Performance = Skill + Luck. When everyone is highly skilled (low skill variance), luck dominates outcomes. This is the "Paradox of Skill" -- as competition improves, luck becomes MORE important, not less. Extreme performers in round 1 tend to regress toward average in round 2, because their extreme result was partly luck.
Watch the scatter plot: with high skill variance, dots cluster near the diagonal (skill predicts performance). With low skill variance, dots scatter everywhere -- luck dominates. The paradox curve shows this systematically. Try "Equal" talent to see pure luck rankings.
Despite being completely random, random walks have predictable STATISTICAL properties: expected displacement grows as sqrt(t), not t. The path looks like it has trends and patterns, but it is pure noise. This is why we see "patterns" in random data -- our brains are pattern-recognition machines that find signal in noise. In 1D and 2D, walks return to the origin infinitely often.
The red dashed envelope shows sqrt(t). Notice how actual paths roughly stay within this envelope. Try adding a slight bias (P up > 0.5) to see drift emerge. In 2D mode, paths trace beautiful random curves that look purposeful but are pure chance.
Informed traders push prices toward fundamental value; uninformed traders add noise. If markets are efficient, prices already reflect all available information, so future price changes are unpredictable (random walk). New information causes instant jumps. Returns should show near-zero autocorrelation -- no predictable patterns.
Try setting informed traders to 0 -- price wanders randomly, disconnected from value. Add many informed traders to see tight tracking. "Inject News" creates surprise events. The autocorrelation chart should show bars within the red bands (insignificant) if the market is efficient.
Each colonel allocates N troops across K battlefields. Each battlefield is won by the side with more troops. Winner takes the most battlefields. There is NO single optimal strategy -- strategies are intransitive (A beats B, B beats C, C beats A). The weaker player should concentrate forces; the stronger player should spread evenly.
Run the tournament to see the heatmap: no strategy dominates all others! "Even" beats "Guerrilla" on average, but "Concentrate" beats "Even," and "Random" can beat "Concentrate." Try asymmetric troops: the weaker side must use high-variance strategies to have any chance.