Market Impact
Why large orders cost more than small ones — in two distinct ways.
Trading moves prices. That sounds obvious, but it splits into two effects that behave completely differently, and conflating them is a classic modelling error.
Temporary impact
You cross the spread and walk down the The Limit Order Book, filling at progressively worse prices. This cost is yours alone and it evaporates once you stop — the book refills, the price recovers. It scales with how aggressively you demand liquidity right now.
Permanent impact
Other participants watch you trade. A persistent stream of sell orders is information: someone knows something, or someone big needs out. They revise their own quotes downward. This shift does not recover, and it's paid by every unit you have left to sell.
Why the distinction matters
It sets up the entire Optimal Execution trade-off. Temporary impact punishes trading fast. Permanent impact — combined with price volatility — punishes trading slow, because you're still holding inventory while the market moves. Classical models write both as (usually linear) functions of trade size, with coefficients $\alpha_t$ for temporary and $\kappa_t$ for permanent impact.
The catch: those coefficients aren't constant. They vary with liquidity, time of day, and market regime. An agent that can't observe the current regime can't condition on it — and will confidently apply a strategy learned in one regime to a market that's in another. Handling that is precisely the motivation for a Mixture of Experts approach.