Math for Business & Business for Math

Summary: Many business puzzles have pure Mathematics behind their solutions, most are embedded in Optimality. To find out how pricing, transportation costs, resource allocation and designing of best auctions are to be done, take refuge in Mathematics.

You are a producer of a range of differentiated products and you are toying with optimal allocation of production resources and prices; Mathematicians have solutions that are optimal, in this case the solution would be to have one high quality product and one low quality, where highly satisfied consumers would be willing to pay much higher than the marginal cost of production, when more consumers are roped into the network the prices can be raised to any level that has no connection with the cost of production. Willingness to pay for quality and that it is good for business is a very fundamental discovery of our current times, this may not have been in practice in business and it came from a Mathematical proof.

Nima Haghpanah & Jason Hartline writes in this seminal Paper, “We prove the optimality of selling only the high-quality product when the base value and value ratio are positively correlated”, here value ratio is the value of low quality divided by the value of high quality. But this is a partial solution as the problem compounds when competitors move in with their strategies and select to operate in both these parts of the product offering, choosing the highest quality segments by the differentiators and the lowest quality by the cost leaders, makes the problem look far more murky.

As higher valued consumers are more sensitive to quality than price and lower valued consumers are more sensitive to price than quality, the solution to this problem is to operate at the Productivity Frontier as propounded by none other than the Economist Michael Porter, although he did not give a Mathematical proof of the optimality.

Or take the Transportation problem where you have to move products from several producing centers across the country to several distribution centers and you want to minimize cost of transportation. If there are A number of producing centers and B number of distribution centers, the problem becomes very difficult and took two centuries to solve. Originally coined by Gasper Monge, the 18th Century French Mathematician, it took two hundred years to be solved by Leonid Kantorovich, the Economist who got the Nobel in 1975.

The Optimal Transport problem has wide spread usage, think of the Gas fields and the distribution centers of gas, or take the more recent Uber taxi service, where you have multiple demands on multiple taxis and how do you minimize the transportation cost by allocating the right driver to the right customer.

This has a revenue maximization connotation as well and we come to pricing problems encountered by businesses.

The very common problem of business is competitive pricing, how does a group of producers find out the right price of its produce given that each has less knowledge of what value it is likely to be assigned by different class of consumers who have more than one option either in terms of products or in terms of producer preference. This is a very difficult problem and only partial solution could be attempted by considering the monopolistic situation of one producer with many buyers.

This problem was first solved by Myerson and his seminal paper written in 1981 stands out as a path breaking initiative in Mathematics, called the “Optimal Auction Design“. Myerson follows Vickery in the design of the auctions, but he brings in the concept of non-cooperative equilibrium under imperfect information and shows that it eventually destroys surplus.

To describe this eventual destruction of surplus let me take the easiest example of a waste generated by a producer which could be of value to buyers, who could use them for alternate usage as inputs to their process. The actual perceived value of this waste to the individual buyer can never be gauged correctly by the producer and he must discover the right price through an auction where he would set a reserve price which should be decided such that none of the buyers would be willing to walk away, so it must be lower than the walk away price. Then through a combination of auctions, the right trigger would be set in for discovering the right price at which a portion of the waste can be taken away by an individual buyer. Through continuation of the auction for the remaining quantity on offer eventually the limit of the auction value could be discovered.

This is the most simple form of price discovery through auctions but it does not capture the full dynamics as we have repetitive auctions, therefore what one side gained in one auction could well be reversed in the next round. This is exactly the point that surplus created could be eventually destroyed in the subsequent rounds.

We have many unsolved problems of pricing and discovery through auctions, as the optimal solutions as shown by Myerson are only partial solutions as the actual dynamics in business are much more multi-faceted.

The Myerson paper ended with a doubt whether buyer actions are independent, in industrial and business applications; in fact they are not. For example in the auction for waste, each buyer has a different utility function and therefore have different value proposition given that the product mix could be different with different proportion of inputs used which also has the particular waste as a variable input.

This leaves us with an unsolved problem that when buyers can shift around their mix depending on the outcome of the auction, how would a repetitive auction design maximize revenue for the seller? A high reserve price in the first round could be detrimental to such a shift in the mix that in the second round there will be lukewarm response to the auction. No wonder these auctions take the surplus out with information asymmetry propagating.

When Mathematicians receive their Field Medals, as recently announced, we must cheer as they keep solving the unsolved problems of business only.

When Apple valuation exceeds $1 Trillion, we must acknowledge that somewhere some obscure Mathematician and his optimality equations were part of this journey of excellence.