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. In the picture, four people share a ladder with six levels, and the red path belongs to the person who started on the second line.

This guide uses four words. A line is one person’s vertical line. A rung joins two neighbouring lines. A level is one height at which rungs can be placed. A place is a position along the bottom, where a result is written.

As long as no two rungs touch the same line on the same level, 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 (14 levels, about 4.7 rungs between each pair of lines) 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 on line 1, medium rungs (14 levels). The red bar is the place straight below the start.

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 levels help, until the group gets large

This table shows how often the person on the end line lands straight below their start. The Few, Medium and Many settings in the tool are 8, 14 and 24 levels.

Red numbers are more than 1.5 times the fair share.

Four people mix well with only a few levels. The cost grows quickly with the size of the group. Take this as the test for “close to even”: whichever line a person starts on, add up how far each place’s chance sits above the even chance (12.5% per place for 8 people), and the total must stay within 5 percentage points. To pass it, a ladder for 4 needs 12 levels, 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 line 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, 14-level ladder but the bottom row shuffled first, the person on line 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 levels 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 lines 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 number of levels using the same code the site runs, with fixed seeds, so rerunning gives the same numbers.
• Ladder rule: call the space between two neighbouring lines a gap. On any level, each gap has the same one-in-three chance of holding a rung. Two gaps side by side never both get a rung on the same level, because two rungs would then touch one line at the same height and the path would fork. A gap that ends up with no rung at all gets one added.
• For 24 levels 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 levels and more people favour nearby places.

Result links, the order of the lines and the bottom row

A ladder result link holds the names in their order, the bottom row and the value the ladder was built from. In a ladder, who stands on which line is part of the result, so the same ladder with the lines in a different order gives different results. The same goes for the bottom row: swap the order of two prizes and the people who get them swap too. When you check a ladder link, compare the order of the names and of the bottom-row items, as well as the names themselves, with what was agreed. The ladder check code under the list covers both orders, so if it was posted before the draw you only need to compare codes. If you would rather draw in a way that does not depend on order, use the slips in the name picker: there the same people get the same slips whatever order the list is in.