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They were robbed! Scoring by the middlemost
to attenuate biased judging in boxing
Stuart Baumann Carl Singleton
June 27, 2024
Abstract
Boxing has a long-standing problem with biased judging, impacting both professional
and Olympic bouts. “Robberies”, where boxers are widely seen as being denied rightful
victories, threaten to drive fans and athletes away from the sport. To tackle this problem,
we propose a minimalist adjustment in how boxing is scored: the winner would be decided
by the majority of round-by-round victories according to the judges, rather than relying
on the judges’ overall bout scores. This approach, rooted in social choice theory and
utilising majority rule and middlemost aggregation functions, creates a coordination
problem for partisan judges and attenuates their influence. Our model analysis and
simulations demonstrate the potential to significantly decrease the likelihood of a partisan
judge swaying the result of a bout.
JEL Codes: D91, L83, Z20, Z28
Keywords: Scoring Rules, Judgement Bias, Contests, Pugilism, Combat Sports
stuart@stuartbaumann.com; Corresponding author: carl.singleton@stir.ac.uk, Economics Division, Stirling
Management School, University of Stirling, Stirling, FK9 4LA, Scotland, UK.
We are grateful for comments and advice from Anwesha Mukherjee.
Declarations of interest: none
1
1. Introduction
Boxing has a reputation for partisan and corrupt judging. At the amateur level, some decisions
in Olympic gold medal bouts have attracted criticism and ridicule, becoming boxing folklore,
such as Roy Jones Jr.’s defeat in the 1988 (Seoul) light heavyweight final to a South Korean
fighter (Ashdown,2012), and Joe Joyce’s defeat in the 2016 (Rio de Janeiro) super heavyweight
final (Ingle,2021;Rumsby,2021). In professional boxing, there is longstanding suspicion about
the integrity of judges (e.g., US Senate,2001). Recent perceived “robberies” include Haney
Vs. Lomachenko (Wainwright,2023) and the first two editions of Alvarez Vs. Golovkin (Reid,
2023).
This short paper models the decisions of boxing judges and proposes an alternative scoring
method that has the potential to significantly attenuate judge bias. Currently, scoring at the
elite level is on a per-judge basis, with three judges usually employed for elite professional bouts
and five at the Olympic amateur level. Judges score each round individually and then award
their entire “vote” to the boxer who wins a majority of rounds.1The bout is then awarded to the
boxer receiving votes from a majority of judges. If neither boxer receives a majority, due to at
least one tied scorecard among the judges, then the bout is a draw. In this system, “aggregation
over rounds and then judges”, or “majority judges rule”, it is relatively straightforward for a
judge to ensure their vote goes to their favoured boxer. They just need to award them half the
rounds (i.e., 7 of 12 for a world championship level men’s professional bout). They can do this
while minimising backlash, by choosing the best rounds for their favoured boxer.
The change we propose, “aggregation over judges and then over rounds”, or “majority rounds
rule”, is for each round to be awarded based on the aggregate scores over all judges. Whoever
wins the majority of rounds wins the bout, rather than whoever wins on a majority of the
judges’ scorecards. This represents a minimalist change to the scoring system in the sport,
so that the aggregation of judges’ scores is first between them within rounds, and then over
rounds, rather than vice versa. The minor nature of this change is sufficient to introduce a
significant coordination problem for a partisan judge, and may be acceptable among fans.2
We focus on modelling the simplest practical case, with three judges, one of whom is biased in
favour of one boxer. Under majority judges rule, a partisan judge can substantially increase the
probability of a boxer winning despite being outnumbered by unbiased judges. Under majority
rounds rule, even if the partisan judge awards a majority of rounds to a favoured boxer, then
this will have no impact on the final outcome unless those rounds align with the decisions of the
other judges. This coordination problem implies that, to achieve a high probability of victory
for their favoured boxer, the partisan judge would have to award more rounds to their favoured
boxer than in the current system. This exposes them to scrutiny and potential backlash, as
1This is a slight simplification as judges can award additional points for a given round based on knockdowns,
fouls, or particularly dominant performances by one fighter.
2Fans tend to scrutinise, oppose and criticise even quite small changes to the rules of their beloved sports. A
notable example from cricket is the LBW rule, which has been continually ‘improved’ over the last century,
often under opposition and criticism (Kumar,2022).
