## 5 Properties of Solution  
Our solution for identifying branching rules can be described as 'greedy', due to the strategy of sequentially making the best possible decision. The alternative of evaluating all possible subsets of CQs that stay within the budget for changed answers is not computationally feasible except for very small surveys. The greedy approach we have taken provides a practical way to reduce survey fatigue by shortening a survey for most respondents. Our solution depends upon the original ordering of the questions in the survey. In some cases, a practitioner may not want to alter the order of the questions for future respondents, having in mind some good reasons for the order they have selected. For example, often the most important questions are asked first in hopes of getting high quality answers from respondents before survey fatigue becomes an issue. However, if the order of the survey questions is up for debate, our approach could be executed on permuted versions of the questions that are thought to be viable alternatives. Each permutation of the questions leads to a changed structure for the starting value of \(\underset{\sim}{N}\). The permutation that yields the largest reduction in the number of questions asked could be identified as the recommended ordering for the questions. When populating \(\underset{\sim}{N}\) with historical surveys, only surveys that have complete data with respect to the questions being analyzed can be used. If the response to any question is missing, uncertainty about which path to place the respondent in arises and it cannot be cleanly resolved. In most instances, branching rules would be developed for subsections of questions within a large survey. This will help keep the number of useable historical surveys large. In some situations, a practitioner may wish to consider data imputation methods. [PERSON] (2012) [PERSON] (2012) provides a general introduction to data imputation methods, while [PERSON] and [PERSON] (2002) [PERSON] and [PERSON] (2002) give a more detailed discussion of the options for creating imputation algorithms that fill in missing data. [PERSON] (1987) [PERSON] (1987) discusses the use of multiple imputation in the context of survey analyses. In our context, multiple imputation could be used to create multiple instances of the initial \(\underset{\sim}{N}\) matrix, and then the branching rules would be derived from an initial \(\underset{\sim}{N}\) obtained by averaging the node frequencies over these instances. The initial \(\underset{\sim}{N}\) matrix may have some pure splits based on what was observed from the historical surveys. As the greedy algorithm processes the data it identifies new pure splits. Sometimes new pure splits will undo a pre-existing pure split, but always the total number of questions saved by the new pure split will exceed whatever was saved by the pre-existing pure splits that become impure. This is  
Figure 3: Modified \(\underset{\sim}{N}\) for after second iteration of illustrative example. because existing pure splits that get undone by the new pure split are always down stream of the new split and thus in aggregate cannot save more questions than a higher level pure split. The branching rules extracted from the final \(\underset{\sim}{N}\) matrix correspond to pure splits formed during the process of spending the budget for the number of allowable changed answers. The changed answers will mostly correspond to the CQs associated with the pure splits. Steps during the sequential process to change answers always create a pure split. However, because later changes at a higher level can undo a previously created pure split, it will be the case that in the end some of the changed answers are related to CQs that are not converted into branching rules. A sensitivity analysis with small upward adjustments to the budget is recommended to see what could be gained in terms of the additional reduction in the number of questions asked. It is unlikely that a precise value for the budget is going to be available.