max ξ , x min p ∈ $_{Ψ}$− ∑ N i =1 p$_{i}$ exp [ − αw$_{i}$ − α ∑ J j $_{=1}$x$_{j}$ ( ( A $^{T}$ξ ) j − A$_{ij}$ )] such that (A) Ψ = { p ∈ R N × $^{1}$| p ≥ 0 , ∑ N i $_{=1}$p$_{i}$ = 1 , ∑ N i =1 p$_{i}$ ln ( p$_{i}$ q$_{i}$ ) ≤ Ω } (B) ξ ≥ 0 (C) ∑ N i $_{=1}$ξ$_{i}$ = 1 (E1) ∀ j ∈ { 1 , 2 , .., J } , x$_{j}$ = 0 if ( A $^{T}$ξ ) j > B$_{j}$b$_{j}$ (E2) ∀ j ∈ { 1 , 2 , .., J } , x$_{j}$ ∈ [0 , Q$_{j}$ ] if ( A $^{T}$ξ ) j = B$_{j}$b$_{j}$ (E3) ∀ j ∈ { 1 , 2 , .., J } , x$_{j}$ = Q$_{j}$ if ( A $^{T}$ξ ) j < B$_{j}$b$_{j}$ (8) The KPM may not be completely pari-mutuel in the sense that the market maker can lose money with positive probability. However, Corollary 1 shows that the KPM subsumes a completely pari- mutuel market. By adjusting the value of Ω, the market designer can fine-tune the extent to which the market is close to being completely pari-mutuel. The larger the value of Ω, the more completely pari-mutuel the market becomes. For example, consider increasing the value of Ω. Problem (8) then models the auctioneer with a large level of ambiguity aversion. The ambiguity-averse DM is very sensitive to the worst-case scenario. Thus, the auctioneer clears the market such that he/she performs moderately even in the worst-case scenario. In other words, the auctioneer does not want to lose too much money even in the worst-case scenario. 5 In the extreme case in which Ω diverges to infinity, the auctioneer becomes so conservative that he/she does not want to lose any money under any circumstances. The market should become completely pari-mutuel. Before further discussion, we introduce new notations: z$_{i}$ = − e − αw$_{i}$ and θ$_{i}$ = q$_{i}$e Ω for each i ∈ { 1 , 2 , ..., N } . In addition, let F be the set of pairs ( ξ , x ) that satisfy the limit order logic constraints (E1), (E2), and (E3). 13 $^{5}$The cost of this strategy is that the market maker may not be able to make a great deal of money on the upside. 4.3 The Market-Clearing Algorithm Proof. See the Appendix.