The synchronous double auction as a stochastic game

  Our double auction can be modeled as a stochastic game, where the state is a vector consisting of all agents' individual endowments. The action is the vector of bids submitted by all agents. At time t, agent i's local state sⁱ_t, is described by its endowment vector, .Thus each local state itself is a vector. Agent i's action at time t, aⁱ_t= (P_t^i,buy,P_t^i,sell), is its buying and selling price offered for one unit of good in the current auction.

The reward for agent i at time t is given by

where

In these state update equations, P_t is the trading price, and $z_t \in \{ -1, 0, 1 \}$ is the quantity the agent trades at time t. When z_t=0, the agent does not trade, and r_t=0.