IJAHP Article: Saaty, Sagir/Ranking countries more reliably in the summer Olympics International Journal of the Analytic Hierarchy Process 589 Vol. 7 Issue 3 2015 ISSN 1936-6744 http://dx.doi.org/10.13033/ijahp.v7i3.341 RANKING COUNTRIES MORE RELIABLY IN THE SUMMER OLYMPICS Thomas L. Saaty Katz Graduate School of Business 322 Mervis Hall, Pittsburgh, PA, USA E-mail: saaty@katz.edu Mujgan Sagir* Eskisehir Osmangazi University Eskisehir, TURKEY E-mail: mujgan.sagir@gmail.com ABSTRACT In this paper we consider the many intangible criteria that influence the outcome of the Summer Olympics by using the Analytic Network Process, and apply the ideas to evaluate the medals won and the country scores in the 2012 London Olympics. Both the categories of games and the events in each game are considered in this weighting process. Different events of the same category game could have different properties. This work shows that the current way of counting the total number of medals is not a bad way of ranking countries. With minor modifications, this systematic approach for ranking countries can be used for any Summer Olympics. Keywords: OR in sports; country ranking; Olympic Games; Analytic Network Process; rating 1. Introduction The modern Olympic Games are a major international event featuring summer and winter sports in which thousands of athletes participate in a variety of competitions. They are considered to be the world's foremost sports competitors and represent nearly 200 nations who may participate. The Games are currently held biennially, with Summer and Winter Olympic Games alternating, thus each occurring every four years. Their creation was inspired by the ancient Olympic Games which were held in Olympia, Greece for more than 1000 years from the 8th century BC to the 4th century AD. There have been a number of studies conductedwhich have focused on the Olympic Games and other Olympic movements. For example, Andrew (2000)’s study researched why countries show different performance in Olympics. This study had three objectives with the key objective being, to examine the influences of factors affecting the Olympic performance.Two specific objectives were (i) to produce a mathematical model facilitating the prediction of the Olympic tally, and (ii) to identify the degree of factors that have influence on the Olympic performance. Bernard (2000) studied different variables in Olympic Games success investigation. On the other hand, Wade (2006) http://en.wikipedia.org/wiki/Multi-sport_event http://en.wikipedia.org/wiki/Summer_Olympic_Games http://en.wikipedia.org/wiki/Winter_Olympic_Games http://en.wikipedia.org/wiki/Winter_Olympic_Games http://en.wikipedia.org/wiki/Ancient_Olympic_Games http://en.wikipedia.org/wiki/Olympia,_Greece IJAHP Article: Saaty, Sagir/Ranking countries more reliably in the summer Olympics International Journal of the Analytic Hierarchy Process 590 Vol. 7 Issue 3 2015 ISSN 1936-6744 http://dx.doi.org/10.13033/ijahp.v7i3.341 studied the prediction of medal winners. Our motivation in this paper comes from the need for a scientific methodology to interpret the number of medals the countries have and a way to rank them. Qualification rules for each of the Olympic sports are set by the International Sports Federations (IFs) that governs that sport's international competition. For individual sports, athletes typically qualify by attaining a certain rank in a major international event and thus gain recognition on the IF's ranking list. National Olympic committees (NOC) may enter a limited number of qualified athletes in each event. In the ancient Olympics, no medals were awarded. First-place winners were given an olive branch to wear on their head, and second and third place winners did not receive anything. In the first modern Games held in 1896 silver and bronze medals were awarded to first and second place winners. In 1900, most winners received cups or trophies instead of medals. In the 1904 Games in St. Louis, gold replaced silver as the medal awarded for first place, followed by silver and bronze medals awarded to second and third place winners. Nowadays, the media decides which country has won the Olympics by adding all the medals won by athletes from that country. However, this kind of practice seems self- defeating because it assumes that all gold, silver and bronze medals should be counted as equal in merit and all games and events are assumed equally important. Nevertheless, it is only an approximate way and as it turns out, not an entirely faulty way of deciding which country is the overall winner of medals. No methodically scientific way to deal with multicriteria ranking involving intangibles has been used to assign appropriate priority weights to each type of sport and medal won in that sport for the Olympic Games. Here we propose using the Analytic Network Process (ANP) for the measurement of intangibles, along with their dependence and feedback, to weight the criteria which we think play an important role in assigning priorities to games and events. Our methodology consists of the following steps: 1) An ANP model is developed to assign weights to different criteria used to prioritize different games and events in each game. 2) Expert knowledge is used to define the criteria and evaluate the games and events. 