Laws & Theorems (TODO)

Electoral & party-system regularities #

  • Duverger’s law (mechanical + psychological effects). Plurality/SMD tends toward two parties; PR toward multiparty systems. Refs: Wikipedia; Cox’s classic book. ( Wikipedia, Cambridge University Press & Assessment)
  • Cox’s M+1 rule. In district magnitude M, ≤ M+1 viable lists/candidates. Refs: Cox (book); discussion/simulation. ( Cambridge University Press & Assessment)
  • Cube law (seats–votes). In two-party plurality, seat share is a nonlinear (often ~cubic) function of vote share. Refs: Wikipedia; classic BJPS treatment. ( Electowiki, Wikipedia)
  • Seat-product model. Assembly size × district magnitude predicts party fragmentation & largest party (“Votes from Seats”). Refs: CUP book page; book frontmatter PDF. ( Cambridge University Press & Assessment, Cambridge Assets)
  • Micro–mega rule. Small institutional tweaks can have large (“mega”) party-system effects. Ref: Wikipedia. ( Electowiki)
  • Cube-root law of assembly size. Lower-house size ≈ cube root of population. Refs: Wikipedia; Taagepera (2008) article. ( Study.com, Wikipedia)
  • Penrose square-root rule. Two-tier voting weights ∝ √population equalize a-priori power. Refs: Wikipedia (method); Wikipedia (square-root law). ( Wikipedia)
  • Second-order election theory. EU & other “SOE” contests show lower turnout/“punish government” patterns. Refs: Reif & Schmitt (1980) EJPR; follow-up EJPR piece. ( Emerging Infectious Diseases Journal, SpringerLink)
  • Apportionment paradoxes & impossibility. Alabama/new-state/etc.; Balinski–Young impossibility. Refs: Wikipedia (apportionment paradox); Wikipedia (Balinski–Young theorem). ( Wikipedia)
  • Effective number of parties; disproportionality. Laakso–Taagepera ENP; Gallagher index. Refs: Wikipedia (ENP); Wikipedia (Gallagher index). ( IIASA PURE, Wikipedia)

Social choice & voting theory #

  • Arrow’s impossibility theorem. No rank-order rule satisfies unrestricted domain, Pareto, IIA, non-dictatorship. Refs: SEP; Wikipedia. ( Stanford Encyclopedia of Philosophy, Wikipedia)
  • Gibbard–Satterthwaite. Any onto, strategy-proof, deterministic rule with ≥3 options is dictatorial. Ref: Wikipedia. ( Wikipedia)
  • Duggan–Schwartz (set-valued rules). For ≥3 options, any onto social choice correspondence that isn’t dictatorial is manipulable (weak dictators otherwise). Ref: arXiv overview. ( Stanford University)
  • May’s theorem (2 options). Anonymity + neutrality + positive responsiveness ⇒ simple majority is unique. Ref: Wikipedia. ( Wikipedia)
  • Condorcet jury theorem. With independent competence >.5, larger juries more likely correct. Refs: Wikipedia; SEP survey. ( Wikipedia, Stanford Encyclopedia of Philosophy)
  • Sen’s “Paretian liberal” paradox. Minimal rights can conflict with Pareto + transitivity. Ref: Sen (1970 JPE, Harvard DASH). ( Harvard Dash)
  • Median voter theorem. With single-peaked preferences in 1-D, the median is a Condorcet winner. Ref: Wikipedia. ( Wikipedia)
  • Judgment aggregation / discursive dilemma. Issue-by-issue majority can yield inconsistent collective judgments. Refs: SEP “Judgment Aggregation”; Wikipedia (discursive dilemma). ( Stanford Encyclopedia of Philosophy, Wikipedia)

