Gross substitutability: An algorithmic survey

The concept of gross substitute valuations was introduced by Kelso and Crawford as a sufficient conditions for the existence of Walrasian equilibria in economies with indivisible goods. The proof is algorithmic in nature: gross substitutes is exactly the condition that enables a natural price adjustment procedure - known as Walrasian tatônnement - to converge to equilibrium. The same concept was also introduced independently in other communities with different names: M♮-concave functions (Murota and Shioura), Matroidal and Well-Layered maps (Dress and Terhalle) and valuated matroids (Dress and Wenzel). Here we survey various definitions of gross substitutability and show their equivalence. We focus on algorithmic aspects of the various definitions. In particular, we highlight that gross substitutes are the exact class of valuations for which demand oracles can be computed via an ascending greedy algorithm. It also corresponds to a natural discrete analogue of concave functions: local maximizers correspond to global maximizers.