On this episode of our podcast My Favorite Theorem, my cohost Kevin Knudson and had the pleasure of talking with Anil Venkatesh, a mathematician at Ferris State University in Big Rapids, Michigan. In addition to his work in math, he is a musician and game developer. You can listen to the episode here or at kpknudson.com, where there is also a transcript.

Dr. Venkatesh told us about a quantity called the Shapley value, developed by a game theorist named Lloyd Shapley in a 1953 paper called “A Value for n-person Games,” part of an anthology available here. Shapley won the 2012 Nobel Memorial Prize in Economics for this and other work.

The Shapley value is a way of allocating value or power to various members of a team based on their contributions. It can be used in many different situations where one might want to quantify the importance of several different factors that work together to create a whole. Dr. Venkatesh used the example of determining how powerful each member of the United Nations Security Council, which has five permanent members and ten rotating members, is. Nine votes are required for a decision, and the five permanent members have veto power. The Shapley value of each permanent member is 19.6 percent of the total power, and each non-permanent member splits the last 2 percent of the power. It can also be used to quantify the power Senators, Representatives, and the President have in the lawmaking process or even the power of individual voters in elections.

Surprisingly enough, Dr. Venkatesh came across the Shapley value and its importance not through his work as a mathematician but through Star Sonata, an online role-playing game he works for as lead content developer. He was trying to figure out a way to quantify the value of various pieces of equipment one might find in the game. If a weapon or piece of armor is too powerful, the game is less fun; everyone will gravitate towards the disproportionately powerful pieces. Dr. Venkatesh discovered that the Shapley value is precisely the way to measure the power of equipment in the game.

My Favorite Theorem is about theorems, and the theorem Dr. Venkatesh shared is that Shapley value is the only sensible way to allocate value to members of the Security Council or weapons in an online game. (Sensible means that it satisfies a few fairly intuitive rules about what such a valuation should look like: an object with no power or contribution should be valued at 0, objects that are interchangeable should have the same value, and so on.) To learn more about the Shapley value and its uniqueness, one resource is A Course in Game Theory by Martin Osborne and Ariel Rubenstein, which can be accessed (after registration) for free online. You can also check out this video from Game Theory Online.

In each episode of the podcast, we ask our guest to pair their theorem with food, beverage, art, music, or other delight in life. Dr. Venkatesh decided to pair it with the experience of going to a nice restaurant that won’t split the bill. You’ll have to listen to the episode to find out why it is the perfect accompaniment to his favorite theorem about the Shapley value.

You can find more information about the mathematicians and theorems featured in this podcast, along with other delightful mathematical treats, at kpknudson.com and here at Roots of Unity. A transcript is available here. You can subscribe to and review the podcast on iTunes and other podcast delivery systems. We love to hear from our listeners, so please drop us a line at myfavoritetheorem@gmail.com. Kevin Knudson’s handle on Twitter is @niveknosdunk, and mine is @evelynjlamb. The show itself also has a Twitter feed: @myfavethm and a Facebook page. Join us next time to learn another fascinating piece of mathematics.

Previously on My Favorite Theorem: