## Senior Independent Study Theses

#### Abstract

Game theory is a mathematical theory that studies situations of conflict between two or more players. These situations are analyzed as games where the players, that are assumed to be rational, always make decisions, also known as strategies, in order to obtain the outcome from which they will benefit the most, or that will provide them with the highest payoff possible. We say that a player behaves rationally if he is fully aware of all the possible choices that can be made and he deliberately chooses an action based on the process of optimization. The games can be thought of as problems, and we are often looking to find an equilibrium within the game. An equilibrium is an optimal outcome of a game where no player has an incentive to deviate from his chosen strategy after considering an opponent’s choice. However, it is not necessarily the best available outcome to the players as I will show in the following sections. This equilibrium is known as the Nash equilibrium in honor of the mathematician and Nobel Prize winner John Nash.

The applications of game theory into real-life situations are vast. Most real-life situations are complicated since many factors can be found that would affect the strategies chosen or the final outcomes. Game theory models these situations in a simple way, however complicated these real-life scenarios may be. In recent years, game theory has been used with increased frequency to model international relations and disputes between countries as situations of conflict where the players, usually countries, big companies or governments, can choose among preemption or deterrence, to impose sanctions or not to impose sanctions, to go to war or not to go to war, etc. Then, these players make decisions, sometimes sequentially (combinatorial game theory) and sometimes simultaneously (classical game theory) to maximize their payoffs. It is important to notice that one player’s payoff is contingent on the strategy implemented by the other player. Each player must take into account the probability of each outcome and use these probabilities to make the best decisions and try to maximize their own outcome or payoff. Mathematical techniques allow us to carry out these computations and calculate these values.

Game theory can provide good models for real-life situations of conflict and draw accurate conclusions from them. When studying game theory, we find that what is best for the collection of players (everyone) often conflicts to what appears to be best for an individual. As John Nash pointed out, this can be extended to the understanding that what is best for society is usually in conflict with what an individual thinks is best for him. Thus, we see that there is a fundamental conflict between optimizing a situation for an individual and optimizing the situation for a group. This conflict is at the heart of the study of game theory.

Pierce, Pamela

Mathematics

#### Disciplines

Other Mathematics

#### Keywords

Math, Game Theory

2021

Bachelor of Arts

#### Document Type

Senior Independent Study Thesis

COinS