3) We prioritize the significance of the three types of medals (gold, silver and bronze), weight them by the overall priorities of the events to obtain the overall priority of a medal won and add these priorities to obtain the priority rank of a country. 4) Our results do not violate intuition about the number and value of the medals and in fact take greater consideration by including the merits of the events in which they are won. Thus, we also show that the ranking of countries produces results that are reasonably close to the current results of adding all medals won but with some important exceptions. http://en.wikipedia.org/wiki/Sport_governing_body http://en.wikipedia.org/wiki/Sport_governing_body IJAHP Article: Saaty, Sagir/Ranking countries more reliably in the summer Olympics International Journal of the Analytic Hierarchy Process 591 Vol. 7 Issue 3 2015 ISSN 1936-6744 http://dx.doi.org/10.13033/ijahp.v7i3.341 2. Criteria to weight the summer Olympic Games The priorities of different kinds of games depend on several factors. Table 1 lists the relevant basic criteria groupings or clusters and the elements in each cluster. Figure 1 represents the top level network of the Analytic Network Process model together with the criteria. Table 1 Criteria and subcriteria Cluster Elements in Cluster Game requirements Required physical characteristics, Required training time Game other factors Risk level involved, Energy spent, Duration of the act People involved Number of competitors, Strength of competitors, Audience reaction, popularity Living environment Effects of daily life on the game, Effects on daily life, Financial resources needed Natural environment Season or climate suitability, Topographic pattern, Absence of pollution Other factors Sport commercialization, Technology, Political factors According to Table 1 and Figure 1 the main cluster of criteria is “Game requirements” which consists of “Required physical characteristics” and “Required training time”. For some sports, one may need physical characteristics that deal with strength. For example, in gymnastics one needs to practice for years starting at a young age. The second cluster of criteria is “Other factors related to the game” which includes “Risk level involved”, “Energy spent” and “Duration of the act”. The “Risk level” involved is a criterion in our analysis because it affects people’s attitudes; some people find it more challenging and encouraging to take risks. We have also included an “Energy spent” criterion because in certain games more energy is needed. For example, wrestling requires a high amount of energy spent, and certain sports also involve higher risk as in some gymnastics events. The duration of an event is another concern because some games last for a relatively long time, and in certain cases as in the marathon, a medal for this game deserves a greater value. A third cluster of criteria is “People involved” which consists of the “Number of competitors”, the “Strength of competitors” and also “Audience reaction, popularity”. Here, we think that there are some games that have greater popularity and this makes these sports more attractive to attend (e.g. football, tennis). On the other hand, to be successful in a game that has many competitors is more difficult. The fourth cluster is “Living environment”, with criteria “Effects of daily life on the game”, “Effects on daily life” and “Financial resources needed”. Some sports can be influenced by the daily life of the competitors; for example football players are usually careful about being involved in too large amounts of social activities and entertainment IJAHP Article: Saaty, Sagir/Ranking countries more reliably in the summer Olympics International Journal of the Analytic Hierarchy Process 592 Vol. 7 Issue 3 2015 ISSN 1936-6744 http://dx.doi.org/10.13033/ijahp.v7i3.341 that may affect their physical strength during the game due to lack of sleep. This is equally true of organizing daily life in a way to support success in the sport, such as being careful about ones diet. Financial resources are also important in some sports both to facilitate a player’s needs and to make it possible to be choosy in meeting basic needs as desired. Tennis, skiing or ice-skating need specific professional environments and specific equipment for training which can be costly. The fifth cluster is “Natural environment” whose criteria are “Season or climate suitability”, “Topographic pattern” and “Absence of pollution”. Climate and topographic patterns have important effects on pollution which is an undesirable factor particularly for events