Solution of a Ranking Problem from Binary Comparisons
Solution of a Ranking Problem from Binary Comparisons
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DOI：
10.1080/00029890.1957.11989117
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发表时间：
1957-10
影响因子：
0.5
通讯作者：
L. R. Ford
中科院分区：
文献类型：
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作者：
L. R. Ford
1.导论.有时候，人们会遇到这样的问题，即根据一些二元比较来对一组物体进行排序，而这些二元比较并没有，也不可能以实验者所希望的方式来选择。标准程序（例如，[1]的第217页）似乎通常要求任何给定对之间的比较次数等于任何其他对之间的比较次数;出于这样或那样的原因，在实践中可能很难在适当的大样本量下实现这种相等。例如，如果一个人试图通过只从实际拥有两种汽车的人那里获得配对比较来对各种汽车品牌进行排名，那么很容易看出，要获得像凯迪拉克和杰苏斯·克罗斯利这样的配对的大量比较可能是极其困难的。另一方面，雪佛兰和福特的比较将大量提供。本文中提出的方法不需要任何特定数量的对之间的比较，因此，可能有一些应用在类似于上述的上下文中。另一个潜在的应用是根据两人游戏的记录对玩家进行排名。例如，在大学橄榄球比赛中，如果比赛没有按照适合实验者的方式安排，就可以使用这种技术。我们假设给定矩阵A=（ai，j），其中a，i表示对象i优先于对象j的次数（a. O）。我们将使用的方法是最大似然法。我们将第i个物体与一个权重w相关联。这些权重将被解释为赔率，在这个意义上，在未来比较中i优于j的概率将被取为w，j（w.+ j）。w;）。有了这些概率，我们可以计算精确获得我们实际上确实获得的结果矩阵（即矩阵A）的先验概率，假设i和j之间发生的每个比较都独立于所有其他比较，并且是从二项分布中随机抽取的，p= w·/（w.+ w1），并且要求ij比较的次数等于a1 + aii。
1. Introduction. Occasionally one is faced with the problem of ranking a collection of objects on the basis of a number of binary comparisons, where these have not been, nor cannot be, chosen in a manner desirable to the experimenter. The standard procedures (eg, p. 217 of [1]) seem to require usually that the number of comparisons between any given pair be equal to the number between any other pair; for one reason or another it may be quite difficult in practice to achieve this equality for a suitably large sample size. For example, if one were attempting to rank various makes of automobiles by obtaining paired comparisons only from persons who have actually owned both makes one sees easily that it might be extremely difficult to obtain a large number of comparisons for such pairs as Cadillac fJersus Crosley. On the other hand, Chevrolet fJersus Ford comparisons would be available in large numbers. The method proposed in this paper does not require any specific number of comparisons between pairs, and for this reason may have some application in contexts similar to the above. Another potential application is in ranking players of a two-person game on the basis of their records against each other. In collegiate football, for example, where matches are not scheduled in a manner suited to the experimenter, this technique could be used. We assume as given a matrix A=(aiJ), where a, i represents the number of times object i has been preferred to objectj (a..= O). The approach we shall use is that of maximum likelihood. We associate with the ith object a weight w,. These weights will be interpreted as odds, in the sense that the probability of i being preferred toj in a future comparison will be taken to be w, j (w.+ w;). With these probabilities we may compute the a priori probability of obtaining precisely the matrix of results which we in fact did obtain (ie, the matrix A), under the assumption that each comparison that takes place between i and j is independent of all other comparisons, and is drawn randomly from a binomial distribution with p= w•/(w.+ w1) and with the number of ij comparisons being required to be equal to a, 1+ aii.