## Game Theory and Introduction to Metagame Shift

Magic: the Gathering players have long been aware of the shifting metagame, or the change in popularity of certain decks with the passage of time and tournaments. This article is an attempt to fuse game theory models with past experience to explain part of how and why metagames shift. (For basic definitions of the metagame and game theory, follow the asterisk.)*

Rather than maximizing utility or expected money payouts, game theory as applied to Magic: the Gathering looks at getting the most wins as its highest goal. (This may not be consistent with other players' goals; for example, some players may play decks on dares, or ignore the “best deck” if it is incompatible with the player's favorite style.) In the quest to get wins, many players of tournament Magic: the Gathering will switch decks if they believe another deck has the advantage in a particular metagame. The result is a churn of popularity; what may be held as a sure road to victory one week could lead to disaster the next. It may be possible to model this shift in deck preference by simplifying the complex conditions of a series of Magic: the Gathering tournament into an easily analyzed package of numbers. As a thought experiment, presume a series of tournaments with simple rules… no sixty-card decks, just probabilities.

For the fictional tournament series that follows, these rules apply:

1. Only three decks are legal to play: Deck A, Deck B, and Deck C. There is no variance in deck lists; that is, one Deck A is the same as any other.

2. All players are of exactly the same play strength and are unaffected by other factors.

3. Winners are determined by the best score after twelve rounds of play; there is no single-elimination portion.

4. All matches are one game each, and draws are not allowed.

5. No one deck will ever make up less than ten percent of the field.

Also for the fictional tournament series, the following winning percentages are going to be presumed:

Deck A will win 75% of the time versus Deck B, but only 40% of the time against Deck C.

Deck B will win only 25% of the time versus Deck A, but 70% of the time against Deck C.

Deck C will win 60% of the time versus Deck A, but only 30% of the time against Deck B.

When identical decks face off (e.g. Deck A versus Deck A) winning percentages are 50%.

With those conditions out of the way, the tournament simulations may begin. For Tournament One, let it be assumed that no player has any special knowledge of which decks are best for a format, and thus the players do not know the percentages outlined above. (An analogous real-world tournament would be States.) In this simplified simulation, presume that equal numbers of players will pick up Decks A, B, and C. The “Expected Value Formula” can provide a good estimate of a deck's relative strength. The Expected Value Formula provides the number of wins a deck playing in the twelve-round tournament can expect to have, hence “Expected Value,” or EV. The expected value of any particular deck for this tournament, in mathematical terms, is:

Expected Value = 12 [(probability of playing Deck A) (probability of winning) + (probability of playing Deck B) (probability of winning) + (probability of playing Deck C) (probability of winning)]

In text, the number of games a deck can expect to win equals the sum of the likelihood of playing a deck multiplied by the likelihood of winning against that deck for each deck, multiplied by the number of rounds in the tournament. The explanation may still be unclear, but inserting numbers may clarify matters. Referencing the percentages above, the expected values are as follows:

Deck A's EV = 12 [(0.333)(0.50) + (0.333)(0.75) + (0.333)(0.40)] = 6.6 games

Deck B's EV = 12 [(0.333)(0.25) + (0.333)(0.50) + (0.333)(0.70)] = 5.8 games

Deck C's EV = 12 [(0.333)(0.60) + (0.333)(0.30) + (0.333)(0.50)] = 5.6 games

Obviously, not all decks will achieve the expected value (as a matter of fact, in this example no one player could exactly match the expected value!), but this expected value is a good estimate of overall strength. While statistical quirks are the difference between winning the tournament and failing miserably, it is intuitive that Deck A will be heavily represented in the upper echelons of the tournament's final standings. Decks B and C will likely have less representation, with Deck C being statistically weakest.

For Tournament Two, players have a good idea that Deck A is the “best deck” in the format, and players willing to play any deck that has the best chance of winning will likely migrate to Deck A. For the Tournament Two simulation, presume that now half the players in the tournament will be using Deck A, and the rest will be equally split between Decks B and C, since the statistical difference between them in Tournament One is likely to have gone unnoticed.

Deck A's EV = 12 [(0.500)(0.50) + (0.250)(0.75) + (0.250)(0.40)] = 6.45 games

Deck B's EV = 12 [(0.500)(0.25) + (0.250)(0.50) + (0.250)(0.70)] = 5.1 games

Deck C's EV = 12 [(0.500)(0.60) + (0.250)(0.30) + (0.250)(0.50)] = 6.0 games

Deck A seems to be an unstoppable juggernaut. Its high winning level, plus its sheer numbers in the tournament, virtually assures its dominance in the standings at the end. Deck C also has a few players mixed in; the overall edge the deck has against Deck A is beginning to manifest, though it is overshadowed by the crushing victory of Deck A. A player playing Deck B in this tournament will likely leave unhappy, with a day of defeats dogging the unfortunate deck-wielder out the door.

