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This paper addresses the issue of developing a widely accepted Olympics ranking scheme based upon the Olympic Game medal table published by the International Olympic Committee, since the existing lexicographic ranking and sum ranking systems are both criticized as biases. More specifically, the lexicographic ranking system is deemed as overvaluing gold medals, while the sum ranking system fails to reveal the real value of gold medals and fails to discriminate National Olympic Committees that won equal number of medals. To start, we employ a sophisticated mathematical method based upon the incenter of a convex cone to aggregate the lexicographic ranking system. Then, we consider the fact that the preferences between the lexicographic and the sum ranking systems may not be consistent across National Olympic Committees and develop a well-designed mathematical transformation to obtain interval assessment results under typical preference. The formulation of intervals is inspired by the observation that it is extremely difficult to achieve a group consensus on the exact value of weights with respect to each ranking system, since different weight elicitation methods may produce different weight schemes. Finally, regarding the derived decision making problem involving interval-valued data, this paper utilizes the Stochastic Multicriteria Acceptability Analysis to obtain a comprehensive ranking of all National Olympic Committees. Instead of determining precise weights, this work probes the weight space to guarantee each alternative getting the most preferred one. The proposed method is illustrated by presenting a new ranking of 12 National Olympic Committees participating in the London 2012 Summer Olympic Games.

The Olympics medal table is a list of National Olympic Committees (NOCs) released by the International Olympic Committee (IOC), which ranks NOCs according to the number of gold medals won by single NOC. The number of silver medals is taken into account next and then the number of bronze medals. Meanwhile, total medal count with respect to each NOC is summed and shown in the Olympics medal table, which is widely accepted as the alternative criterion to sort NOCs. These two ranking mechanisms are defined as the lexicographic ranking system and the sum ranking system, respectively. However, the lexicographic ranking system is criticized as overvaluing gold medals. In particular case, the NOCs that won a large quantity of silver and bronze medals but without gold medal are ranked below the NOCs that won only one gold medal. For instance, the 2012 Summer Olympics Medal Table ranks India with zero gold medal, two silver medals, and four bronze medals behind Venezuela with only one gold medal. On the other hand, the sum ranking system takes into consideration of the total sum of the medals won by the NOCs, which inevitably fails to reveal the real value of gold medals and thus fails to discriminate NOCs that won equal number of medals. For instance, the sum ranking system at 2012 London Summer Olympics ranks Uzbekistan with one gold medal and two bronze medals and Thailand with two silver medals and one bronze medal at the same position. This is definitely full of conflict in real life.

Although the IOC publishes the quasi-official medal table during each Olympic Game, the IOC itself has not officially recognized and endorsed any ranking system. The former then-president of IOC, Jacques Rogge, says during the 2008 Beijing Summer Olympic Game:

The present study is motivated by his viewpoint and seeks to provide a comprehensive ranking system while simultaneously considering the preferences between the lexicographic and the sum ranking systems. Very few studies in literature have addressed the issue that measures the NOCs’ performance based only on the number of medals won. Sitarz [

Apart from the previous studies about assessing Olympic achievements, this paper provides new research directions and more method options for ranking construction, based upon the methodology developed by Song et al. [

We modify the Olympics medal table published by IOC through jointly considering the lexicographic and the sum ranking systems, while almost all of the extant literature is interested in dealing with the lexicographic ranking system. The weights associated with gold, silver, and bronze medals are obtained through a well-designed mathematical approach to aggregate the lexicographic ranking system.

Different preferences between the lexicographic and the sum ranking systems are proposed and investigated to obtain a holistic ranking. Regarding certain preference, a sophisticated mathematical transformation is developed to support the decision maker generating interval measurement with respect to each NOC. This gives rise to an interval decision matrix for aiding ultimate ranking.

SMAA-2 is applied to determine holistic ranking acceptability index for the proposed interval decision matrix, by which all NOCs could be fully ranked taking into account both the lexicographic and the sum ranking systems.

