Towards a Unified Theory of Resources: Exploring Mechanisms of Exchange
In my last article, I discussed the concepts of expected value and utility as they relate to card advantage. This article will apply those concepts to other resources, completing what I intend to be a framework for understanding resources in the game of Yu-Gi-Oh. Before going on, you should read the preceding article about Sixth Sense and expected utility if you haven’t already. As this article will pick up where the last one left off, it will assume the reader is already acquainted with the concepts introduced and explained in the last article.
The project of modeling resources
You’d be a shrewd duelist to know that card advantage is one of many resources in the Yu-Gi-Oh card game. The concept is initially a little difficult for most duelists to grasp, as unlike most other card games, Yu-Gi-Oh does not have a resource system “built-in” to the game’s design (e.g. lands in Magic: the Gathering, energies in Pokemon, and so on). As a result, our theories regarding resources are underdeveloped compared to those of most other card games. What exactly are resources, and how are they present in Yu-Gi-Oh? Furthermore, why should we care about them?
The short answer is that resources are the things you use to do the things that make you win. But even the concept of “things” within a card game is a loaded one, and one that may be difficult to fully grasp at first glance. When we inquire into the abstract nature of resources, we are embarking on a very special sort of theoretical endeavor unlike the “theory-oh” most duelists are used to. We will not be talking about the positioning of Mermails or Fire Fists in the metagame, and we will not be arguing over how many copies of Maxx “C” to sidedeck at the next big premier event. Our focus is much more rudimentary: we are looking for first principles, or basic truths that extend to all games of Yu-Gi-Oh. As Chapin writes in The Theory of Everything, we are looking for an account of what is “actually going on” in a game of Yu-Gi-Oh.
Resources are scalar values that give the player the ability to take actions in-game to attempt to secure a winning outcome. At any given point in the game, each player has a certain number of each resource, and through the player’s possession of these resources, he or she is able to initiate certain in-game actions pursuant to the ultimate goal of winning the game. These actions include, but are not limited to, effects, summons, and attacks.
Therefore, utility as a measure of a player’s benefit inheres in resources; it is in the essence of resources to grant some utility to the player, in that resources give the player the ability to win. This is the function of all resources: the only end resources serve is to help the player win. At any given point in a game, each player has a certain amount of utility dependent on their utility functions for each of their resources. It thus follows that there is always some ratio of any two resources such that the player is indifferent between the two. No matter the game state, there is some amount of life points that a player would be willing to trade for an extra card, and likewise, there is some number of cards that a player would be willing to trade for extra life points. While these resources are not perfectly fluid in that players do not always have the ability to exchange one for the other, it is nonetheless an exchange that players can make, and often do.
We have already seen that cards are subject to diminishing marginal returns. It seems fair to assume the same for any other resources we might allow. It is inconceivable to me that any resource’s marginal utility could have constant marginal returns that never decayed. If this were the case, there would exist some resource that the player truly could “never have enough of.” But resources only have utility to the player as a “means” to the “ends” of winning the game. Therefore, there will always be some amount r of a resource sufficient for the player to win the game with. If this is the case, the player must be indifferent between having r and r+1 of that resource, since the additional resource can’t help the player win the game any more in that scenario. Just as we label some cards as “win-more” when they only benefit the player in winning situations, so too do all resources eventually become “win-more” when the player amasses infinitely large quantities of them.
With this, we have described two qualities that inhere in the essence of resources, such that all resources must possess these qualities by their very nature. These assumptions will guide the models to be demonstrated later in the article. First, resources grant the player the ability to take actions that result in that player winning the game. If having some nonzero amount of W does not give the player access to some sort of action or mechanic by which he or she could conceivably win the game, W must not be a resource. Second, resources are subject to diminishing marginal returns. If W’s marginal utility to the player does not decay with the addition of more units in some way, W must not be a resource.
Life points as a resource
Life points are an underappreciated resource in the game of Yu-Gi-Oh. When discussing cards like Solemn Warning, Upstart Goblin, and Seven Tools of the Bandit, players will often say things like “the life just doesn’t matter.” But these players are missing the minute ways in which life totals affect decisions and how games unfold. Indeed, this seems to be the case with some cards, like Return from the Different Dimension. Often, it may feel that your life total becomes irrelevant when your field is hyper-developed and your opponent has no way of reaching your life points. But this is always due to an overwhelming dominance of one player over the other in some other resource, such as card advantage. It does not follow that life points do not matter to the player at all, only that there are some quantities of other resources that can outweigh a deficiency in life points in terms of the utility of the player, and vice versa.
