The Economics of Sixth Sense, or, Using Expected Value and Utility to Understand Card Advantage and Trade-offs

The Economics of Sixth Sense, or, Using Expected Value and Utility to Understand Card Advantage and Trade-offs




The release of Legendary Collection 4: Joey’s World in October of last year marked the beginning of a turbulent period of time in the competitive Yu-Gi-Oh scene. When it was announced that Sixth Sense would not only be released in the TCG for the first time since its OCG printing in 2003, but also be legal for TCG tournament play, previous expectations of a quiet, fair format mostly free of format-warping staple “power cards” went out the window.
Combining an unparalleled potential for card draw with a variant effect dependent on a die roll, the card almost seems like an expose on “what not to do in card design.” The competitive community immediately and near-unanimously recognized the card’s high power level. Joe Giorlando even campaigned, albeit unsuccessfully, to have the card banned in the first ARG Circuit Series tournament where it would be legal.
This is not an article about the degeneracy of Sixth Sense. It is not another rant about the banlist and modern card design and the state of the game and how Konami is out for everyone’s money and will “break the game” as many times as it needs to to obtain it. My concern is with another controversy of which Sixth Sense was also at the center, a controversy of card theory rather than game balance. I believe that in the debates that occurred over the use of Sixth Sense, primarily in Dragons, the Yu-Gi-Oh community discovered an area of theory that had gone unexplored for the entirety of the game’s history up until then. This discovery has repercussions for the ways we think about our in-game resources, including cards and life points.
As time went on and players began to play more with Sixth Sense in their decks, the card became the focus of another controversy, this time regarding the card’s optimal use. Despite the card being overwhelmingly more powerful than most of the legal card pool at the time, players found themselves polarized over what numbers to select before rolling the die for its effect, and indeed, many superstar duelists of the time were brought to fisticuffs arguing over this very issue.
Conventional wisdom dictated that you should call 5 and 6 every time, with any deck. The reasoning is fairly straightforward: no matter what numbers you select, the odds that the roll matches one of the numbers you selected remain the same, so why not select the numbers that yield the most benefit for the player, so to speak, when this happens? And indeed, when the card first became legal, this seemed to most players to be the intuitive and optimal way to play Sixth Sense.
However, by the time November rolled around, more than a few prominent players had begun to rail against this notion, especially in the Dragon Ruler variants that dominated the game at the time. In decks with access to nine to twelve Dragon Rulers that can summon themselves and function almost entirely out of the graveyard, the milling effects of Sixth Sense became much more attractive. Sure, drawing cards is always awesome, but a lot of the time, even when the Dragon player lost the roll on Sixth Sense, he’d find some benefit in the cards he sent from the top of his Deck to the Graveyard. While drawing five or six cards will often end games in the Dragon mirror, drawing three or four cards will generally provide enough of a swing in advantage to do the same. Very rarely do the two extra cards between those outcomes make the difference between winning and losing a Dragon mirror. Milling, however, is a different story. Since Dragon Rulers could Special Summon themselves out of the Graveyard as if they were in the hand, milling five or six cards could often yield huge benefit to the Dragon player, either by fueling a Graveyard already full of Dragon Rulers with monsters to be banished for their effects, or by sending Dragon Rulers to the Graveyard that weren’t previously there, opening up new lines of play. The basic idea was that by calling 3 and 4, you minimized variance and maximized your chance at getting some “good” outcome. Excluding the ones involving a roll of 1 or 2, which are common between both strategies, with a call of 5 and 6, your worst outcomes are milling three and four cards, and your best outcomes are drawing five and six cards. By calling 3 and 4, you improve your worst outcomes and worsen your best ones. The question is, when and why do you want to do this?
Duelists quickly learned that the optimal call depended on the game state. They began to call 3 and 4 in certain game states, 5 and 6 in others, and even 2 and 3 on some occasions. This depended on the number of Dragon Rulers a player already had available in his grave, the number of Dragon Rulers remaining in the player’s deck, and even the number of cards in each player’s hand. It is this approach to Sixth Sense that I want to draw attention to in discussing the expected value and utility of cards.
The easiest way to understand the benefits Sixth Sense yields to the player that activates it is by starting with a raw consideration of card value, i.e. how many cards Sixth Sense’s controller gains by resolving it. This number is how many cards we say Sixth Sense is “worth,” in an abstract sense. Furthermore, in the case of Sixth Sense, this number isn’t constant. Whether or not you draw cards depends on the result of a dice roll. Remember that the “milling” outcomes have value to us as well. Think of having a Dragon Ruler in the grave as having it in your hand, minus one effect. Outside of the discard effects, you can do everything with a Dragon Ruler in the grave that you can with the same Dragon Ruler in your hand. Therefore, milled cards can be valued as fractions of cards.
