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# Correlated Binary Decision Rules

There are rules mapping a set of factors onto a binary outcome. This Demonstration shows how a set of rules can be generated in which the mappings are correlated with each other. This process is useful in generating, among other things, synthetic parameterized judiciaries, which can be compared to real-world judicial bodies.
You choose the number of factors the "judges" collectively take into account in determining how to decide a case. You choose the number of "judges" on the "court". These parameters generate a set of random and uncorrelated rules. You then correlate the rules by choosing (a) the number of clusters into which the judges fall, (b) the degree of correlation of rules within each cluster, and (c) the method by which correlation is to occur. When the "mean" method is selected, the system responds by moving the decision rules within the clusters toward the mean decision rule within that cluster. When the "leader" method is selected, the system responds by moving the decision rules within the clusters toward the first decision rule within that cluster. You can then visualize the results in one of three ways. With the "distance matrix" view, you see the Hamming distance between all of the judges. Green dividers show the clustering of the judges. With the "binary rules" view, you see arrays of the raw decision rules themselves, with green dividers again delineating the different clusters. With the "decision tree" view, you can use the advanced controls to see visualizations of the decision trees of each of the judges.

### DETAILS

With "decision tree" view selected and with the advanced controls open, you select the cluster and the judge within each cluster. You also select the form for the the visualization of the decision tree. The factors used by the judges are labeled from "a" to "z". The logical and is represented as ∧; the logical or is represented as ∨; and logical nand is represented as ⊼. Decision tree labels are suppressed when the number of decision factors is five or greater, unless the "force vertex labeling" control is checked.
If on a two-judge court, Judge 1 makes decisions on the basis of factors A, B and C, while Judge 2 makes decisions on the basis of factors B, C and D, there would be four factors that the judges collectively use to decide a case.
Political scientists have attempted to predict the decisions of United States Supreme Court justices on the basis of binary decision trees. See A. Martin, "Competing Approaches to Predicting Supreme Court Decision Making," Perspective on Politics, 2, 2004 p. 763.

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