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Consider games such as Team Fortress 2 or Counter Strike Global Offensive on steam, there we can obtain certain virtual goods only by means of opening "crates" through the usage of "keys". The only possible way to obtain keys other than bartering with other players would be to use real money. The key cost remains fixed.

By luck the crate, upon usage of key, would give items of varying rarity. Higher rarity means typically higher price.

However, something I notice is, the price of the common items which can only be obtained by crates is almost always lower than that of the actual key+crate. Why is this so?

tryst with freedom
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Assuming people are rational and risk neutral the cost of crate + key would be the expected value of items that the crate could drop plus the fixed cost of key. So

$C= E[V] +K$

where $C$ is the total cost, V is the random variable - value of item that drops, and $K$ value of key to crate.

However, once you open the crate value of the item is no longer random, by opening the crate you force $v_i$ to be drawn from $V$ distribution. If you get $v_i$ from bottom of the value distribution (e.g. suppose you get lowest $\underline{v}$. its completely possible it is smaller than the value of key plus crate $ \underline{v} < E[V] +K$.

Moreover, note the expected value can be skewed by very rare but valuable items. Suppose you can get item costing 1000 with probability 0.1 and then with probability 0.9 you get some common item costing 100, expected value of crate for a risk neutral person will be 190, despite that you are unlikely to actually get item with value that is higher than 100.

If we assume people are risk averse values would change, but there are still parameters for which the result above will hold, just with different value/price.

This is similar to lottery, you buy some lottery ticket, even if expected value of lottery ticket is 10 bucks it does not guarantee you get ten bucks. Ex ante, before the ticket is scratched the value is given by expected value of the lottery (if you are risk neutral), ex post - once you scratch it has the value that is on the lottery ticket which could even be zero.

1muflon1
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  • I mean in lottery it's either win or lose. Here it's like you achieve goods whose opportunity cost is opening a crate – tryst with freedom Sep 14 '23 at 21:23
  • Answer is not clear. Math is clear but not sure how it says something new – tryst with freedom Sep 14 '23 at 21:24
  • @ReineAbstraktion no, this is dynamic problem. Ex ante you open the crate if value of what could be in crate is above cost of procuring crate including opportunity cost. However, ex post once you open the crate its just value of an item. If I sell you real crates that can contain either gold or dirt, then ext ante the value of the crate will be somewhere between value of gold and dirt, whereas once you open the crate and discover it only contains dirt then that is the value you got out of the crate ex post – 1muflon1 Sep 14 '23 at 21:29