility or mere illusion) but draws more profit for the developers?
I talked with my maths and finance friends. Their general response is also what I agree: it is (much) more effective to insert attractive content than to create a better gacha logic. Better illustrations, better characters, better gaming system...improve the game itself, not means of profitization.
Yet, I found the above a fine little problem that is laid clear by the means of mathematics. This is only the first step and the rest is still largely unexplored. For those interested you can consider the following variations:
- The isoelastic utility can be used instead of the power utility. This is defined by $u(x) = \frac{x^{1-\eta} - 1}{1-\eta}$ for $\eta \geq 0$ and $\eta \neq 1$ (for $\eta = 1$ the utility if $u(x) = \ln x$ by continuous extension). Repeat the above recover the formula $V+P = g_{x,p}(\eta)$. With the help of Taylor series, try to recover a similar statement like the Proposition above.
- In reality gacha provides multiple prizes. To deal with this we can think about the case of two prizes -- you will find that the derivation is quite similar, without the need to use matrix algebra like multi-asset portfolios.
