Two candidates start at opposite ends of the spectrum, then edge toward the center, and toward each other, until both sound almost the same. Voters complain there is no real choice. That drift is not cowardice; it is the median voter theorem, and it is a Nash equilibrium.
Political economy
Game theory reaches politics too. Downs and Hotelling explain why two candidates converge on the median voter, why that breaks down with abstention or three candidates, and why lobbying is an all-pay contest where money is burned even in defeat.
The median voter theorem predicts two candidates converge on the...
In an all-pay contest, a loser...
Predict, then play
Drag two candidates along a spectrum of seven voters. First keep everyone voting and find the median convergence. Then let voters abstain when both candidates are too far away and watch the clean result break.
Lobbying is a separate game. Choose spending against a stated rival bid and check what both sides pay.
Why did that happen?
Take a guess, then reveal the reason
Candidates want votes, and on a single left-right dimension each voter picks the nearer candidate. If one candidate sits away from the median, the other can move just inside them, closer to the middle, and win everyone on that side plus the median, a majority. So any position but the median is beatable, and both-at-median is the only stable outcome. This is why rivals often sound alike: they are chasing the same pivotal voter, not their base. The convergence breaks when extremists abstain if a candidate drifts too far center (pulling platforms apart to hold turnout), when there are three or more candidates, or when the policy space is multi-dimensional, where a median generally does not exist. Lobbying is a different game: an all-pay contest where spending is sunk whether you win or lose.
The real strategy
Downs (1957), building on Hotelling, showed that with single-peaked preferences on one dimension, two office-seeking candidates converge to the median voter’s position under the lesson’s vote-maximizing assumptions. The result is fragile: it can fail with more candidates, multiple policy dimensions, policy-motivated candidates, or turnout effects. Lobbying can be modeled separately as an all-pay contest (Baye, Kovenock and de Vries, 1993): contestants spend to win a prize, but all spending is forfeit, so equilibrium expenditure can dissipate much of the prize’s value.
Take the seven voters in the exhibit, at positions ; the median is . If A stands at and B at , B can slide toward the median and improve its vote share. Under the demonstration’s tie rule and office-only objective, both at the median is the stable outcome. Add turnout costs, policy motives, more candidates, or more policy dimensions and that conclusion need not survive. Lobbying is a separate all-pay model: firms and groups spend on influence, and losers do not get their expenditure back. Rational ignorance means that acquiring political information can cost an individual voter more than the small chance that it changes an election outcome. That idea does not follow from the all-pay model, so it is a separate explanation for limited monitoring.
Work it out
Convergence, and when it breaks
With voters at , the median is . Suppose A is at and B at . B moves to : now B is nearer to the median voter and to everyone at , and , winning four of seven, a majority. A can only respond by moving to too. Once both sit at , neither can profitably deviate, since any move loses the median to the other. That is the unique Nash equilibrium, and it explains platform convergence.
Now let voters abstain if the nearest candidate is too far from them. A candidate at dead center now loses their own extreme supporters to abstention, so shifting slightly toward one’s base can gain more turnout than it loses at the middle. The equilibrium platforms pull apart, and full convergence fails, matching real elections where candidates court their base as well as the center. With three candidates the logic collapses entirely: a third entrant near the median can leapfrog, and no stable median position exists. Meanwhile lobbying runs as an all-pay auction, where each side’s spending is sunk, so competition can burn resources up to the value of the favour sought.
Key idea
Two vote-maximizing candidates converge on the median voter, the unique Nash equilibrium, which is why platforms look alike. The result breaks with abstention, three candidates, or multiple dimensions. Lobbying is an all-pay contest where spending is sunk even in defeat.
Check yourself
Sources
- Downs, A. (1957). An Economic Theory of Democracy. Harper and Row.
- Baye, M. R., Kovenock, D., and de Vries, C. G. (1993). Rigging the lobbying process: an application of the all-pay auction. American Economic Review, 83(1), 289-294.