The market maker has two decision variables for his/her optimal clearing of the market: the equilibrium state prices ξ and the order fill vector x . Like other market makers in the financial markets, the market maker of the KPM also has an inventory of contingent claims. If the i th state is realized in the future, the inventory subjects the market maker to the monetary payoff of w$_{i}$ . Let α denote the market maker’s risk aversion coefficient. Suppose that the constant absolute risk aversion (CARA) utility function u ( x ) = − e − αx characterizes the market maker’s risk appetite. The market maker has Knightian ambiguity toward the random future event on which the claims are written. Let the set Ψ define the set of probability distributions that the market maker considers. q ∈ R N × 1 is the market maker’s pivot probability distribution. Assume that every element in q is strictly positive. Any probability distribution for which the Kullback-Leibler from q is no greater than Ω is acceptable for the market maker. Ω quantifies the market maker’s level of ambiguity aversion. A large value of Ω implies that the market maker has strong ambiguity aversion. The i th elements of p and q describe the market maker’s probabilistic belief about the i th outcome. We assume that the market maker adheres to the standard decision-making theory under Knightian ambiguity aversion. The market maker’s optimization problem can be framed as (8). Unlike the CPCAM, the KPM asks the market participants to pay the market-clearing prices of the claims instead of the bid prices they submitted. This optimization problem does not make any arbitrary assumptions. The problem is a corollary of the standard theory of decision making under ambiguity aversion. However, the constraints (E1) - (E3) and the objective function causes the problem to be non-convex. Finding a global optimal solution to a non-convex optimization problem is extremely difficult. 12 4.2 The Market-Clearing Optimization Problem Ψ = { p ∈ R N × $^{1}$| p ≥ 0 , N ∑ i =1 p$_{i}$ = 1 , N ∑ i =1 p$_{i}$ ln ( p$_{i}$ q$_{i}$ ) ≤ Ω } (7)