Flogging the Data: Determinants of Card Prices, Part 2

1 Introduction
This article is a continuation of the last one, the focus here is on elements of tournament and game design that affect prices. The full article is available here.
2 Game-Related Factors
There are two main questions with regard to game balance: card types and card color. For example, the introduction of the Planeswalker card type led to criticism that it was too powerful; the same was said of Equipment cards. For a long time Blue and Black were held to be strongest, with Green and White held to be the weakest colors. Assuming that all buyers prefer powerful cards to weaker cards, but do not have systemic preferences for a color or card type, we can assume that in a balanced game, card attributes should have an insignificant impact on card prices. To hunt for this, we are required to drop the variables that capture power level, ea sunt, Cube Power Ranking, Vintage restricted, and the Legacy and Modern banned lists.

2.1 Color Balance
Table 8 is evidence that Blue is not the most powerful color, and if there is a runt color, it is Red. On average Black cards are worth about 18% more than Red. This could be because many of the powerful Red cards are slotted as Commons (Lightning Bolt, Chain Lightning). Whereas many of the most powerful Black cards are Rares (Vampiric Tutor, Yawgmoth's Will).

Other things being equal, colorless cards demand about a 31% premium over colored cards, which could be due to the fact that these types of cards fit into any deck.

Color Balance and Price
Card is Red = +0% price
Card is White = +11% price
Card is Blue = +13% price
Card is Green = +17% price
Card is Black = +19% price
Card is Gold = +23% price
Card is Colorless = +43% price

The weakness of this type of analysis is that card power is concentrated at the upper end of the price segment with the vast majority of cards being "unplayable'', so any color-related power level might be concentrated in the highest priced cards. To ensure that the color results are robust to this aspect of card prices, the sample was also truncated in a few different ways: 1. the regression for cards with prices in the top 50th and 95th percentile; 2. by removing all the sets before Homelands. In general, Black remained slightly stronger and Red slightly weaker, which is essentially what is visible in Table 8.

When taking this logic even further, by sampling on the most 1-3% most valuable cards, some regressions reveal, in a statistically significant way, that: 1. Blue is about 56% more valuable than average; Green is worth about 29% less. While worth noting as a hint for future research and completeness, the results were not comparable with the general model used here, and thus omitted.

2.2 Card Types
The non-color attributes of cards seem more salient than color for determining card prices. The playing public generally recognizes how bad Aura cards are in actual play; this is because they are conditional on a given game state such as a permanent in play, and are prone to generate card disadvantage. The way Aura cards are valued by players indicates that WotC has not taken this inherent variance into account when designing such cards. Although about nigh 6% of all cards are Auras, there are perhaps only 3-4 that are playable across all tournament formats. Creatures sell at a 25% discount, which is not surprising given that the company prints legions of unplayable filler creature cards for its Limited format.

Card Attributes and Price
Card is Artifact = -8% price
Card is Aura = -23% price
Card is Creature = -24% price
Card is Enchantment = +6% price
Card is Equipment = +25% price
Card is Instant = +6% price
Card is Land = +67% price
Card is Legendary = +48% price
Card is Planeswalker = +0.0% price
Card is Sorcery = +11% price
1 mana increase in converted mana cost = -23% price
1 mana increase in colored converted mana cost = +14% price

The fact that Legendary cards are worth more on than others stands out. While Legends do have strong appeal to casual players and are even required to play the Commander format, these cards are generally not overpowered. Moreover, players usually do not need four copies because of the battlefield singleton rule, this makes their premium of 45% seem both a bit high and inexplicable.

Table 9 also reveals that there is a strong bias towards lower casting cost spells, and that this effect is also slightly non-linear. The model assumes that lower casting costs are a better thing in all formats. Going from 0 to 1 converted mana cost lowers a card's price by about 23% whereas going from 1 to 2 to lowers the price only by about another 18%.

If Wizards of the Coast had an appropriately tuned mana curve (or game design), converted mana cost should have little bearing on the price of a card. The positive coëfficients for the converted colored mana cost (CCMC) suggest that: either these spells are too aggressively costed in the game; or access to mana fixing is too prevalent. Given that lands, essential cards to play the game, are worth a 67% premium, the easy access to color fixing is likely to be the culprit.

