Each round, an odd number of players pick 0 or 1 and whoever ends up on the smaller side wins. Walk through the plain game first, then connect the players so they can copy their most successful neighbour, and watch leaders emerge.
Based on Sang Hoon Lee and Hawoong Jeong, "Effects of substrate network topologies on competition dynamics", Phys. Rev. E 74, 026118 (2006), which builds on the follower model of M. Anghel, Zoltán Toroczkai, Kevin E. Bassler, and G. Korniss, "Competition-Driven Network Dynamics: Emergence of a Scale-Free Leadership Structure and Collective Efficiency", Phys. Rev. Lett. 92, 058701 (2004).
Players only see the last m winning sides (the history). Each holds S fixed strategies: lookup tables that map every one of the 2m possible histories to a choice. A player always uses the strategy with the best virtual score, and after each round every strategy that would have picked the minority gains a point. Click any player to open her strategy tables.
Now the players sit on a substrate network. Every round each player first computes her own choice, then looks at her neighbours’ points (total wins so far). If a neighbour has more points than she does, she drops her own choice and copies that neighbour’s. The “who copies whom” arrows form the follower network. Node size shows how many followers a player has.
Share of players with at least k links (log–log). A straight, fat tail in the follower curve is the paper’s scale-free leadership.
Each dot is a player: substrate degree against win rate. In the paper (Fig. 4), well-connected players tend to lose, because their followers pile onto the same side.
The paper’s main measurement: volatility σ²/N (variance of the number of players choosing 1, divided by N) across α = 2m/N. Lower is more efficient; 0.25 is what pure coin tossing gives. This sweep runs full games for each memory length, with and without networked copying, so you can reproduce the shapes in Figs. 2 and 3.
What the paper reports: on the random graph the curve keeps the original shape; on the 1D lattice volatility is huge except at the smallest m; scale-free substrates sit in between, rising again at small α. Networks mostly make the crowd less efficient by amplifying herding.
Implementation notes: points are the number of rounds a player has actually won with her final choice. A player copies her neighbour’s own-strategy choice (one step, no chains); ties between equally good neighbours keep the current leader, otherwise pick at random. The paper’s text defines A(t) as a difference but quotes σ²/N = 1/4 for coin tossing, which matches the variance of the count of one side, so volatility here is measured on that count.
Demo built with Claude Opus 5.5 (Anthropic). The model and results belong to the paper’s authors.