The rest of this paper proceeds as follows. Section

The majority of existing approaches to measuring Olympic achievement are related to Data Envelopment Analysis (DEA), evaluating input (i.e., GDP per capita and population) and output (i.e., the number of medals) efficiency using a family of DEA models and their variants. The pioneering work in this domain is presented by Lozano et al. [

There are also some complementary approaches that rank NOCs solely considering the number of medals, i.e., Copeland method [

Initiated by Lahdelma et al. [

SMAA has been efficiently applied in the domain of decision making, i.e., facility location [

To the best of our knowledge, almost all existing studies have ignored the fact that different NOCs may have different preferences between the lexicographic and the sum ranking systems. Even under typical preference, it is significantly difficult to achieve a group consensus on the exact weights with respect to each ranking system. Therefore, this paper pioneers the adventure to formulate intervals to represent the NOC-specific performances and then apply SMAA-2 to holistically rank NOCs in the presence of interval input data.

Regarding the lexicographic system, this section determines a system of points with respect to various medals. This is in line with the work performed by Sitarz [

Gold medal should be assigned more points than silver medal, while silver medal should be given more points than bronze medal [

The difference between a gold medal and a silver medal is larger than that between a silver medal and a bronze medal [

These conditions definitely make sense in real life and could be mathematically expressed by a convex cone as follows:

Inspired by the observations that MCDM and statistical problems usually use the mean value to support decision making [

Following the numerical methods developed by Henrion and Seeger [

Based upon the results derived from aggregating the lexicographic system, we modify the Olympics medal table as shown in Table

Modified Olympics medal table.

NOC | Lexicographic ranking system | Sum ranking system |
---|---|---|

| | |

| | |

| | |

| | |

Due to the fact that the preferences between the lexicographic and the sum ranking systems may change across NOCs, we exhaustively denote all possible preferences as follows:

Previous studies have proposed a large number of objective and subjective as well as integrated approaches to implement weights determination [

The optimal evaluation result of NOC

We denote

Similarly, the least favorable performance of each NOC under preference

The optimal evaluation result of NOC

In a word, the process to determine the lower and upper bounds of NOC-specific intervals is simple-to-understand and easy-to-execute, which could be effectively solved without the elicitation of the exact values of weights. Meanwhile, the results under preference

As for the MCDM problem with unknown, inexact, or partially missing information, SMAA denotes a set of approaches for support to find solutions. The logic of SMAA is discovering the weight space to obtain the preferences that make individual alternative the most preferred position or ensure a specified ranking position for a certain alternative. Lahdelma et al. [

With respect to the mathematical formulation presented in Section

The SMAA-2 approach is completely dependent on investigating the sets of preferable rank weights

Some useful indices developed by SMAA-2 approach will be introduced in the present subsection. The first is

The weight space associated with the (

The third index is the

Additional information about the proposed indices could be found in the paper published by Lahdelma and Salminen [

Based upon the mentioned

Barron and Barrett [

In this section, we will measure the performance of NOCs using the data at the London 2012 Summer Olympic Games. To illustrate the effectiveness of applying SMAA-2 to rank NOCs, we select a set of 12 NOCs from the 2012 Olympics medal table and present them as shown in Table

2012 London Summer Olympics medal table.

Ranking | NOC | Gold | Silver | Bronze | Total |
---|---|---|---|---|---|

1 | USA | 46 | 28 | 29 | 104 |

2 | CHN | 38 | 27 | 23 | 88 |

3 | GBR | 29 | 17 | 19 | 65 |

4 | RUS | 24 | 25 | 32 | 81 |

5 | KOR | 13 | 8 | 7 | 28 |

6 | GER | 11 | 19 | 14 | 44 |

7 | FRA | 11 | 11 | 12 | 34 |

8 | ITA | 8 | 9 | 11 | 28 |

9 | AUS | 7 | 16 | 12 | 35 |

10 | JPN | 7 | 14 | 17 | 38 |

11 | KAZ | 7 | 1 | 5 | 13 |

12 | NED | 6 | 6 | 8 | 20 |

Source:

The published ranking is obtained according to the lexicographic ranking system. However, the sum ranking system generates different ranking from the lexicographic ranking system. This remains a controversy and definitely complicates the formulating of ranking. For the purpose of applying our method, we first aggregate the lexicographic ranking system using medal points derived from (

Normalized Olympics medal table.