like canoeing and marathon. The sixth cluster is “Other factors” related to political, economic and social issues. Political factors can determine whether the games would be attended by some countries. Technology affects performance and in the long run new technology can change performance in a sport very significantly. Figure 1 gives a screen view from the SuperDecisions software with the clusters and their criteria from Table 1 along with their interconnections. Figure 1. ANP top level model As an illustration, Table 2 gives a view of paired comparisons related to the effect of “Risk level involved” and “Duration of the act” on the criterion “Energy spent”. “Duration of the act” has two times greater influence on the “Energy spent” criterion than “Risk level” does. Table 3 presents the criteria weights. These judgements have been obtained by interviewing different experts who have been a judge or competitor in different Olympic games, and geometric mean is used. IJAHP Article: Saaty, Sagir/Ranking countries more reliably in the summer Olympics International Journal of the Analytic Hierarchy Process 593 Vol. 7 Issue 3 2015 ISSN 1936-6744 http://dx.doi.org/10.13033/ijahp.v7i3.341 Table 2 Comparisons between “Risk level involved” and “Duration of the act” on “Energy spent” Energy spent Risk level involved Duration of the act Priorities Risk level involved 1 1/2 0.3333 Duration of the act 2 1 0.6667 Table 3 Criteria priorities Criterion Priority Criterion Priority Required physical characteristics 0,0606 Effects of the daily life on the game 0,0935 Required training time 0,0735 Effects on the daily life 0,0170 Risk level involved 0,0107 Financial resources needed 0,0136 Energy spent 0,0692 Season or climate suitability 0,0805 Duration of the act 0,0740 Topographic pattern 0,0921 Number of competitors 0,0539 Absence of pollution 0,0673 Strength of competitors 0,0735 Sports commercialization 0,0463 Audience reaction, popularity 0,0760 Political factors 0,0296 Technology 0,0679 3. How to evaluate different games and different events involved in each game Based on the previous discussion, it appears that declaring a winning country by adding all medals may not reflect the quality of the games that are won by the athletes from that country. The difference in the quality of the sports themselves is an important factor. By using the Fundamental Scale of absolute numbers of the AHP given in Table 4, one can compare the importance of different games and the importance of the events involved in each game (Saaty, 2004). IJAHP Article: Saaty, Sagir/Ranking countries more reliably in the summer Olympics International Journal of the Analytic Hierarchy Process 594 Vol. 7 Issue 3 2015 ISSN 1936-6744 http://dx.doi.org/10.13033/ijahp.v7i3.341 Table 4 Fundamental scale of absolute numbers Intensity of Importance Definition Explanation 1 Equal importance Two activities contribute equally to the objective 2 Weak or slight 3 Moderate importance Experience and judgment slightly favor the dominance of one activity over another 4 Moderate plus 5 Strong importance Experience and judgment strongly favor the dominance of one activity over another 6 Strong plus 7 Very strong or demonstrated importance An activity is favored very strongly in dominating over another activity; its dominance may even be demonstrated in practice 8 Very, very strong 9 Extreme An activity extremely domiantes another activity The experts who provided the judgments were a group of four people who have taken part in sports games as a referee, a coach and/or an athlete for over 12 years. We interviewed them for several days with regard to the criteria to be considered in the evaluation of the games and their events. We prepared a questionnaire to obtain the weights for each game and event, and then used the geometric mean to aggregate their judgments into a representative judgment for the group. When an inconcistency was discovered, we discussed the possibility of changing a judgment with the relevant person in order to reduce the inconsistency to an acceptable level and be closer to consensus on that set of judgments. We also did a literature review to obtain detailed data and information related to the criteria and the games themselves. Finally, we applied our approach to rank the winning countries for the 2012 London Summer Olympics. Table 5 lists all the games for this particular Olympics. Table 5 2012 London Summer Olympic Games Archery Cycling Gymnastics Shooting Triathlon Athletics Diving Handball Swimming Volleyball Badminton Equestrian Judo Synchronized swimming Water polo Basketball Fencing Modern pentathlon Table tennis Weightlifting Boxing Field hockey Rowing Taekwondo Wrestling Canoe Football Sailing Tennis IJAHP Article: Saaty, Sagir/Ranking countries more reliably in the summer Olympics International Journal of the Analytic Hierarchy Process 595 Vol. 7 Issue 3 2015 ISSN 1936-6744 http://dx.doi.org/10.13033/ijahp.v7i3.341 We categorized