Where Impossibility Bites: A Threshold and Influence Theory of Paradoxes and Manipulation

Kalai & Safra (2005) show how Boolean threshold phenomena rigorously capture when small parameter changes trigger large outcome shifts. We import these tools to social choice, encoding events like Condorcet cycles, nontransitivity, and profitable manipulations as Boolean properties of random preference profiles (extending Penn, 2011). For families of rules (e.g., plurality, STV, Condorcet-consistent, approval, proportional ranked methods; cf. Benade et al.), we derive noise sensitivity, influence spectra, and critical curves that delineate robust vs. fragile regimes. We validate with participatory budgeting field data (Geobey et al., 2023) to produce practical “phase diagrams” guiding rule selection: e.g., when correlation among voter blocs exceeds a threshold, rule X becomes highly cycle-prone but approval-based variants remain stable. This program quantifies the practical scope of Arrow/GS by identifying parameter regimes where their constraints are negligible in practice and others where they dominate—shifting from binary axiomatic verdicts to measurable robustness margins. The innovation is to unify paradox and manipulation analysis under a common probabilistic-influence framework, creating predictive tools for institutional design.

References:

1. Impossibility Theorems And Voting Paradoxes In Collective Choice Theory. E. M. Penn (2011).
2. Threshold Phenomena and Influence, with Some Perspectives from Mathematics, Computer Science, and Economics. G. Kalai, S. Safra (2005).
3. Proportionality for ranked voting, in theory and practice. Gerdus Benade, Christopher Donnay, Moon Duchin, Thomas Weighill (None).
4. Testing participatory budgeting voting design: Two cases from the city of Kitchener. Sean Geobey, Sean Campbell, Norman Kearney (2023). Local Development & Society.