Ranking | NOC | Lexicographic ranking system | Sum ranking system |
---|---|---|---|

1 | USA | 0.2013 | 0.1799 |

2 | CHN | 0.1695 | 0.1522 |

3 | GBR | 0.1257 | 0.1125 |

4 | RUS | 0.1260 | 0.1401 |

5 | KOR | 0.0560 | 0.0484 |

6 | GER | 0.0668 | 0.0761 |

7 | FRA | 0.0558 | 0.0588 |

8 | ITA | 0.0430 | 0.0484 |

9 | AUS | 0.0490 | 0.0606 |

10 | JPN | 0.0491 | 0.0657 |

11 | KAZ | 0.0267 | 0.0225 |

12 | NED | 0.0312 | 0.0346 |

Based upon Theorems

Interval decision matrix.

Ranking | NOC | LS | SL |
---|---|---|---|

1 | USA | | |

2 | CHN | | |

3 | GBR | | |

4 | RUS | | |

5 | KOR | | |

6 | GER | | |

7 | FRA | | |

8 | ITA | | |

9 | AUS | | |

10 | JPN | | |

11 | KAZ | | |

12 | NED | | |

In addition, the metaweights to construct the

In the present subsection, the interval input data in Table

Holistic acceptability indices and rank acceptability indices (uniform distribution).

NOC | | | | | | | | | | | | | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

USA | 1.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 1.00 |

CHN | 0.00 | 1.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.68 |

GBR | 0.00 | 0.00 | 0.00 | 1.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.41 |

RUS | 0.00 | 0.00 | 1.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.52 |

KOR | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.08 | 0.16 | 0.76 | 0.00 | 0.00 | 0.00 | 0.14 |

GER | 0.00 | 0.00 | 0.00 | 0.00 | 1.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.33 |

FRA | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.46 | 0.45 | 0.08 | 0.00 | 0.00 | 0.00 | 0.00 | 0.23 |

ITA | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 1.00 | 0.00 | 0.00 | 0.09 |

AUS | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.03 | 0.16 | 0.63 | 0.18 | 0.00 | 0.00 | 0.00 | 0.17 |

JPN | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.51 | 0.30 | 0.13 | 0.06 | 0.00 | 0.00 | 0.00 | 0.23 |

KAZ | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 1.00 | 0.03 |

NED | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 1.00 | 0.00 | 0.06 |

Rank acceptability indices (uniform distribution).

The new ranking when we apply SMAA-2 and assume that the interval input data are uniformly distributed is

We assume that the interval data in Table

Therefore, the obtained results on the rank acceptability indices and the holistic acceptability indices are reported in Table

Holistic acceptability indices and rank acceptability indices (normal distribution).

NOC | | | | | | | | | | | | | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

USA | 1.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 1.00 |

CHN | 0.00 | 1.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.68 |

GBR | 0.00 | 0.00 | 0.00 | 1.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.41 |

RUS | 0.00 | 0.00 | 1.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.52 |

KOR | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.07 | 0.15 | 0.77 | 0.00 | 0.00 | 0.00 | 0.14 |

GER | 0.00 | 0.00 | 0.00 | 0.00 | 1.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.33 |

FRA | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.48 | 0.48 | 0.04 | 0.00 | 0.00 | 0.00 | 0.00 | 0.23 |

ITA | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 1.00 | 0.00 | 0.00 | 0.09 |

AUS | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.07 | 0.74 | 0.19 | 0.00 | 0.00 | 0.00 | 0.16 |

JPN | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.52 | 0.39 | 0.07 | 0.03 | 0.00 | 0.00 | 0.00 | 0.23 |

KAZ | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 1.00 | 0.03 |

NED | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 1.00 | 0.00 | 0.06 |

Rank acceptability indices (normal distribution).

