The trading community is buzzing with the rumor that the upcoming Star Wars movie will offer a glimpse of the capital markets that finance the Galactic Empire’s expansion. One scene reportedly takes place on the floor of the Imperial Stock Exchange (ISE), an oceanless planet whose entire surface has been covered with sturdy blue carpeting and fluorescent lighting. Scattered randomly on the surface of this planet are outposts of the galaxy’s various securities firms which function as dealers; there are no specialists on the ISE. Due to the complete lack of natural topography, any location on the planet is equally likely to be the site of a dealer’s post. Dealers trade with each other by dispatching order-carrying robots. Trading electronically would be much more efficient, but that would leave the Hollywood special effects wizards with little to depict.

Consider the post of Dealer A, who is sending out his robots in random directions in search of counterparties. Because the other dealers’ posts are small (for the purposes of this problem you may assume they are geometric points), the robots risk circumnavigating the planet many times before blundering into one. Accordingly, they are programmed with the following rule - the post closest to the robot must at all times be either Dealer A (its origin) or the post at which it will arrive if it continues without turning, which we will call Dealer B. If the robot detects a third post (Dealer C) which is closer than either of these - call this event a close encounter - it returns home to Dealer A, registers Dealer B as an unacceptable destination to prevent other robots from wasting their time on it, and sets off for Dealer C. The effect of this rule is that one dealer will trade with another if and only if a robot traveling directly between them would never find itself closer to a third dealer than to the closer of the two.

Each dealer trades with several other dealers, the exact number of which depends on how the posts happen to be arranged in his neighborhood. If the posts are randomly and independently situated, what is the average number of counterparties with whom each dealer trades? You may assume that the average distance between dealers is much smaller than the size of the planet, so that the curvature of the planet may be neglected. In other words, a flat map is an adequate representation of any part of the globe.

Bonus Questions

1. Imagine that Dealer A finds he has too few counterparties to effectively work his trades. He reprograms his robots so that they ignore the first close encounter but turn back upon the second. What happens to his expected number of counterparties? What if he allows his robots to ignore two close encounters?

2. If trading took place in a three-dimensional space rather than on a two-dimensional planetary surface, what would happen to the expected number of counterparties?