A ∈ R N × J represent the payoff structure of those orders. The column matrix A$_{·}$$_{j}$ ∈ R N × 1 is the payoff structure of the contingent claim that the j th order attempts to transact. For example, suppose that the second order attempts to buy three shares of the contingent claim that pays $1 per share if and only if state 1 is realized. In such a case, the column matrix A$_{·}$$_{2}$ is [ 1 0 0 ... 0 ] $^{T}$. When determining the market equilibrium price of each order, the market maker first determines the equilibrium price for each state. Let ξ = [ ξ$_{1}$ ξ$_{2}$ ... ξ$_{N}$ ] T denote the equilibrium state prices. Then, the market maker determines the market-clearing price of each contingent claim by taking the dot product between the payoff vector and ξ . For example, consider the contingent claim with payoff structure [ 1 0 1 0 ] $^{T}$. Suppose that the equilibrium state price vector ξ is [ ξ$_{1}$ ξ$_{2}$ ξ$_{3}$ ξ$_{4}$ ] $^{T}$. Then, the market-clearing price of this contingent claim is ξ$_{1}$ + ξ$_{3}$ . Therefore, to determine the equilibrium price of each order in the limit order book, the market maker only has to determine the value of ξ . The binary variable B$_{j}$ is 1 if the j th limit order is a buy order and − 1 if the j th limit order is a sell order. Let b$_{j}$ and Q$_{j}$ denote the limit price and the limit quantity associated with the j th order, respectively. Let x$_{j}$ denote the actual number of the shares of the claim that the submitter of the j th order is allowed to trade. We call x$_{j}$ the ”order fill” for the j th order. Once the equilibrium price of each order is determined, the market maker decides x$_{j}$ , ∀ j ac- cording to the limit order logic. Consider a buy order. If the market-clearing price of an order is strictly higher than the limit price, x$_{j}$ is exactly equal to 0. If the market-clearing price is strictly lower than the limit price, x$_{j}$ is set to Q . In these two cases, the limit order logic automatically determines the order fill. In contrast, if the market-clearing price of an order is exactly equal to the limit price, the value of x$_{j}$ can be any number in the closed interval [0 , Q$_{j}$ ]. The logic works similarly for a sell order. Define the vector x ∈ R J × 1 such that the j th element of x is x$_{j}$ . Similarly, define Q ∈ R J × 1 such that the j th element of Q is Q$_{j}$ . Define b ∈ R J × 1 such that the j th element of b is b$_{j}$ . Definition 2 (Limit Order Logic) N ∑ i =1 A$_{ij}$ξ$_{i}$ is the market-clearing price of the j th order. 11 x$_{j}$ = 0 if N ∑ i =1 A$_{ij}$ξ$_{i}$ > B$_{j}$b$_{j}$ x$_{j}$ ∈ [0 , Q$_{j}$ ] if N i =1 A$_{ij}$ξ$_{i}$ = B$_{j}$b$_{j}$ x$_{j}$ = Q$_{j}$ if N ∑ i =1 A$_{ij}$ξ$_{i}$ < B$_{j}$b$_{j}$