Steam’s trading cards are an interesting phenomenon in Steam’s wider Community Market economy. Valve’s Gabe Newell has said that his interested in creating markets where players can exchange items cross-games, he would like to see the value that the player has created one game to carry over to another.

His example is a game where the player has spent a lot of time and has top-tier equipment but wants to start to play another game; in the new game he’s poor and has poor equipment and he needs to spend a lot of time to be powerful. Currently, the Steam platform gives the player two possibilities¹: he can trade items between other players directly, or he can go to the real-money Steam Community Market where he can buy and sell items in cash.

One could argue that this is the point of playing the game in the first place and part of the natural progression - “buying” your way instantly to top means the player will miss a lot of the intended experience. However, Newell would like that the player can sell his old equipment and use the money to buy similar value items in the new game: like you would sell your tennis racket to buy a golf set.

Clearly there is value in the virtual items. High-level characters in many MMORPGs go for substanial amount of money. This value is usually derived from the value of time it would take for the player to acquire the items through normal play. Similarly, in Valve’s Team Fortress 2 or Dota 2, the random drops mean that a player would eventually get the item he wants, but he might be inclined to take a shortcut.

Steam trading cards are a bit more difficult to explain, because there seems to be very little meat around the metagame but a lot of complexity. Some of the games on Steam come with a set of trading cards that the player unlocks by just playing the² game - the card drops are linked to playtime, not any achievements within the game, a player would get all the cards by just idling on the main menu for enough time. However, the player can only acquire up to half of the cards in the game’s set. For example Half-Life 2 has 8 cards, 4 of which the player gets by playing the game. The rest he needs to get by trading. And, like with the old school physical trading cards, it’s entirely possible to get duplicates instead of getting 4 distinct cards of the 8 card set. Also, the complicate things further, there are also foil versions of each card. They are very rare (and also more valuable) and do not count towards completing a normal set and instead are a set of their own.

In the Half-Life 2 example, this means that the player has the following probabilities of having n duplicates³:

0 duplicates	\(\frac{7}{8} \cdot \frac{6}{8} \cdot \frac{5}{8} = \frac{210}{512} \approx 41,0\%\)
1 duplicate	\(\frac{7}{8} \cdot \frac{6}{8} \cdot \frac{3}{8} + \frac{7}{8} \cdot \frac{2}{8} \cdot \frac{6}{8} + \frac{1}{8} \cdot \frac{7}{8} \cdot \frac{6}{8} = \frac{252}{512} \approx 49,2\%\)
2 duplicates	\(\frac{7}{8} \cdot \frac{2}{8} \cdot \frac{2}{8} + \frac{1}{8} \cdot \frac{7}{8} \cdot \frac{2}{8} + \frac{1}{8} \cdot \frac{1}{8} \cdot \frac{7}{8} = \frac{49}{512} \approx 9,6\%\)
3 duplicates	\(\frac{1}{8} \cdot \frac{1}{8} \cdot \frac{1}{8} = \frac{1}{512} \approx 0,2\%\)

In other words, in 59% of the cases (assuming a set has 8 cards) the player has duplicate cards and clear incentive⁴ to sell or trade them away. This is good for the market, because it means there will be supply to match the inevitable demand. If the duplicate likelihood was too low, there would not be enough supply as people would be more hesitant to sell off their non-duplicate cards⁵.

So, what’s the point of collecting these full sets? For one, each collected (and “crafted”) set gives the player 100 XP and potentially increases his level in the Steam Community. The player will also get a game-specfic emoticon and a profile background (both almost always entirely worthless in the economy). The player can also get a discount coupon for a game or game content⁶. Once a full card set is “crafted” and the rewards given to the player, the cards are removed from the economy. If owning a game was the only source of cards, in an ideal situation only 50% of the playerbase could ever get a full set. Fortunately, once a player has exhausted his initial drops, he becomes eligible for booster packs. A booster pack is a random set of 3 cards for a set, so if the player is really lucky (2% chance), he could have 7 distinct cards out of the 8 Half-Life 2 cards after opening his first booster pack.

How does one get booster packs? They drop totally randomly for players who have exhausted their initial card drops. As mentioned before, the cards are removed from the economy once a full set is crafted, so booster packs are a way of recycling them back to the economy. The global booster pack availability is linked to the amount of crafted sets (or, badges) so that Valve/Steam can control the amount of cards in the economy and a player’s chance of getting a pack is linked to his Steam Community level. This is a neat way of making the rich richer and rewarding active participants of the Steam Community market economy to keep on participating. And, what’s the most important, these booster packs can be sold unopened just like any other item.

In the ideal and simplest case, the worth of a booster pack would equal the worth of three cards. It’s quite realistic to assume that each card has the same exact properties (drop rate) and so have the same supply, demand and price so it doesn’t matter which cards are in the booster pack… unless you’re worried about duplicates. Also, to complicate things, Steam takes a fee of each transaction made on the Steam Community Market⁷. This fee is usually 15% (or at least 0,02€) of which 10% goes for the game’s developer and 5% to Steam.

