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in the fourth column implies that he/she is willing to pay at most 0.18 dollars per share. The payoff matrix in the last five columns shows how the person will be paid by the market maker. For example, the market maker will pay $ 1 to the person who submitted the first order if and only if the first outcome is realized. The person who submitted the fourth order will receive $ 1 if and only if the fourth outcome is realized. 

Based on this limit order book, we can determine the number of distinct limit prices associated with each Arrow-Debreu security. For example, for the first Arrow-Debreu security, there are three distinct limit prices: 0.18, 0.20 and 0.25. Thus, n$_{1}$ should be 3. Likewise, n$_{2}$ = 1 , n$_{3}$ = 1 , n$_{4}$ = 1 , and n$_{5}$ = 0 . 

Next, we sort the limit prices in ascending order. For example, for the first Arrow-Debreu security, we have B 1 = 0 . 18 , B 2 1 = 0 . 20 , and B 3 1 = 0 . 25 . In addition, B 1 2 = 0 . 18 , B 1 3 = 0 . 18 , and B 1 4 = 0 . 18 . Because there is no limit order associated with the fifth Arrow-Debreu security, B 1 5 is undefined. ■ 

Suppose that there are n$_{k}$ distinct limit prices. Let E denote the N -dimensional space defined as (12). We define n$_{k}$ + 1 convex subsets of E such that if ξ is restricted to one of those subsets, (8) becomes a convex optimization problem. The intuition is as follows. The limit order logic constraints (E1) - (E3) are non-convex because we do not know which of the three conditions - ( A $^{T}$ξ ) j > b$_{j}$ or ( A $^{T}$ξ ) j = b$_{j}$ or ( A $^{T}$ξ ) j < b$_{j}$ - hold at an optimal solution. We define subsets to ensure that such ambiguitiy is resolved within each set. As a result, the limit order logic constraints can be replaced by x$_{j}$ = 0 or x$_{j}$ ∈ [0 ,Q$_{j}$ ] or x$_{j}$ = Q$_{j}$ . 

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