Optimal Enfranchisement Explorer

Five agents each have a preference (0 or 1) on a binary decision and a stake — how much they care. A planner chooses a voting committee; the committee's majority wins. The question: who should be on the committee?

A planner picks an odd-size franchise of N=5 agents; that subset votes by strict majority on a binary decision. Each agent i has stake Δᵢ ≥ 0 and the planner maximises E[Σ Δᵢ · 𝟙(Xᵢ = Y_C)].

Two natural conjectures: (1) some franchise dominates any constant-rule dictatorship; (2) the optimum is “high-stake” (insiders all weakly higher stake than outsiders). Both fail under full support alone. Play with the controls to find out when, and why.

Joint distribution of opinions

p is the probability each agent independently favours outcome 1. ρ (rho) controls correlation: ρ=0 means fully independent opinions, ρ=1 means all agents always agree.

Marginal P[Xᵢ=1] (p)0.50

Pairwise correlation (ρ)0.00

Exchangeable model: with prob ρ all opinions equal one shared Bern(p) draw; otherwise iid Bern(p). Gives marginal p and pairwise correlation exactly ρ.

Each row is a possible opinion profile — a combination of 1s and 0s for each of the 5 agents. Toggle bits to set who votes 1. The weight controls how often this profile occurs. A small uniform noise ensures every profile has positive probability (full support).

Uniform noise (full-support mix)0.010

Expected welfare across all 16 odd franchises 16 possible committees — click/hover for details

Each bar is one odd-size voting committee (1, 3, or 5 members out of 5 agents). Height = expected welfare. ★ = the best committee. The dashed lines show the best constant dictatorship — if it reaches the bars, conjecture 1 fails.

strict high-stake weak high-stake not high-stake optimal — — best constant dictatorship

Per-agent decomposition

For the highlighted committee, each agent's contribution to expected welfare. The Δ column compares to the runner-up committee (or the best dictatorship when C1 fails).

Insights

Failure diagnostics

Scenario contributions

Each row is one of the 2⁵=32 possible opinion profiles, sorted by probability. "Selected" and "comp." show each rule's weighted welfare in that state. Rows where the two rules produce a different outcome (different Y) are what drive the total welfare difference.

Swap diagnostic

What if we replaced one committee member with an outsider, keeping committee size the same? Each cell shows the welfare change. Green = improvement. Red = worse. An orange border marks a counterintuitive swap: the outsider has higher stake than the insider, yet swapping hurts welfare — this is the proxy effect (see CE2).

Phase diagram counterexample sweeps

Each pixel is a different opinion distribution. Colors show which conjectures hold. Drag inside the diagram to load that distribution into the explorer above.

Sweep from balanced iid to the CE1 sparse distribution. The vertical axis is the share replaced by the uniform distribution over all 32 opinion profiles.

0.05

mixture toward CE1

uniform profile share ε

Both conjectures holdoptimum is high-stake and beats every constant dictatorship

C1 fails: dictatorship winssome constant rule beats every franchise

C2 fails: optimum not high-stakea lower-stake agent is included over a higher one

Both fail