2
boxing pundits and fans will often criticise poorly awarded rounds on judges’ scorecards.3Our
analysis and simulations of the model demonstrate that the scoring rule change could be highly
effective in diminishing the incentives for biased judging in boxing and its influence.
Our proposed scoring rule is an application of majority rule and the middlemost aggregation
function from social choice theory, which minimise the effective manipulability of outcomes
by graders (e.g., Arrow,1963;Balinski and Laraki,2007;Young,1974b,a). This principle
is already applied somewhat to the scoring in boxing, since the decision of the middlemost
judge now determines the bout result. We suggest awarding bouts based on the aggregated
middlemost round-by-round votes instead.
The prevalence of judging bias in combat sports has been documented in a growing literature
of empirical academic papers (e.g., Lee, Cork and Algranati,2002;Holmes, McHale and
Zychaluk,2024). This behaviour was also described vividly in the recent judge-led independent
investigation McLaren (2022) report, which examined unethical conduct in Olympic boxing
after being commissioned by the Association Internationale de Boxe Amateur (AIBA). While
the report did propose improved appointment processes and training of judges, it did not
explore how to make the incentives inherent in the judging process more resilient to biases and
corruption.
Our goal of improving the incentives of judges in boxing is closely related to the focus of
Frederiksen and Machol (1988), who analysed the judging in sports like figure skating and dance,
where judges need to decide between multiple competitors, a setting where Arrow’s (1963)
theorem implies that all possible ways to combine judge preferences have some undesirable
characteristics. Frederiksen and Machol proposed a new method for aggregating judge scores
for such situations that attenuates some of these issues. Their context though faced the problem
of the Arrow Impossibility Theorem (social choice paradox), given there were more than two
alternative outcomes in the contest. That theorem does not apply here for a boxing bout since
it consists of just two competitors, only one winner, and potentially biased judges.
In general, this paper contributes to the vast operational research literature that either post
hoc analyses changes to scoring rules and laws in sports or proposes new changes (for recent
surveys see Wright,2014;Kendall and Lenten,2017). Our work falls into the latter type
of study, particularly where minimalist changes have been proposed that could still in theory
substantially improve the fairness of sports outcomes. For instance, in the world’s most popular
sport, association football, recent contributions have used simulations to explore whether
incentives and outcomes could be altered significantly under different tie-breaking rules in
round-robin tournaments (Csat´o,2023;Csat´o, Molontay and Pint´er,2024), whether dynamic
sequences in penalty shootouts could be fairer (Csat´o and Petr´oczy,2022), and whether the
allocation system for the additional slots of the expanded FIFA World Cup could be improved
3Criticism of judges in social and sports media is generally fiercest after bouts where a robbery is perceived to
have happened. It often focuses on specific rounds where a judge’s decision appears to be particularly poor.
While authorities seldom intervene to order a rematch, judges may be stripped of their status and not employed
again (e.g., Slavin,2017).
3
according to the stated goals of the organisers (Krumer and Moreno-Ternero,2023). Finally
this paper builds on a growing sports economics and management literature studying various
incentive issues in boxing and other combat sports (Akin, Issabayev and Rizvanoghlu,2023;
Amegashie and Kutsoati,2005;Butler et al.,2023;Butler,2023;Duggan and Levitt,2002;
Dietl, Lang and Werner,2010;Tenorio,2000). However, to the best of our knowledge, the
incentives of boxing judges have not yet been studied, given the scoring rules they face, despite
a well-developed literature on the influences and implications of biased decision making by
the referees and judges in other sports (e.g., Dohmen and Sauermann,2016;Bryson et al.,
2021;Reade, Schreyer and Singleton,2022), including other combat contests (Brunello and
Yamamura,2023)).
The remainder of our short paper proceeds as follows. In Section 2, we setup a styled model of
potentially biased judging in a boxing contest. Section 3 describes our analysis and discussion
of the model. The detailed proofs of the main propositions regarding the scoring rules are
presented in the Online Appendix, as are variations on the main results from simulating the
model.
2. The Model
Consider a contest between two boxers of equal ability, in the Blue and Red corners. We assume
each sequential round t∈ {1,2, . . . , N}of the contest has a true result, τt∈ {B, R}, which is
a binomial random variable with equal probability.