Legislatures, bargaining, and agendas #

  • McKelvey–Schofield chaos theorem. In multidimensional majority rule, agenda control can reach (almost) any outcome; instability without structure. Ref: Wikipedia. ( Wikipedia)
  • Plott’s necessary conditions. Characterizes when a multidimensional majority-rule equilibrium can exist. Ref: Plott (1967 AER, Caltech record/PDF). ( authors.library.caltech.edu)
  • Structure-induced equilibrium (Shepsle/Weingast). Institutions/committees/agenda rules create stability where pure preferences don’t. Refs: Public Choice article; PDF reprint. ( IDEAS/RePEc, edegan.com)
  • Romer–Rosenthal “setter model.” An agenda setter vs. status quo can tilt outcomes under referenda. Refs: Public Choice (1978); accessible PDF. ( IDEAS/RePEc, edegan.com)
  • Riker’s size principle. Office-seeking coalitions tend to be minimal winning. Refs: Wikipedia (book); full book scan. ( Wikipedia, ia600708.us.archive.org)
  • Veto players theory (Tsebelis). More/ideologically distant veto players → greater policy stability. Refs: Wikipedia (book); seminal BJPS article. ( Wikipedia, Cambridge University Press & Assessment)
  • Pivotal politics (Krehbiel). Gridlock zones from filibuster/override pivots explain policy inertia beyond party control. Refs: UChicago Press book page; Stanford GSB page. ( University of Chicago Press, Stanford Graduate School of Business)

Collective action, parties, and regimes #

  • Olson’s logic of collective action. Free-riding in large groups; selective incentives/organization as solutions. Ref: Wikipedia. ( Wikipedia)
  • Tullock’s paradox of revolution. High social returns but low individual incentive to revolt. Ref: Tullock (1971, Public Choice). ( JSTOR)
  • Michels’ iron law of oligarchy. Organizations—even democratic—tend toward oligarchic control. Ref: Wikipedia. ( Emerging Infectious Diseases Journal)
  • May’s curvilinear disparity (intra-party). Activists often more extreme than leaders/electorate. Ref: Wikipedia. ( Wikipedia)
  • Selectorate theory. Leader survival depends on winning-coalition size vs. selectorate; predicts policy/regime patterns. Ref: Wikipedia. ( Academia)
  • Democratic peace program. Democracies rarely fight each other; mechanisms debated. Ref: Wikipedia. ( SpringerLink)

Federalism & policy dynamics #

  • Tiebout sorting. Mobility + local public goods ⇒ “vote with your feet.” Ref: Wikipedia. ( Wikipedia)
  • Oates’s decentralization theorem. With heterogeneous preferences/no spillovers, decentralization (weakly) dominates uniform central provision. Ref: Wikipedia. ( disp.web.uniroma1.it)
  • Thermostatic public opinion. Public preferences move opposite to recent policy change; policymakers respond. Ref: Wlezien (1995 AJPS). ( Wikipedia)
  • Richardson’s conflict regularities. Event sizes of war/violence are heavy-tailed. Ref: Wikipedia. ( Emerging Infectious Diseases Journal)

Voting power & measurement #

  • Banzhaf & Shapley–Shubik power indices. Measures of a-priori power in weighted voting games. Refs: Wikipedia (Banzhaf); Wikipedia (Shapley–Shubik). ( Wikipedia, SciSpace)
  • Penrose limit/square-root results. Individual power ~ 1/√N under simple majority; links to square-root rule. Ref: Wikipedia (square-root law). ( Wikipedia)

Mechanism design & incentives for governance #

  • Holmström’s Informativeness Principle. In moral-hazard contracts, use any performance signal informative about the agent’s action (balancing risk/measurement costs). Ref: Holmström (1979, Bell J. Econ.). ( Wikipedia)
  • Holmström’s “Moral Hazard in Teams.” With unobservable individual effort in teams, can’t get budget balance + Nash equilibrium + Pareto efficiency simultaneously. Refs: JSTOR entry; author PDF. ( arXiv, SpringerLink)

Opinion dynamics / learning in networks (bonus) #

  • DeGroot learning. Iterated averaging → consensus/learning conditions. Ref: Wikipedia. ( Wikipedia)
  • Friedkin–Johnsen model. DeGroot with stubbornness (initial-opinion anchoring). Ref: Wikipedia. ( Study.com)
  • Persuasion bias. Social influence can overweight repeated signals; network structure matters. Ref: DeMarzo–Vayanos–Zwiebel (QJE 2003). ( X (formerly Twitter))
  • Golub–Jackson “wisdom of crowds.” Sufficient conditions (e.g., bounded influence) for learning in large networks. Ref: AER 2010. ( People Math Wisconsin)
  • Hegselmann–Krause bounded-confidence. Opinion clusters with local averaging within confidence bounds. Ref: Wikipedia. ( Wikipedia)