Flogging the Data: Determinants of Card Prices, Part 1

1 Introduction
Card prices are common and hotly debated topic in the Magic community, but until now there has been very little comprehensive analysis about which factors drive individual card prices. Prices are more than just costs. Prices contain information about players' and collectors' preferences. This two-part article shows how to extract some of that information. Part 1 introduces the data, methodology, and looks at what drive prices. Part 2 uses those same methods and data to better understand the game's design.

If you'd like to flog the data a bit harder, the full paper with footnotes, equations, and statistical details is available here.

2 Research Design and Data
The basic design here relies on regression analysis of price data. Recall that regression analysis establishes a relationship between a variable to be explained (price) and one or more explanatory variables (card color, set, rarity, etc.). More importantly, regression analysis can help us understand how a card's price changes when any one of the explanatory variables, such as the card's rarity changes whilst the other explanatory variables remain constant. This design allows us partial out individual factors such as converted mana cost, color, or card type. There are a couple assumptions that my reader needs to understand (read: swallow) before proceeding.

2.1 Assumptions
The first and rather trivial assumption is that players and collectors pay more for cards that have favorable effects on game outcomes. If you look at the price of Jace, the Mind Sculptor at USD 59.99, a card that was so good that it was banned, and compare it to Geth, Lord of the Vault, a tournament-unplayable card of the same rarity and priced at only USD 1.49, this seems like a trivial assumption indeed. However, upon closer inspection, we realize that this assumption, while good, does not hold up entirely: foil cards are case in point. Foil cards exhibit systematically higher prices than regular cards despite the fact that their shiny finish has no bearing on game outcomes. The effect here can be attributed to collectors, who purchase cards for reasons other than tournament use. A similar logic applies to slightly and moderately played cards. Another assumption is that vendors do not collude or have systematic correlations in their inventory. That first assumption is fairly weak; the second one is a bit stronger because there is evidence that by using discrete rather than continuous pricing vendors do not optimally manage their inventory levels (Table 1, Table 2, Figure 1). We shall assume that on average these inventory inefficiencies cancel out on average whereby some vendors have too little stock and others have too much. Inventory effects could play a role where cards are thinly traded (e.g. Beta or Portal editions), many of the vendors do not carry stock and thus have correlated inventories.

A vendor's inventory may not follow an ``optimal'' natural distribution.

2.2 Data
Unlike stocks, sold primarily through a single market, Magic cards are sold through a wide variety of channels: retail, specialty shops, digitally, mail order, etc. Aggregating this price information is difficult. Online sales are highly competitive, and are generally the cheapest means of procuring medium to large quantities of cards. Given this (and the readily available amount of data), the design here uses the average prices of certain online venders. Specifically, the price data were polled on February 23rd, 2012 using Decked Builder, and they come from the www.TCGPlayer.com. The data include average price information for each card. These average prices include cards of all conditions. This price information was combined with the card attributes from the Gatherer database, again via Decked Builder. Some print run and rarity information was incorporated from www.crystalkeep.com. These data neither distinguish between foil and non-foils, nor various conditions.

Normalized results of MTG Salvation's 2011 Cube Power Ranking survey were used as an approximate measure of card power levels.

To get a better idea of a typical online vendor, a sample of price and stock data was pulled from Star City Game's website on 21-22.06.2011 for the sets from M11 through it Mirrodin Besieged.

Then some card attributes had to be manually coded to make them compatible. Phyrexian mana is treated as half colored and half colorless mana. Colorless alternatives in a card such as Spectral Procession are treated as two-thirds colorless and half colored. X is treated as 2 because it seems to be the most plausible value for most cards.

The data set is available here.

2.2.1 Rarity
The basic assumption here is that rarer cards are worth more due to that fact alone, but getting a complete model of absolute rarity in the population of cards available turned out to be a bit involved. The older sets, such as Arabian Nights, employed a very different rarity system than that of the most recent sets; moreover, the set sizes are quite different. Thus comparing an Arabian Nights uncommon to that of Innistrad is misleading due to the definition of uncommonness, their relatives sizes, and the rarity composition of the sets. Furthermore, Magic cards are not released through a single channel with a single rarity structure: there are starter decks, promotions, different foil insertions, etc. The simplifying assumption here is that the entire card population solely be defined by booster packs.

