Articles on the 100th Anniversary

Analyzing Three-Way
Playoffs in the Sumo
World
Fairness and Excitement

Yasuhiro Sakai

Professor Emeritus / Shiga University

120

I

Sumo as an old andpopular sport in Japan

Sumo is one of the oldest and most popular sports in Japan. Two big sumo wrestlers
clap their hands before they start fierce wrestling on the small circle with a diameter of lessthan five meters. It is really exciting to see howeach of the wrestlers exerts all his energy defeats his opponent. Usually, he engages in one
bout in a day, and one tournament consists offifteen bouts. If he succeeds in wining all fifteen bouts, he will no doubt be declared thetournament victor. Suppose a special occasion
in which there are three wrestlers, say Adam,
Bill, and Carl, with 13 wins and two losses, sothat each can claim a contender. Then, we haveto invent a specific method to make three-way
play-offs to determine the final victor. Although there might be several methods
conceivable for such play-offs, it is a sort ofsmall wonder that only one specific method,
called a tomoesen, has long been adopted in the
traditional sumo world.
According to this tomoesen, sumo fights arecarried out in a step-by-step fashion. At thefirst step, only two wrestlers, say Adam andBill, are picked up, with Carl being left for ashort rest. Suppose that Adam defeats Bill on
the first counter. Then, there appears the second stage, in which Adam has to encounter
Carl, the third remaining wrestler. Adam may
or may not defeat Carl. If Adam succeeds inbeating Carl, then it is the end of the story:
Adam is declared the final winner, receiving thevictory trophy. If Carl defeats Adam, however,another set of sumo bouts would be necessary.
It is now Carl's turn to have a sumo match with
Bill. There are two possibilities, Carl may or

THE HIKONE RONSO

Summer / July 2024 / No.440

may not defeat Bill. If Carl defeats Bill, thenCarl is declared the tournament winner because Carl is proud of winning two successive
bouts, against both Adam and Bill. If Bill beatsCarl, however, another possibility would appear. This would imply an extra complication.
Then, sumo matches would continue until
there appears a strong wrestler who successively
downs the two opponents.
Theoretically speaking, sumo bouts would
go on and on, with no end in sight. Fortunately
or unfortunately, reality is different from theory. According to the long history of sumo
records, three-way playoffs have been rarely
seen. Besides, even if tomoesen happens in thesumo world, two or even more bouts would beneeded to determine the final winner. The historical, physical or psychological reasons why
this is so will be discussed later in greater details.
The outline of this paper is as follows. Section 1 serves as an introduction, discussing
sumo as one of the oldest and most popularsports in Japan. The tomoesen or a specific formof three-way playoff is heuristically describedwith recent historical data. Section 2 turns attention to a more theoretical analysis with thesupport of mathematical probability theory. It
is shown that mathematically, tomoesen is not afair game even though it has been a very favorite sport from the good old days to the present
time. Section 3 attempts to reply to the question why a mathematically unfair game such as
sumo has been overwhelmingly loved by theJapanese people. Then, in Section 4, we cometo the conclusion that sumo is a sort of humandrama in which excitement sometimes tends
to overwhelm rationality. After all, human being is a complex animal, always seeking adelicate balance between reason and emotion.

II

A Historical Analysis ofTomoesen : Previous
Records Outlined

Concerning the study of the sumo world,so many approaches are available. Amongthem, it would be very reasonable to think that
historical and theoretical approaches play key
roles in such study. Remarkably, tomoesen characterizes one of the cores of Japanese sumo
world. The question of why and how it distinguishes from other sports seems to be quite
intriguing, thus reserving careful investigation.
While a compact historical approach to tomoesen will be taken in this section, a moredetailed theoretical approach will be developedin the next session.
The first historical approach should bebased on the accumulation of real sumo dataover so many years. It is well-known that sumo
has a very long history, being firmly embeddedin the natural and cultural features of Japan.
Although no man is 100 percent sure of the exact origin of sumo, several myths about thecontests of physical strength, vividly written in
Kojiki (Records of Ancient Matters) and Nihonshoki (Chronicles of Japan) in the 8th
century, are believed by a number of researchers to be such origin. Whether this is truly
correct or not, it is a historical fact that Classical Sumo contests have been carried out inagricultural villages as the formal and religious
rituals celebrating rice harvest of a year in question. 1)
Modern Sumo began in the Edo period,
more than 400 years ago, and uninterruptedly
continues until today. At every fight, two big
wrestlers and a referee step up into the rather
small ring with the diameter of around 4.5 me1) For details, see Ohno-Yasumaro (ed.) (712) and Fuhito

et al. (720). Note that the records of those classicalbooks are not perfectly reliable.