Olympic games in terms of the “Risk levels involved”, “Energy spent” and “Duration of the act”. Tables 6, 7 and 8 present these categories, noticing that games could fall into different categories for different criteria. Table 6 Summer Olympic Games according to the “Risk level involved” HIGH MEDIUM LOW Modern pentathlon Football Boxing Fencing Equestrian Diving Basketball Canoeing Taekwondo Volleyball Weightlifting Water polo Cycling Badminton Field Hockey Gymnastics Sailing Tennis Table tennis Judo Athletics Rowing Archery Wrestling Triatlon Shooting Badminton Swimming Synchronized Swimming Handball Table 6 implies that sports like diving and weightlifting have higher risks. They are usually considered dangerous sports that may cause harmful injuries and even death. On the other hand, according to the classification of “Energy spent” in Table 7, we obtain a different grouping of sports, i.e. fencing is a game that needs less energy when compared with other games like football. Table 7 Summer Olympic Games according to the “Energy spent” HIGH MEDIUM LOW Weightlifting Modern pentathlon Sailing Badminton Archery Athletics Basketball Volleyball Table tennis Fencing Swimming Football Equestrian Diving Shooting Wrestling Water polo Canoeing Synchronized Swimming Field Hockey Tennis Rowing Handball Cycling Badminton Gymnastics Triatlon Boxing Judo Taekwondo Similarly, Table 8 groups the games according to “Duration of the act”. Football and volleyball have long durations while others, like Taekwondo and wrestling, generally take shorter times. IJAHP Article: Saaty, Sagir/Ranking countries more reliably in the summer Olympics International Journal of the Analytic Hierarchy Process 596 Vol. 7 Issue 3 2015 ISSN 1936-6744 http://dx.doi.org/10.13033/ijahp.v7i3.341 Table 8 Summer Olympic Games according to the “Duration of the act” HIGH MEDIUM LOW Athletics Triatlon Canoeing Field Hockey Archery Fencing Basketball Voleyball Gymnastic Rowing Boxing Shooting Cycling Swimming Sailing Badminton Taekwondo Football Handball Diving Water polo Modern pentathlon Table tennis Equestrian Weightlifting Tennis Synchronized Swimming Judo Wrestling There were 29 games in the 2012 London Olympics. In order to prioritize them we used the ANP ratings module by evaluating them one at a time. The categories “low, medium, high” or “low, medium, high, very high” were chosen for the 17 criteria with an appropriate adjustment for the number of categories in each group (3 or 4). Tables 9(a) and 9(b) show a screen view of the rating module from the Super Decisions software. Five different experts from different professions were consulted. Table 9 (a) Screen view of the rating module to weight the games (for the criteria 1-7) IJAHP Article: Saaty, Sagir/Ranking countries more reliably in the summer Olympics International Journal of the Analytic Hierarchy Process 597 Vol. 7 Issue 3 2015 ISSN 1936-6744 http://dx.doi.org/10.13033/ijahp.v7i3.341 Table 9(b) Screen view of rating module to weight the games (for the criteria 8-17) Table 10 summarizes all the game priorities. Table 10 2012 London Summer Olympic Games weights Olympic game Weight Olympic game Weight Olympic game Weight Olympic game Weight Archery 0.019 Equestrian 0.024 Rowing 0.042 Triathlon 0.045 Athletics 0.053 Fencing 0.024 Sailing 0.038 Volleyball 0.038 Badminton 0.022 Field Hockey 0.027 Shooting 0.023 Water polo 0.029 Basketball 0.045 Football 0.054 Swimming 0.046 Weightlifting 0.032 Boxing 0.023 Gymnastics 0.048 Synchronized swimming 0.030 Wrestling 0.029 Canoe 0.043 Handball 0.021 Table tennis 0.021 Cycling 0.049 Judo 0.024 Taekwondo 0.024 Diving 0.021 Modern pentathlon 0.037 Tennis 0.061 IJAHP Article: Saaty, Sagir/Ranking countries more reliably in the summer Olympics International Journal of the Analytic Hierarchy Process 598 Vol. 7 Issue 3 2015 ISSN 1936-6744 http://dx.doi.org/10.13033/ijahp.v7i3.341 Tables 11 and 12 below present examples of category comparisons from the rating scale for the criteria “Energy spent” and “Physical characteristics”, respectively. Table 11 Category comparisons on the criteria “Energy spent” Energy spent High Medium Low Priorities High 1 2 3 0.5396 Medium 1/2 1 2 0.2970 Low 1/3 1/2 1 0.1634 Table 12 Category comparisons on the criteria “Physical characteristics” Energy spent Very high High Medium Low Priorities Very high 1 2 4 6 0.4990 High 1/2 1 3 5 0.3129 Medium 1/4 1/3 1 2 0.1202 Low 1/6 1/5 1/2 1 0.0679 Table 13 presents the priorities of the events. Table 13 2012 London Summer Olympic Events Priorities Games Events Event priorities Games Events Event priorities Archery Archery Modern Pentathlon Modern Pentathlon 0,011862 Athletics Decathlon 0.014525 Rowing Double Sculls 0,012572 Heptathlon 0.013967 Eight 0.012273 Jump 0.012536 Four 0.01159 Marathon 0.016925 Lightweight Four 011405 Pole Vault 0.012636 Pair 0,012273 0.012273 Relay 0.009546 Quadruple Sculls 0,012273 Shot Put 0.012382 Single Sculls 0,013401 0,012572 Throw 0.0091 