Public Choice & Political Economy
Riker-Ordeshook Calculus of Voting Formula
Calculate Expected Instrumental Pivotal Payoff p * B, Downsian Net Voting Return p * B - C, Riker-Ordeshook Expressive Turnout Utility R = p * B - C + D, and Breakeven Pivotal Probability.
Resolve Anthony Downs’s Paradox of Voting with Riker & Ordeshook’s R = pB - C + D!
In An Economic Theory of Democracy (1957), Anthony Downs posed the famous Paradox of Voting: if a rational citizen cares only about casting the decisive tie-breaking ballot (Pivotal Probability p) for Differential Policy Stake B at Time/Travel Cost C > 0, then in any large electorate where p ~ 10^-4 to 10^-7, Expected Instrumental Benefit p * B is pennies while C is $10–$30, yielding p * B - C < 0 (zero predicted turnout)! William Riker and Peter Ordeshook (1968, American Political Science Review) augmented the calculus with Expressive / Civic-Duty Benefit D (the intrinsic satisfaction of fulfilling democratic norms or expressing partisan identity), yielding Net Turnout Reward R = p * B - C + D > 0 whenever D + p * B > C!
Verify on Pivotal_Probability p = 0.0001 (1 in 10,000 close local race, B2), Differential_Policy_Stake B = $20,000 (B3 -> Expected Instrumental Value p * B = $2.00 in E2), Voting_Cost C = $16.00 (B4 -> Pure Downsian Payoff = $2.00 - $16.00 = -$14.00 in E3!), and Civic_Duty_Benefit D = $25.00 (B5): Riker-Ordeshook Net Utility is R = 2.00 - 16.00 + 25.00 = +$11.00 (Rational Turnout!), while Pure Downsian Breakeven Pivotal Probability without civic duty would require p_Downs* = C / B = 16 / 20,000 = 0.0008 (8x higher than 0.0001)! Test an election-day storm raising Voting Cost C to $30.00 in cell B4: R drops to -$3.00, predicting rational abstention on the margin!
| Turnout Calculus Metric (p=0.0001, B=$20,000, C=$16, D=$25) | Formula | Verified Output |
|---|---|---|
| Expected Instrumental Value ($) | p * B | $2.00 |
| Pure Downsian Payoff ($) | p * B - C | -$14.00 (Downs Paradox) |
| Riker-Ordeshook Net Reward R ($) | p * B - C + D | +$11.00 (Voter Turns Out) |
| Downsian Breakeven Pivotal Prob p* | MIN(1, C / B) | 0.0008 (1 in 1,250) |
Riker-Ordeshook Net Voting Reward R Formula
Validate 0 <= p <= 1, B > 0, C >= 0, and D >= 0!
=LET(pr_pv, B2, bn_df, B3, cs_vt, B4, dv_cv, B5, IF(OR(pr_pv < 0, pr_pv > 1, bn_df <= 0, cs_vt < 0, dv_cv < 0), "Invalid Voting Calculus", ROUND(pr_pv * bn_df - cs_vt + dv_cv, 2)))Downsian Breakeven Pivotal Probability Formula
Compute C / B to see how impossibly small an electorate must be for purely instrumental motives (D = 0) to sustain turnout!
=IF(ISTEXT(E4), E4, ROUND(B4 / B3, 4))Try the example
Use the original values or change them to test your understanding.