2.2.2 Print Run Size
One of the essential pieces of information is the number of cards printed for each set. While there is some information on print runs for 22 of the early sets (presumably because this information used to be released), it is not available for 73 of the more recent sets. Hence, I contacted the game's manufacturer, Wizards of the Coast (WotC), to obtain the size of the print runs for each of the sets. Its brand manager said that such data could not be made available. I proposed several statistically viable alternatives for addressing the issue of set size, whilst protecting WotC's interests, but I received no reply. Given the theoretical importance and apparent empirical salience of the print run size, players and collectors should insist on this information. It has real monetary implications for the investments in their collections. Due to the lack of real print run sizes, the logged number of reprints was used a proxy variable in its stead.

For those sets that do have print run numbers, the variable was highly salient. An (extremely naïve) bivariate linear regression revealed that a 100% increase in the print run lowers prices by 88%, and print run size describes about 40% of the price's total variance! This correlation between set size and card prices is evident in Figure 1a.

The point banding shows the range of card prices within a set (and its corresponding print run size).

3 Interpretation
The interpretation of the price analysis falls into two broad categories. The first is what game-independent features drive pricing. The second aspect uses prices to assess the overall balance and design of the game.

3.1 Price Drivers
This section looks at those factors that drive prices, but which are not necessarily related to game play, these include: rarity, time trends, number of cards in circulation, and tournament legality.

3.1.1 Rarity
There is not too much of a surprise that card rarity drives prices up. However as pointed out above, rarity in Magic is not typical measured in absolute terms. Consequently, dealers uses a heuristic short-cut when pricing inventory---they use standardized prices based on nominal rarity. This violates two assumptions underlying the model: 1. prices reflect the absolute rarity of cards; 2. vendors price that rarity efficiently on average. Star City Games' pricing model reveals a lot of discreteness as can be seen in Table 1, and is a potential violation of the first assumption.

Apart from nominal rarity, we assume absolute rarity also plays a role. Absolute rarity is how scarce a given card is in a stack of random cards; the inverse of this concept is commonality. Table 2 reveals that a percent increase in "true commonality" lowers the price by about a percent.

Table 2: Rarity and Price (vs. Common)
Card is Common = +0% card price
Card is Uncommon = +153% card price
Card is Basic land = +200% card price
Card is Rare = +1233% card price
Card is Mythic = +9300% card price
+1% "true commonality" = -0.92% card price

The next thing to note is that the estimate for Basic land is higher than the Common rarity. In our sample, basic lands tended to sell for more than most Common cards. Mythic rarity is estimated to increase the card price by about 93 times that of a Common. Given the median price of a Common at 3 cents, this leads to a predicted median Mythic of about USD 2.79, underpredicting the actual sample median Mythic price of USD 3.11 by 10%.

3.1.2 Time
Do cards increase in value over time? Can Magic be considered an "investment''? The answer is probably not. The average age of a Magic card is around nine and a half years old. Table 3 implies that if we add a year to that average card's age, a card worth a dollar today is only worth about 96 cents after a year; if inflation is taken into account, that loss rises to about 7% per year. Assuming demand is stable, this phenomenon is a bit disconcerting because investments in real assets should at least maintain pace with inflation.

Table 3: Relation between Card Age and Price
Full sample | +1% card age = 0.39% card price
Full sample w/o ABU | +1% card age = 0.15% card price
Urza's Saga onward | +1% card age = 0.26% card price

To test whether the findings were being driven mainly by the sample, the same model as above was tested to two subsamples. The first sub-sample excluded the Alpha, Beta, and Unlimited editions because the cards are universally seen as primarily collectors' sets, the time losses are slightly lower, but very close to significant.

Another supposition is that foil cards, introduced in Urza's Legacy and with higher prices on average, bias the estimates for the trend variable making it more negative. Indeed, the trend effect is lower and insignificant when the sample includes only editions with foil cards. It's difficult to attribute this to the bias or simply the fact that the sample is both smaller (more error) and the temporal dimension is shorter (less information).

There are a few alternative explanations for what could be driving both the bias and the negative time trend: 1. power creep; 2. collector tastes or familiarity; 3. card condition. Power creep, refers to the inherent bias that the game manufacturer has to print higher powered cards over time. This has the effect of devaluing all current card holdings, an effect which is independent of the cyclical demand for tournament cards. Power creep and card condition seem the most likely culprits. Power creep has been accounted for to a certain extent using the Cube Power Ranking, card condition has not. The average card from Arabian Nights has seen considerable more usage than the average card from the more recent Dark Ascension.