It is evident that life points are subject to diminishing marginal returns. Games have a finite number of turns, as a player will eventually deck out if each player keeps passing, and there is a finite maximum amount of damage a player can do in a turn. It thus follows that there exists some life point value that is so large that your opponent will never be able to reduce it to zero before the game naturally ends. This means that life points beyond that value are useless to the player. Furthermore, the probability that an opponent will be able to deal a certain amount of damage throughout a game is lower for larger amounts of damage. The last life point before that upper bound is less valuable because there is a much lower probability that you would ever need it to not lose.
Like cards, players start with a nonzero amount of life points, but unlike cards, life points do not naturally replenish themselves. Indeed, in modern formats, life points are often not replenished at all, as there are few life gain effects in the current card pool that are powerful enough for constructed-level play. Like cards, we use life points to “do things” in a general sense, but these things aren’t as obvious to the naked eye as the benefits offered by cards. Just like cards, players always have some utility function for life points and some marginal utility of life points.
Utility calculations with life points work much the same as they do with cards. A player has the utility function of life points U(lp) at any given point in a game. The marginal utility of life points U’(lp) is found in the derivative of the utility function. As is true for all resources, the marginal utility of life points will decay as the player’s life points increase (and grow as they decrease). To no one’s surprise, marginal utility has the same meaning here as it did with cards: it represents the player’s benefit from an additional life point gained or lost. It may help to think of life points in multiples of 100 or 1000, as there are few real-world game scenarios where a player gains or loses a single life point.
Complication, or, life points for cards
Uniting the utility models for cards and life points gives us a glimpse into the metaphysical side of the game and reveals basic mechanisms for the exchanging of resources present in the game’s design. The one I will discuss in this article deals with the player’s ability to exchange life points for cards drawn. In order to understand this mechanism, we must turn our eye to the concept of attacking, one of the most basic notions in Yu-Gi-Oh.
What changes when your opponent knocks you from 8000 life points to 6200 with a direct attack from Machina Gearframe? In order to understand, we must first look at what life totals mean in a broader sense. A player loses the ability to play the game without life points. When your opponent connects with his first direct attack of the game, you have come closer to not being able to play the game. You are on what is called the “clock” in Magic: in a certain number of turns, given neither player plays any other cards and your opponent just attacks every turn, you will be dead in four more turns. This amount of time is derived directly from your life total.
Players gain cards as time goes on by drawing during their Draw Phases. While it may seem intuitive, keep in mind that when your life points hit zero, you lose the ability to gain more resources. By the same token,you can give up a number of life points by taking another direct attack and draw a card in return, or you can try to develop the board and attack your opponent’s cards. Your incentive to do this is directly proportional to your marginal utility of cards relative to both your life total and your opponent’s marginal utility of cards. You want to trade resources with the opponent when you need your cards less than he needs his, and you want to wait and draw more cards when you need your cards more than he needs his.
Any of the head designers at Konami will be able to tell you that at a fundamental level, Yu-Gi-Oh is a game about attacking. This is evident in the built-in interactions between cards and life points. The most fundamental means by which the player exchanges life points for cards is by taking attacks. This is made possible by the existence of the Draw Phase, which imbues “time” with a value in resources. As time passes, each player receives a steady flow of cards drawn through their normal draw for each turn. In almost any scenario, the player will always have a normal draw waiting for them in their next Draw Phase, and that new card is always going to yield the player some benefit. Every decision made by each player in the Battle Phase relates in some way to this exchange. There are certainly other ways for the player to exchange cards for life points and vice versa, generally through the use of certain card effects, but this method is unique in that it is built into the very design of the game. It depends on the presence of no specific card or effect, nor does it necessarily require initiation by the player through some line of play. As long as turns are passing and draw phases are occurring, this exchange is always happening in some way at every point of every game through the fundamental interactions of monsters and reactive cards in the Battle Phase. Therefore, it behooves the player to always be aware of this tradeoff, and adjust his or her play decisions accordingly.
The concept of utility allows us to weigh cards and life points against each other by extracting from the concepts of each resource the player’s benefit in its purest form. The question, then, is how to maximize that benefit through exchange. If you need one resource more than another, it obviously follows that you’ll want to trade the less valuable resource for the more valuable one given the chance. Suppose a player is posed with the choice of using Book of Moon to block an opponent’s direct attack. The player is essentially choosing between losing one card (by activating the Book) and losing life points equal to the ATK of the attacking monster (by taking the attack and keeping the Book). The correct choice depends on your marginal utility for each resource. Intuitively, you’re going to want to give up whatever you benefit less from: the Book or the life points. Given a player’s specific utility functions for cards and life points and a specific complete game state, we would be able to mathematically determine an optimal distribution of cards and life points for that player. This gives us an objective metric to evaluate the gains or losses that result from a player giving up life points or cards. In the example with Book of Moon, we know that one option is always worse than the other, even though we aren’t sure which one it is. In these scenarios, there will generally be one best play, and that play will involve giving up whichever quantity of resources is of less value to the player.
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