Analyzing Three-Way Playoffs in the Sumo World

Yasuhiro Sakai

121

ters. The ring is regarded as a very sacred place
on which each wrestler is asked to pitch ahandful of pure salt before the sumo fight begins. The fight is speedy and exciting, usually
being finished in a short time. If a wrestler either knocks down his opponent on the ring or
pushes him out of the ring, then the successful
player is declared by the referee a sure winner.
All spectators at each sumo match are naturally
very excited and applaud the excellent performance of the wrestlers.
At present, there are six sumo performances a year: they are carried out in January,
March, May, July, September, and November.
Each performance has its own feature, reflecting the change of place and season. Besides,
the final victor is awarded the Emperor Cup
together with many commemorative items. In
short, all sumo wrestlers make strenuous efforts
to become the Cup winner at every performance.
Every healthy wrestler is asked to take partin all 6 performances, and every performance
consists of 15 bouts. If a certain wrestler, say
Adam, wins all of those bouts in a tournament,
then he is naturally declared to be the victor,
winning the Emperor Cup. If Adam cannot
win all bouts, the determination of a victor is
not an easy job. The most difficult job to decide a victor occurs when there exist three
wrestlers, Adam, Bill and Carl, who has won 13
bouts but lost 2 bouts at the end of the tournament. On this occasion, Historical Tradition
plays a critical factor in seeking a reasonable
way out, teaching us that the tomoesen whichsounds classic yet romantic is indeed an effective and exciting method to find the solution.
In what follows, let us scrutinize the essence of tomoesen. According to this tomoesen,

122sumo fights are carried out in a step-by-stepfashion. At the first step, only two wrestlers,say Adam and Bill, engage in their fight, withthe third wrestler, say Carl, being left for ashort rest. At the second stage, the winner, say
Adam, of the first bout has to engage in his second bout against Carl, a resting third wrestler.
Adam may be a winner again, demonstrating
the two successive wins, and he is declared atournament victor. In short, such two boutswith win◯-lost● results will be shown as follows.
Adam ◯ ― ● Bill
Adam ◯ ― ● CarlThis is merely one possibility, however.
Another possibility is that at the second bout,
Adam may be unfortunately defeated by Carl.
Then, Carl can continue his sumo fight against
Bill. Carl may be either a winner or a loser. If
Carl defeats Bill, then Carl has beaten the two
successive bouts, winning the Emperor Cup.
Such successive fighting results will be illustrated as follows.
Adam ◯ ― ● Bill
Adam ● ― ◯ Carl
Bill ● ― ◯ Carl
However, this is not the end of the story.
We have to consider another possibility that
Carl may unfortunately be a loser against Bill.
Then, Bill, the man who has lost bout against
Adam before, can get a second chance to clinch
the crown. If Bill is lucky enough to beat Adam, Bill is a trophy winner in spite of the loser
of the first bout. Summing up, the four bouts
are played as shown below.

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Adam ◯ ― ● Bill
Adam ● ― ◯ Carl
Bill ◯ ― ● Carl
Bill ◯ ― ● Adam

(1) 1956, March
The following wrestlers were played in tomoesen: Wakanohana ( Waka for short),
Wakahaguro (Hagu for short), and Asashio(Asa for short).

At this point, another question might occur to our mind. Let us instead assume that at
the final contest, Adam is strong enough tobeat Bill. Then, we will face the opposite situation to be depicted below.
Adam ◯ ― ● Bill
Adam ● ― ◯ Carl
Bill ◯ ― ● Carl
Bill ● ― ◯ Adam
As is easily seen, Adam's record is "twowins and one loss". Bill's record is "one win andtwo losses" whereas Carl's record is "one win
and one loss". None can claim to be two successive winners among those three players. We
wonder who will be the final trophy winner.
Under such complex situations, we have no
choice but stick to the original rule by which
any wrestler winning two successive matches
must be celebrated as the glorious Cup winner.
Perhaps, it may take time and pains to find a final reasonable solution. We do not care for itat all. In our opinion, this is no more than theessence of tomoesen of Japanese sumo world.
Now, we are in a position to document thereal records of tomoesen in recent times. As far
as our memory tells us, the March Tournamentin 1956 is the first recorded tomoesen after the
Pacific War. Let us pay attention to it in details
below.

Analyzing Three-Way Playoffs in the Sumo World

Waka ◯ ― ● Hagu
Waka ● ― ◯ Asa
Hagu ● ― ◯ Asa
Asa was a young and ambitious wrestler.
He sat down out of ring first, closely watching
the match between Waka and Hagu. After
knowing that Waka beaten Hagu, Asa succeeded in defeating Waka and Hagu successively.
This demonstrated Asa's impressive victory intomoesen. Even now, we can easily guess that all
sumo fans got very much excited by such fantastic achievement by Asa.
In what follows, we will briefly outline othertomoesen results and make their characteristics.
(2) 1961, September
The following wrestlers are played in tomoesen: Kashiwado (Kashi for short), Myobudani(Myo for short), and Taiho (Tai for short).
Kashi ◯ ― ● Myo
Kashi ● ― ◯ Tai
Myo ● ― ◯ Tai
Tai was the final victor of the tournament.
His winning pattern was as the same as (1)above: namely, he sat out of the ring and successively defeated his two opponents. This
must excite all sumo fans in Japan. No doubt,
it symbolized the arrival of the Golden Age ofGrand Champion Taiho.

