Awhile back I had a post on triangulation, and although I had received a number of questions on how this might be depicted, I never got around to showing it. The attached picture below is pretty good and is motivated by an engagement involving wireless spectrum valuation. The basic idea is that we used three methods to get the range of potential dollar values for each wireless spectrum property. The chart below is the summary chart that ties the results of separate analyses together. The first method of valuation on the left (historic comparables) involved looking at past government auctions for similar properties and coming up with high, low, and in-between values. The second method of valuation (economic model) used a statistical regression model trained on a priori auction conditions (e.g., number of bidders, amount of money ante) and ex-post results (e.g., property values), and here we basically used the model to predict what might be the outcome of a new, unseen auction. The third model (financial metrics) basically took a look at recent market transactions (e.g., outright asset sales) or values imputed by the stock market value of wireless-heavy companies (e.g., by taking enterprise value and normalizing by spectrum holding amount).
 

When one plots these three different valuation methods on the same chart, it becomes possible to "triangulate" and see what a competitive values for the wireless properties might be. We basically added two horizontal lines to depict the expected minimum amount of money to pay to be in the game and also the maximum amount of money that should be spent (e.g., based on balance sheet of company and financial capacity).