Projects
Research: Stability of dining clubs in the Kolkata Paise Restaurant Problem with and without cheating
Abstract
We introduce the idea of a dining club to the Kolkata Paise Restaurant Problem. In this problem,
agents choose (randomly) among restaurants, but if multiple agents choose the same restaurant, only one will eat. Agents in the dining club will coordinate their restaurant choice to avoid choice collision and increase their probability of eating. We model the problem of deciding whether to join the dining club as an evolutionary game and show that the strategy of joining the dining club is evolutionarily stable. We then introduce an optimized member tax to those individuals in the dining club, which is used to provide a safety net for those group members who do not eat because of collision with a non-dining club member. When non-dining club members are allowed to cheat and share communal food within the dining club, we show that a new unstable fixed point emerges in the dynamics. A bifurcation analysis is performed in this case. To conclude our theoretical study, we then introduce evolutionary dynamics for the cheater population and study these dynamics. Numerical experiments illustrate the behaviour of the system with more than one dining club and show several potential areas for future research.
Conclusions and future directions
In this paper, we studied the Kolkata Paise Restaurant Problem (KPRP) with dining clubs. Agents in a dining club mutually agree to visit separate restaurants, thereby increasing the probability that they eat (obtain a resource). An evolutionary game model was formulated describing the choice to join the dining club. We showed that joining the dining club is an evolutionarily stable strategy, even when members are taxed (in food) and resources are distributed. When cheating was introduced to the non-dining club members, i.e. the non-dining club members could deceptively benefit from the communal food collected by the dining club, a new unstable fixed point appears. We analysed this bifurcation as well as the decision to cheat using the resulting replicator dynamic. Numeric experiments on two dining clubs show that the behaviour in this case is similar to the case with one dining club, but may exhibit richer dynamics.
There are several directions for future research. Studying the theoretical properties of two (or more) dining clubs is clearly of interest. Adding many groups (i.e., so that the number of groups is a proportion of the number of players) might lead to unexpected phenomena. Also, allowing groups to compete for membership (by varying tax rates) might create interesting dynamics. As part of this research, investigation of the dynamics on the boundary both in theory and through numeric simulation would be of interest. Exploring multiple dining clubs with a spatial component might also lead to interesting results. If we introduce a diffusion component to the evolutionary dynamic with multiple groups, we should expect to see travelling wave solutions spreading out from the group origin points, as in the Fisher–KPP equation [31], [32], [33]. However, how these waves interact when multiple equivalent clubs meet is unclear. In two dimensions, it may lead to fractal geometry, like that seen in compact Apollonian packing. In this case, characterizing the resulting spatial structures would be of great interest. A final area of future research would be to investigate the effect of taxing cheaters who are caught, thus allowing them to eat, but discouraging them from cheating. Determining the impact on the basins of attraction in this case would be the primary research objective.
Capstone Project: Modular Hybrid Power System for UAVs
Overview
Lockheed Martin handed off the design of a previous semester’s capstone project, which was created to implement a power system for a UAV drone using a generator and battery. The team was tasked with modifying the previous design to implement a hybrid controllable load-sharing system in which a propeller is powered by both a generator and a battery simultaneously.
Objectives
– Study the previous semester’s design to gain a full understanding of the power generation system.
– Modify the previous semester’s design to support load sharing between the generator and the battery.
– Provide real-time data to the user through an easy-to-read graphical user interface (GUI). Data output should include current and voltage output from each source.
– Run multiple test cases with the hybrid system and provide data to ensure that power output between the generator and the battery is shared and can be controlled.
Approach
– Simulate a power-sharing system with two parallel diodes in Multisim. This is done to prove the design’s feasibility before physical implementation.
– Connect both the generator and battery to the fan with two parallel diodes. Control the load sharing by adjusting the generator’s voltage output. This is done by utilizing an electronic speed controller connected to the generator and an Arduino. Using this, the user can control which diode is providing power by increasing and decreasing the generator’s voltage.
– Control the power supplied by the generator and the propeller’s power demand through two potentiometers.
– Modify the previous semester’s GUI code to provide data to the user without internet access, lower latency, and updated data readings.
Outcomes
– The hybrid system successfully provides power to the propeller through both a generator and a battery. The power split shared between the two sources is easily controlled via a potentiometer.
– Testing data shows that the battery will respond rapidly to a high-power demand followed by the generator taking over.
– Real-time data such as generator/propeller speed, generator/battery current, and voltage are provided to the user through a GUI.