Each judge, j∈ {1,2,3}, gets an i.i.d. signal, xt,j ∈ {B, R}, about the result of a round. With
probability α∈(0,1
2) this signal is the incorrect result, xt,j =τt, while with probability 1 −α
it is correct, xt,j ≡τt.
Judges have a utility of:
U=S1Blue wins +G1Red wins −L , L =PN
t=1 1st,j =τt
N,(1)
where S≥0 and G≥0 represents the a judge’s value from Blue or Red winning respectively.
Lrefers to a backlash cost.
We consider the case of two fair judges have S=G= 0. As these judge’s utility does not
depend on the bout’s winner their optimal behaviour is to minimise backlash by awarding
fairly, defined by choosing a round score of st,j =B⇐⇒ xt,j ≡B.4The third judge is
partisan in favour of Blue and so has S > 0 and G= 0.
Under majority judges rule, the middlemost judge scorecard determines the bout. Under
majority rounds rule, the middlemost judge determines each round’s winner, and then the
middlemost round determines the bout. Judges award rounds separately and simultaneously.
4We analyse the case where all three judges are fair in Online Appendix B.
4
3. Analysis, Results, and Discussion
The partisan judge (j= 1) can minimise backlash by awarding rounds fairly.5Under majority
judges rule, they can maximally increase the chance of Blue winning the bout, while minimising
backlash, by awarding st,1=Bin more than N
2rounds. Under majority rounds rule, their
problem is more complex; a judge could award more than N
2rounds to a boxer who then does
not win them because the other judges disagreed.
If Sis low, however, then the expected backlash can be sufficient for the partisan judge to
award rounds fairly. We can characterise the critical ˆ
Swhere the partisan judge is indifferent
between awarding fairly or gifting an additional round to Blue. We find that this critical value
is higher under majority rounds than majority judges rule, indicating that the former is more
resilient to judge bias.
Proposition 1. For three-round bouts, in which Red won a majority of rounds according to the
true realisations, τ= [τ1, τ2, τ3], the critical ˆ
Sis higher under majority rounds than majority
judges rule, ∀α∈(0,1
2).
Sketch of Proof: We can calculate the probability of each fair judge awarding a round for Blue,
denoted by q, conditional on the signal seen by the partisan judge:
q|(xt,1≡B) = (1 −α)2+α2, q|(xt,1≡R)=2α(1 −α).(2)
Under majority rounds rule, the number of fair judges awarding for Blue in a particular round
can be represented as drawing from a binomial distribution with probabilities as in Equation
2. The survival function of this binomial, in conjunction with the decision of the partisan
judge, is sufficient to infer the probability of Blue winning the round. From the probabilities
for each round, we can derive the optimal number of rounds for the partisan judge to award for
Blue. Under majority judges rule, we can infer the probability of another scorecard being in
favour of Blue by combining the probabilities in Equation 2across rounds. We can use these
probabilities to evaluate whether the partisan judge should award additional rounds such that
Blue wins on their card. Then we can derive, for each scoring rule, expressions for the critical
ˆ
Sbelow which the partisan judge will award rounds fairly (see Online Appendix A). We find
that there is a higher ˆ
Sunder majority rounds rule, giving us the proposition.
Proposition 1establishes that the majority rounds rule is more robust to partisan judging than
the majority judges rule for three-round bouts. We numerically solve the model to establish
the robustness of this result in longer bouts.6We use a benchmark parametrisation of α= 0.1,
S= 0.8, three judges (one of whom is partisan), and N= 12 rounds.7
5If S≡0, then this judge’s actions will be congruent to the fair judges.
6This is done along the lines discussed in the sketch proof for Proposition 1. The code for calculating partisan
judge best responses and bout simulations are included in the online supplementary material.
7Importantly, Online Appendix Bshows that when all three judges are fair, the majority rounds rule is still more
accurate than the majority judges rule in generating a deserving winner of the bout. Intuitively this occurs as
5
To demonstrate a partisan judge’s decision making, Figure 1shows the probability of Blue
winning the bout, given they truly won 6 rounds, for each number of rounds the partisan judge
awards them. Under majority judges rule, there is a sharp increase in the probability of Blue
winning if the partisan judge awards them more than 6 rounds. If Blue truly deserved to win
4 or 5 rounds, then, to award Blue the win, the partisan judge only needs to risk the backlash
associated with giving them 3 or 2 more rounds on their scorecard. In contrast, Figure 1shows
that under majority rounds rule, a judge cannot secure a sharp increase in the probability of
Blue winning by giving them a small number of extra rounds; more rounds only gradually
increase Blue’s chances.