In any case, the natural maximum price⁸ for a booster pack is

\[P_{booster} = 3((1-p_f)P_{card} + p_f \cdot P_{foil})\]

where p_f is the foil drop chance (p_f > 0). If we further assume that considering their rarity, the cards are at least⁹ as valuable as normal variants (ie. \(P_{foil} \ge MP_{card}\), where \(M \ge 1\)), this means that

\[(1-p_f)P_{card} + p_f P_{foil} \ge (1-p_f)P_{card} + p_f M P_{card} \ge P_{card}\]

It’s also quite easy to assume that the markets are liquid and otherwise somewhat perfect for the normal variants or at least more so than for the rare foils.

We do not know the foil drop rate, but we can observe it in the marketplace. Currently the normal Half-Life 2 cards have a range of 0,13€ - 0,15€ (with around 5 000 for sale) and the foils range around 0,51€ - 1,08€ (with 100 of each for sale) which would imply M of somewhere around 3.6 - 7.2 and p_f of 2% (\(\approx \frac{100}{5100}\)). Based on this, it would seem that the foils are trading at a premium as

\[(1-0.02) + 0.02 \cdot 5.5 = 1.09 > 1\]

And the maximum price for a booster pack to be

\[P_{max} = 3((1-0.02)\cdot 0.1375€ + 0.02 \cdot 0.725€) = 0.45€\]

However, the price as of writing for a booster pack is 0,39€. There are many things that could explain the difference between the maximum price and the current minimum price at the market:

1. Low liquidity of the booster card market, it’s a buyer’s market as he can always go and buy the cards he wants. He would only buy a booster pack if the price was worth the risk.
2. The drop reate estimate might be wrong, 2% would mean one foil in 50 cards which sounds high. It’s very likely that the foils are overrepresented in the marketplace.
3. Simple average in the pricing formula assumes risk-neutrality, but people are risk-averse and want a discount for not knowing what’s in the box. Assuming each normal card costs exactly the average of 0,14€, the price of buying three specific cards from the market is 0,42€. Are people willing to pay 0,03€ for the chance of getting a foil?
4. Duplicates do matter and reselling cards incurs transaction fees, lowering their value

Assuming no transaction fees, it would be profitable to purchase booster packs at 0,39€, open them, and sell the cards for 0,14€ each netting a 0,03€ profit - not including the extra profits when happening to get a foil card! However, at 0,14€ price, Steam’s transaction costs would be 0,02€ per sale so the player would only get 0,36€ (+ extra profits from foil cards) or make a loss of 0,03€.

We can use this to calculate the minimum price for a booster pack, as we just assume the player resells everything he encounters.

\[P_{min} = (1-p_f) Pr_{card} + p_f Pr_{foil}\]

Where \(Pr_x = P_x - C_x(P_x)\), and \(C_x(P_x)\) are transaction fees at price P_x. At 0,73€, the transaction feees are 0,09€ so the net profit for a foil card would be, on average, 0,64€. This would put the price at which anyone should buy a Half-Life 2 booster pack as they could always¹⁰ open it up and break-even by selling the cards individually at

\[P_{min} = (1-0.02) \cdot 0.12€ + 0.02 \cdot 0.64€ = 0,39€\]

However, because of transaction fees, this is not the lowest price someone would be willing to sell. Because the seller wants at least 0,39€ in profit, the price including transaction fees would be 0,04€ higher, or 0,43€, for the buyer. Yet, the current lowest buying price is even less than that. What’s going on?

The sellers are risk-averse. There is a probability that the contents are worth less than that. The lowest price at which he would always be better off selling the cards individually is three times the profit one would get from selling the cheapest card in the set, in this case 0,33€ or (or 0,37€ including fees).

So, we end up with three prices for booster packs from the seller’s point of view, assuming he has no use for any of the cards other than selling. 0,37€ is the lowest price even the most regret-minimizing seller would put his booster pack for sale at. A seller from a probability textbook who looks at expected revenue would be ready to sell from 0,43€, with repetition he would break-even at this price. The highest price a buyer from a probability textbook would be willing to pay is 0,45€, but if he is regret-minimizing, he would only pay up to 0,33€ - the profit he’s guaranteed from reselling even the cheapest cards.

We can see that in a world of regret-minimizers and transaction fees, there is no acceptable price range for both parties (selling begins from 0,37€ but buying stops at 0,33€). In a more risk-neutral world, we would expect price ranges between 0,43€ - 0,45€. Yet, in the real world the sellers are ready to sell lower than what expected value is, forgoing value to make the sale because the buyers are risk-averse.

Another interesting aspect is to find out how much you would be willing to pay for a booster pack. Given the state of your inventory of cards, what is the price you’d be willing to pay for a booster pack instead of buying the cards from the market. This is interesting, because in the real world you’re not at either extreme: duplicates matter, but only to some extent.

The funny thing is that the expected difference between the decisions is just cents, but when the cards cost on average 0,12€, it can be 10-20% of a card’s price. Yet, this is what makes Steam Community Market so interesting - it’s a bit like stock market but the stakes are very low yet there is real money on the line. On top of everything, the market participants are people who play a lot of games, so one would expect them to make rational decisions! What could be a better little¹¹ laboratory for a decision theorist or an economist?