3.3 Card Stocks & Tournament Legality
One aspect of investing in Magic has been speculation on the effects of reprints. Increasing the supply of a card should lower the price. For some cards, there is an argument to be made that reprints can be neutral or even positive for certain cards as a reprint allows the card to be rotated into new formats and makes players aware of the power of cards: Mana Leak from Stronghold, a neglected counterspell, got a big boost when it was reprinted and became legal in Standard. However, this argument is more about tournament legality than reprints per se. Because the coëfficient for ln[Reprints] is insignificant in Table 4, the evidence that reprints lower prices is weak for cards not legal in Block, Standard, or Modern. However, the number of reprints is correlated with but not equal to the print run size. Hence, there is a good chance that using the number of reprints rather than absolute print run sizes is subject to measurement error.

Reprints, Tournament Legality, and Price
1% increase in reprints = -8% price
Card is Block legal = -38% price
Card is Standard legal = +42% price
Card is Modern legal = +11% price
1% increase in reprints for Block legal cards = -14% price
1% increase in reprints for Standard legal cards = +16% price
1% increase in reprints for Modern legal cards = 5% price
1% increase in reprints for a 1% newer card = +2% price
1% increase in reprints for a 1 rank in draft pick = -3% price

To disentangle the tournament legality question from the issue of reprints, several interaction effects were included in the model. What this means is that the number of reprints for a card are evaluated conditional on the card being legal in a tournament format. The term ln[Reprints]*Standard captures the effects of reprints if the card is legal in Standard tournaments. The estimate above tells us that an additional reprint for an average Standard card lowers the price of the card by about 18%.

The multiplication of the logged number of reprints and power level tells us that reprints are especially detrimental to high-powered cards. Specifically, each reprint lowers the value of a card exponentially: an average first-pack first-pick Cube card like Sol Ring would lose about 36% of its value if it were to be reprinted; a reprint for a 10th pick Cube card like Regrowth, will lower the price only by about 13%. For cards that do not even making onto the power ranking list, this additional effect on price disappears.

2.4 Tournament Bannings and Restrictions
Cards get banned or restricted for different reasons: for being too powerful (Black Lotus); for being unfun (Trinisphere, Mental Misstep), or not fitting within the modern rules system (Darkpact, Chaos Orb). The table below reveals the impact that tournament legality has on prices. Keep in mind the model measures the effects of each class as if they were mutually exclusive.

Less tournament demand should drive prices down. So by "power-adjusting'' the estimates (i.e. removing the card power effects), estimates should capture the pure market effects of being on the list. One would thus then expect that all the coëfficients be zero or negative. Table 5 evaluates this hypothesis.

Tournament Restriction and Price
Card is banned in Vintage = -57%
Card is banned in Modern = +246%
Card is banned in Legacy = +48%
Card is restricted in Vintage = +517%
Card is restricted in Vintage and banned in Legacy = -81%
Card is banned in Legacy and banned in Modern = -82%

This hypothesis is not rejected for the Vintage banned list. Vintage-banned cards consequently are worth about 57% less on average. This is not surprising since the list covers those cards that do not fit with the modern game, and these cards see practically no play, even in casual play.

The estimates for the Modern and Legacy are not negative as expected, but power-adjusting them does reduce the estimated tournament-legality effect. The fact that it is not negative raises the question of whether about the Cube Power Ranking is capturing all of the card power effects. Some statistical tests, presented in the main paper, provide additional evidence that the Cube Power Ranking is not an entirely accurate measure of power.

The coëfficients for the joint Vintage & Legacy and joint Legacy & Modern lists being negative and significant is in line with our hypothesis. And by these estimates, being on either of those lists lowers the value of the cards by about 82%. In simple terms, where there is no real tournament market for the card, card price suffers.

3 Conclusions
By decomposing the various effects that contribute to card prices, we now better understand card pricing. First, we measured the effects of the intuitive price drivers such as rarity and reprints, and then examined how game-related factors affect card prices.

Moreover, we have found no substantial evidence that there is an inherent color bias in the game. Card attributes do play an important role in the prices of cards, with Legends, Lands, and Colorless cards earning substantial pricing premiums; Aura cards are steeply discounted. Higher converted mana costs tend to drive prices down.

4.1 Future Research
Before undertaking a more refined analysis, it would be important to know the exact volumes of cards in circulation. Considering that print runs could explain up to 40% of the overall variance, obtaining or somehow inferring these numbers should be a research priority.

Future research would do well to incorporate price information from digital card sales as these comprise a substantial part of the card market and also represent a substitute for paper sales. A more detailed model might also include more card attributes like card condition or whether it is foil.

So get out there and write Wizards (Jennifer Meyen, Assistant Brand Manager) that you want to know just how many cards are in circulation. Your collection's value and my research will be æternally grateful.


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