Introduction
The focus of this topic is the card game called Three Kingdoms Kill (3KK for in short), which is a popular Chinese card game based on the Three Kingdoms period of China and the semi-fictional novel Romance of the Three Kingdoms (ROTK for in short). The rules of 3KK originally come from the Italian card game Bang!
I am going to talk about the relationship between this game, 3KK, and mathematics. Whether people can use the science of probability and statistics to win the game?

The features of the
characters
The different
characters are the most attractive part of 3KK. Everyone who knows this game
must have his or her own favorite characters. However, some of the characters
have more supporters and some have less; one important reason is because
different characters also have different strengths. I can calculate the benefit
of the characters per round so that I can measure them in the “scientific” way.
For example, there are the abilities of the character called Zhen Ji
who has 3 health points.
Character
ability 1: Impetus for War 倾城Every hand that is a "club" or a "spade"
suit can be used as a dodge.
Character ability 2: Goddess Luo 洛神At the beginning of your turn, you can flip a judgment card.
If the judgment card is a black-suited card ("clubs" or
"spades"), you can keep the card in your hand and can continue to
flip over another judgment card. Once the judgment card is a red-suited card,
the ability ends and you cannot keep that red-suited card.
The first ability of Zhen Ji is to
transform the cards to a dodge.
According to the calculation in the last paragraph, we already know that the value
of using a dodge is more than 1
unit. That means this ability can transform a lower value card to a higher
value card. The value of this ability depends on whether the other characters
use attack on you or not. According
to the percentage of attack cards in
the deck, I assume that Zhen Ji will be attacked once per round. Therefore,
there will be one lower value card transformed to a higher value card. The
value-difference is 2 - 1 = 1 unit. However, as I said before, because the dodge cannot be played proactively, the
value of this ability is lower than what the data shows.
The
second ability is a drawing ability. From observing, I can tell that the value equation
of this ability is a geometric progression. Therefore, I can use the geometric
sequence summation equation:
S = a1 * q^ (n - 1) / 1 – q
Set
a1 = 1;
q = ½;
n = infinite
S = 1 * ½^ (n – 1) / 1 - ½
S = 2 * ½^ (n – 1)
∵ lim n→∞ [½^(n – 1)] = ∵ ½ * lim n→∞ (½ ^n)
∴ S = 2 * ½ = 1
Then, I get the average value of the
second ability is 1 units/ round.
Consequently,
the benefit of Zhen Ji is available now. If I assume the value of ability
“impetus of war” is 0.5 units, the average of Zhen Ji’s benefit is 1.5 units/
round.
For
comparison, there is another character named Sun Quan, who has 4 health points.
His abilities are shown below.
Character
ability 1: "Balance of Power 制衡" During
your action phase, you can discard any number of cards and draw the same number
of new cards from the deck. This skill is limited to once per round.
Character
ability 2: [Ruler ability, Enforced ability] "Rescued 救援" When you are rescued from the brink of
death by other Wu characters via the use of peach, you recover 1 additional unit of
health (from zero health to two units of health).
Because the second ability
is not used often, I will simplify by skipping it. Seemingly, the value of the
first ability is easy to count. The cost = the benefit, the value of this ability
equals zero. However, this character is the strongest at the “1 versus 1” mode.
The reason is the existence of high value cards. Sun Quan
can transform the low value or non-value (because it cannot be use immediately)
cards to random cards. According to the calculation I did above, once Sun Quan
draw 4.45 cards, he can get 1 unit of benefit. I assume that the value of every
non-value card is 0.5 units. Sun Quan on average transforms 1 non-value card (0.5
units) and 1 low value card (1 unit) to 2 random cards (2.45 units) per round.
The cost is about 0.5 + 1 = 1.5 unit, therefore:
2.45–1.5
= 0.95 units/ round
The average benefit of Sun Quan is around 0.95
units per round.
The character Zhen Ji
has higher benefit than Sun Quan, so the way to balance it is by using via
different health points. The characters that have the weaker ability usually
have higher health points.
I can use the same method
as the examples above to get the average benefit of all the characters. However,
that will be a lot work, so I searched for the information on the internet. Someone
else has figured that out. He stated that the average benefit of characters
with 3 health points is around 1.5 united, and the benefit of characters with 4
health points is around 0.7 units.
As you see, the
character Zhen Ji has the benefit equal to the average. Thus the strength is in
the normal level. Sun Quan has the benefit higher than the average. Therefore he
is what we call a “strong character”.