Say a jury is going to decide who wins a competition. First, each member evaluates all the competitors by grading them; then, for each competitor, a collective grade is derived from all the judgments of all the members; finally, the jury chooses as the winner the competitor with the highest collective grade. This is collective grading. The grades that are used might typically be numerical scores, or evaluative expressions of a natural language, such as “good,” “fair,” and “bad.” They could be any signs at all, though, that come in a “top” to “bottom” order: thumbs up and down; happy, neutral, and sad emojis; or cheering, clapping, booing, and angry hissing at public events. Panels, boards, and committees throughout society evaluate all manner of things by grading them. Thus risks are prioritized, research proposals are funded, and candidates are shortlisted for jobs. Apart from acclamation in special cases, collective grading is not a usual way to pick winners in political elections.

This article takes up a question about the quality of judgments and decisions made by collective grading: under which conditions are outcomes likely to be right? An answer comes in the form of a jury theorem for median grading. Here, the collective grade for a thing is the median of its individually assigned grades—the one in the middle, when all of them are listed from top to bottom. Section III prepares the ground for this theorem by discussing different senses in which grades can be the right ones for things, or the wrong ones as the case may be, independently of which grades are assigned in the end. These notions of right and wrong are relevant to judgments of different kinds of things: risks, research proposals, job candidates, options in referendums and elections. The grading‐jury theorem in Section V identifies conditions on the grading competence of individual people under which median grades, and decisions that follow them, are likely to be, independently, right.

A second objective of this article is to suggest a solution to problems of voter ignorance in democracies. The idea developed here is to use voting methods that make more of people's limited knowledge than do traditional methods such as majority voting. Grading holds promise, because voters can express themselves more fully by grading the options on their ballots rather than simply choosing one or ranking them all. To count the one option “good” and the other “bad” is, for instance, to rank the first above the second; but there is more information in these grades than just this order, because while counting the first option instead “fair” would put them in the same order, the expressions “good” and “fair” mean different things. By tapping into the richer information carried by graded ballots, collective grading methods could, in principle, allow more of voters' knowledge to find its way into collective decisions than traditional voting methods do.

Median grading sometimes does make more of voters' knowledge than majority voting possibly can. Section VI draws from the grading‐jury theorem the consequence that median grading is—in a special sense, presently explained—forgiving of the incompetence of voters who, perhaps as a result of their ignorance and prejudices, are not likely to make right decisions on their own. This agreeable nature of median grading is on display, in Section VI, in the example of an assembly that, going by median grades, reliably picks out the better option from a pair, even though the individual members, in their bemusement, are far more likely to vote for the wrong one. Under these circumstances, as Condorcet warned long ago, and as is explained in the next section, letting the majority decide is likely to make things worse. It would expose the assembly to a risk of making false decisions. The upshot is that, in theory anyway, and perhaps also in practice, median grading can enable unenlightened assemblies to “track the truth”—even as majority voting would run them off the rails.

假设陪审团将决定谁赢得比赛。首先，每个成员通过对所有竞争对手进行评分来对其进行评估；然后，从所有成员的所有判断中得出每个参赛者的集体评分；最后，评审团选择集体成绩最高的竞争对手作为获胜者。这是集体评分。所使用的等级通常是数字分数或自然语言的评价表达，例如“好”，“一般”和“不好”。但是，它们可能完全是从“上”到“下”顺序出现的任何迹象：大拇指向上和向下；快乐，中性和悲伤的表情符号；或在公共场合欢呼，鼓掌，嘘声和生气的嘶嘶声。整个社会的小组，董事会和委员会通过对事物的等级进行评估。因此，将风险放在优先位置，为研究计划提供资金，并为候选人入围职位。除了特殊情况下的鼓掌之外，集体评分并不是在政治选举中挑选胜者的通常方法。

本文讨论了有关通过集体评分做出的判断和决策的质量的问题：在什么条件下结果可能是正确的？答案以陪审团定理的形式出现，用于中位评分。在这里，事物的集体等级是其单独分配的等级的中位数，即从上至下列出所有等级的中间等级。第三节通过讨论坡度可能是正确的不同意义，为该定理奠定了基础。事与愿违，视情况而定是错误的，与最后分配的成绩无关。这些对与错的概念与对各种事物的判断有关：风险，研究建议，求职者，公民投票和选举中的选择。第五节中的等级陪审定理确定了个人等级能力的条件，在这些条件下，中位等级及其后的决策很可能是独立的。

本文的第二个目的是提出一种解决民主国家中选民无知问题的解决方案。这里开发的想法是使用投票方法，使人们的有限知识比传统方法（如多数投票）更多地利用人们的有限知识。评分具有希望，因为选民可以通过对选票进行分级来表达自己，而不是简单地选择一个选票或对所有选票进行排名，从而更加充分地表达自己的意见。例如，将一个选项计算为“好”，将另一个选项视为“差”，则是将第一个选项排在第二个选项之上；但是这些等级的信息不仅仅只是这个顺序，因为在计算第一个选项时，“公平”将使它们处于相同的顺序，“好”和“公平”这两个词的含义不同。通过利用分级投票所携带的更丰富的信息，集体分级方法原则上可以实现以下目标：

有时，中级评分确实比多数投票所能使更多的了解选民的知识。第六节从评分-陪审定理得出以下结论：中位数评分（在特殊意义上，目前已进行了解释）是宽容的选民的无能，他们可能是由于他们的无知和偏见而无法自行做出正确的决定。在第六节中，以中位数等级为例，显示了中位数等级的这种令人满意的性质，该组件按中等级等级可靠地从一对中选择了更好的选择，即使单个成员在其困惑中所占的比重更大。可能会选错一票。在这种情况下，正如Condorcet早就警告过的那样，正如下一节将要解释的那样，让大多数人决定可能会使情况变得更糟。这将使大会面临做出错误决定的风险。结果是，无论从理论上还是在实践中，中位数评分都可以使开明的集会“追踪真相”，即使多数投票会使他们脱轨。