For Tournament Three, virtually nobody is playing deck B anymore. Most of the deserters will shift to Deck A; Deck A will be representative of sixty percent of decks. Deck C will also win converts; these are likely players who simply want to play a deck other than Deck A, though players do have other reasons for converting.

Deck A's EV = 12 [(0.600)(0.50) + (0.100)(0.75) + (0.300)(0.40)] = 5.94 games

Deck B's EV = 12 [(0.600)(0.25) + (0.100)(0.50) + (0.300)(0.70)] = 4.92 games

Deck C's EV = 12 [(0.600)(0.60) + (0.100)(0.30) + (0.300)(0.50)] = 6.48 games

Simply put, Deck A has become a victim of its own success. The number of Deck A versus Deck A matches (or “mirror matches”) combined with the loss of its favorite prey has led to it having an under-six (or losing) expected value. Deck C, while not having as much representation in the overall field, is likely well-represented (and almost certainly disproportionately represented) in the Top 8 places in the standings. Deck B still seems to be a lost cause, with monuments to Saint Jude** appearing in tournament halls nationwide.

For Tournament Four, many players keep with Deck A, but some defect to Deck C. The proportion of players playing Decks A and C are now equal at forty-five percent. Deck B retains its stalwarts at ten percent.

Deck A's EV = 12 [(0.450)(0.50) + (0.100)(0.75) + (0.450)(0.40)] = 5.76 games

Deck B's EV = 12 [(0.450)(0.25) + (0.100)(0.50) + (0.450)(0.70)] = 5.73 games

Deck C's EV = 12 [(0.450)(0.60) + (0.100)(0.30) + (0.450)(0.50)] = 6.3 games

Tournament Four is Deck C's coronation. Statistically the weakest at Tournament One, nobody is laughing at Deck C's power. With its predator in hiding, Deck C seems ready to take on all comers.

For Tournament Five, even more players desert Deck A, basking in the newfound glory of Deck C. Deck A retains only thirty percent of the players' allegiance, with Deck C capturing a full sixty percent of the tournament scene. Deck B remains at ten percent.

Deck A's EV = 12 [(0.300)(0.50) + (0.100)(0.75) + (0.600)(0.40)] = 5.58 games

Deck B's EV = 12 [(0.300)(0.25) + (0.100)(0.50) + (0.600)(0.70)] = 6.54 games

Deck C's EV = 12 [(0.300)(0.60) + (0.100)(0.30) + (0.600)(0.50)] = 6.12 games

While it is quite likely that Deck C was the eventual winner of this tournament, due to the low turnout for Deck B, Deck B's disproportionately strong showing has probably attracted some attention. Deck C's players are not necessarily disappointed, but they may be wanting something more.

For Tournament Six, the metagame's former wallflower, Deck B, has a small turn in the sun. Thirty percent of players have now adopted Deck B, with the same sixty percent toting Deck C. The now-maligned Deck A has only its devotees, with ten percent of the metagame.

Deck A's EV = 12 [(0.100)(0.50) + (0.300)(0.75) + (0.600)(0.40)] = 6.18 games

Deck B's EV = 12 [(0.100)(0.25) + (0.300)(0.50) + (0.600)(0.70)] = 7.14 games

Deck C's EV = 12 [(0.100)(0.60) + (0.300)(0.30) + (0.600)(0.50)] = 5.40 games

How the mighty have fallen! Deck C has fallen to a crushing last place, with a vast majority of the Deck C players bewailing their various fates as Deck B dominates the tournament from a minority standing in the metagame.

For Tournament Seven, Deck B revels in its glory and players flock to it, eager to cash in on its success. The percentages for Decks B and C transpose, with Deck B commanding sixty percent of the metagame and Deck C holding only thirty percent. Deck A keeps its loyal following at ten percent.

Deck A's EV = 12 [(0.100)(0.50) + (0.600)(0.75) + (0.300)(0.40)] = 7.44 games

Deck B's EV = 12 [(0.100)(0.25) + (0.600)(0.50) + (0.300)(0.70)] = 6.42 games

Deck C's EV = 12 [(0.100)(0.60) + (0.600)(0.30) + (0.300)(0.50)] = 4.68 games

The severely wounded Deck C is percolating in its own misery after this tournament. While Deck B makes up the bulk of the top ranks of the tournament standings, the ease with which Deck A plowed through its matches with Deck B has doubtless attracted notice for the next tournament.

