Ladder lottery (Ghost Leg): how it works and how fair it is

A ladder lottery is fair in one sense and unfair in another. It always hands every person exactly one result. But with the number of rungs people normally draw, you are much more likely to land near the place you started. That is why the Pickboard ladder game shuffles the bottom row before it builds the ladder. The numbers below are the reason.

How a ladder lottery works

The game goes by several names: Ghost Leg in English, Amidakuji (あみだくじ) in Japan and sadaritagi (사다리타기) in Korea. Draw one vertical line per person, write the names along the top and the results along the bottom, then add horizontal rungs between neighbouring lines. Each person travels down their line and must cross every rung they meet. Where they come out is their result.

As long as no two rungs touch the same line at the same height, each path is unambiguous and no two people can end at the same place. Our tests check this on thousands of generated ladders by comparing the traced paths with a second method that simply swaps neighbouring lines rung by rung.

Where does the person on the end land?

We built 400,000 ladders for 8 people with a medium number of rungs (about 4.7 per gap) and recorded where the person starting on the far left came out. A fair draw would give 12.5% for each of the eight places.

Plain ladder, bottom row not shuffled. Start at place 1, medium rungs.

Landing straight below the start happened 24.8% of the time, and reaching the far end only 1.9%. That is about 13 times less likely. If the forfeit sits at the far right, the person on the far left is almost safe.

More rungs help, until the group gets large

This table shows how often the person on the end lands directly below their start. Height is the number of levels where rungs can go; the Few, Medium and Many settings in the tool are 8, 14 and 24.

Four people mix well with only a few rungs. The cost grows quickly with the size of the group. To bring one person’s spread of finishing places within 5% of an even spread, a ladder for 4 needs a height of 12, for 8 it needs 49, for 15 it needs 173, and for 30 it needs 696, which is roughly 6,728 rungs in total. Nobody can draw or follow that.

The cause is simple. A rung only swaps two neighbouring lines, so each level moves you one place sideways at most. Getting from one edge to the other takes many lucky rungs in a row, with rungs pulling you back along the way.

What shuffling the bottom row does

With the same 8-person, medium ladder but the bottom row shuffled first, the person at place 1 received each of the eight results between 12.4% and 12.6% of the time, which is the size of difference chance alone produces. A row that is already in a uniformly random order stays uniformly random whatever ladder you put on top of it, so the number of rungs no longer matters.

The same idea works on paper. Write the bottom row out of sight, and let people choose their line at random or draw lots for it. If the starting places are random, the result is fair even when the ladder mixes poorly.

How these numbers were produced

• We generated between 20,000 and 400,000 ladders for each group size and height using the same code the site runs, with fixed seeds, so rerunning gives the same numbers.
• Ladder rule: on each level, working from the left, every gap gets a rung with probability one third, and two neighbouring gaps never both get one on the same level. A gap that ends up with no rung at all gets one added.
• For heights of 24 and above, where that last rule is rarely used, the measured values agree with an exact calculation to within 0.5 percentage points.
• Hand-drawn ladders follow looser rules, so their numbers will differ a little. The direction is the same: fewer rungs and more people favour nearby places.