x$_{j}$ = 0 if A T · $_{j}$ε > b$_{j}$ x$_{j}$ ∈ [0 , Q$_{j}$ ] if A T · $_{j}$ε = b$_{j}$ x$_{j}$ = Q$_{j}$ if A T · $_{j}$ε < b$_{j}$ (4) max x , s $_{,M}$b$^{T}$x − M + ∑ N i $_{=1}$δ$_{i}$ log( s$_{i}$ ) such that (A) ∑ J j $_{=1}$A$_{i,j}$ x$_{j}$ + s$_{i}$ = M for each i ∈ { 1 , 2 , ..., N } (B) 0 ≤ x ≤ Q (C) s ≥ 0 (3) 8 Let u : R → R denote the market maker’s utility function. Suppose that u is an increasing function. Unlike Peters et al. (2005), we suppose that the market maker uses the uniform starting order. That is, δ$_{i}$ is the same constant δ for ∀ i . Let ε ( δ ) denote the state price vector associated with (3) when δ$_{i}$ = δ for ∀ i . Let x ( δ ) denote an optimal value of x for (3). The ∑ N i $_{=1}$δ$_{i}$ log( s$_{i}$ ) term ensures the existence of a unique state price vector. However, the starting order subjects the market organizer to potential financial loss when the claims mature. To minimize organizer’s potential loss, Peters et al. (2005) suggest making the magnitude of δ very small. The person who submitted the j th order pays the premium worth b$_{j}$x$_{j}$ to the market maker. If the i th outcome is realized, the market maker pays the person A$_{ij}$x$_{j}$ . The Karush-Kuhn-Tucker (KKT) optimality condition for (3) implies the limit order logic (4) for each j . The market maker can exercise his/her discretion if the bid price is exactly equal to the market-clearing price of the order. 3.2 Equivalence with the Ambiguity-Averse Market Maker