FIGURE (1) Simulated Probability of Blue winning, when both boxers truly won 6 of the 12
rounds, and 1 of the 3 judges favours Blue
20%
40%
60%
0 1 2 3 4 5 6 7 8 9 10 11 12
Number of rounds biased judge awards to blue boxer
Probability of blue boxer winning bout
Majority Rounds Rule Majority Judges Rule
Figure 2shows the impact of these differing incentives for the partisan judge, from running
a series of simulations and counting the proportion of times each boxer wins under the two
scoring systems, conditional on the true number of rounds won by Blue. When deciding the
contest by majority judges, there is a high probability of erroneous results when Blue truly
won only 4-6 rounds. When Blue truly wins the most rounds, the partisan judge unduly helps
to lock in a deserved victory, so there is not a large difference in the number of incorrectly
awarded bouts.
Finally, in the majority rounds case, it can be noted from Figure 2that the probability of the
there are more combinations of rounds that could be flipped to change the result. Consider a three-round bout
with three fair judges and a τrealisation of [B,B,R]. Consider that two mistakes happened in judging the
bout (in that for two round-judges the xt,j realisation differs from τ). There are six possible pairs of xt,j values
that can be flipped to change the result of the bout with the majority rounds rule. For one of the rounds where
Blue won, we need to flip two xt,j values and there are six combinations that achieve this. But there are twelve
possible pairs of xt,j values that can be flipped to change the result of the bout with the majority judges rule.
We need to flip two of the xt,j values awarded to Blue on two different scorecards and there are twelve pairs of
values that achieve this.
6
Blue boxer winning always increases when the biased judge awards them more rounds. This is
in contrast to the majority judges case, when a judge ceases to impact the result at the point
at which they award a majority of their card to a boxer. For instance, consider a bout where
the fair judge sees 10 rounds with xt,1≡Band only two rounds are seen to be won by red.
In this case, the biased judge may award additional rounds to Blue to lock in a Blue victory,
while they would have no such incentive under majority judges rule.
FIGURE (2) Probability of a “correct” result depending on the number of rounds truly won
by Blue and how judges’ scores are aggregated
40%
60%
80%
100%
0123456789101112
Number of rounds blue boxer won (excl noise)
Probability of correct decision for bout
Majority Judges Rule Majority Rounds Rule
This effect, however, does not tend to lead to a greater probability of an erroneous result
under the majority rounds rule. The main reason for this is that the effect occurs in a context
where Blue has likely won a large majority of rounds and is likely to win the bout. The more
important case is when a bout is more even and there is a sharp increase, under the majority
judges rule, in the winning probability at the 7 round level in Figure 2.
This point can be seen in Figure 3, which shows the probability of each possible outcome on the
y-axis and the number of rounds Blue truly won (excluding noise) on the x-axis. Under majority
judges rule (top panel), in evenly matched bouts, where the true result is a draw, Blue wins
47.0% and Red wins 11.2%. When evenly matched bouts are awarded under majority rounds
rule (bottom panel), Blue wins 13.4% and Red wins 19.3%.
Figure 3also shows the frequencies where one boxer wins despite the other deserving outright
victory, e.g., the blue area to the left of the vertical black line. Under majority judges rule, it is
more likely for an erroneous victory to be in favour of Blue than Red; in this parametrisation,
a robbery in favour of Blue is 12.5 times more likely than a robbery in favour of Red. Under
majority rounds rule, the likelihood of a robbery is still in Blue’s favour, by a multiple of 1.99,
because there is still some incentive for the partisan judge to favour Blue. But this scoring
system can substantially attenuate Blue’s advantage from the presence of a partisan judge.
7
There are also fewer robberies in absolute terms.
