Discussed possible topics with my mentor. Several ideas involved sustainability, and we settled on what that dealt with the forests. The goal for the duration of the REU will be to develop a model for forest sustainability using game theory. Game theory is something new to me so I have spent time reading about the concepts and will continue to do so for the next few weeks. I now understand the basic terminology that goes with game theory as well as the simplest game: the two-person, zero-sum game with equilibrium points.

This week was filled with research. When it comes to the game theory aspect of my work, I worked on the general, two-person, zero-sum game as well as utility theory. The former included the von Neumann's Minimax Theorum, as well as strategies for solving mixed strategy games. Utility theory deals with a utility function, which is simply a "quantification" of a person's likes and dislikes with reguards to the outcome of the game. On the topic of research, I left the week off looking into biomass as a possible direction for my project. Biomass is critical for forest sustainability, since it poses a fire hazard when there is too much, but removing it can hurt biodiversity. The issues associated with biomass seem to be removing it, which is something interesting to look into. However, at this point in time there are no scientific ways of accuratelly measuring biomass. I do not want to work on something like this.

The week started out by reading and understanding the two-person, non-zero-sum game. This game, while more realistic than those previously studied, is complicated and depends heavily on the individual players wants and desires. I also began to develop my own model for forest sustainability based on the game-theoretical model developed for fisheries. While the setup and approach is the same, the details are what need to be looked into. This was done by researching the outcome of both cooperative and non-cooperative games in the field of fishery management and attempting to put similar implications into the forestry aspect. One idea I discovered was based on a biological model created by M.B. Schaefer. By utilizing this, and possibly Pontryagin's Maximum Theorum, a result for the maximum payoff for the players can be reached...if I could just figure out how the theorem works.

This week started out by developing a payoff matrix for my problem. Here, the players were loggers and conservationists. The goal of the loggers is to make money, while the conservationists want to not only prevent the loggers from working, but planting new forests to grow as well. I have made 3 simple assumptions to this model: 1) If the loggers do nothing, they get no payoff. 2) Conservationists get nothing if they do nothing while the loggers work. 3) If both do nothing, the conservationists will win since the forests natural growth rate will benefit them. By creating "utils" I was able to construct the following payoff matrix with loggers representing the rows and conservationists the columns:

Payoff Matrix | Work | Do Nothing |
---|---|---|

Work | (80,30) | (100,0) |

Do Nothing | (0,50) | (0,20) |

Good week this week. I was finally able to develop the equations for all three players. Imputting these into the payoff matrices was next, and has given me some useful insight into the development of the model. Some kinks still need to be worked out, though. This week really just dealt with manipulating equations, so not much else to discuss.

This week started off by altering the model I developed. Up to this point, much of the power was in the hands of the Conservationists and Loggers. By manipulating the model, I have given more power to the Land Owners. This slight alteration also led to some interesting results. Most notibly, an equilibrium point. I also noticed an error in one of my equations. This was easily corrected, however, and the model is now working smoothly. I am happy with the results I've obtained and my mentor has expressed similar satisfaction.

The final week here consisted of creating the final presentation and beginning the writeup of the final paper.