For Tournament Eight, Deck B retains half the metagame, but Deck A runs a close second with forty percent of the decks in the tournament. Deck C keeps the home garrison at ten percent.

Deck A's EV = 12 [(0.400)(0.50) + (0.500)(0.75) + (0.100)(0.40)] = 7.38 games

Deck B's EV = 12 [(0.400)(0.25) + (0.500)(0.50) + (0.100)(0.70)] = 5.04 games

Deck C's EV = 12 [(0.400)(0.60) + (0.500)(0.30) + (0.100)(0.50)] = 5.28 games

The ascendant Deck A reclaims the throne it last held between Tournaments Two and Three. With the overwhelming dominance of Deck A, neither Deck B nor Deck C seems like a viable alternative, despite the fact that Deck A has fallen in the past; doubtless the continued presence of Deck B is a contributing factor.

For Tournament Nine, Deck A takes seventy percent of the metagame. Deck B keeps a few stragglers beyond the usual suspects, maintaining twenty percent of the vote. Nobody sees any point to transferring to Deck C, so it stays at ten percent.

Deck A's EV = 12 [(0.700)(0.50) + (0.200)(0.75) + (0.100)(0.40)] = 6.48 games

Deck B's EV = 12 [(0.700)(0.25) + (0.200)(0.50) + (0.100)(0.70)] = 4.14 games

Deck C's EV = 12 [(0.700)(0.60) + (0.200)(0.30) + (0.100)(0.50)] = 6.36 games

Besides its obvious strength in numbers, Deck A maintains the highest expected value, beating out Deck C, albeit by a small margin. It would be unsurprising if all the top five spots were occupied by Deck A, and not all that improbable that all top ten would be Deck A as well. Deck B continues to be punished by the metagame.

For Tournament Ten, Deck B loses all but its most hardcore proponents. They split evenly between Decks A and C, leaving seventy-five of all decks being played as Deck A, fifteen percent as Deck C, and the remainder as B.

Deck A's EV = 12[(0.750)(0.50) + (0.100)(0.75) + (0.150)(0.40)] = 5.82 games

Deck B's EV = 12[(0.750)(0.25) + (0.100)(0.50) + (0.150)(0.70)] = 4.11 games

Deck C's EV = 12[(0.750)(0.60) + (0.100)(0.30) + (0.150)(0.50)] = 6.66 games

By expected values, only Deck C could expect to have a winning record over the course of the day, despite the fact that the Deck C players were only fifteen percent of the field. Doubtless Deck A has some representation at the top, but it would not be a surprise at all to see copies of Deck C mixed in. Deck B is still wallowing in losses, the unfortunate victim of the rest of the metagame.

How can these results be distilled? Assuming the form of “top deck” > “middle deck” > “bottom deck” for expected values, here are the shifts in deck placement over time (ignoring the proportional representation in the metagame.)

A > B > C

A > C > B

C > A > B

C > A > B

B > C > A

B > A > C

A > B > C

A > C > B

A > C > B

C > A > B

Though this metagame is completely fictitious, the patterns are not, and while real metagames are far more complicated (the model gets very messy with even four decks), the principles remain the same: find the big players, position oneself to *beat* the big players and most of the rogues, and let the rest come down to skill and luck.

John Beety

* Definitions:

The “metagame” is the pattern of decks found at a tournament. For example, suppose only three deck types existed at a tournament (excluding minor differences in card choice) and that half the players were playing Deck A, while the other players were evenly split between Deck B and Deck C. The metagame for that tournament would be 50% Deck A, 25% Deck B, and 25% Deck C. The metagame is very important to most tournament players; if one knows that 90% of players are taking Deck A to the tournament, for example, and Deck B loses three out of four times to Deck A, common sense says Deck B is probably not the best choice to take to the tournament. If Deck C wins three out of four times, it is a much better choice.

“Game theory” is a branch of economics that uses logic and mathematics to determine a person's best (or rational) behavior. This takes the form of viewing an economic problem as a game with moves, decisions, and strategies. A game is also defined by strategic interdependence; translation, each player's activity and strategy is defined, at least in part, by the opponent. (For example, a creature played by one player is bound to influence the other's strategy, leading that player to either deal with the creature or ignore it at his or her own peril.)

** Saint Jude is the patron saint of lost causes, such as the author's Constructed rating.