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Here is my first article on the subject, in the event you are unfamiliar with the basic system I am talking about.
So how did I come up with the idea for Marginal Mana? What was the realization that put everything together, the epiphany? The answer lay in a simple fact – Magic is a game of resource utilization. Once that realization stuck, the system came together almost immediately. I realized that what matters is not how any individual utilizes their resources, but that they utilize them better than other players. This angle of approach is the single reason I feel Marginal Mana improves upon the two existing candidate theoretical constructs for "The Grand Unified Theory of Magic." Here's the reason why.
What are the implications of reducing Magic to a game of resource utilization? What happens when you expose this core mechanism? Let's take a look. The best way to see this is to look at aspects of practical resource utilization and apply them to Magic.
Firstly, resources are designed to be converted into other things, and these potential options are represented in the value of a resource. As an example, let's consider a practical example of this phenomenon – money. The value of money is that it can be exchanged for other things. Money can be traded for luxuries as well as necessities, and can even be converted to more money via investment.
Thus resources have some inherent value that just arises from possessing the resource, not even utilizing it. As far as Magic is concerned, this is mostly applicable to cards as a resource, thus any systemic theory of Magic must represent the inherent value of possessing a card. A card has value beyond what it does, because the resource of a card is worth something. I'll elaborate on this later.
Secondly, the value of any given resources is proportional to the amount of it remaining, at least in practice. While in theory the value of a resource should not be dependent on the amount of it remaining, this is not true in practice. For example, if you only had $5 in your name you would probably feel less comfortable spending $1 on a candy bar than if you had one million to your name. While this is a rather extreme example, it showcases the idea that human beings attach increased value to the same resource in situations of scarcity.
The value of the candy bar, in the objective sense, is the same in both cases – $1. However, the subjective value of the bar is far greater to the person with only $5 to his name. This phenomenon relates to the idea of resource allocation. What is important is not the inherent value of the resource, in this case, the $1, but the portion of your available resources represented by that value. In terms of our example, it is not important that you are spending $1 on the candy bar. It is important that you are spending 1/5 or 1/1000000 of your total money on the candy bar.
Now, expand this example further to the concept of deficit spending – spending more than you earn. In the real world, this results in debt, which is unsustainable on a permanent basis. However, deficit spending is sustainable on a short-term basis, and is frequently used to accomplish specific objectives. For example, most people who go to college engage in deficit spending. They take out loans to pay for their education, assuming debt in exchange for education, which they hope will lead to a better job and increased profitability in the future. If this happens, the person is successful. If not, then the debt becomes a liability.
Magic operates in much the same way. A game of Magic operates completely within the realm of deficit spending. This is because Magic players start the game with a set number of resources which never increases. At the beginning of the game you have your library (and in some cases your sideboard), and 20 life points. There is absolutely no way to increase your total resources. You can augment one resource at the cost of another (spending a card to gain life, for example), but you can't increase the total pool of resources available to you.
The objective of a game of Magic then is to win the game before your deficit spending catches up to you. This is where Chapin's "ability to continue playing the game" resource comes in. In effect you are using deficit spending to trade your starting resources (cards and life) for your opponent's "ability to continue playing the game" resource, with the final objective being to deplete your opponent's resource before your resources are spent; your opponent, of course, is doing the same.
Thus, the value of a single element of your set resources in Magic has to increase as the total pool diminishes. In other words, the value of both a life point and card goes up as your life total and the number of cards in your library decrease. This is the reason for the exponential function for life total. As I said before, the function is not perfect, but it does illustrate the principle.
So why is a card only worth 2 and not represented by an exponential function? In reality, it is represented by a function, but for most practical applications thinking about it as 2 will suffice. This valuation is not true in two cases – when your opponent is actively attacking your cards in library (milling), and when your library becomes sufficiently depleted that decking naturally becomes a threat. The first is rare in tournament Magic (although far more common in limited than in constructed) and the second typically only happens in control mirrors or in limited when board stalls happen. In essence, the value of a card is effectively two in most cases and thus, for simplicity's sake, it is simply easier to think of it as "2" instead of some function.
These are two of the fundamental aspects of resource utilization that Marginal Mana attempts to incorporate. While it's not perfect, Marginal Mana as a system at least successfully reflects both of these aspects, which no existing system (to my knowledge) for the "Grand Unified Theory" of Magic does. These two aspects are absolutely crucial to Magic as a whole, and any system that attempts to unify our disparate heuristic concepts of the game must represent these ideas as well.
Projections and the utilization of Marginal Mana
The second thing that is crucial to remember about Marginal Mana is that it is an instantaneous measurement. In other words, it is only useful to determine, ceteris peribus, who is winning the game after any given action. Any projected value that comes from a permanent after the play in question must be looked at as a separate instance. The reason that this is true is because projected value is not always obtained. Projected value is, in effect, potential value and must be balanced against the likelihood of obtaining it.