FIGURE (3) Probability of each outcome depending on the number of rounds truly won by
Blue and how judges’ scores are aggregated
Majority Judges Rule
Majority Rounds Rule
0 1 2 3 4 5 6 7 8 9 10 11 12
0%
25%
50%
75%
100%
0%
25%
50%
75%
100%
Number of rounds blue boxer won (excl noise)
Probability of outcome
Blue wins Draw Red wins
For robustness, the Online Appendices demonstrate extensions and checks on our analysis.
Appendix Cconsiders simulations with alternative parametrisations of the benchmark model,
and, in Appendices D-Fwe repeat the analysis for setups consistent with women’s professional,
men’s Olympic, and women’s Olympic boxing, respectively (i.e., different numbers of rounds
and judges). The results of all these extensions support our key findings: deciding bouts by
majority rounds, compared with by majority judges, makes it less likely that a partisan judge
sways the outcome of a bout.
8
Declarations
The authors did not receive financial support from any organisation for the submitted work.
The authors have no competing interests to declare that are relevant to the content of this
article.
9
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11
They were robbed! Scoring by the middlemost
to attenuate biased judging in boxing
Online Appendix
Appendix A. Full proof of Proposition 1
To simplify, we assume there are three rounds. Consequently, there are four information sets
that the partisan judge can receive before they choose how many rounds to award to Blue.
These are {BBB, BBR, BRR, RRR }, which are the signals of each round of the bout after
sorting round results.2
A.1 Majority Judges Rule
We will first analyse the majority judges case. We will denote that the number of rounds that
Blue truly won is bT. Conditioning on bT, the number of rounds that the Blue boxer wins on
the fair judge’s scorecare is x+y, where x∼Binomial(bT,1−α) and y∼Binomial(N−bT, α).
The probability that the Blue boxer wins on a fair judge’s card is, therefore:
Prob(Blue wins|bT) =
3
X
x=0
fx(x)Gy(2 −x) (A.1)
where fx(·) is the probability mass function of Binomial(bT,1−α) and Gy(·) is the survival
function of Binomial(N−bT, α).
The partisan judge cannot condition on bT, however, as they cannot observe it. They can assign
probabilities to the various possibilities of bTafter conditioning on their observation of b. We
can work out Prob(b|bT) using the following expression that is similar to Equation (A.1):
Prob(Blue wins brounds|bT) =
3
X
x=0
fx(x)gy(b−x) (A.2)
where gy(·) is the probability mass function of Binomial(N−bT, α). We can use this expression
in conjunction with Bayes rule to derive an expression for Prob(bT|b).
We use these to calculate the probability of another judge’s scorecard being in favour of Blue
for each information set that the partisan judge observes. We use Cto denote this probability:
C|b=
3
X
bT=0
Prob(bT|b)×Prob(Blue wins|bT) (A.3)
The utilities available at each information set (where the first column shows the information
set for each row) and action (the second to fifth columns) are shown in Table A1.
2Note, in both scoring systems, no distinctions are made as to when in the bout a particular round result
occurred. Therefore, a bout with true result BRB is the same as a bout with true result BBR, and we reorder
round results to simplify the analysis.
12
Award BBB Award BBR Award BRR Award RRR
BBB S(1 −(1 −C|BBB)2)−α S(1 −(1 −C|BBB)2)−2α+(1−α)
3S(C|BBB)2−α+2(1−α)
3S(C|BBB)2−(1 −α)
BBR S(1 −(1 −C|BBR)2)−2α+(1−α)
3S(1 −(1 −C|BBR)2)−α S(C|BBR)2−2α+(1−α)
3S(C|BBR)2−α+2(1−α)
3
BRR S(1 −(1 −C|BRR)2)−α+2(1−α)
3S(1 −(1 −C|BRR)2)−2α+(1−α)
3S(C|BRR)2−α S (C|BRR)2−2α+(1−α)
3
RRR S(1 −(1 −C|RRR)2)−(1 −α)S(1 −(1 −C|RRR)2)−α+2(1−α)
3S(C|RRR)2−2α+(1−α)
3S(C|RRR)2−α
Table (A1) Utilities for each action and information set in the basic model
The best responses to seeing BBB and BBR are to award BBB and BBR, respectively. This
minimises backlash while still awarding the card of the partisan judge to the favoured Blue. If
the partisan judge sees BRR, however, then they need to choose between BBR (which awards
the card to Blue) and BRR (which minimises expected backlash). The condition for awarding
rounds BRR as preferred to awarding rounds BBR is:
b
SMaj. Jgs, BRR ≤1−2α
6(C|BRR)(1 −(C|BRR))
≤(1 −2α)÷"192α(1 −α)32α(α−1) 0.5α4−1α3+ 0.875α2−0.375α+ 0.125+ 1
×0.5α4−1α3+ 0.875α2−0.375α+ 0.125#(A.4)
For a simple example, when α= 0.1, this expression becomes approximately 1.09.