Yasuhiro Sakai

123

(3) 1965, September
The following wrestlers were played in tomoese n: Ka sh iwado (Ka sh i for shor t),
Myobudani (Myo for short), and Sadanoumi(Sada for short).
Kashi ◯ ― ● Myo
Kashi ◯ ― ● Sada
Kashi defeated his two opponents easily
and impressively. This was undoubtedly thesimplest result of tomoesen, indicating Kashi asa dominant winner.
(4) 1990, March
The following wrestlers were played in tomoesen: Konishiki (Koni for short), Kirishima(Kiri for short), and Hokutoumi (Hoku for
short).
Koni ◯ ― ● Hoku
Koni ● ― ◯ Kiri
Hoku ◯ ― ● Kiri
Hoku ◯ ― ● Koni
This tomoesen was quite unique in that four
matches were needed to decide the final victor.
Such complication was "historically first" (andprobably "historically last").
(5) 1993, July
The following wrestlers were played in tomoesen : Akebono (Ake for short),
Wakanohana (Waka for short), and Takanohana (Taka for short).
Ake ◯ ― ● Waka
Ake ◯ ― ● Taka

Ake was the first grand champion from
Hawaii. It was quite impressive to see that Ake
defeated the popular brothers Waka and Takain this tomoesen. No doubt, the "Golden Ageof the Sumo World" began with the brilliant
arrival of those three wrestlers.
(6) 1994, March
The following wrestlers were played in tomoesen: Takanonami (Nami for short),
Takatouriki (Riki for short), and Akebono(Ake for short).
Nami ◯ ― ● Riki
Nami ● ― ◯ Ake
Riki ● ― ◯ Ake
Ake was again the final winner of tomoesen.
This showed his explosive power over all his
opponents.
(7) 2022, November
The following wrestlers were played in tomoesen: Takayasu (Yasu for short), Ahbi (Abifor short), and Takakeisho (Kei for short).
Yasu ● ― ◯ Abi
Kei ● ― ◯ Abi
This demonstrated how exciting sumo was
as a sport. Abi was physically a rather small
wrestler, but technically a very skillful player.
In the sumo world, power was, and is, not everything: namely, skill was, and is, sometimes
more important than mere power. This eloquently taught us another attraction of
tomoesen. 2)2) Remarkably, among those seven cases of tomoe-sen,

three exciting cases (namely, (1), (2) and (6) ) represent
what we may call the Lucky Cases, in which the third
man is lost in the first fight but finally wins the crown ofvictory. We think that the probability of "three out ofseven" is amazingly a large number. This clearly shows
how exciting tomoe-sen is in the sumo world.

124

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III

A Theoretical Analysis ofTomoesen : Rationality
versus Excitement

We now turn our attention to a theoreticalanalysis of tomoesen in the sumo world. The
problem of greatest interest is whether and towhat extent tomoesen is a "fair game" from atheoretical and mathematical viewpoint. 3)
It is not a plain and easy job to determinethe "degree of fairness" in the sumo world.
Suppose that three players are engaged in tomoesen. Since three persons are involved, witheach person seeking the winning trophy, it
seems that tomoesen should be a "fair game" if
the probability of winning the trophy is justthe same, i.e. one third, among the three players. Reality may entirely be different fromappearance, however. The central question to
ask is whether and in which way tomoesen canbe regarded as a "fair game".
Imagine the scene that three big wrestlers
step up into the rather small ring with the diameter of around 4.5 meter. Is it practicallypossible that those three wrestlers together
start sumo wrestling simultaneously? Of
course, it is completely out of our imagination.
On reality, only two wrestlers are allowed to beon the ring, a third wrestler being idle out of
the ring. And then, the winner of the firstmatch turns to the next match against a third
resting wrestler. This is clearly nothing but theessence of tomoesen.
Following the previous example we used
before, let us continue to assume that there arethree wrestlers: They are Adam, Bill and Carl.
Practically, let us consider the following two
successive games. At the first step, Adam andBill are supposed to fight together on the ring.

And at the next step, the winner of the game,
say Adam for instance, is destined to fightagainst Carl, a third and resting wrestler. Adam may or may not defeat Carl. Conceivably,
several extra fights will be followed until thedominant winner of two successive fights appears. Theoretically speaking , even an infinite
number of fights would be required to settle
the issue. But, practically speaking, it is quite
convenient for us to separately scrutinize twodifferent patterns toward the final victory. The
first pattern is a sort of "Straight Pattern" inwhich Adam beats Bill in the first bout, andsomehow wins the victory trophy. The second
pattern is distinct from the first pattern in thesense that although Adam is lost in the firstfight against Bill, he somehow wins the trophy
in a roundabout fashion. This may be named"Roundabout Pattern" toward final victory. In
what follows, both Straight and Roundabout
Patterns will theoretically be discussed in moredetails.
3-1. The First Straight Pattern: Adam wins the first battle andproceeds to capture to the cup
There are three wrestlers, namely Adam,
Bill and Carl. At the first bout, Adam beatsBill, with Carl taking a short rest. When Adam defeats Carl at the second bout, Adam will
surely win the victory trophy. The data of such
two battles will be illustrated as follows.

