Statistics and Detection of Corruption 

 Duggan and Levitt's (2002) article on   "corruption in sumo wrestling"  demonstrates how statistical analysis may be used to detect crime and corruption. Sumo wrestling is a national sport of Japan. A sumo tournament involves 66 wrestlers participating in 15 bouts each. A wrestler with a winning record rises up the official ranking, while a wrestler with a losing record falls in the rankings. An interesting feature of sumo wrestling is the existence of a sharp nonlinearity in the payoff function. There is a large gap in the payoffs for seven wins and eight wins. The critical eighth win garners a wrestler roughly four times the value of the typical victory.  

 Duggan and Levitt uncover striking evidence that match rigging takes place in the final days of sumo tournaments. They find that the wrestler who is on the margin for an eighth win is victorious with an unusually high frequency, but the next time those same two wrestlers face each other, it is the opponent who has a very high win percentage. This suggests that part of the currency used in match rigging is promise of throwing future matches in return for taking a fall today. They present a histogram of final wins for the 60,000 wrestler-tournament observations between 1989 and 2000, in which a wrestler completes exactly 15 matches. Approximately 26.0 percent of all wrestlers finish with eight wins, compared to only 12.2 percent with seven wins. The binomial distribution predicts that these two outcomes should occur with an equal frequency of 19.6 percent. The null hypothesis that the probability of seven and eight wins is equal can be rejected at resounding levels of statistical significance. They report that two former sumo wrestlers have made public the names of 29 wrestlers who they allege to be corrupt and 14 wrestlers who they claim refuse to rig matches. Interestingly, they find that wrestlers identified as "not corrupt" do no better in matches on the bubble than in typical matches, whereas those accused of being corrupt are extremely successful on the bubbles. 

 A similar kind of empirical study of corruption dates to 1846, when Quetelet documented that the height distribution among French men based on measurements taken at conscription was normally distributed except for a puzzling shortage of men measuring 1.57–1.597 meters (roughly 5 feet 2 inches to 5 feet 3 inches) and an excess number of men below 1.57 meters. Not coincidentally, the minimum height for conscription into the Imperial army was 1.57 meters (recited from Duggan and Levitt 2002). These examples show that detection of statistical anomaly can give compelling evidence of corruption.  

 Corruption in conscription has been a big political issue in South Korea. Examination of anomaly in the distributions of height, weight, eyesight at each physical examination site for conscription may provide evidence of cheating and/or corruption. This kind of statistical evidence will fall short of proving crime or corruption, but will make a sufficient case for thorough investigation.