8.6 The Proof of Theorem 3 Note from our pseudo-code that the number of times we need to solve the problem (27) is Π K k $_{=1}$n$_{k}$ In addition, it is well known that, in principle, the interior-point method can solve any convex optimization in polynomial time of the problem dimension. Thus, (27) should also be solvable in polynomial time. Let τ ( ℓ$_{1}$, ℓ$_{2}$, ..., ℓ$_{K}$ ) denote the time required to solve problem (27). Let T denote the time required to execute the entire pseudo-code. max ℓ$_{1}$,ℓ$_{2}$,...,ℓ$_{K}$ τ ( ℓ$_{1}$, ℓ$_{2}$, ...ℓ$_{K}$ ) is bounded above by a polynomial function of J . Thus, T is also bounded above by a polynomial of J . Please see Baron and Lange (2007) or Lange and Economide (2005) for a more thorough discussion. In this subsection, we briefly introduce some of the strengths of pari-mutuel auctions and our insights. The market maker can reduce his/her inventory holding cost by being involved in more than one market. This lower inventory holding cost allows the market maker to supply liquidity to each market at lower cost. To illustrate, consider an exotic derivative market with the Consumer Price Index (CPI) as the underlying variable. Suppose that there are two types of options: a call option with the strike 0% and a put option with the same strike. For example, if the CPI is 1%, the call option pays $1, while the put option does not pay. Imagine that there is an overwhelming demand for both options. First, consider the case in which two markets are fragmented. There is one dealer for each market. Overwhelming demand for each option forces the market maker to take a large short position. The inventory of each market maker becomes highly unbalanced, exposing him/her to 50 8.7.1 Liquidity Aggregation 8.7 Other Well-Known Strengths of the Pari-mutuel Auction Π K k $_{=1}$n$_{k}$ ≤ Π K k $_{=1}$J = J K (87) T = n$_{1}$ ∑ ℓ$_{1}$ =1 n$_{2}$ ∑ ℓ$_{2}$ =1 ... n$_{K}$ ∑ ℓ$_{K}$ =1 τ ( ℓ$_{1}$, ℓ$_{2}$, ...ℓ$_{K}$ ) ≤ J K max ℓ$_{1}$,ℓ$_{2}$,...,ℓ$_{K}$ τ ( ℓ$_{1}$, ℓ$_{2}$, ...ℓ$_{K}$ )