Track 0.011834 Walk 0.010107 Badminton 0.007193 Sailing 470 0.008177 IJAHP Article: Saaty, Sagir/Ranking countries more reliably in the summer Olympics International Journal of the Analytic Hierarchy Process 599 Vol. 7 Issue 3 2015 ISSN 1936-6744 http://dx.doi.org/10.13033/ijahp.v7i3.341 49 Er 0.009312 Elliot 0.009312 Finn 0.011755 Laser 0.012341 Rs-X 0.009312 Star 0.009312 Basketball Basketball 0.014477 Shooting Pistol 0.007328 Rifle 0.004107 Rifle 25m 0.004107 Rifle 50m 0.007328 Skeet 0.007328 Trap 0.007328 Boxing Bantam 0.003306 Swimming Backstroke 0.008692 Fly 0.003919 Breaststroke 0.009397 Heavy 0.004534 Butterfly 0.010527 Light 0.003428 Freestyle 0.009397 Light Fly 0.003638 Marathon 0.014701 Light Heavy 0.00362 Medley 0.009397 Light Welter 0.00362 Middle 0.004044 Super Heavy 0.007356 Welter 0.006076 Canoeing Slalom 0.013815 Synchronize d Swimming Synchronized Swimming 0,00976 Sprint 0.011023 Cycling Bmx 0.013819 Table tennis Table tennis 0.006955 Mountain 0.01568 Road 0.010855 Track 0.01568 Diving Platform 0.0069 Taekwondo 49 0.004095 Springboard 0.00481 57 0.004095 58 0.004095 67 0.006577 68 0.007224 80 0.007657 Equestrian Dressage 0.004108 Tennis Tennis 0,01949 Eventing 0.007657 Jumping 0.005944 Fencing Epee 0.007842 Foil 0.006532 Sabre 0.004283 Triathlon Triathlon 0,014624 0.014624 IJAHP Article: Saaty, Sagir/Ranking countries more reliably in the summer Olympics International Journal of the Analytic Hierarchy Process 600 Vol. 7 Issue 3 2015 ISSN 1936-6744 http://dx.doi.org/10.13033/ijahp.v7i3.341 Field Hockey Field Hockey 0,00848 Volleyball Beach 0.012032 Indoor 0.010881 Football Football 0,017058 Water Polo Water Polo 0,009229 Gymnastic Artistic 0.015236 Weightliftin g 48 0.00652 Rhythmic 0.013007 53 0.00652 Trampoline 0.01403 56 0.00652 58 0.00652 62 0.00652 63 0.00652 69 0.00652 75 0.00652 +75 0.006704 77 0.006704 85 0.006704 94 0.009347 105 0.010491 105+ 0,010491 Handball Handball 0,00675 Wrestling Freestyle 0.009463 Greco-Roman 0.005714 Greco-Roman 120 0.005714 Greco-Roman 60 0.005994 Greco- Roman84 0.008799 Judo J1 48 0.005035 J2 52 0.005035 J3 57 0.005035 J4 60 0.005546 J5 63 0.005546 J6 66 0.006 J7 70 0.006287 J8 73 0.006287 J9 78 0.006287 J10 81 0.005777 J11 90 0.007659 J12 100 0.007659 J13 100+ 0.007659 IJAHP Article: Saaty, Sagir/Ranking countries more reliably in the summer Olympics International Journal of the Analytic Hierarchy Process 601 Vol. 7 Issue 3 2015 ISSN 1936-6744 http://dx.doi.org/10.13033/ijahp.v7i3.341 4. How to obtain medal weights and their priorities for each event under different games The relative values of gold, silver and bronze medals were studied in an earlier work as follows (Saaty, 2008). Thirteen sets of comparisons and their actual outcomes are shown in Tables 14-19. From the pairwise comparison judgments between different types of medals, one derives the priorities of different types of medals under 13 possible situations and then averages them to obtain the priorities of gold, silver and bronze medals. In Table 14, a gold medal is very slightly favored over a silver medal and is not dependent on whether it is moderately or extremely favored over a bronze medal. Table 14 Gold slightly over Silver Gold Silver Bronze Relative Values Gold Silver Bronze Relative Values Gold 1 2 3 0.55 Gold 1 2 9 0.61 Silver 1/2 1 3/2 0.27 Silver 1/2 1 5 0.32 Bronze 1/3 2/3 1 0.18 Bronze 1/9 1/5 1 0.07 Table 15 shows that the gold medal is moderately favored over the silver medal and from very strongly to extremely over the bronze medal, and is noy dependent on whether a silver medal is moderately or strongly favored over a bronze medal. Table 15 Gold moderately over Silver Gold Silver Bronze Relative Values Gold Silver Bronze Relative Values Gold 1 3 7 0.64 Gold 1 3 7 0.65 Silver 1/3 1 3 0.26 Silver 1/3 1 5 0.28 Bronze 1/5 1/3 1 0.10 Bronze 1/7 1/5 1 0.07 Gold Silver Bronze Relative Values Gold 1 3 9 0.67 Silver 1/3 1 3 0.27 Bronze 1/9 1/3 1 0.06 In Table 16, the strength of a gold medal over a silver medal increases even more to between moderately and strongly and a gold medal is favored nearly very strongly to extremely over a bronze medal, while a silver medal is only moderately favored over a bronze medal in both cases. IJAHP Article: Saaty, Sagir/Ranking countries more reliably in the summer Olympics International Journal of the Analytic Hierarchy Process 602 Vol. 7 Issue 3 2015 ISSN 1936-6744 http://dx.doi.org/10.13033/ijahp.v7i3.341 Table 16 Gold between moderately and strongly over Silver Gold Silver Bronze Relative Values Gold Silver Bronze Relative Values Gold 1 4 6 0.69 Gold 1 4 9 0.73 Silver 1/4 1 3 0.22 Silver 1/4 1 3 0.20 Bronze 1/6 1/3 1 0.09 Bronze 1/9 1/3 1 0.07 In Table 17, a gold medal is strongly favored over a silver medal and very strongly to extremely favored over a bronze medal. Table 17 Gold strongly over Silver Gold Silver Bronze Relative Values Gold Silver Bronze Relative Values Gold 1 5 7 0.72 Gold 1 5 9 0.74 Silver 1/5 1 4 0.21 Silver 