Adam ◯ ― ● Bill
Adam ◯ ― ● CarlThis indicates possibly the shortest straightpattern case for the determination of the victorof tomoesen. Note that such Adam's sumo data

3) To our knowledge, only a few papers on tomoe-sen

have been published so far. We believe that this paper
distinguishes from them, sheding a new light on it.

Analyzing Three-Way Playoffs in the Sumo World

Yasuhiro Sakai125

is recorded as Case 1A: ○○ in Table 1. The
winning probability is (0.5)2 as is written in theright of Table 1.
There are some other ways for Adam to
win the final victory. Even if Adam is defeatedby Carl in the second battle, Adam should not
be discouraged so easily. If Carl happens to bebeaten by Bill as indicated by the striking sharp
sign (#), Adam may have a second chance to
win the trophy. As is carefully shown below, if
Adam succeeds in successively defeating Billand Carl, Adam is declared the final victor
Note that such sharp sign (#) indicates the fightin which Adam is not directly participated.
Adam ◯ ― ● Bill
Adam ● ― ◯ Carl
Bill ◯ ― ● Carl ( # )Bill ● ― ◯ Adam
Carl ● ― ◯ Adam
If we focus on Adam's records, his sumo records can be shown in Table 1 as Case 1B: ○●＃○○, together with the probability of (0.5)5.
Compared with the previous case 1A, Adam's
effort to clinch the trophy is very great here:
Indeed, he has to take part in four painful battles.
Now, let us gaze with peering eyes at thefollowing score sheet.
Adam ◯ ― ● Bill
Adam ● ― ◯ Carl
Bill ◯ ― ● Carl (#)
Bill ● ― ◯ Adam
Carl ◯ ― ● Adam
Carl ● ― ◯ Bill (#)
Adam ◯ ― ● Bill
Adam ◯ ― ● Carl126

Here, there are as many as eight bouts including the two bouts which are not directly
related to Adam. This clearly shows Adam's
suffering history toward the final goal. Summing up, Adam's records are shown in Table 1
as Case 1C: ○●#○●#○○ , together withthe probability of (0.5)8.
Fortunately or unfortunately, this not theend of story. If Adam happens to be lostagainst Carl at the eighth boat, the relevant tomoesen must continue until the final glorious
victor comes on the stage. Then, as Table 1
teaches us, we will have to take care of Case 1D
or Case 1E, or ... until the very final victor go
up to the special commendation ring.
Now, suppose that Adam got lost in thefifth fight against Carl. Adam neverthelessshould not be out of hope, however. Indeed, if
Carl is a loser against Bill, Adam's hope is backagain. If the reviving fighter Adam successively
defeats Bill and Carl, then Adam can be the final victor after struggling. Then, Adam's longfighting records can be illustrated below.
Adam ◯ ― ● Bill
Adam ● ― ◯ Carl
Bill ◯ ― ● Carl (#)
Bill ● ― ◯ Adam
Carl ◯ ― ● Adam
Carl ● ― ◯ Bill (#)
Adam ◯ ― ● Bill
Adam ● ― ◯ Carl
Bill ◯ ― ● Carl
Bill ● ― ◯ Adam
Carl ● ― ◯ Adam
Note that there are as many as eleven bouts
including the three bouts which are not direct-

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Table 1 Requirement for a sumo wrestler to win
the tomoesen
Case 1 (Rightful Case): Won the first bout, and also the crown of victory
Win○-Loss● Data

Case 1A
Case 1B

Probability: ○○: ○●＃○○(0.5)2(0.5)5(0.5)8

Case 1D: ○●＃○●＃○○: ○●＃○●＃○●＃○○

Case 1E・・: ○●＃○●＃○●＃○●＃○○
・
･
・
・
･
・

(0.5)14・・Case 1C(0.5)11

Case 2 (Lucky Case): Lost in the first bout, but won the crown of victory
Win○-Loss● Data

Case 2A

Probability(0.5)4

Case 2B(0.5)7: ●＃○○: ●＃○●＃○○
Case 2C : ●＃○●＃○●＃○○(0.5)10

Case 2D : ●＃○●＃○●＃○●＃○○
Case 2E : ●＃○●＃○●＃○●＃○●＃○○
・
・
･
・
・
・
･
・

(0.5)13(0.5)16・・

Remark. The sharp mark "＃" indicates the occurrence of the situation
under which the wrestler, who won the previous bout, is beaten himself inthe present bout.

