Expected Value
There are 658,008 distinct 5-card opening hand combinations in a 40-card deck. Add one card to the deck, and the total combinations increase by 91,390. Add five more cards and the total combination balloons well over a million. Every card matters when deck building with consistency in mind. However, there is a magic number here. If a duelist plans to play this game at competitive events, then he or she needs to understand that they need to win only two out of every three duels in each match. Two divided by three provides 0.66, or 66%. From here, one barrier to entry for building a competitive deck is established.
For example, Amy is interested in constructing a deck of cards; however, she wants to ensure that she has at least one trap card in hand to protect her monsters, and most importantly, her life points. There exists a variety of trap cards capable of achieving this task, however, she as well as every duelist is limited by two factors; the rules of the game, and any existing ban list, further restricting card use. Unfortunately, Amy is unable to use more than three copies of Mirror Force. However, there are similar cards that exist, in which she can pool together into one group of cards. She calls them “defensive traps”.
How would Amy go about determining the required number of cards to meet her requirement of maximizing the opportunity of obtaining at least one defensive trap card in a 5-card opening hand? There is a method known as calculating the expected value. This process involves a series of simple multiplication and addition to determining the average outcome of an event based on the probability of a set number of outcomes. Look at calculating the expected value for Amy if she were to include 8 defensive traps” in her 40-card deck.
Expected Defensive Cards Drawn
Defensive Traps Drawn (X) | Probability of Drawing # of Defensive Traps (P) | Expected value (X * P) |
0 | 30.6% | 0.00 |
1 | 43.72% | 0.44 |
2 | 21.11% | 0.42 |
3 | 4.22% | 0.13 |
| 4 | 0.34% | 0.01 |
| 5 | 0.01% | 0.00 |
| Total | 69% | 1 |
The total expected value here equals exactly one. Paraphrasing the data here, Amy can expect to draw “defensive trap” card on average, per five-card opening hand. Almost out of pure coincidence, the sum of the probabilities of obtaining at least one card of this category is equal to 69.40%, which satisfies the condition of being “match consistent”. What if Amy wants at least one defensive trap card, yet does not want to risk drawing multiples? What is the minimum?
Minimum Expected Defensive Cards Drawn
Defensive Traps Drawn (X) | Probability of Drawing # of Defensive Traps (P) | Expected value (X * P) |
0 | 57.65% | 0.00 |
1 | 35.8% | 0.36 |
2 | 6.5% | 0.13 |
3 | 0.4% | 0.01 |
4 | 0% | 0.00 |
5 | 0.00% | 0.00 |
| Total | 43% | 0.50 |
Reducing the total of defensive cards to four illustrates the absolute minimum to ensure at least one copy of a card without opening multiples. However, Amy should only expect to see one copy once in a match in contrast to twice in a match with eight cards.
Immediate Benefits
Applying the calculation of expected values allows deck builders to construct decks with respect to knowing exactly what the average opening hand is. Additionally, calculating the expected value allows duelists to know what he or she will draw in subsequent turns. This is crucial information to help duelists round out their deck after establishing desired opening hand combos.
Take for instance the average duel lasts for about 5 turns, which means 2.5 one-card draws per player. If the duelist opens with one of eight targets in a 40-card deck, seven targets will remain in a 35-card deck. There is a flat 1 in 5 chance of drawing another target on the next turn. This presents an expected value of 0.20, which effectively means the player will most likely not draw into another target for the remainder of the duel. Duelists can use this information to compensate for the inability to draw into target cards.
Again, take for instance a deck, which has a very consistent opening hand. However, the deck presents poor top-decking options, as there is a one in three chance of drawing a card capable of providing additional support. With this information, the duelist may consider altering the deck’s defensive card lineup to ensure the user will last long enough for three draw phases. This could mean the difference between deciding between more floodgates, or simple disruption with low cost. Alternatively, decks featuring highly consistent opening hands and top-decks may illustrate the duelist does not need many cards to slow down the opponent, and can invest in other cards capable of winning faster. Conversely, such information provides significant value to users of mill decks, interested in knowing the likelihood of what specific cards will exist in the graveyard after milling a specific number of cards.
The Magic Number
Outside of knowing the expected value of drawing cards, should every deck always consist of 40 cards? Better yet, how many cards should a player have in a deck? The answer is relative to the magic number of 66%. The total number of cards in a deck is irrelevant for as long as the total probability of possible first turn plays is or above 66%. However, after examining the decks of the 2016 Yugioh World Championship finals, it is possible competitive duelists sit anywhere between 60% and 66% concerning opening hand consistency.
Furthermore, opening hand probabilities are dependent on the duelist opening first or second. Current world champion, Shunsuke Hiyama, expertly constructed a 42-card Blue-Eyes deck for the sole purpose of going second in a duel. Blue-Eyes is an archetype known for not being as consistent as smaller decks such as Xyz-based archetypes. However, opening up with an extra card mitigates this weakness to a serviceable degree. This topic also plays into using the side-deck, and knowing how many cards are available for siding out without ruining consistency. If a deck is at or above a consistency rate of 66% with a five-card hand, the user can afford to side out essential cards if they expect to go second.
Some readers may wonder about how to calculate opening hand probabilities. Combinations are a mathematical method used to calculate the probability of possible outcomes for an event such as drawing cards from a deck. There are several calculators available for single cards, as well as multiple cards. A personal favorite is Michael B. Moore’s Deck-u-lator. However, spreadsheet tools such as Google Sheets and Microsoft Excel provide much more flexibility in calculating opening hand ratios.
Conclusion
In conclusion, understanding the numbers behind opening hand consistencies of decks provides several advantages for duelists. Duelists can playtest their decks with a focus on the actual plays on the field instead of working out card ratios. In addition, he or she will know exactly what cards to side out without disrupting consistency. Lastly, duelists will have a strong idea of how often the ideal opening hand will occur. However, consistency is not everything. There are several consistent decks lacking real board presence, power, or disruption to compete with the current metagame. It is important to find a balance between all elements of a competitive deck, relative to what opponents might field against oneself.