1/5 1 4 0.19 Bronze 1/7 1/4 1 0.07 Bronze 1/9 1/4 1 0.07 In Table 18, a gold medal is considered strongly more important than a silver medal and extremely more than a bronze medal while a silver medal is first moderately and then between moderately and strongly more important over a bronze medal. Table 18 Gold very strongly over Silver Gold Silver Bronze Relative Values Gold Silver Bronze Relative Values Gold 1 7 9 0.79 Gold 1 7 9 0.78 Silver 1/7 1 3 0.15 Silver 1/7 1 4 0.16 Bronze 1/9 1/3 1 0.06 Bronze 1/9 1/4 1 0.06 In Table 19, a gold medal is extremely important over a silver medal and a bronze medal while a silver medal is first strongly important and then extremely important than a bronze medal. Table 19 Gold extremely over Silver Gold Silver Bronze Relative Values Gold Silver Bronze Relative Values Gold 1 9 9 0.80 Gold 1 9 9 0.78 Silver 1/9 1 5 0.15 Silver 1/9 1 9 0.18 Bronze 1/9 1/5 1 0.05 Bronze 1/9 1/9 1 0.04 The tables given above give the priorities of different types of medals under 13 different situations shown in Tables 14-19, and then the priorities of different types of medals are IJAHP Article: Saaty, Sagir/Ranking countries more reliably in the summer Olympics International Journal of the Analytic Hierarchy Process 603 Vol. 7 Issue 3 2015 ISSN 1936-6744 http://dx.doi.org/10.13033/ijahp.v7i3.341 obtained by taking the geometric mean of the priorities derived from the judgment matrices above as seen in Table 20. Table 20 The 13 vectors of priorities from Tables 10-15 and their average Total GEO. MEAN G* 0.55 0.61 0.64 0.65 0.67 0.69 0.73 0.72 0.74 0.79 0.78 0.80 0.78 9.15 0.6900 S 0.27 0.32 0.26 0.28 0.27 0.22 0.20 0.21 0.19 0.15 0.16 0.15 0.18 2.86 0.2000 B 0.18 0.07 0.10 0.07 0.06 0.09 0.07 0.07 0.07 0.06 0.06 0.05 0.04 0.99 0.0060 *G: Gold, S: Silver, B: Bronz Now we re-rank the countries that won medals in the 2012 London Summer Olympics according to a our method which considers not only the total medals won but also the weighted priority of each game and event under each game. The rank of the countries when simply counting medals won is shown in the sixth column of Table 21. Following the traditional way of counting the total number of medals won, the USA is the top ranked country followed by China and Russia. However, the ranking of the countries that won medals in the 2012 London Summer Olympics (shown in the last column of Table 21) is different when based on our finer approach. For example, the Ukraine won 20 medals in boxing, canoeing, fencing, gymnastic, rowing, shooting, weighlifting and wrestling. This includes 6 gold medals 5 silver medals, and 9 bronze medals. When counting the total number of medals won, the Ukraine is ranked 10 th place (if we use the total number of gold medals as a second criterion for the countries when the total number of medals are the same, then Ukraine ranked as 12th place in the current methodology, South Korea is 9 th , Italy is 10 th , Netherland is 11 th and Ukraine is 12 th ), but when considering the priorities of the events and the games by proposed methodology the Ukraine’s rank moves to 15 th place. This is because gymnastics is one of the important games considered in this research, and the Ukraine won a bronze medal in gymnastics. On the other hand, they won 5 medals in boxing and 2 medals in fencing events which have relatively lower priorities. Another example is Latvia which won only two medals, one in cycling (gold) and the other in beach volleyball (bronze). When considering the number of medals won, Latvia is one of the lowest ranked countries by current ranking system. As we explained above in the Ukraine example, the countries that have an equal number of medals are ranked as the same. Latvia, Bulgaria, Indonesia, Dominican Republic and so on are ranked 25 th by the current ranking. However, although Latvia is ranked 25 th , it is actually 57 th if we use the number of gold medals as the second criterion (when we just count the countries above Latvia in the current order, Latvia is 57 th not 25 th ). When we look at it like this, we can more easily interpret the difference from the proposed rank and say that Latvia deserves a better rank even though it has just two medals because the priorities of those games are high. Our methodology ranked Latvia 42 th because the priorities of the events in which Latvia won medals are relatively high. A similar example is Tunisia which won only three medals. These medals were won in the marathon, track and swimming, all events with higher priorities as compared to other games. IJAHP Article: Saaty, Sagir/Ranking countries more reliably in the summer Olympics International Journal of the Analytic Hierarchy Process 604 Vol. 7 Issue 3 2015 ISSN 1936-6744 http://dx.doi.org/10.13033/ijahp.v7i3.341 Table 21 2012 London Olympics medals current and proposed ranking comparisons Current Methodology Proposed Methodology Country Gold Silver Bronze Total Medal Count Current Ranking Total