This is actually one of the central problems of Marginal Mana. It has no built in method of determining the likelihood of obtaining projected values of a play. I'm not confident this is even possible in an objective sense, and thus the system leaves it to the player to determine how likely it is to get full value out of a play. Let's look at an example.
Player A is at 5 life, has six lands on the table and Wurmcoil Engine, alongside other cards, in hand. He is facing a traditional red-based aggro deck. Assume the player has made the decision to play Wurmcoil Engine. Let's examine this play from a Marginal Mana standpoint.
Player A immediately obtains value out of his play. The card moved from his hand to in play, and thus was neutral in that sense. However, Wurmcoil Engine is a 6/6 (worth probably 6.5 mana) and has deathtouch and lifelink. It's graveyard trigger probably adds about two mana worth of value as well, bringing the total for Wurmcoil Engine to about 9 mana. This is the immediate value of playing Wurmcoil Engine.
However, Wurmcoil Engine is also very well capable of generating potential value. If Wurmcoil Engine manages to block the player will gain a base of 4 points of Marginal Mana (Wurmcoil Engine turns into two creatures and takes one with it, causing a 4-point swing just based on cards). He will add to that six points of life, which is worth about 22.5 according to the Marginal Mana life equation. This brings the total value of the block to 26.5 Marginal Mana plus whatever the value of the creature Wurmcoil Engine kills is.
Wurmcoil Engine could very well also generate potential value by attacking. Assuming it went unblocked (the worst case scenario for the player), the opponent will suffer 6 damage and Player A will gain 6 life. Assuming Player A's opponent is at 20 the 12-point life swing is worth 24 Marginal Mana. Thus, if Wurmcoil Engine realizes even its next turn's potential value Player A stands to gain quite a bit. This is a potential-heavy play indeed!
But there are no guarantees that Player A will achieve that value. The game might end before Wurmcoil Engine ever gets to attack or block. Of course, the likelihood of this happening changes from situation to situation, but nonetheless in most situations it is at least possible Player A will lose the game before Wurmcoil Engine realizes its full potential.
For example, if Player A's opponent, whom we will call Player B, has six cards in hand it is very likely Player A is just dead. However, if Player B has zero cards in hand and no creatures on the board Player A is very likely to realize the full potential of his Wurmcoil Engine, and thus win the game. At this point, the Wurmcoil Engine is a strong play because of the potential it can generate.
Thus it is important to look at both the immediate and potential value of a play when determining its viability. Some plays only have a static value. For example, casting Blightning on your opponent, for example, is worth whatever the 3 life is worth plus 2 Marginal Mana, which you gain from the Mind Rot part of the card. Other plays, however, have both static and potential values. Evaluating the amount of the potential plus the likelihood of achieving it is important.
Returning to our Wurmcoil Engine example, assume Player B has Lightning Bolt in hand. Ostensibly that Lightning Bolt is worth 50 Marginal Mana (third, fourth, fifth life points). However, if Player B has only the Lightning Bolt in hand then he is still likely to lose the game, even if plays the Bolt. Playing the Bolt grants the Wurmcoil Engine's lifelink part a 67.5-point potential, thus raising the value of Wurmcoil Engine's attack far beyond that of the Lightning Bolt. Because Player A is more likely to achieve the full potential of his Wurmcoil Engine, in this situation the Wurmcoil Engine is likely worth more than the Lightning Bolt.
Aside: This is the other major imperfection in Marginal Mana. The life total function increases a bit too quickly, but for simplicity's sake it works reasonably well. If you prefer a more gradual function, the 28-(x/2) function I suggested in the first article might be better for you. You are also welcome to use a more complicated function if you want, but I believe that for ease of calculation the function should be simple. 2x is the simplest exponential function and the easiest to calculate, and thus I believe it is the proper baseline.
However, if we change the situation to one where Player B has, say, five cards in hand that Lightning Bolt rises in value, because one of those five cards is probably a burn spell, thus allowing Player B to end the game with this burn spells before Player A achieves the full potential of his Wurmcoil Engine. In this situation Player A probably believes he will lose the game, whereas in the previous situation Player A probably thinks he will win the game, despite his play being the same. This disparity is caused by what he feels is the probability his Wurmcoil Engine will live up to its full potential.
An excellent example of this "potential" phenomenon at work is Umezawa's Jitte. By itself, Umezawa's Jitte does very little. In fact, the value of the card is almost entirely stored in potential. It is a two-mana equipment that simply gains charge counters. What those charge counters are worth is highly situational. At any given moment a charge counter on Umezawa's Jitte has no objective value. It has three separate abilities the player can utilize, and it is only on the utilization that the counter obtains value, whether it is negating a removal spell, killing a creature, gaining two life, or dealing an opponent two damage.
