### General Information

Student: Sheamus Larkin 407 United States Air Force Academy sheamus.larkin@gmail.com Effect of Incentives on Venue Security: A Game Theoretic Approach

### Project Description

This project explores the effects of implementing various promotional incentives in attempt to reduce the influx of patrons as they enter a stadium or venue. Parkinson's Law shows that work expands or contracts to fill the time available for its completion. Therefore when security staff visibly sees longer lines when entering a stadium or venue, they will decrease the quality of their security searches to increase the rate of customers entering the stadium. This phenomena poses a security concern which will lead to substandard security and a higher probability for missing prohibited items. This problem is modeled using a game theoretic approach with the first player being the customer and the second player being the venue.

### Weekly Log

Week 1:
Took a trip to Met-Life Stadium to collect data for service rates of security screeners. We performed various statistical tests on this collected data and fit various theoretical distributions to the service time histogram. It was utlimately determined that a variation of the erland distribution was the best fit, but this fit did not have statistical significance after performing a chi-squared test.
Week 2:
We met with Dr. Cozzens to formulate potential project ideas. We ultimately decided to take on the venue security problem using a game theoretic approach. Resultingly, we spent most of the week reading documents on game theory and how this could be applied to our model. We defined the players of the game (Venue and Customer), and the two strategies. It was ultimately determined that we were going to use a sequential game with the venue making the first move in the game, and the customer making their decision based upon the gained information.
Week 3:
An econometric analysis was performed on a dataset that we recieved to motivate why incentives get people into stadiums earlier than they otherwise would. Also, we built an initial model with over 64 branches. The decisions model the sequential game that occurs every time a promotional incentive is offered at a baseball game.
Week 4:
Started learning Lie Algebra/Representation theory to extend Dr. Coulson's thesis to an overpartition identity. Wrote an almost-proof of Jongwon's identity (hereafter J) that has a final step that needs to be checked further (by expanding many (but finite) terms and checking some base cases) but must be true. We found an analogue to J; started trying to prove it and find similar analogues of other identities.
Week 5:
Discovered that Jongwon's identity, along with several other known identities, have a remarkable property in common that I believe can easily be proven by analyzing a newer proof of one of the other identities. Once this conjecture is proven, it links Jongwon's and the other identities together in a manner that makes them all corollaries of each other (which must be unknown by the world or else J would have been discovered) and explains all the strange initial conditions. It also immediately proves the analogues of J discovered last week and leads to a whole new class of analogues of J and gives the analogues of the other theorems. Once the proof is complete, I will be less cryptic here.

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