Networks

Videogame Lootcrate Keys – Buy or Sell?

Rules and Assumptions of the Game

This post will be talking about the crate system in Applecraft – an online Minecraft server using the concept of risk in betting in Chapter 22.

1. At the start of the month, every player gets a certain number of keys
2. Every key represents an opportunity to take part in a lucky draw, where they can win an item worth a certain number of diamond blocks (DBs, the game’s currency).
3. Every player has their own individual wealth and can either use the key or buy/sell the key with other players. In this month, if the player uses their key:
   1. The jackpot is worth 8 * 64 DBs and everything else is worth on average 16 DBs.
   2. There is a 2/67 chance the player gets the jackpot (worth 8 * 64 DBs)
   3. There is a 65/67 chance the player gets something else (worth 16DBs)
4. If the player sells their key: they’re guaranteed to get the expected value of ~32DBs

For a more detailed explanation, please see https://wiki.applecraft.org/wiki/crates-and-rares

Key Claim: When faced with the option of whether to use their key, a poorer player’s dominant strategy is to sell their key rather than use it, even if the expected payoffs are equal (between buying versus using)

Modeling Risk

Looking at these numbers, the price of 32DBs per key seems fair and there’s no difference between whether the player sells or opens their key. However, the key is that poorer players are more risk averse than rich players, and so follow a more logarithmic utility of wealth.

Logarithmic Utility of Wealth – Textbook Page 697

If a poor player opens all 31 of their keys this month, there is a 60.9% chance of them winning the jackpot (P(X>=1) in a Binomial distribution where n = 31, p=2/67). That means there’s a 39.1% chance they’ll not win a jackpot.

If you’re an ordinary player and end up with no Jackpots after 31 keys, their payoff per key is -16DBs. However, if they’re rich, they have the option of buying another 31 keys. Perhaps this time, they’ll get lucky and get 3 Jackpots. You’ve now turned a profit (made more money than had you sold all your keys). If you’re still unlucky, you can buy more keys (so on and so forth) until you get lucky. How probability works is that the average value you get per key will get closer and closer to 32DBs the more keys you buy (aka Central Limit Theorem). This is why rich players can afford to take risks.

The basic idea is that to a poor player, you’ll either get very lucky or (most likely) lose a bit of value on your key. If you’re rich, getting unlucky a few times is ok, and you can keep buying keys until you get ~32DBs on average of value out of each key from winning jackpots. Therefore, if you’re a poor player, you should take the less risky option and sell your keys.

When Should Poor Players Open Their Key

1. Other players are offering to pay much less than what the key is worth. This is unlikely because the market for keys is usually the largest (high supply high demand), so the prices are close to their true value.
2. The crate is *not too risky* – e.g. if the price for a key is 64DBs and most items in the crate are worth ~64 DBs

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