If the partisan judge sees RRR, then they need to choose between BBR (which awards the card
to Blue) and RRR (which minimises backlash). The condition for RRR to be perferred to BBR
is:
b
SMaj. Jgs, RRR ≤1−2α
3(C|RRR)(1 −(C|RRR))
≤(1 −2α)÷"48α2(1 −α)21−16α2(1 −α)2·α2−α+ 0.75α2−α+ 0.75#
(A.5)
For a simple example, when α= 0.1, the right-hand-side here becomes approximately 12.14.
It makes intuitive sense that the critical level of Blue winning utility will be higher here than
in Equation (A.4), as they need to award two more rounds than they believe Blue won (so
higher backlash) and there is lower odds of at least one other judge awarding in favour of Blue
(so less chance that partisan judging will deliver a victory).
13
A.2 Majority Rounds Rule
We can derive the following probabilities of Blue winning a round conditional on what the
partisan judge observes and does:3
Partisan judge observes B and does B: q1= 1 −α(1 −α) (A.6)
Partisan judge observes R and does B: q2= 3α−3α2(A.7)
Partisan judge observes B and does R: q3= 1 −3α+ 3α2(A.8)
Partisan judge observes R and does R: q4=α(1 −α) (A.9)
At this point, we define the function fto denote the probability of getting at least two
realisations from 3 binomial distribution trials with probabilities p1, p2, p3:
f(p1, p2, p3) = p1p2(1 −p3) + p1(1 −p2)p3+ (1 −p1)p2p3+p1p2p3(A.10)
Using this function, we can write the utilities available at each information set (where the first
column shows the information set for each row) and action (the second to fifth columns), shown
in Table A2.
Award BBB Award BBR Award BRR Award RRR
BBB Sf (q1, q1, q1)−3α+0(1−α)
3Sf (q1, q1, q3)−2α+1(1−α)
3Sf (q1, q3, q3)−1α+2(1−α)
3Sf (q3, q3, q3)−0α+3(1−α)
3
BBR Sf (q1, q1, q2)−2α+1(1−α)
3Sf (q1, q1, q4)−3α+0(1−α)
3Sf (q1, q3, q4)−2α+1(1−α)
3Sf (q3, q3, q4)−1α+2(1−α)
3
BRR Sf (q1, q2, q2)−1α+2(1−α)
3Sf (q1, q2, q4)−2α+1(1−α)
3Sf (q1, q4, q4)−3α+0(1−α)
3Sf (q3, q4, q4)−2α+1(1−α)
3
RRR Sf (q2, q2, q2)−0α+3(1−α)
3Sf (q2, q2, q4)−1α+2(1−α)
3Sf (q2, q4, q4)−2α+1(1−α)
3Sf (q4, q4, q4)−3α+0(1−α)
3
Table (A2) Utilities for each action and information set in the basic model under majority
rounds rule
Similar to the case in Table A1, the utility on the diagonal is better than the utility from
awarding more rounds than this to Red. This implies that if the partisan judge sees BBB, then
they should award rounds as BBB. We can solve for the levels of Sat which the partisan judge
prefers to award rounds fairly rather than giving additional rounds to Blue. Starting at the
BBR information set:
b
SMaj. Rds, BBR ≤1−2α
12α2[α4−3α3+ 4α2−3α+ 1] (A.11)
For the BRR information set, the partisan judge could do BBR or BBB rather than the fair
result of BRR. We derive the critical Sfor both and can determine that a partisan judge will
deviate to BBR at a higher Svalue than they would deviate to BBB. Hence, we below report
the threshold above which the partisan judge will not deviate to BBR:
b
SMaj. Rds, BRR ≤
1−2α
6α[−2α5+ 6α4−8α3+ 6α2−3α+ 1] (A.12)
3For instance, we can work out q1and q3as follows. If we see B, then there is (1 −α) chance that B is the true
state and αchance that Ris the true state. If B is the true state, then there is a chance αthat a fair judge sees
R and a 1 −αchance they see B. If R is the true state, then there is a chance αthat a fair judge sees B and a
1−αchance they see R. The chance that B wins on at least one other scorecard (and thus wins the round) is
therefore (1 −α)
1−α2
| {z }
Btrue state
+α
1−(1 −α)2
| {z }
Rtrue state
= 1 −α(1 −α). We can work out q2,q3and q4analogously.
