Abstract
In “The Logic of Campaigning”, Dean and Parikh consider a candidate making campaign statements to appeal to the voters. They model these statements as Boolean formulas over variables that represent stances on the issues, and study optimal candidate strategies under three proposed models of voter preferences based on the assignments that satisfy these formulas. We prove that voter utility evaluation is computationally hard under these preference models (in one case, \(\#P\)-hard), along with certain problems related to candidate strategic reasoning. Our results raise questions about the desirable characteristics of a voter preference model and to what extent a polynomial-time-evaluable function can capture them.
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Notes
- 1.
Some sources, particularly in the belief revision literature, use the term belief base.
- 2.
Note that we modify our notation from Dean and Parikh’s. In particular, they represent this preference function using two quantities — a weight in [0, 1] and a truth-value preference in \(\{ -1, 0, 1 \}\) — which we combine into the single function \(p_v\).
- 3.
Dean and Parikh assume that for the purpose of determining “expected value” of the worlds, all worlds are considered equally likely.
- 4.
One consequence of this definition is that the utility of an empty or otherwise tautological theory is 0 for any expected-value voter.
- 5.
In Grice’s account of implicature [7], participants in a conversation assume each other to be obeying certain maxims of cooperativity (for instance, illustrated here is the maxim of Quantity — roughly, give as much information as necessary, and do not give more information than necessary); they interpret each other’s statements in light of this mutual assumption.
- 6.
The name “weighted satisfiability” (WSAT) has been used by different sources to refer to two different groups of problems — one where an instance consists only of a propositional formula and the value of a solution is the number of true variables (the Hamming weight), and the generalization we use here where the instance includes weights for the variables. The maximization/minimization versions of the former are sometimes called “maximum number of ones” (MAX-ONES) / “minimum number of ones” (MIN-ONES), and are complete for NPO -PB [12], a subclass of NPO where the magnitude of a solution’s value is polynomially bounded by the size of the input.
- 7.
We divide through by the maximum weight so that the \(p_v(x_i)\)’s are in [0, 1].
- 8.
From Revolution PAC’s 2012 interview (https://www.youtube.com/watch?v=9jKszduiK8E)
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Acknowledgments
The authors thank three anonymous reviewers for their helpful feedback and thank Alec Gilbert for catching errors in a late draft. All remaining errors are the responsibility of the authors. This material is based upon work partially supported by the National Science Foundation under Grants No. IIS-1646887 and No. IIS-1649152. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
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Siler, C., Miles, L.H., Goldsmith, J. (2017). The Complexity of Campaigning. In: Rothe, J. (eds) Algorithmic Decision Theory. ADT 2017. Lecture Notes in Computer Science(), vol 10576. Springer, Cham. https://doi.org/10.1007/978-3-319-67504-6_11
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