Priority Score Proposed Ranking USA 46 29 29 104 1 0.4348 1 China 38 27 23 88 2 0.2515 3 Russian 24 25 32 81 3 0.2237 4 Great Britain 29 17 19 65 4 0.2868 2 Germany 11 19 14 44 5 0.1288 5 Japan 7 14 17 38 6 0.0756 8 Australia 7 16 12 35 7 0.0899 7 France 11 11 12 34 8 0.1248 6 South Korea 13 8 7 28 9 0.0710 9 Italy 8 9 11 28 9 0.0686 11 Netherlands 6 6 8 20 10 0.0502 13 Ukraine 6 5 9 20 10 0.0404 15 Hungary 8 4 6 18 11 0.0687 10 Canada 1 5 12 18 11 0.0356 20 Spain 3 10 4 17 12 0.0402 16 Brazil 3 5 9 17 12 0.0256 27 Cuba 5 3 7 15 13 0.0258 26 Kazakhstan 7 1 5 13 14 0.0396 17 New Zeland 6 2 5 13 14 0.0536 12 Belarus 2 5 5 12 15 0.0384 18 Iran 4 5 3 12 15 0.0300 23 Jamaica 4 4 4 12 15 0.0450 14 Kenya 2 4 5 11 16 0.0345 21 Czech Republic 4 3 3 10 17 0.0278 24 Azerbaijan 2 2 6 10 17 0.0202 32 Poland 2 2 6 10 17 0.0209 31 Romania 2 5 2 9 18 0.0238 29 Denmark 2 4 3 9 18 0.0358 19 IJAHP Article: Saaty, Sagir/Ranking countries more reliably in the summer Olympics International Journal of the Analytic Hierarchy Process 605 Vol. 7 Issue 3 2015 ISSN 1936-6744 http://dx.doi.org/10.13033/ijahp.v7i3.341 Colombia 1 3 4 8 19 0.0190 34 Sweden 1 4 3 8 19 0.0190 34 Ethiopia 3 1 3 7 20 0.0343 22 Mexico 1 3 3 7 20 0.0177 37 Georgia 1 3 3 7 20 0.0074 50 North Korea 4 0 2 6 21 0.0195 33 Croatia 3 1 2 6 21 0.0215 30 South Africa 3 2 1 6 21 0.0275 25 India 0 2 4 6 21 0.0146 40 Mongolia 0 2 3 5 22 0.0043 56 Turkey 2 2 1 5 22 0.0163 38 Lithuania 2 1 2 5 22 0.0094 45 Ireland 1 1 3 5 22 0.0040 57 Trinidad And Tobago 1 0 3 4 23 0.0095 44 Switzerland 2 2 0 4 23 0.0189 35 Norway 2 1 1 4 23 0.0186 36 Slovenia 1 1 2 4 23 0.0048 59 Argentina 1 1 2 4 23 0.0098 43 Serbia 1 1 2 4 23 0.0063 53 Malaysia 0 1 3 4 23 0.0021 64 Finland 0 1 2 3 24 0.0037 58 Tunisia 1 1 1 3 24 0.0254 28 Uzbekistan 1 0 2 3 24 0.0085 47 Armenia 0 1 2 3 24 0.0160 39 Belgium 0 1 2 3 24 0.0087 47 Thailand 0 2 1 3 24 0.0024 63 Indonesia 0 1 1 2 25 0.0020 65 Dominic Republic 1 1 0 2 25 0.0083 48 Latvia 1 0 1 2 25 0.0109 42 Chinese Tapei 0 1 1 2 25 0.0004 73 Estonia 0 1 1 2 25 0.0027 61 Puerto Rico 0 1 1 2 25 0.0031 60 Bulgaria 0 1 1 2 25 0.0068 51 IJAHP Article: Saaty, Sagir/Ranking countries more reliably in the summer Olympics International Journal of the Analytic Hierarchy Process 606 Vol. 7 Issue 3 2015 ISSN 1936-6744 http://dx.doi.org/10.13033/ijahp.v7i3.341 Egypt 0 2 0 2 25 0.0025 62 Moldova 0 0 2 2 25 0.0016 67 Qatar 0 0 2 2 25 0.0020 64 Greece 0 0 2 2 25 0.0019 66 Singapore 0 0 2 2 25 0.0014 69 Algeria 1 0 0 1 26 0.0083 47 Bahamas 1 0 0 1 26 0.00668 52 Botswana 0 1 0 1 26 0.0024 63 Guatemala 0 1 0 1 26 0.0020 65 Grenada 1 0 0 1 26 0.0083 49 Uganda 1 0 0 1 26 0.0118 41 Slovakia 0 1 0 1 26 0.0050 55 Montenegro 0 1 0 1 26 0.0014 69 Venezuela 1 0 0 1 26 0.0055 54 Bahrain 0 0 1 1 26 0.0090 46 Gabon 0 1 0 1 26 0.0015 68 Morocco 0 0 1 1 26 0.0012 70 Portugal 0 1 0 1 26 0.0022 64 Tajikistan 0 0 1 1 26 0.0003 74 Cyprus 0 1 0 1 26 0.0025 62 Afghanistan 0 0 1 1 26 0.0007 71 Hong Kong 0 0 1 1 26 0.0016 67 Saudi Arabia 0 0 1 1 26 0.0006 72 Kuwait 0 0 1 1 26 0.0007 71 * The rows in bold show that the rank of a country obtained by the proposed method is the same as its current rank 5. Conclusion Training for competition in the Olympics requires time and resources with different types of events having different characteristics. Individual games require more concentration while team games require more cooperation. To become successful in gymnastics, one usually has to start training at a very young age (five or six). The duration of a volleyball game is usually about one to two hours, and the marathon takes about two hours and requires more energy when compared to other events. On the other hand, archery takes only a few seconds. Thus, a medal should be given a different value depending on which game it is won for. We propose that ranking countries in the Olympics should not only be decided by counting the total medals won, but also by the type of game in which the medal was won. In this study we prioritized different games and the events under each IJAHP Article: Saaty, Sagir/Ranking countries more reliably in the summer Olympics International Journal of the Analytic Hierarchy Process 607 Vol. 7 Issue 3 2015 ISSN 1936-6744 http://dx.doi.org/10.13033/ijahp.v7i3.341 game. Our elaborate approach to the Olympics shows that counting the total numbers of medals won is not a bad way of ranking countries. Finally, while it is known that multi- criteria decision making is very important in optimal allocation of limited resources, it may not always produce radically better results than much simpler existing ways of ranking. For the last winter Olympics (2014) in Sochi, Russia, there were more noticeable differences in the two rankings methods so that Norway which ranked third according to the number of medals won, actually ranked first according to prioritization of the different kinds of games. This is a significant finding because ranking first would have been a very distinguished