As far as Jitte is concerned, the only cards capable of "dealing" with it while preventing it from generating any of its potential are Krosan Grip, Take Possession, Trickbind, Wipe Away, and Word of Seizing. Any other card in Magic would allow the controller of Jitte to at least get one counter off of it. The strength of Jitte is almost entirely in the potential value it generates, but that fact doesn't prevent people from playing the card because the likelihood of realizing that potential value is so high.
A good example of this is betting and odds. You have five dollars to bet, and can bet them in the following situations:
- 1:2 odds with 10% chance of success
- 2:1 odds with 20% chance of success
- 3:1 odds with 25% chance of success
- 4:1 odds with 50% chance of success
- 5:1 odds with 75% chance of success
I think given these options, everyone would take situation 5, as it has the highest EV in the long run. Thus, while in each of these above situations you generate a static value (five dollar bet on an event), the potential value of the situations differs, and thus they have greatly different expected values.
It is important to remember that Marginal Mana is a system that measures instantaneous advantage. Of course, Magic is not a game about individual moments. Most plays have both an immediate and a potential value. The value of projected plays can be assessed to some degree, but their expected value as a whole must factor in the likelihood that the full potential of the play will be realized. This probability is left to the player to determine.
Interactivity and the value of threats
Magic is a game of imperfect information. This is why the threat of a play in Magic can be so powerful. As a system Marginal Mana assumes that at any given time the evaluator has perfect information, which is not true of most game states. However, Marginal Mana as a system is capable of handling this discrepancy via the simple addendum of educated guessing.
Based on their knowledge of the opponent's deck and play style, along with the card pool, a player is asked, in game, to make an educated guess as to what their opponent's likely play is. In many situations the player's play is obvious, in others, not so much. Marginal Mana provides a way of evaluating the effectiveness of plays in situations where it is not obvious which play is stronger.
Many of these situations involve a specific threat. Situations like "it's correct to play Jace TMS here if he does not have Spell Pierce, but it is correct to play Jace Beleren here if he does" arise on a regular basis during games of Magic. These situations demonstrate the inherent value of the threat.
Often the threat of a card is enough to make the opponent change plays. This happens in situations where the potential loss is far outweighed by the potential gain. Counterspells frequently put players in this type of situation, and it is one of the reasons for their strength.
Marginal Mana can be used to evaluate how swingy these plays are. Returning to the Jace example above, assume, under worst case scenario, that you play your Jace, the Mind Sculptor and get it Spell Pierced, allowing your opponent to untap and play his own Jace, the Mind Sculptor with Mana Leak back-up. Marginal Mana is capable of determining the value you lost on this exchange.
On the other hand, assume you play Jace Beleren here with Negate or Mana Leak in hand, allowing you to potentially stop your opponent's attempt to remove your Jace. Marginal Mana can also be used to calculate the value of this play.
The difference between these plays shows you how much you are risking on the determination of whether or not your opponent has Spell Pierce. Sometimes, this value is negligible and you should go for the stronger play. Other times, it is quite significant and you should play around the threat. Which situation you happen to be in is, once again, left to the player to determine.
In some respects this is similar to the concept of pot odds in poker. For those of you unfamiliar with the concept, the basic idea is simple. If you know/suspect you are behind in poker, but have a draw that will win you the hand, you should call any bet where the amount you are risking in relation to the size of the pot is lower than your chance of making the draw.
As a quick example, let's say you are playing Texas Hold'em heads-up, and you have A,7 of spades as your hole cards. The flop comes K Hearts alongside J,3 of spades. Here you have a flush draw, which you have between a 29% and 36% chance of hitting (depending on how many spades your opponent is holding). Thus, you should definitely call any bet that is less than 29% of the pot and you should strongly consider calling any bet that is less than 36% of the pot. This is because if you make this call every time you will make money in the long run.
Marginal Mana cannot provide you with as clear a decision point as pot odds, but it does begin to give you an idea of how much you are risking on the chance that your opponent has the specific card he is threatening. This is another aspect where the system is superior to existing systems because while it does not reduce these decisions to an easy decision point it at least provides a tool and metric to assist the player in determining whether the risk is worth it or not.
Marginal Mana is by no means a perfect system. However, I strongly believe that it is a significant step in the right direction toward constructing a unified framework for all our concepts related to Magic. Marginal Mana provides methods for viewing Card Advantage, Tempo, Velocity, and the Philosophy of Fire. It is the only system (to my knowledge) that has the ability to represent these four major concepts. The system is a bit unwieldy (a problem I am working on), but it is a unified metric that can evaluate any situation in Magic. These systems have always proved to be useful in games.
Players who want to improve need a way to evaluate the strength of plays. This necessitates some sort of metric or scale on which players can judge the strength of individual plays. These metrics exist for most strategy games (they are what computers use to play the game). They vary in strength and complexity from game to game, but they all are useful for analysis. By finding stronger plays and understanding what makes them stronger, players improve. I believe that Marginal Mana is a step in the right direction for developing this sort of metric for Magic.
Chingsung Chang
Conelead most everywhere and on MTGO
