value of the objective function of (35), which is (37), should be at least as large as 1. (38) is equivalent to (39). ∑ J j $_{=1}$x ∗ j ( ( A $^{T}$ξ $^{∗}$) j − A$_{ij}$ ) is the monetary payoff for the market maker if the i th outcome is realized. Therefore, min i ∑ J j $_{=1}$x ∗ j ( A $^{T}$ξ $^{∗}$) j − A$_{ij}$ ) is the worst possible monetary payoff that the market maker can ever receive. Inequality (39) shows that the market maker never loses money even in that worst-case scenario. The market is completely pari-mutuel. (40) is the inner minimization problem isolated from (8). The optimal value of this inner optimiza- tion problem is an implicit function of ξ and x . Because the objective function is linear in p and Ψ is a convex set, this problem is a convex optimization problem. x and ξ should be treated like constants when solving (40). To make notations simpler, we introduce new constants. d$_{i}$ = z$_{i}$ exp [ − α ∑ J j $_{=1}$x$_{j}$ ( A $^{T}$ξ ) j − A$_{ij}$ ) ] for ∀ i ∈ { 1 , .., N } (41) d = ( d$_{1}$ d$_{2}$ ... d$_{N}$ ) T (42) min p ∈ Ψ ∑ N i $_{=1}$p$_{i}$ z$_{i}$ exp [ − 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 } (40) min i ∑ J j $_{=1}$x ∗ j ( A $^{T}$ξ $^{∗}$) j − A$_{ij}$ ) ≥ 0 (39) min i − exp [ − α ∑ J j $_{=1}$x ∗ j ( A $^{T}$ξ $^{∗}$) j − A$_{ij}$ ) ] ≥ 1 (38) 8.4 The Proof of Lemma 1 8.4.1 The Dual Problem of the Inner Minimization Problem 34