3.1.5 The Reserved List
Wizards of the Coast promised never to reprint some cards, these cards comprise the "Reserved List''. It was put in place to boost WotC's credibility after a series of overprinting débâcles. Wizards has not been entirely consistent. The List has changed a few times, and has reserved substantial carve outs for digital cards and functional reprints. It has since become a millstone around the neck of Vintage and Legacy tournament play. For example, a typical Vintage deck requires 5-13 cards on the Reserved List in order to be competitive, and these run from 400-1000 USD each, posing an extremely high barrier to entry. Consequently, the Reserved List is a highly controversial topic amongst players and collectors. Because abolishing the list would constitute either a violation of an implicit contract or promissory estoppel; litigants would thus have to show material harm to prevail in a lawsuit. Is there a way to abolish the list without harming collectors, to the benefit WotC and players? Possibly.

Being on the List raises the price of the card by about 40% merely taking them off the list would cause substantial harm, regardless of whether the cards would be reprinted. Recall though that being Standard legal raises the price of cards by about 42%. Hence, were WotC to reprint the cards once in Standard, card prices could even go up for cards currently on the Reserved List.

Table 6: Reserved List and Price
Is on Reserved List = +40% card price
Is Standard legal = +42% card price
+1% increase in reprints of a Standard legal card = -0.18% card price

The problem here is that the first reprint doubles the existing stock of cards (at least according to the construction of our proxy variable), and would engender an 20% loss of value. Speculators err...collectors would howl. The question becomes how could WotC mitigate the downside of the reprint. One part of the solution could be Mythic rarity---print runs at Mythic rarity must be much much larger before the volume reaches those of the old rarity structure of the cards on the Reserved List.

Another part of the solution would be to disband the reprint list, but not reprint immediately; this would not be legally actionable, but some collectors/speculators would unwind their positions hopefully lowering the premium for such cards. This would, with a bit of smart deployment, allow for reprints in Standard that would have higher prices. Collectors could then get out from underneath their golden burden, and repurchase after the rotation price drop.

3.1.6 Expansion Sets
Expansion sets are an integral part of the collectable card game genre. A set itself does not necessarily have a bearing on the game, but many attributes that affect price, such as general power level, mechanics, artwork, print run, are strongly correlated with a set. They are also a proxy for the size of the print run. But because such effects do not necessarily have a direct measure, the set coëfficients do not have unambiguous interpretation.

Portal Three Kingdoms does stand out as being systematically higher priced; it was conceived as a beginners set, but it has several cards that are overpowered. It also has strong collector's appeal for its distinctive oil and acrylic artwork and Chinese Warring States Period rather than a fantasy world setting. Common cards sell for about 10 times the normal price of a common; several rare cards sell for over USD 100.

Table 7: Reprints, Sets and Prices
| Price vs. Average Set |Median # Card Reprints | Print Run
Portal Three Kingdoms = +1612% | 1 | 16.7m
Beta = +470% | 6 | 7.3m
Starter 2000 = +385% | 5 | ?
Arabian Nights = +274% | 2 | 5m
Portal Second Age = +175% | 2 | ?
Alpha = + 171% | 6 | 2.6m
Unlimited = +19% | 6 | 40m
Ice Age = -69% | 1 | 500m
Chronicles = -73% | 2 | 180m
Fallen Empires = -73% | 1 | 363m
Mirrodin Besieged = -74% | 1 | 434m
Revised = -74% | 6 | 155m
Imputed print run
Limited Edition Beta is also a set that commands a significant premium; cards sell for about five times more than average. Beta and Unlimited present somewhat of an interesting case because we have both the print run information, and they are identical in terms of cards. Using the naïve regression from above, we can substitute the actual print run values for Beta and Unlimited to show Unlimited should be worth about 22% of Beta based on print run size; put another way Beta prices should be about 5.55 times those of Unlimited. According to Table 7 (of the full version), the results of a much more detailed model, Beta is worth about 4.8 times Unlimited. These comparative results again reäffirm the importance of print run size in that the estimate for a set is likely capturing a some of the effects coming from the underlying and missing print run size information.

Given that there is some information on set sizes, Table 7 also shows some guestimates of some of the print runs based on an imputed value using the full model.

We've seen how econometrics can be used to infer something about the card attributes and their prices. So far these attributes have been fairly objective determinants of price. In Part II of this article, we'll be taking a look at different aspects of play aspects of cards, such as color, converted mana cost, and the role they have on card prices.

Until next time, keep flogging that data.


Posts Quoted:
Clear All Quotes