Push the Button
Everyone in the world has to take a private vote by pressing a red or blue button. If more than 50% of people press the blue button, everyone survives. If less than 50% of people press the blue button, only people who pressed the red button survive. Which button would you press?
The final blue share X is modelled as Normal(μ, σ²). In direct mode you set μ (expected blue share); in detailed mode you set P(blue wins) and the conditional loss share bloss independently. The expected death toll if blue loses is D = bloss · N.
Asks: what is the chance your one vote is the one that tips the outcome? With X ~ Normal(μ, σ²), a single vote shifts μ by 1/N, so pivotality = φ((0.5−μ)/σ) / (σ · N). Expected harm (red) or benefit (blue) = pivotality × D.
Under this view your vote only matters if it changes the outcome. With billions of voters that probability is astronomically small.
Aggregate note. Per-vote pivotality is a derivative; naively multiplying it by the size of a bloc would linearise the response and overshoot the death toll. For a bloc as a whole the natural counterfactual is "everyone in the bloc voted the other way." In both directions expected deaths collapse to ~0 (a unanimous-blue world has no losing scenarios; a unanimous-red world has no blue voters to die). So the bloc-level harm or benefit equals the current expected death toll D · P(lose). The two frameworks therefore agree at the bloc level and only diverge in how they slice that total among individuals.
Asks: what share of the collective outcome do you bear responsibility for? Total expected deaths in the world = D · P(lose). Red voters share the harm equally across the red bloc; blue voters share the prevention equally across the blue bloc.
Per red voter: D · P(lose) / ((1−μ) · N) = bloss · P(lose) / (1−μ). Per blue voter: bloss · P(lose) / μ.
Each row compares casting that vote against the opposite. If you vote blue and blue loses, you die: expected personal cost = P(lose) × 1 life. If you vote red instead of blue, you avoid that fate: expected personal benefit = +P(lose) × 1 life.