rgb(216, 216, 240)

Zen Mahjong: Part I - Perfect

Zen Mahjong is a thought experiment that arose from a simple question: "Is a perfect game of
Tenhou possible?" As it turns out, that question has a very simple null hypothesis answer.
Since Tenhou uses a mahjong ruleset a ruleset where busting (having negative points) ends the game,
and results in Tenhou are based solely on placement, there is a nice and easy solve for this
problem. A Tenhou + 1 Double Yakuman tsumo on the first turn would simultaneously bust the other
players, ending the game while also giving us the best possible outcome (first place).

With that out of the way, we can begin asking ourselves the real tough questions. "When does a
perfect game cease to be perfect?" and "What constitutes a perfect play?" To answer these
questions, we need to dive into and explore exactly what we mean when we say "perfect" when talking
about games. To do that, we need to define what perfect is; this will prove to be the pillar and
foundation for Zen Mahjong.

-- "Perfect" doesn't always mean perfect --

When I was a young lad, I was fascinated to death with a mini-game NPC that exists in the video
game Tales of Phantasia. If my memory serves me correctly, this certain NPC appears in the past
city of Alvanista, and challenges you to a game he calls "Ishi-Tori". The premise of the game is
simple; starting with a randomised number of counters, take turns removing either one, two, or
three counters from the community pot. The person who removes the last counter is the loser.

Beating the so called "Ishi-Tori" master was quite challenging for my younger self. The NPC seemed
to play flawlessly every time, and I somehow found myself always ending up as the one to take the
last counter. Frustrated, I swallowed my pride and went online to find a perfect strategy to defeat
the NPC. As it turns out, the strategy is roughly as follows:

"The number of counters at the beginning of the game will always be of the form (X * 4) + 1. When
challenging the Ishi-Tori master, always elect to go second. When the Ishi-Tori master takes his
turn, note how many counters Y he takes. Then, simply take (4 - Y) counters from the community
pot. The Ishi-Tori master will be forced into taking the last counter every time."

This was a mind blowing revelation to my infant mind. If you played the game of Ishi-Tori perfectly,
there was a surefire way of winning everytime. In technical terms, Ishi-Tori is a "solved game." So
long as the starting number of counters is of the form (X * 4) + 1, the player who goes second can
always force a win.

But there is something that I didn't bother to think about as a kid; is there a "perfect" strategy
for the player going first? No matter what the first player does, the second player can always
force a win. I believe in chess, they refer to the first player's situation as a "zugzwang."
Assuming that the second player knows the winning strategy, whatever move the first player makes
will be a losing move. The EV of each move that the first player makes is strictly the same: 0.
Does this mean that any play that the first player follows can be considered a "perfect" play?

(Interestingly, there is a winning strategy for the player who goes first if the starting number of
counters is any number that does not produce a remainder of one. For example, if the starting
number of counters is 10, the player that goes first can choose to take a single counter. This
creates a game state where the "first" player is now second to act, and the "second" player is now
in zugzwang.)

-- "The game of Chess" and "That game of Chess" --

There's a video out there that shows Magnus Carlsen, the current chess champion at the time of this
post, checkmating multi-billionaire Bill Gates in only 9 moves. Perhaps most impressively is that
Carlsen accomplished this feat with a meager time budget of 30 seconds, against Gates's 2 minutes.

Needless to say, when Magnus Carlsen checkmated Bill Gates in 9 moves, he didn't play a perfect
game of chess. There are numerous videos on the internet that explain how, if Gates had made a few
different moves, he would have easily attained the upper hand against Carlsen. Carlsen himself
admitted that he essentially relied on a "trick"; he presented Bill Gates with a problem that he
didn't believe Gates would be able to solve.

Let's twist the question a little bit. Clearly, Carlsen didn't play "the game of chess" perfectly.
But did Carlsen play "that game of chess" perfectly? I think that it's arguable that he did. Given
his knowns, Carlsen had ample reason to believe that Gates would undoubtedly fall for his trick.
Gates, while an obviously intelligent individual, is relatively unfamiliar with the game of chess.
He knows the rules of course, but not much else about the game. Since Carlsen only has 30 seconds
on his clock, he needs to find a checkmate sequence fast. As Gates is unfamiliar with the game, and
has himself only 2 minutes on the clock, Carlsen made a pretty safe bet that the sequence of moves
he made would result in his victory. Gates fell for the trick, Carlsen converted that mistake into
an advantage, and the rest is history.

This might seem like a stretch, but chess players play like this all the time. A lot of chess
basically lends itself to putting the opponent in the most uncomfortable position possible. When
playing against weaker players, stronger players put themselves in "theoretically" weaker positions
simply to take the weaker player out of their comfort zone. If the strong human player was playing
against a supercomputer that could read the entire game tree, from the current board state all the
way to a forced checkmate sequence, then the human player would obviously lose. But, because the
weaker player doesn't know how to play that board state, the stronger player is advantaged even if
they are in a theoretically disadvantaged state.

Of course, this can't work all the time. The reason it doesn't work is because it's impossible to
accurately predict someone's thoughts. Imagine a scenario where you're playing Rock-Paper-Scissors
with a child who stubbornly refuses to throw anything but Rock. You've played 100 games with them,
and they've thrown rock everytime. If we assume that child will throw Rock for their next move,
there is a "perfect play" to win the next round: throw Paper! All too predictably, the child has a
sudden change of heart and throw Scissors.

The reason that your "perfect play" was not perfect, is because what you initially considered was
a "known" (your opponent's action/response), was actually an unknown. This happens all the time. In
chess, the weaker player might just end up randomly finding the correct way to deal with your
trick. In mahjong, a player is in guarenteed 1st place, unless they play into your mangan hand.
It is in their best interest to fold, but they brazenly push multiple dangerous tiles. They end up
hitting you with their mangan hand, dropping you into 4th place.

-- "The Axiom of Perfection" -- [WIP]

This is the train of thought that led me to the first axiom of Zen Mahjong -- the Axiom of
Perfection.

"A perfect play is the play with the highest scoring Expected Value given all known relevant
variables."

Observe:

In the example where we are playing Ishi-Tori, there are really two relevant variables:

EV of Ishi-Tori = -1 * P((4 * X) + 1 | First to act) + 1 * P((4 * X) + 1 | Second to act)

Where P((4 * X + 1) | First to act) = Probability that the are (4 * X) + 1 counters in the
community pot given that you are first to act.

In the example when we were playing Rock-Paper-Scissors with the child, there are three relevant
variables, and we can model the game as follows:

EV of Throwing Paper = -1 * P(Child(Scissors)) + 0 * P(Child(Paper)) + 1 * P(Child(Rock))

If we are given that the child will always throw Rock, then our model becomes:

EV of Throwing Paper = -1 * 0 + 0 * 0 + 1 * 1 = 1

And thus, if we are given that the child will always throw Rock, the perfect play is to throw
Paper.