The Grays projectSeven agents, 100 points, and a few liars
Ask seven independent sources the same question, see how honest sources disagree, then find out how many dishonest ones the group can survive.
The learner lays a grid of 10 by 10 points, 250 m apart, over Grays and asks seven public elevation models, served by OpenTopoData, for the height of the ground at every point. Each model plays the part of an agent. Even with nobody lying, the answers differ: the gap between the highest and lowest report at a point averages 8.4 m and ranges from 2.0 m to 23.0 m. At the worst point, the seven agents said 17, 22, 28.6, 29, 29, 29 and 40 m. Two of the agents, the 30 m SRTM model and the Mapzen tiles, gave identical heights at all 100 points, which is what you would see if both drew on the same source here. Seven voices, then, but only six independent ones.
Average distance of each agent from the median of all seven, over 100 points in Grays (our Python run)| Agent (elevation model) | Average distance from the median |
|---|
| EU-DEM, 25 m | 0.46 m |
| SRTM, 90 m | 0.50 m |
| SRTM, 30 m | 0.63 m |
| Mapzen | 0.63 m |
| GEBCO 2020 | 2.00 m |
| ASTER, 30 m | 3.89 m |
| ETOPO1 | 4.92 m |
Then some agents turn Byzantine and report whatever does the most damage. If a trusted collector combines the reports, the choice of rule decides everything. One agent claiming 1,000 m drags the mean of seven up by 139.8 m on average. The median barely notices: across every choice of which agent lies, and in either direction, one liar shifted it by 1.07 m on average and 5.00 m at most, two liars by 2.66 m and at most 8.86 m, and three liars by 6.32 m and at most 14.86 m. A fourth liar gives the liars a majority, and then the median goes wherever they send it.
Without a trusted collector, the agents must agree among themselves, relaying what they were told. We coded the Oral Messages algorithm from the 1982 paper by Lamport, Shostak and Pease, with one traitor who changes a height by 50 m. With three agents, the loyal pair ended up wrong or split in 2,000 of 3,000 runs, every run in which the traitor was one of the relaying agents. With four agents, every one of 4,000 runs ended with the loyal agents in agreement, and on the commander's value whenever the commander was loyal. That is the rule the paper proves: tolerating f traitors without a referee takes at least 3f + 1 agents.
Ages 8 to 11
Five pupils report a measurement, two secretly fib, and the class compares averaging with picking the middle value.
Ages 11 to 15
Fetch the seven sets of heights in Python, find the median at each point, then add a liar and measure the damage.
Ages 15 and up
Code Oral Messages for three and four agents, run thousands of trials, and explain why three is not enough.
Sources and limits
Heights come from the EU-DEM (produced using Copernicus data and information funded by the European Union), SRTM, ASTER GDEM, Mapzen terrain tiles, ETOPO1 and GEBCO 2020 datasets, all read through the public OpenTopoData API. The coarse global models were never meant for street-level work, so their larger gaps are expected. None of the models is ground truth, so we measure disagreement, not error. The grid, the liars, the 50 m lie and the tie rule are our choices, and other choices give other numbers.