ly related to Adam. This clearly shows Adam's
suffering history toward the final goal. Summing up, Adam's records are shown in Table 1
as Case 1D: ○●#○●#○●#○○ , together with the probability of (0.5)8 .
Fortunately or unfortunately, this should
not be the end of story. If Adam happens to belost against Carl at the eighth boat, the relevant
tomoesen must continue until the final glorious
victor comes on the stage. Then, as Table 1
teaches us, we will have to take care of Case 1D
or Case 1E, or ... until the very final victor go
up to the special commendation ring.
Summarizing those cases of victory or defeat, we obtain the upper half of Table 1. If wefocus on Case 1, the total probability for Adam
to win the victory cup, written as P (Case 1),

Analyzing Three-Way Playoffs in the Sumo Worldshould be calculated as the sum of the probabilities of Cases 1A, 1B, 1C, 1D, 1E and otherpossible cases. Therefore, it is simply computed
as follows.
P(Case1) = (0.5)2 + (0.5)5 + (0.5)8 + (0.5)11+ (0.5)14 + …
= (0.5)2 ×【1 + (0.5)3 + (0.5)6 +(0.5)9 + (0.5)12 + …】(1)Note that the term in the square bracket【…】means the geometric series of the firstterm 1 and the common ratio (0.5)3. Clearly,
such series must converge, and the limit is 1 /[1 − (0.5)3], namely 8/7. Hence, in the light of
Eq. (1), we obtain the following result.

Yasuhiro Sakai

127

P(Case1) = (1/4) × (8/7) = 2/7.

(2)

In plain English, the probability that Adam wins the first boat and proceeds to win thecrown of victory is just 2/7. Needless to say,Bill's probability of winning the victory shouldbe the same as Adam's probability.
3-2. The Second Roundabout Pattern: Adam loses the first battle
yet manages to capture to the cup
In the above, we have scrutinized the "First
Straightforward Pattern", showing how the respectable wrestler Adam, after winning the first
battle, proceeds to win the victory trophy. In
what follows, we will carefully examine the"Second Roundabout Pattern", demonstrating
how in spite of the loss of the first battle, Adam
somehow manages to become the final victor.
We start our analysis with the unlucky situation under which Adam got lost against Billin the first match. The urgent problem is now
how Adam deals with this adversity toward thefinal victory. Presumably, the easiest way out
for him is that after patiently waiting out of thering and watching Carl defeating Bill, he makes
a swift recovery by successively beating the two
wrestlers, Carl and Boll. Such a come-from-behind victory will be demonstrated below.

Adam ● ― ◯ Bill
Carl ◯ ― ● Bill (#)
Carl ● ― ◯ Adam
Bill ● ― ◯ Adam
Adam's fighting records are also confirmed
in Table 1 as Case 2A : ●＃○○. The winningprobability is as small as (0.5)4.

128

Note that there remain some other ways
for the indomitable wrestler Adam to win the
final trophy. When Adam got lost against Billat his fourth battle, he may take a short breakout of the ring and wait for Bill's defeat by Carlat the next match. In fact, if Adam is a lucky
person, he will still have a chance to become
the final victor as indicated below.
Adam ● ― ◯ Bill
Carl ◯ ― ● Bill (#)
Carl ● ― ◯ Adam
Bill ◯ ― ● Adam
Bill ● ― ◯ Carl (#)
Adam ◯ ― ● Carl
Adam ◯ ― ● Bill
Note that after the loss-win-loss records in
his earlier bouts, Adam have to successively defeat Carl and Bill in his later bouts. This isreally a nice achievement for the tireless man
Adam. His fighting records are confirmed inTable 1 as Case 2B :●＃○●＃○○ , the winning probability
being as small as (0.5)7.
Adam ● ― ◯ Bill
Carl ◯ ― ● Bill (#)
Carl ● ― ◯ Adam
Bill ◯ ― ● Adam
Bill ● ― ◯ Carl (#)
Adam ◯ ― ● Carl
Adam ● ― ◯ Bill
Carl ◯ ― ● Bill (#)
Carl ● ― ◯ Adam
Bill ● ― ◯ Adam
Amazingly, from a pure theoretical viewpoint, Adam can be beaten a couple of times

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Summer / July 2024 / No.440

before winning the crown. As the very long series of battles seen above tells us, however, thisshould really be a very tiresome ordeal. According to Case 2 in Table2 , as far as theaccumulation of exhaustion on the side ofwrestlers can entirely be ignored, such extremely long battles will continue so that even Case
2D or Case 2E may be realized. In fact, Case2D can be illustrated below, with focus on Adam's win-loss table.
Adam ● ― ◯ Bill
Carl ◯ ― ● Bill (#)
Carl ● ― ◯ Adam
Bill ◯ ― ● Adam
Bill ● ― ◯ Carl (#)
Adam ◯ ― ● Carl
Adam ● ― ◯ Bill
Carl ◯ ― ● Bill (#)
Carl ● ― ◯ Adam
Bill ◯ ― ● Adam
Bill ● ― ◯ Carl (#)
Adam ◯ ― ● Carl
Adam ◯ ― ● Bill
As is shown below, the demonstration ofCase 2E will be more complicated than that ofCase 2D. Take a close watch on the fierce alternation of win ◯ and loss ●.
Adam ● ― ◯ Bill
Carl ◯ ― ● Bill (#)
Carl ● ― ◯ Adam
Bill ◯ ― ● Adam
Bill ● ― ◯ Carl (#)
Adam ◯ ― ● Carl
Adam ● ― ◯ Bill
Carl ◯ ― ● Bill (#)
Carl ● ― ◯ Adam

