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Counting votes in coupled decisions An efficient method for counting votes in coupled decisions with multiple inequality restrictions

机译：计数联合决策中的票数一种有效的方法来计算具有多个不平等限制的联合决策中的票

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摘要

We consider scenarios with distributed decision processes, e.g., coupled majorities and personal union in parliament chambers, supranational decisions and supervisory boards. When computing the adoption rate for reaching a decision in these scenarios, multiple linear inequality restrictions in combinatorial countings are present. These rates cannot be computed in closed form. We introduce a general method for incorporating multiple inequality conditions in multiple majority decisions, which significantly reduces the number of involved summations and removes restrictions on the summation indices. Exact solutions are provided through (a) integral representations which can be evaluated numerically, and (b) unrestricted, contracted sums over discrete events. Further, we provide methods to reduce the number of necessary summations by splitting or recurring the original problem to easier sub-problems. For five dedicated scenarios, full results are given which indeed require a single unrestricted summation only.
机译：我们考虑具有分布式决策过程的场景，例如，多数席位和议会会议厅，超国家决策和监督委员会的个人联合。在这些情况下计算采用率以做出决策时，组合计数中会存在多个线性不平等限制。这些费率不能以封闭形式计算。我们介绍了一种在多个多数决策中纳入多个不等式条件的通用方法，该方法可显着减少涉及的求和数并消除对求和指数的限制。精确的解决方案是通过（a）可以通过数字方式评估的积分表示形式，以及（b）离散事件上不受限制的收缩总和提供的。此外，我们提供了一些方法，可通过将原始问题拆分或重现为更简单的子问题来减少必要总和的数量。对于五个专用方案，将给出完整的结果，而这些结果实际上仅需要单个无限制的求和。

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