The index of a completely mixed equilibrium in a game—a topological invariant that counts how the equilibrium sits in the space of strategy profiles—can be any integer. Positive, negative, zero, arbitrarily large. This seems abstract until you learn what index means for robustness: in a restricted class called monogenic equilibria, having a non-zero index guarantees the equilibrium survives payoff perturbations. You can't destroy it by slightly changing the game's rewards. The equilibrium persists not because the payoffs are finely tuned, but because its algebraic structure makes it indestructible.
For monogenic equilibria, the index is restricted to -1, 0, or +1. The ones with index +1 or -1 are payoff-robust. Small changes to the game might shift the equilibrium's location in strategy space, but they can't eliminate it. The equilibrium is structurally stable. Index zero equilibria, by contrast, can vanish under perturbation. The topological invariant predicts which equilibria are brittle and which are robust.
This makes equilibrium selection less arbitrary. Not all equilibria are equally real. Some exist only because the payoffs are exactly what they are; nudge the parameters and they disappear. Others are locked in by the game's structure. The index tells you which is which before you perturb anything.
The pattern extends beyond game theory. Any system with multiple steady states faces this question: which ones are structural and which are coincidental? Topological invariants provide the answer. Phase transitions, organizational attractors, market structures—when a steady state has a non-zero index (or its analog in the domain's topology), it's robust to perturbations. The system can shake, the parameters can drift, but that state remains accessible. Some equilibria exist because of the details. Others exist because of the shape.