content_image/296881.jpg content_image/296880.jpg content_image/296879.jpg content_image/296877.jpg content_image/296878.jpg
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. 

## 8.4 The Proof of Lemma 1 

## 8.4.1 The Dual Problem of the Inner Minimization Problem 

<img src='content_image/296881.jpg'>

<img src='content_image/296880.jpg'>

(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. 

<img src='content_image/296879.jpg'>

x and ξ should be treated like constants when solving (40). To make notations simpler, we introduce new constants. 

<img src='content_image/296877.jpg'>

<img src='content_image/296878.jpg'>