Since each possible die roll outcome has an equal probability (barring rigged dice), the player’s expected value from each possible “call” for Sixth Sense is simply the average of each possible outcome for that call:
c represents whatever fraction of the value of a card we assign to a milled card. This number depends on the probability that one of the cards milled is a Dragon Ruler, and therefore, the amount of Dragon Rulers left in the deck. Therefore, it is not static: it changes as the contents of the Deck change. During the format where Sixth Sense was legal, this was certainly a loaded concept that played into Sixth Sense decisions in many ways. Milling outcomes are more attractive when the player’s deck is thinner, or when there are more Dragon Rulers in the player’s deck(or other cards that the player could benefit from milling, such as monsters to be used as “fuel” for the Dragon Rulers, or cards with effects that can activate in the Graveyard, such as Breakthrough Skill or Spore). However, these concerns are fairly format-specific, so they won’t play too much into the concepts we’ll be exploring later on. For our purposes, just know that the milling outcomes of Sixth Sense could have some value, on average, to the player. If we valued all of the milling outcomes the same, the decision in playing Sixth Sense would always be completely binary: calling 5 and 6 would always yield the most value on average, since all of the worst outcomes are constant and calling 5 and 6 yields the best best outcome.
Recall that expected value is a raw measure of cards as resources. We are looking at expected value because we want to understand the impact Sixth Sense will have on our resources, specifically, our card presence. In order to model the player’s “benefit,” in a general sense, from certain allotments of resources, I posit we use the abstract economic concept of utility. In economics, utility is used to represent consumers’ satisfaction from quantities of goods. Given that resources are that which gives the player the ability to take actions that ultimately alter the game state, a player’s utility of cards will represent his or her benefit from the ranges of actions offered by his or her cards. This is obviously something we would never be able to actually objectively quantify; rather, it’s something we can only estimate. However, the actual value of a person’s utility is not our concern. It follows from the definition of utility that a player is always better off with more utility rather than less. It is here that our focus lies. We will be isolating specific factors that affect the player’s utility of cards. Armed with this knowledge, a player can identify these factors in real-world game states and use them to determine optimal plays.
Our utility from cards is distinct from the mere tally of a player’s in-hand and on-field cards. Insofar as our goal in building any resource is to ultimately win the game, cards as a resource must be subject to diminishing marginal returns. There is some point at which an additional card would have no meaningful impact on whether or not I won or lost the game — namely, the point at which I win the game. This means that the fifteenth card I add to my hand is less helpful to me than the fourth. The marginal utility of cards, the utility gained from one additional card, decays as the player amasses more and more cards.
This brings us to the concept of expected utility. An action yields expected utility U to the player where U is the sum of the utility of each possible outcome multiplied by its probability. Expected utility will be used here to calculate how better or worse off a player is made by making a certain play over another. Consider the following sample utility function for cards:
Suppose c is equal to 0.3:
Given this function and value of c, a player is better off calling 5 and 6 when activating Sixth Sense. But this is not always going to be the case. As duelists discovered when they learned to adjust their calls based on specific factors in the game state, the optimal call depends on the player’s utility function.
Under the law of diminishing marginal returns, expected utility functions have decaying rates of change. This gives rise to risk aversion. Uncertainty tends to reduce utility, insofar as you would likely rather there be a 0% chance of your opponent having Mirror Force to punish your overextension as opposed to a 10% chance. Your risk aversion determines the magnitude of this impact on your utility. In turn, the rate of decay of your utility function for cards determines how risk-averse you are because it determines how much utility you lose for losing a card, and consequently, how much utility you lose when there is some possibility of you losing a card. The marginal utility of cards is given by taking the derivative of the utility function of cards:
This function represents the utility you gain from drawing one additional card. It is the slope of the graph of the utility function. This function is different for each player at any given point in the game, and while most of the time players may not even notice it, it is the foundation on which many of the most basic play decisions are made.
Just as an example, imagine you’re debating over whether or not to attack directly with a Thunder King Rai-Oh into some number of set cards. If your opponent is at 1900 Life Points, you’re going to have much more incentive to make that attack, and you’re going to be willing to accept a lot more risk in order to do so. Likewise, if your opponent is at full life, and the Thunder King is your last monster, you’re going to be much less willing to make the attack and risk losing your only monster. This makes intuitive sense given the importance of potential outcomes in which the player wins. The impact of any single outcome on a player’s expected value and utility is proportionate to its size, insofar as an outcome becoming better will drive the player’s expected value and utility up. A possible outcome in which the player wins has infinite utility to the player. When such an outcome is present, the player’s expected utility will be much higher. The marginal utility of cards will also be higher, affecting the player’s utility function and incentivizing riskier behavior.
