Visualising the Dynamics

The arguments in the previous two sections were algebraic. Here we make them visual.

The Stability Threshold

The diagram below plots the expected payoffs $V_C$ and $V_D$ against the discount factor $\delta$.

$$V_C = \frac{b - c}{1 - \delta}, \qquad V_D = g$$

At $\delta^* = 1 - \frac{b - c}{g}$ the two curves intersect. To the right of $\delta^*$, patient agents prefer cooperation; to the left, short-term defection wins. For our reference parameters $(b = 3,\ c = 1,\ g = 2)$ the crossover sits at $\delta^* = 0.5$.

Communication Capacity as Catalyst

Once cooperation is established it generates a second dynamic: a richer shared vocabulary lowers encoding costs and raises detection of deception. The interactive chart below traces how payoffs shift as communication capacity $\kappa$ grows.

Drag the cursor over the chart to read precise values. The vertical marker shows $\kappa^*$, the threshold beyond which cooperation dominates even at moderate discount factors.

The Topology of Belief Space

There is a deeper geometric point lurking behind the payoff analysis. The space of communicable mental states is not Euclidean. Identifying indistinguishable states under the equivalence relation of pragmatic synonymy produces a quotient space with non-trivial topology.

The simplest compact non-orientable surface is the real projective plane $\mathbb{RP}^2$. The Roman surface below is one classical immersion of $\mathbb{RP}^2$ in $\mathbb{R}^3$ — realised via the map $(x, y, z) \mapsto (xy,\ xz,\ yz)$ that collapses each pair of antipodal points on $S^2$ to a single point. The self-intersections along three line segments are not defects but signatures of non-orientability.

Whether the actual belief space of language users has this topology is an open empirical question. The visualisation serves as a reminder that the attractor argument requires a topology theorem, not just an economic one.

Phase Portrait of Cooperation Stability

The diagram below shows the advantage of cooperation $V_C - V_D$ across all combinations of patience $\delta$ and communication capacity $\kappa$, derived from a parameter sweep computed offline. Green cells indicate regions where cooperation wins; red cells where defection wins. The dashed boundary traces the critical patience threshold $\delta^*(\kappa)$ above which cooperation is always preferable. As $\kappa$ increases, this threshold drops to zero — cooperation becomes the dominant strategy regardless of patience.

Vocabulary Growth as Network Formation

A communication network can be thought of as a graph whose edges represent active cooperative links. The animation below tracks how this network forms as shared vocabulary $\kappa$ grows from zero to its equilibrium value. Agents begin as defectors or silent observers; as $\kappa$ rises and cooperation becomes rational, edges appear and strategies shift from $D$ and $S$ toward $C$.

Use the controls to step through frames manually or let the animation run. The transition from a sparse, defection-dominated network to a fully connected cooperative one mirrors the attractor argument: given sufficient communication capacity, cooperation is individually rational and structurally stable.