Arjun Pahwa Math Research Paper The Application of the Nash Equilibrium in Game Theory to Microeconomics ! One of the most challenging problems a business owner comes across is the
amount of a certain item he or she should stock and the price at which to sell it. Many factors play into ﬁnding this appropriate price. These include the cost of stocking the item, the projected demand, and what the competition is pricing the same item at. The latter of the three factors is considered to be the most challenging to consider. When attempting to tackle this problem we must consider three factors. First, we must be able to accurately predict the outcome of our decisions. Second, we must be able to accurately predict our competitionʼs decisions. Finally, we must be able to predict the outcome of our competitionʼs decisions. If all of these three requisites can be met our solution will be able to provide us the necessary information needed to price our item. ! The solution to this problem is game theory. Game theory is a branch of
mathematics that is used to predict the actions of an opponent or competitor in a certain “game.” Game theory has many applications including war, macro and microeconomics, and even biology. Game theory is a relatively new concept created by John von Neumann and Oskar Morgenstern in 1944. Over time it has evolved into a very complex ﬁeld of mathematics that has applications in many other ﬁelds. John von Neumannʼs contribution to game theory was speciﬁc to economics. He contributed the minmax theorem in 1928. This theorem stated that in certain zero sum games all players will be able to pick a strategy that will reduce the potential losses for all players. This principle was groundbreaking in all ﬁelds of economics, especially in microeconomics. Oskar
Morgenstern helped John von Neumann with his research. Other major contributors are John Nash, Reinhard Selten, and John Hersanyi. This paper will focus on the contributions of John Nash and his Nash equilibrium. The Nash equilibrium is a solutions concept of a game that assumes that all players will take into account each other playerʼs actions. Simply put, the Nash Equilibrium is a way solving a certain game where all players beneﬁt from a certain decision. ! Before learning about the Nash equilibrium we must be familiar with one other
way of solving a certain game. When using the game theory there are four fundamental questions we need to ask. 1. Who are the participants or players? 2. What are all of the choices available to each player? 3. When does each player take their “turn”? 4. How much can each player possibly earn? After assessing these fundamental questions using game theory becomes very simple. ! There are two formal ways of presenting the concept of a game: extensive and
strategic form. To represent these two forms I will be using the prisoners dilemma. Prisonerʼs Dilemma in Strategic Form ! Two prisoners, Prisoner 1 and Prisoner 2, have been arrested. They are located
in different cells and canʼt communicate with one another. While they are being interrogated they are presented the following table.
Table 1: Each colored number corresponds to the prisoner of the same color. Each number represents the number of years each prisoner will be sentenced to.
Confess No Confession Partly
Prisoner 2 Confess No Confession 2, 2 0, 5 5, 0 1/2, 1/2 3, 1 1/4, 4
Partly 1, 3 4, 1/4 1, 1
The table is read
as follows. If Prisoner 2 1 confesses, they will in prison. If Prisoner 2 Prisoner 1 does, than sentenced to no years 2 will be sentenced to
confesses and Prisoner each receive two years doesnʼt confess and Prisoner 1 will be in prison while Prisoner 5.
Prisonerʼs Dilemma in Extensive Form The following is the same dilemma shown above, but represented in extensive form: ! ! ! ! ! Prisoner 2
Prisoner 1 !
! ! When applying game theory to economics one of the ﬁrst...
Please join StudyMode to read the full document