The rule that decides what it costs to outrank someone, and the reasoning behind picking it. This page is not linked from the navigation — you found it on purpose.
α = 1 is a flat percentage: taking the top costs the same fraction at every size, forever. α > 1 turns the leader into a fortress — the higher they climb, the proportionally harder they get to touch. α < 1 does the opposite: raises still grow in dollars, but shrink as a share of the bid, so the top stays reachable however high it goes.
We use α = ½. The raise is √B.
Required raise as a share of the bid being beaten. Both axes are logarithmic, so a power law is a straight line and α is its slope.
Hover the plot to probe the √B rule.
It is not a tuned constant. The system contains exactly two natural quantities: the smallest unit of money ($1) and the current bid (B). √B is √(1 × B) — the geometric mean of the two, the exact midpoint between them on a multiplicative scale. Every other value of α is a preference. This one is a derivation.
It needs no floor. √B never falls below $1 for any bid of $1 or more, so the rule is a single expression with no special case bolted on. A flat percentage collapses below $50 and has to have a minimum patched in, which is exactly where arbitrary numbers creep into a system.
It is the counterweight to bids being permanent. Nothing here expires, so the standing risk is a category going dead under somebody who parked high early. Under α < 1 the leader’s wall grows in dollars but shrinks as a share — $71 to take a $5,000 position is 1.4%. Big money buys real distance from the pack. It never buys immunity.
√B is harsher than a flat 2% below about $2,500 and gentler above it — the two cross exactly where √B = 0.02B, at B = 2,500. So the bottom of the board is not churned for pennies, and the top never becomes unreachable. The rule corrects itself in both directions.
| To beat | Flat $1 | 2% rule | √B rule | New total |
|---|---|---|---|---|
| $2 | +$1 | +$1 | +$2 | $4 |
| $25 | +$1 | +$1 | +$5 | $30 |
| $100 | +$1 | +$2 | +$10 | $110 |
| $300 | +$1 | +$6 | +$18 | $318 |
| $1,000 | +$1 | +$20 | +$32 | $1,032 |
| $5,000 | +$1 | +$100 | +$71 | $5,071 |
Climbing from $2 to $5,000 one raise at a time takes 4,998 raises under a flat $1, 395 under 2%, and 139 under √B.