Second, Agrawal et al. (2011) show that many important pari-mutuel markets in the literature (e.g., the Market Scoring Rule mechanism, cost-function based market makers, utility-based market makers, and the Sequential Convex Pari-mutuel Mechanism) can be understood under a common theoretical framework. The Sequential Convex Pari-mutuel Mechanism (SCPM) (Peters et al., 2007) is one of the pari-mutuel markets that Agrawal et al. (2011) analyze. In addition, the CPCAM and the SCPM are very closely related to one another. The only major difference is that the CPCAM is a call auction and the SCPM is a continuous market. Therefore, the CPCAM and other important pari-mutuel markets are closely related to one another. Given this close relationship between different market designs, our analysis of the CPCAM may also apply to other pari-mutuel markets. However, we leave that extension to future work. 

## 4 The Knightian Pari-mutuel Mechanism (KPM) 

In this section, we design a new market called the Knightian Pari-mutuel Mechanism (KPM). 

## 4.1 The Market Setting 

The limit order logic and the basic trading environment are similar to those of the PDCA (Lange and Economide, 2005). However, the algorithm through which the market maker clears the market is original. In particular, the probabilistic treatment of the market maker’s optimization problem is original. 

Like the CPCAM, the KPM allows the common market maker to handle multiple types of contingent claims written on the same random event. There are N possible states of the uncertain event, each of which is indexed by i ∈ { 1 , 2 , ..., N } . 

The KPM allows traders to submit both market orders and limit orders. Traders can submit market orders just as if they were submitting limit orders simply by making the limit price extremely high or low. Therefore, throughout the rest of the paper, we assume that people trade only limit orders. When submitting each limit order, the trader indicates the limit price, the limit quantity, and if the order is a buy or a sell. 

For the sake of simplicity, we describe the setting in which the market is run as a call auction. However, the setting can be easily adjusted to accommodate continuous trading in the same manner as the CPCAM (Peters et al., 2005) is changed to the SCPM (Peters et al., 2007). 

Suppose there is a total of J limit orders outstanding in the limit order book. Let the matrix 