Since your utility function is not constant, there will be some game states in which your marginal utility for cards is very high, and some game states in which it is very low. We already know that cards are more valuable when you have less of them from the law of diminishing marginal returns. This is reflected in conventional wisdom; look no further than the principle of simplification employed by early Gadget decks maxing on one-for-one removal cards (cards which have a constant expected value of 1). Cards that create card advantage, like the Gadgets, are typically more powerful when each player has less cards; when this is the case, their marginal utility is necessarily higher.
Forget Sixth Sense now. This is February. Let’s talk about cards you’re actually playing right now. Drawing effects make for the easiest calculations because they allow for uncertainty; while the value of destruction effects is partially dependent on what cards they’re destroying, draw effects will always get you some random cards from the cards remaining in your deck, allowing for a probabilistic approach to which expected value and utility lend themselves well. So with that in mind, let’s talk about Cardcar D.
Cardcar D has a constant expected value of 2. If its effect resolves, it will always draw you two cards. (You may either choose to ignore that Cardcar D tributes itself to activate this effect, or pretend that its expected value is 1, adjusting for the “cost” in cards of Cardcar itself.) However, it carries the steep cost of your ability to Normal or Special Summon or attack for a turn. Often, players run into a quandary when playing decks with Cardcar when they must decide whether or not to develop the board by Normal Summoning a “real” monster or develop their hand with Cardcar instead.
We can use risk aversion to understand how to make this decision. You want to use “+1” effects like Cardcar’s when you have a low number of cards and when the impact of those additional cards is high. This is the same principle by which we “save” certain power cards before finally activating them for the most possible value. In a game state where each player has seven or eight cards available to them, you aren’t going to want to waste a Torrential Tribute on a single monster. Torrential Tribute has a negligible impact when it’s destroying a single monster in a complex game state, and there is a substantial reward (the possibility that you use Torrential Tribute later on for more value) in taking on additional risk by waiting.
Remember the difference in value and utility of cards drawn and cards destroyed. Just as we added a coefficient to the value of cards to represent the possibility of milling a Dragon Ruler with Sixth Sense, we can take the value of drawn cards based on probability in situations where a player is searching for a certain out or range of outs to win the game. In such scenarios, a player’s marginal utility depends on the probability of drawing those outs. Obviously, we also have to account for the relevance of all of the other cards in the player’s deck, since there are likely some that will have some impact on the game. Note, however, that this relevance diminishes when a player is on a faster clock; if my opponent can kill me next turn unless I draw one of five cards in my deck this turn, and no other cards in my deck will kill him or stop him from killing me, I care only about drawing those five cards.
Cards destroyed are a different animal. When I spend a card to destroy one of my opponent’s cards, I’m giving up one of my resources (a card) to attack my opponent’s resources (their cards). It is beneficial for me to do this when the cost in utility of my card is less than the loss in my opponent’s utility that results from the destruction of their card. In other words, the value of trading cards for cards depends on the relative marginal utility of cards. If my opponent’s marginal utility of cards is greater than my own, in general, it benefits me to trade my cards for theirs. If I have six cards and my opponent has two, I’ll gladly trade two of my cards to get rid of my opponent’s last two. But if I have three cards to my opponent’s seven, I might reconsider. Furthermore, if my opponent is more willing to accept risk than me, trading my cards for theirs is beneficial for me. If my opponent is playing Hieratic Dragon Rulers and is in the middle of a combo that could win him the game, and I have a bunch of face-down one-for-one removal spells and traps, I want to be trading my cards for his much more aggressively in order to survive. Conventional principles of Magic theory teach the tried-and-true mantra that threats are always better than answers. The trades are beneficial for me because my opponent’s cards are essentially worth more than mine. His cards are going to win him the game if they aren’t dealt with, while mine are cards that simply help me not lose; I’m never going to be able to kill my opponent with a Dimensional Prison, at least not under the current official TCG rulings for the card. Also note that to initiate a trade of resources, you do not necessarily need to activate a card or effect that destroys another card. We initiate trades of the same sort whenever we summon monsters and attack into multiple backrows with the purpose of “baiting” them.
The single most important thing to take away from this article is how the relative marginal utility of cards should affect your in-game decisions. By learning to identify the factors that cause your opponent to tend towards or away from risk aversion, you can optimize the ways in which you use your own cards. Insofar as players’ marginal utilities of cards are not equal, there is always one winner and one loser in each one-for-one trade of resources.
Looking at resources and play decisions in this way leads to many interesting parallels between areas of card theory. Who would have thought that deciding the optimal call for Sixth Sense had anything to do with the use of one-for-one removal cards? The economic analysis of resources as they pertain to game decisions is a criminally underdeveloped area of Yu-Gi-Oh theory, and I hope that in writing this article I can inspire some other bright minds to explore it as I have to find even more previously unacknowledged truths about the relationships between game concepts like resources.