Operations Research (OR) is often referred to as “the science of decision making.” Applications of OR span all domains, from scheduling Major League Baseball games to planning the modernization of airline fleets; from designing the optimal configuration of lines at Disney World attractions to assigning nurses to patients in emergency rooms; from streamlining factory floor operations to optimizing large-scale transportation logistics.
One unit of MATH 358: Topics in Operations Research is devoted to studying Markov Chain models.
Markov chains allow us to study and predict the long-term behavior of systems that evolve over time in uncertain ways. Popular applications include queueing systems, inventory management, and animal migration patterns. Perhaps the most famous Markov chain model is Google’s PageRank algorithm, which is used to help determine which web pages appear at the top of a Google search.
Many board games, including Monopoly, Chutes and Ladders, and Hi-Ho Cherry-O, can be analyzed using Markov chain models. MATH 358 students, taught by Prof. Lucas Waddell this semester, first warmed up by reviewing the math behind the game of Craps (also a Markov chain). They rolled some dice and kept track of their wins and losses.
Top left: Prof. Waddell in front of the class
Top middle: Bret Cozby (’28, Mathematical Economics major) and Olivia Seward (’26, Mathematical Economics major)
Top right: Michael Amante (’27, Mathematical Economics major and Italian Studies major), Will Lindquist (’27, Statistics major), and Kerry O’Day (’28, Mathematical Economics major)
Bottom first row: Skyler Le (’27, Mathematics major), Roger Chen (’28, Applied Mathematics major), Cornelius Nichilo (’27, Mathematical Economics major), Zach Selfon (’27, Mathematical Economics major)
Bottom second row: Samuel Berry (’27, Applied Mathematics major), Reed Nester (’27, Mathematical Economics major)
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To get some more hands-on experience with Markov chain modeling, the students went on to investigate the classic game, Cootie. In the game Cootie, players race to construct a Cootie bug, which consists of a head, body, one eye piece, one tongue/teeth/lips, one antennae/hat/bow, and six legs. The order in which pieces get added to the partially completed Cootie bug is randomly determined by the roll of a die.
Top right: Skyler Le (’27, Mathematics major), Kerry O’Day (’28, Mathematical Economics major)
Bottom left: Bret Cozby (’28, Mathematical Economics major), Roger Chen (’28, Applied Mathematics major), John Rich (’27, Computer Science and Engineering Major)
Bottom right: John Occhipinti (’27, Mathematical Economics Major), Samuel Berry (’27, Applied Mathematics major), Zach Selfon (’27, Mathematical Economics major)
Students were tasked to play a game of Cootie, and then use their intuition about probability to guess the expected number of turns it would take to somebody to completely construct the Cootie bug.
After guessing, students created a Markov chain model of the game. This required them to determine that there are 30 unique states that the Cootie bug can be in on its journey from the beginning of the game to the end, and to calculate transition probabilities between these states. For example, if the Cootie bug currently has just a body and a head, there is a 1/6 probability that after the next turn, the Cootie bug will have a body, a head, and one leg.
The students were then able to calculate that the expected number of turns in a game of Cootie is (spoiler warning) 48.2694 turns.
A homework assignment asked them to think about how the model would change if the game was played with a weighted die. Another spoiler warning… it changes significantly.
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