and celebrated outcome for Norway. We also performed Compatibility Index calculations to compare actual ranking and the estimated one as shown in Appendix. Since the ranks are slightly different, the index is obtained was 1,22 which could be acceptable and reasonable. IJAHP Article: Saaty, Sagir/Ranking countries more reliably in the summer Olympics International Journal of the Analytic Hierarchy Process 608 Vol. 7 Issue 3 2015 ISSN 1936-6744 http://dx.doi.org/10.13033/ijahp.v7i3.341 REFERENCES Andrew, B. B., Meghan, R. B., (2000). Who wins the Olympic Games: Economic development and the medal total, National Bureau of Economic Development, NBER working paper series, No. 7998, JEL No. O10, L83. Bernard, A. B., Meghan R. B., (2000). Who wins the Olympic Games: Economic development and medal total, NBER Working Paper, 7998. Wade, D. P., (2006) Predicting the medal wins by country at the 2006 winter Olympic Games: An econometrics Approach, National Graduate Institute for Policy Studies, Tokyo, Japan. Forsyth, S., China 2008 Olympic medal tally by population, and China 2008 Olympic medal tally by gross domestic product (GDP), http://simon.forsyth.net/olympics.htm. Saaty T.L., (2004). Decision making- The Analytic Hierarchy and Network Processes (AHP/ANP), Journal of Systems Science and Systems Engineering, 13(1), 1-35. doi:10.1007/s11518-006-0151-5 Saaty, T. L., (2008). Who Won the 2008 Olympics? Journal of Systems Science and Systems Engineering, 17, 4, 473-486. doi: 10.1007/s11518-008-5092-8 The World Factbook 2013-14. Washington, DC: Central Intelligence Agency, 2013. https://www.cia.gov/library/publications/the-world-factbook/index.html http://www.andrew.bernard.org/ http://www.kellogg.northwestern.edu/Faculty/Directory/Busse_Meghan.aspx http://simon.forsyth.net/olympics.html http://simon.forsyth.net/olympics.htm https://www.cia.gov/library/publications/the-world-factbook/index.html IJAHP Article: Saaty, Sagir/Ranking countries more reliably in the summer Olympics International Journal of the Analytic Hierarchy Process 609 Vol. 7 Issue 3 2015 ISSN 1936-6744 http://dx.doi.org/10.13033/ijahp.v7i3.341 APPENDIX COMPATIBILITY INDEX ANALYSIS Pairwise Comparison Matrix from Actual Data A1 A2 A3 A4 A5 A6 A7 A8 A1 1 1,14433 1,32934 1,59712 2,00000 2,67470 3,96429 7,92857 A2 0,8738739 1,00000 1,16168 1,39568 1,74775 2,33735 3,46429 6,92857 A3 0,7522523 0,86082 1,00000 1,20144 1,504504505 2,01205 2,98214 5,96429 A4 0,6261261 0,71649 0,83234 1 1,25225 1,67470 2,48214 4,96429 A5 0,5 0,57216 0,66467 0,79856 1 1,33735 1,98214 3,96429 A6 0,3738739 0,42784 0,49701 0,59712 0,747747748 1 1,48214 2,96429 A7 0,2522523 0,28866 0,33533 0,40288 0,504504505 0,674699 1 2,00000 A8 0,1261261 0,14433 0,16766 0,20144 0,252252252 0,337349 0,5 1 Pairwise Comparison Matrix from Estimated Data A1 A2 A3 A4 A5 A6 A7 A8 A1 1 1,743141 1,959768 1,528591 3,403727 5,798942 4,876529 3,512821 A2 0,573677 1,00000 1,12427 0,87692 1,95264 3,32672 2,79755 2,01522 A3 0,510265 0,88946 1,00000 0,77999 1,736801 2,95899 2,48832 1,79247 A4 0,654197 1,14036 1,28207 1 2,22671 3,79365 3,19021 2,29808 A5 0,293796 0,51213 0,57577 0,44909 1 1,70370 1,43270 1,03205 A6 0,172445 0,30060 0,33795 0,26360 0,265569 1 0,84093 0,60577 A7 0,205064 0,35746 0,40188 0,31346 0,697981 1,189153 1 0,72035 A8 0,284672 0,49622 0,55789 0,43515 0,968944 1,650794 1,388209 1 Transpose of Comparison Matrix from Estimated Data A1 A2 A3 A4 A5 A6 A7 A8 A1 1 0,573677 0,510264599 0,654197 0,29379562 0,172445 0,20506387 0,284672 A2 1,7431412 1,00000 0,88946 1,14036 0,51213 0,30060 0,35746 0,49622 A3 1,9597675 1,12427 1,00000 1,28207 0,57577 0,33795 0,40188 0,55789 A4 1,5285914 0,87692 0,77999 1 0,44909 0,26360 0,31346 0,43515 A5 3,4037267 1,95264 1,736801242 2,22671 1 0,265569 0,69798137 0,968944 A6 5,7989418 3,32672 2,95899 3,79365 1,70370 1 1,18915344 1,650794 A7 4,8765295 2,79755 2,48832 3,19021 1,43270 0,84093 1 1,388209 A8 3,5128205 2,01522 1,79247 2,29808 1,03205 0,60577 0,72035 1 Result of Hadamard (Cell-wise) Multiplication of Previous Two Matrices A1 A2 A3 A4 A5 A6 A7 A8 A1 1 0,6564758 0,678315814 1,044833 0,587591241 0,461239 0,81293176 2,257039 A2 1,52329 1 1,03327 1,59158 0,89507 0,70260 1,23833 3,43811 A3 1,47424 0,96780 1 1,540334 0,86625 0,67998 1,19846 3,32742 A4 0,95709 0,62831 0,649209948 1 0,56238 0,44145 0,77805 2,16019 A5 1,70186 1,11723 1,154400826 1,778163 1 0,35516 1,38350 3,84117 A6 2,16807 1,42329 1,47064 2,26527 1,27394 1 1,76250 4,89342 A7 1,23012 0,80754 0,83441 1,28527 0,72281 0,56738 1 2,77642 A8 0,44306 0,29086 0,30053 0,46292 0,26034 0,20436 0,36018 1 IJAHP Article: Saaty, Sagir/Ranking countries more reliably in the summer Olympics International Journal of the Analytic Hierarchy Process 610 Vol. 7 Issue 3 2015 ISSN 1936-6744 http://dx.doi.org/10.13033/ijahp.v7i3.341 TOTAL 10,49773 6,89150 7,12077 10,96837 6,16837 4,41215 8,53394 23,69377 Cell sum of previous matrix = 78,286 Number of Alternatives (n) = 8 Saaty Compatibility Index = Sum/n 2 1,22