Analyzing Three-Way Playoffs in the Sumo World

Bill ◯ ― ● Adam
Bill ● ― ◯ Carl (#)
Adam ◯ ― ● Carl
Adam ● ― ◯ Bill
Carl ◯ ― ● Bill (#)
Carl ● ― ◯ Adam
Bill ● ― ◯ Adam
No doubt, it would be a sort of miracle
that Adam comes out as the very final winnerof such complicated battles. Mathematicallyspeaking, no matter how slight it seems to be,
it should be counted as a mere possibility .
When we summarize those cases of victoryor defeat, we can obtain the lower half of Table
1. If we focus on Case 2, then the total probability for Adam to win the victory cup, namely
P (Case 2), should be computed as the sum of
the probabilities of Cases 2A, 2B, 2C, 2D, 2E
and other possible cases. Overall, it is computed as follows.
P(Case 2) =(0.5)4 + (0.5)7 + (0.5)10 +(0.5)13 + (0.5)16 + …
= (0.5)4 ×【1 + (0.5)3 + (0.5)6 +(0.5)9 + (0.5)12 + … 】(3)As was already stated above (see Eq. (1)),
the term in the square bracket【…】means thegeometric series of the first term 1 and the common ratio (0.5) 3. Clearly, such series must
converge, and the limit is 1 / [1 − (0.5) 3],
namely 8/7. Therefore, in the light of Eq. (3),
we are nicely led to the following result.
P(Case 2) = (1/16) × (8/7) = 1/14

Yasuhiro Sakai

129

(4)

3-3. Analyzing Adam's probabilityof winning the victory cup
We are ready to mathematically computethe probability for Adam to win the victory
cup. In the light of meticulous preparations wehave made so far, this is a relatively easy job. As
the saying goes, a danger is not as great as it
seems in advance.
Frankly speaking, Adam is a lucky personin the sense that his possibility never vanishes
even if he is lost in the first match against Bill.
Generally speaking, there are two patterns toward the final victory ―― the "First Straight
Pattern" and the "Second Roundabout Pattern".
If Adam wins the fist battle, the First Straight
Pattern occurs. Then, he may proceed to finally capture the victory cup. If he loses the first
battle, however, the Second Roundabout Pattern begins to take place. Then, in spite of his
unhappy start, he may nevertheless manage tofinally capture to the cup.
Adam's victory probability, denoted by P
(Adam wins the cup), should be calculated as
the sum of the probabilities of those two patterns. Hence, in the light of Eqs. (2) and (4),
we have the following result.

P(Adam wins the cup) = P(Case 1) + P (Case 2)= (2/7) + (1/14)= 5/14(5)
A little reflection would tell us that Bill'schance of winning the cup is the same as Adam's chance. Hence, Bill's probability ofwinning the cup can be computed as follows.
P(Bill wins the cup) = (2/7) + (1/14)= 5/14
(5)

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3-4. The third man Carl 's probability of winning the cup
Tomoesen is a special form of three-way
playoffs in which at the first battle, only two
wrestlers, say Adam and Bill, are allowed to beon the ring, with the third man, say Carl, muststay out of the ring and patiently wait for the
result of the battle between Adam and Bill.
Such a non-symmetric treatment of the three
men may suggest that tomoesen is not a perfectly fa ir g ame : in fact, Carl may be in a
disadvantageous position in comparison with
Adam and Bill. We will show that this guessing turns out to be true. We also have tounderstand, however, that mathematical disadvantage might not directly lead to psychologicaldisadvantage.
Let us more carefully analyze the fundamental difference between the first two men
and the third man. All three wrestlers aim at
winning the tournament. Then, either Adamor Bill may be lost in the first fight. The defeat
at the first battle looks unlucky, but not quite
so. This is because, as the "Lucky Case" in Table 1 tells us, either man is guaranteed topossibly come back and finally win the final
victory. In short, things will eventually startgoing his way.
Now, let us compare the destiny of thethird man Carl and the one of the other twomen. Alas, Carl must stick to only the "Second
Lucky Case", not being allowed to have the"First Rightful Case". Let us explain why andhow Carl is destined to be in such an unfair
position. On the one hand, suppose that Adam defeats Bill at the first feat. Then, if Carl islost against Adam, then Adam should be thevictor of the tournament. On the other hand,
suppose that Bill is instead a winner. Then, if

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Carl is defeated by Bill, then it is Bill who is the
final victor. In those cases of "sudden death",the third man Carl is in a miserable position,
having no chance at all to win back. In otherwords, in contrast to Adam and Bill, if Carl islost in the first feat, then he will never besmiled by the Goddess of Fortune.
Summing up, Carl is destined to deal with
only the "First Rightful Case", having nothing
to do with the "Second Lucky Case". Therefore, the probability for Carl to win the cup isexpressed as follows.
P(Carl wins the cup) = P(Case 1)= 2/7