14
BRR
RRR
0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5
0.01
0.1
1
10
100
Alpha
Critical S
Majority Judges Rule Majority Rounds Rule
FIGURE (A1) Critical Sbelow which partisan judges do award fairly for three-round bouts
with three judges in the BRR and RRR information sets
Finally, for the RRR information set, the partisan judge could do BRR, BBR or BBB, rather
than the fair result of RRR. We can establish that if αis near zero, then the partisan judge
will deviate to BBB at a higher Sthan they would deviate to the other options. When αis
near (but below) 0.5, then they will deviate to BBR at a higher Sthan they would deviate to
the other options. As a result, we have the critical Svalue:
b
SMaj. Rds, RRR ≤min[
1−2α
4α2(13α4−39α3+ 45α2−25α+ 6),
1−2α
6α2(4α4−12α3+ 15α2−10α+ 3)] (A.13)
Now we can summarise Equations (A.4), (A.5), (A.11), (A.12) and (A.13) with a chart of α
against the critical Sratio below which the partisan judge does not mis-award rounds. This is
shown in Figure A1.
In all cases where Red wins more rounds, the majority rounds rule has a higher Svalue at
which the partisan judge is indifferent to awarding fairly and giving more rounds to Blue. This
indicates that the majority rounds rule is more robust to partisan judging.
15
Appendix B. No judges are partisan
When no judges are partisan, the proposed change, from awarding bouts by majority judges
to majority rounds, reduces the probability of bouts being wrongly decided. This can be seen
in Figure B1, which is comparable to Figure 2, with the same parametrisation, but reflects the
case where all three judges are fair.
FIGURE (B1) Probability of a “correct” result depending on the number of rounds truly won
by Blue and how judges’ scores are aggregated: the case of no partisan judges
60%
70%
80%
90%
100%
0123456789101112
Number of rounds blue boxer won (excl noise)
Probability of correct decision for bout
Majority Judges Rule Majority Rounds Rule
16
Appendix C. Other parametrisations
We consider different model parametrisations, to demonstrate the extent to which the
qualitative results of this paper may vary.
High disagreement between different judges
We increase the noise variance, such that the different judges disagree more often about the
outcome of a round. Specifically, we increase α= 0.2 and leave the other parameters as
they were in the main body of the paper. Figures 1-3are reproduced below for this new
parametrisation as Figures C1-C3.
FIGURE (C1) Probability of favoured boxer winning in one bout where both boxers won 6
rounds (in the absence of noise)
20%
40%
60%
80%
0 1 2 3 4 5 6 7 8 9 10 11 12
Number of rounds biased judge awards to blue boxer
Probability of blue boxer
winning bout
Majority Rounds Rule Majority Judges Rule
FIGURE (C2) Probability of correct result for bout
25%
50%
75%
100%
0123456789101112
Number of rounds blue boxer won (excl noise)
Probability of correct
decision for bout
Majority Judges Rule Majority Rounds Rule
17
FIGURE (C3) Probability of each outcome for bout
Majority Judges Rule
Majority Rounds Rule
0 1 2 3 4 5 6 7 8 9 10 11 12
0%
25%
50%
75%
100%
0%
25%
50%
75%
100%
Number of rounds blue boxer won (excl noise)
Probability of outcome
Blue wins Draw Red wins
18
High degree of favouritism
We increase Sto 1.0 and leave other parameter values as they are in the main body of the
paper. Figures 1-3are reproduced below for this new parametrisation as Figures C4-C6
FIGURE (C4) Probability of favoured boxer winning in one bout where both boxers won 6
rounds (in the absence of noise)
20%
40%
60%
0 1 2 3 4 5 6 7 8 9 10 11 12
Number of rounds biased judge awards to blue boxer
Probability of blue boxer
winning bout
Majority Rounds Rule Majority Judges Rule
FIGURE (C5) Probability of correct result for bout
40%
60%
80%
100%
0123456789101112
Number of rounds blue boxer won (excl noise)
Probability of correct
decision for bout
Majority Judges Rule Majority Rounds Rule
FIGURE (C6) Probability of each outcome for bout
Majority Judges Rule
Majority Rounds Rule
0 1 2 3 4 5 6 7 8 9 10 11 12
0%
25%
50%
75%
100%
0%
25%
50%
75%
100%
Number of rounds blue boxer won (excl noise)
Probability of outcome
Blue wins Draw Red wins
19
Appendix D. Women’s professional boxing
In women’s professional boxing, there are generally 10 rounds and 3 judges. Figures 1-3are
reproduced below for women’s professional boxing as Figures D1-D3, with otherwise identical
parametrisations.