The ranking in this subsection is the same as that in Section

We summarize and compare the rankings according to the lexicographic ranking system, the sum ranking system, and SMAA-2, the results of which are presented in Table

Ranking comparisons.

NOC | Lexicographic ranking | Sum ranking | SMAA-2 ranking |
---|---|---|---|

USA | 1 | 1 | 1 |

CHN | 2 | 2 | 2 |

GBR | 3 | 4 | 4 |

RUS | 4 | 3 | 3 |

KOR | 5 | 9 | 9 |

GER | 6 | 5 | 5 |

FRA | 7 | 8 | 6 |

ITA | 8 | 10 | 10 |

AUS | 9 | 7 | 8 |

JPN | 10 | 6 | 7 |

KAZ | 11 | 12 | 12 |

NED | 12 | 11 | 11 |

Note:

Ranking comparisons.

Compared with the lexicographic ranking system, SMAA-2 increases the ranking positions of RUS, GER, FRA, AUS, JPN, and NED and decreases that of GBR, KOR, ITA, and KAZ. Both USA and CHN simultaneously keep their status. Compared with the sum ranking system, SMAA-2 increases the ranking positions of FRA and decreases that of AUS and JPN. The rest of these NOCs keep their positions. Among these three ranking systems, only USA and CHN stay at their ranking positions. This reveals that the rankings of USA and CHN are robust and acceptable. Meanwhile, the ranking positions of FRA, AUS, and JPN change across three ranking systems. That is to say, the ranking positions of them are unreliable and full of conflict.

The modern Olympic Games are the leading international sporting event and featured in terms of summer and winter sports competition, with the involvement of over 200 NOCs and thousands of athletes. However, measuring Olympics achievement still remains a controversy and is full of conflict. This paper comprehensively measures the NOC-specific performance in the following three steps. First, we use a sophisticated mathematical method based upon the incenter of a convex cone to aggregate the lexicographic ranking system. Second, we abstract the fact that the preference between the lexicographic and the sum ranking systems may change across NOCs and develop a well-designed mathematical transformation to obtain the NOC-specific evaluation results under certain preference. However, it is extremely difficult to achieve a group consensus about the exact weights associated with each ranking system, since different weight determination approaches may generate different weight results. Therefore, we formulate intervals to represent the NOC-specific achievement under typical preference. Third, regarding the proposed stochastic decision making problem with interval input data, we use SMAA-2 to provide a holistic ranking of all NOCs. Our analysis is illustrated by measuring the performance of 12 NOCs participating the London 2012 Summer Olympic Games. We find out that the final ranking is robust, irrespective of the distribution functions. In addition, comparisons with the lexicographic ranking and the sum ranking are performed to show the difference among these mechanisms. We notice that the majority of NOCs display different ranking positions among them.

Future research is suggested to investigate more options for aggregating the lexicographic ranking system and also study the comprehensive performance of NOCs by taking into account both Summer and Winter Olympics.

(1) This paper provides a new Olympic ranking scheme by considering different preferences between the lexicographic and the sum ranking systems. (2) The lexicographic ranking is aggregated using the concept of incenter of a convex cone. (3) An interval decision matrix is formulated to support Olympics ranking. (4) A Stochastic Multicriteria Acceptability Analysis is employed to obtain final ranking.

The author declares that there are no conflicts of interest regarding the publication of this paper.

This work is supported by the National Natural Science Foundation of China (NSFC no. 71661011).