(6)

Interestingly enough, this is definitely lessthan 1/3. Compared with this, the probabilities for both Adam's victory and Bill's victoryshould be the same, being expressed as 5/14,
which is clearly greater than 1/3.
3-5. Considering the rationality of
tomoesen
We are in a position to discuss the rationality of tomoesen from many angles. Let us
carefully argue the problem of what "rationality" is all about in the sumo world.
Tomoesen is a specific form of three-way
playoffs in which the three wrestlers, Adam,
Bill and Carl, are fighting on the ring. If it is astrictly rational game, each player's probabilityof winning the victory cup must be the same.
As a result, the victory probabilities should
equally be distributed among the three persons
in the following fashion.

1/3, 1/3, 1/3Analyzing Three-Way Playoffs in the Sumo World(7)

As we discussed above in great detail, however, the victory probabilities for the three
persons are not equal at all, showing a sort of"distributive bias" as follows.
(8)

5/14, 5/14, 4/14

Such bias clearly exists in the sense that it isinclining a bit toward the first two wrestlers,
Adam and Bill. As a result, the third man Carlis forced to be in a slightly disadvantageous position. Exactly speaking, the size of such bias iscomputed as 1/14 or 0.0714286... . The question of much interest is how an ordinary
person evaluates this fraction. It depends onthe character and feeling of a person. Whilesome people might think that it is large enoughand cannot be neglected, other people might
not think so. Perhaps, it is a psychologicalproblem rather than a mathematical one. In
human sport, "warm heart" may work strongly
than "cool head".
We have thus far extensively discussed how
a three-way playoff in the sumo world shoudnot be regarded as "a fair game" in a narrow
sense. Such a conclusion, however, may becompromised if we are allowed to introduce asort of "third factor" such as "decision by drawing lots" into the playoff. In reality, before theplayoff is conducted, the order in which three
wrestlers face one another is decided by drawing lots. As a result, the first wrestler has asumo bout with the second wrestler, ant thethird wrestler sits out of the sumo ring andwaits for the chance to have another bout withthe winner of the previous bout.
Then, the probability of each wrestler to
win the victory trophy seems to be calculated
as follows.

Yasuhiro Sakai131

(5/14)(1/3)+(5/14)(1/3)+(4/14)(1/3)=(14/14)(1/3)=1/3
We have to keep in mind that such a conclusion is no more than the result of the strange
combination of the two different items, namelya three-way playoff in pure form and drawing
lots in practice. Besides, if we completely rely
on drawing lots per se, playoffs on the ring
would be regarded as unnecessary sport shows.
We believe that theory is just theory. No
practiocal compromise such as drawing lots or
tarot reading should be desirable.

IV The charm of tomoesen :the spiritual climateof JapanIn the above, we have carefully discussed
tomoesen in both historical and theoretical perspectives. It is a specific form of three-way
playoff that is observed only in Japan in theworld. Historically speaking, while almost 80
years have passed after the end of the PacificWar, tomoesen has been played only seventimes. Therefore, tomoesen is neither a commonly seen event nor a very rare event: it canbe seen almost once in ten years.
We have also done a detailed theoreticalanalysis with the support of mathematicalprobability theory. As has been seen above,from a mathematical point of view, tomoesen
cannot be said to be a fair game, even though it
has been a very favorite sport from the oldgood days to the present time. Then, the interesting question which might naturally come toour mind is why such a mathematically unfair
game such as tomoesen of sumo has been over-whelmingly loved by the Japanese people formore than a thousand years.
Perhaps, there would be a certain group ofpeople who say that excitement sometimes
tends to overwhelm rationality. After all, human being is a complex animal, always seeking
a delicate balance between reason and emotion. Famous economics professors George A.
Akerlof and Robert J. Shiller have jointly written a very exciting yet a bit controversial bookin order to show how human psychology tends
to derive the economy and why it matters for
global capitalism. For details, see Akerlof andShiller (2009).
In order to understand the charm of tomoesen, it is quite useful to discuss the problem ofwhether and to what extent "mathematics inEurope" is similar to, or different from, "mathematics in Japan". Although mathematics seems
to be an highly abstract and global entity, we
believe that it cannot escape from a delicate
product of local cultures and traditions.
First of all, it should be noticed that thefollowing three mathematicians were outstanding contemporaries in the seventeenth century,
making great contributions to the development
of differential and integral analysis. 4)
Isaac Newton (1642 - 1727), Great Britain
Gottfried W. Leibniz (1646 - 1716), Germany
Takakazu Seki (1649? - 1708), Japan
Although the exact birth date of the Japanese genius Seki remains unknown to us, it
should be a sort of miracle that in almost similar periods yet in different countries, those
three geniuses were actively involved in abstract
mathematics, especially in the difficult subjectof differential and integral calculus.
4) For the men of mathematics, see Bell (1937) and Yano
(1966).

