Non-spatial Probabilistic Condorcet Election Methodology

There is a class of models for pol/mil/econ bargaining and conflict that is loosely based on the Median Voter Theorem which has been used with great success for about 30 years. However, there are fundamental mathematical limitations to these models. They apply to issues which can be represented on a single onedimensional continuum, like degree of centralization of a government, or cartel members negotiating the price to ask for their commodity. They represent fundamental group decision process by a deterministic Condorcet Election. There has been some extension to multidimensional issue sets, but they are not well documented and are limited to cases where the difference between policies is well approximated by a Euclidean distance and each actor’s utility is monotonically declining in distance. This work provides a methodology for addressing a broader class of problems. The first extension is to continuous issue sets where the consequences of policies are not well-described by a distance measure or utility is not monotonic in distance. Simple one-dimensional examples are discussed. The difficulty is more acute in multidimensional policy spaces. An example is the negotiations over national economic policies, where the effects differ by region, by industrial sector, and by social group. Further, even a weighted sum over the effects can be non-monotonic or multi-peaked. The second fundamental extension is to inherently discrete issue sets. The discussion will focus on subset selection problems, though the methodology is not limited to them. Two examples are the selection of which subset of competing parties will form a parliament (and which will be excluded), or the selection of a portfolio of defense projects. In the parliament formation case, the utility of a potential parliament to each actor can be modeled as a look-ahead by that actor in order to consider the various policies which that parliament might choose. This models a two-stage process, where the uncertainty in the second stage (choices on issues) is an important factor in the first stage (choices of parliaments). Because there are generally more issues than parties, the discrete choices in the first stage embody trade-offs in the second stage. Because the options cannot easily be mapped into a multidimensional space so that the utility depends on distance, we refer to it as a non-spatial issue set. The third, but most fundamental, extension is to represent the negotiating process as a probabilistic Condorcet election. This provides the flexibility to make the first two extensions possible; this flexibility comes at the cost of less precise predictions and more complex validation. Because the analyses are inherently probabilistic, this provides a smooth "response surface" for expected utility, thus simplifying strategy optimization even in discrete issue sets. Some common bargaining algorithms are inapplicable in the more general issue sets addressed here. For example, power-weighted interpolation between the positions of two actors is often inapplicable in continuous but multimodal issue sets. More fundamentally, interpolation is unusable in discrete issue sets because continuous interpolation is not even defined between discrete options. We provide motivation and overview of the general methodology followed by mathematical details. The methodology has been implemented in two proof-of-concept prototypes which address the subset This work was performed in 2013 when the lead author was employed at BAE Systems