content_image/296527.jpg
<img src='content_image/296527.jpg'>

Proof. See the Appendix. 

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. 

## 4.3 The Market-Clearing Algorithm 

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). 