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A bit earlier than those calculus geniuses,
another branch of mathematics, named thetheory of risk and probability, was created andlargely developed by the following two mathematicians. 5)
Pierre Fermat (1601 - 1666) , France
Blaise Pascal (1621 - 1662) , France
French mathematicians seem to distinguishthemselves from other people in the sense thatthey devoted their lives to developing uniquesubjects such as risk and probability, which areclosely related to society and common sense.
For example, Pierre Simon Laplace (1749 1827), a successor of the French tradition ofmathematical science, remarked as follows. 6)"We see that the theory of probability is atbottom only common sense reduced to
calculation; it makes us appreciate withexactitude what reasonable minds feel bya sort of instinct, often without being able
to account for it. It is remarkable that thisscience, which originated in the consideration of games of chance, should havebecome the most important object of human knowledge."(Laplace (1812 ))According to the superman Laplace, thetheory of probability is a common sense reduced to calculation, being originated in theconsideration of games of chance. Looking
back to tomoesen, a unique way of three-way
playoffs in Japan, it has been originated in thelong tradition of sumo, i.e. unequaled sport
game of chance. It is really a kind of mystery
why the mathematical consideration of Japa-

nese tomoesen has been rather neglected so far.
We sincerely hope that this paper will shed a
new light on such a problem of great importance.
[Acknowledgement]This is a completely revised English version
of Sakai (1995a, 1995b). I am indebted to Kazuya Mizushima (Professor Emeritus, Kobe
University) and many others for helpful comments and discussions. All remaining errors
are solely my responsibility.
References
⦿Akerlof G. A, Shiller R. J. (2009) Animal spirits: howhuman psychology drives the economy, and why it matters for global capitalism, Princeton University Press.

⦿Bell, E.T. (1937) Men of mathematics, Penguin Books

Ltd., New York.

⦿Fuhito et al. (eds.) (720) Nihonshoki ( Chronicles of Ja-pan) , 30 volumes, in Japanese. [Exact records unknown]⦿Gillies, D. (2000) Philosophical theories of probability,

Routledge, London & New York.

⦿Laplace, P. S. (1812) Théorie analytique des probabilité ,Paris, in French.

⦿Ohno-Yasumaro ( ed. ) (712) Kojiki ( Records of An-

cient Matters) , 3 volumes, in Japanese. [ Exact records
unknown]

⦿Sakai, Y. (1995) On Mathematics and Insurance, withspecial reference to Japanese Culture, in Japanese. Contained in Mizushima, K. (ed.) (1995) Insuranceand Culture: Risk and the Japanese People, Chikura Publishers,
Tokyo, in Japanese .

⦿Sakai, Y. (1995) On tomoesen or three-way playoffs inthe sumo world, in Japanese. Contained in Mizushima,
K. (ed.) (1995) Insurance and Culture: Risk and the Japanese People, Chikura Publishers, Tokyo, in Japanese .

⦿Sakai, Y. (2019) J.M. Keynes Versus F. H. Knight: Risk,

Probability, and Uncertainty, Springer Nature, Singapore.

⦿Sen, A.K. (1977) Rational fools: a critique of the behav-

ioural foundations of economic theory, Philosophy andPublic Affairs, 6-1, 317-344.

⦿Yano, K. (1966) Men of mathematics, Koudansha Pub-lishers, Tokyo, in Japanese.

5) For the history of the theory of risk and probability,

see Gillies (2000) and Laplace (1812).

6) For this point, see Akerlof and Shiller (2009), Sakai(2019) and Sen (1977). All of those works indicate that ahuman being is neither a "mere economic animal" nor "arational fool". We should keep in mind that there should
be a reasonable balance between fairness and excitement.

Analyzing Three-Way Playoffs in the Sumo World

Yasuhiro Sakai

133

Analyzing Three-Way Playoffs in the Sumo World
Fairness and Excitement

Yasuhiro Sakai

This paper is concerned with tomoesen, aspecial method of three-way playoffs in the Japanese sumo world. In the tomoesen, only two
wrestlers fight each other, with the third man
resting for a while. Then, the winner of the firstfight has to fight against the resting man. The
former may be either a winner or a loser. Thiskind of fight will continue until there appears
some one who bravely wins the two successive
fights. Since the Pacific War ended, while morethan 450 fights have been carried out, we havewitnessed special tomoesen merely seven times.
From a theoretical viewpoint, we can show thateach of the men who come up to the ring has
the victory probability of 5/14, and the thirdman, the probability of 4/14. The difference
between those two numbers is 1/14 or 7.14286,,, . Although tomoesen is not a fair game in astrict sense, it has nevertheless been acceptedby the public as an exciting sport game. It
seems that there exists a reasonable balance between fairness and excitement.
Key words: tomoesen, three-way playoffs, fairness versus excitement

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Analyzing Three-Way Playoffs in the Sumo World

Yasuhiro Sakai

135