FIGURE (D1) Probability of favoured boxer winning in one bout where both boxers won 6
rounds (in the absence of noise) - Women’s professional boxing
20%
40%
60%
0 1 2 3 4 5 6 7 8 9 10
Number of rounds biased judge awards to blue boxer
Probability of blue boxer
winning bout
Majority Rounds Rule Majority Judges Rule
FIGURE (D2) Probability of correct result for bout - Women’s professional boxing
60%
80%
100%
012345678910
Number of rounds blue boxer won (excl noise)
Probability of correct
decision for bout
Majority Judges Rule Majority Rounds Rule
FIGURE (D3) Probability of each outcome for bout - Women’s professional boxing
Majority Judges Rule
Majority Rounds Rule
012345678910
0%
25%
50%
75%
100%
0%
25%
50%
75%
100%
Number of rounds blue boxer won (excl noise)
Probability of outcome
Blue wins Draw Red wins
20
Appendix E. Men’s Olympic amateur boxing
In men’s Olympic amateur boxing, there are generally 3 rounds and 5 judges. Figures 1-3are
reproduced below for men’s Olympic amateur boxing as Figures E1-E3.
10%
15%
20%
25%
30%
0 1 2 3
Number of rounds biased judge awards to blue boxer
Probability of blue boxer
winning bout
Majority Rounds Rule Majority Judges Rule
FIGURE (E1) Probability of favoured boxer winning in one bout where the Blue boxer won
1 round and the Red boxer won 2 (in the absence of noise) - Men’s Olympic amateur boxing
96%
97%
98%
99%
100%
0123
Number of rounds blue boxer won (excl noise)
Probability of correct
decision for bout
Majority Judges Rule Majority Rounds Rule
FIGURE (E2) Probability of correct result for bout - Men’s Olympic amateur boxing
Majority Judges Rule
Majority Rounds Rule
0 1 2 3
0%
25%
50%
75%
100%
0%
25%
50%
75%
100%
Number of rounds blue boxer won (excl noise)
Probability of outcome
Blue wins Red wins
FIGURE (E3) Probability of each outcome for bout - Men’s Olympic amateur boxing
21
Appendix F. Women’s Olympic amateur
In women’s Olympic amateur boxing, there are generally 4 rounds and 5 judges. Figures 1-3
are reproduced below for women’s Olympic amateur boxing as Figures F1-F3.
FIGURE (F1) Probability of favoured boxer winning in one bout where both boxers won 6
rounds (in the absence of noise) - Women’s Olympic amateur boxing
5%
10%
15%
20%
01234
Number of rounds biased judge awards to blue boxer
Probability of blue boxer
winning bout
Majority Rounds Rule Majority Judges Rule
FIGURE (F2) Probability of correct result for bout - Women’s Olympic amateur boxing
90.0%
92.5%
95.0%
97.5%
100.0%
0 1 2 3 4
Number of rounds blue boxer won (excl noise)
Probability of correct
decision for bout
Majority Judges Rule Majority Rounds Rule
FIGURE (F3) Probability of each outcome for bout - Women’s Olympic amateur boxing
Majority Judges Rule
Majority Rounds Rule
01234
0%
25%
50%
75%
100%
0%
25%
50%
75%
100%
Number of rounds blue boxer won (excl noise)
Probability of outcome
Blue wins Draw Red wins
22
