This was an exciting and mind-opening research project, and I really learned a lot from it. While difficult to choose what to say, and more importantly how to say it, I found the web site gametheory.net, which had a ton of information and where I got most of my information. I will be using quite a lot of passages from this site however, as it is worded better then I ever could. These passages will be following by parenthetical references that site exactly where I got them. I don’t know if this is necessary for a math research paper, but it is how the AP style book requires it, so that is how I will be doing it.

**Game Theory:**

Game theory is an awesome yet distinctively hard to explain method of mathematics that has been used for thousands of years but has only recently come to our attention as a great type of math to use when approaching decisions that are otherwise impossible to reach. The actual first well-know and publicized book was written by John Von Neumann, who was a great mathematician, and known for the founding of the theory, named The Theory of Games and Economic Behavior. He wrote this in collaboration with another great mathematician who was also an economist by the now well-known name of Oskar Morgenstern in 1944.

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How to describe game theory? It is very hard to describe it, so the following is the best explanation I can give. Game theory is a distinct and interdisciplinary approach to the study of human behavior. The disciplines most involved in game theory are mathematics, economics and the other social and behavioral sciences. (Strategy and Conflict: An Introductory Sketch of Game Theory. Taken on March 13th, 2003.

Now the theory of games in no one theory at all, but many, now in game theory, like in regular games, there are players, always more then one, and these players must make decisions that will affect the outcome, whether it is a reward or punishment. With just two people involved in the game, it is rather simple to predict the outcome, the more players get included, the more complex the game grows, and it becomes almost impossible to give answers to the questions that the game is imposing. These questions generally are, how should the players behave? And what should the ultimate outcome of the game? To answer these questions of course, is the solution of the game. So there we have the basic outline of what game theorists have to play around with.

Game theory is not only related to war and games, it is also linked largely to economics. This shouldn’t be so surprising since one of the original theorists mentioned above was one of the leading economists of his time. People ask how math is related to our economy. This is tricky, as there are several problems that make it difficult to put many economic problems in game farm. Firstly, it is hard to specify precisely the strategy that is supposed to be made available to the players, and the resolutions of the final problem is obviously debatable, as what one player might consider good and well, the other player or players might consider unfair. Better put "these are among the "circumstances" that the person takes into account in maximizing rewards. The implication of property rights, a money economy and ideally competitive markets is that the individual needs not consider her or his interactions with other individuals. She or he needs consider only his or her own situation and the "conditions of the market." But this leads to two problems. First, it limits the range of the theory. Where-ever competition is restricted (but there is no monopoly), or property rights are not fully defined, consensus neoclassical economic theory is inapplicable, and neoclassical economics has never produced a generally accepted extension of the theory to cover these cases. Decisions taken outside the money economy were also problematic for neoclassical economics." (Strategy and Conflict: An Introductory Sketch of Game Theory. Taken on March 13th, 2003.

Now an interesting fact: we all know that logic is the foundation of mathematics, and game theory has been proved useful in highly technical areas, of which logic is one of the chief ones. Another technical area to which game theory can be applied is in differential games, which have applications in control and optimization. This includes games of pursuit and evasion, warfare, and can even spread as far as profit maximization. A lot of military leaders and historical figures have applied this. In my presentation I will be including such examples as when Cortez landed in Mexico, he only had a small force with him, but he burned his ships after he landed. This was used for multiple reasons, one, his soldiers now knew that they had no way of going home, so they had better win the fight or they would all die, 2, the Aztecs saw the ships being burnt so they figured that the Spaniards had some kind of reason and must be invincible, so they fled to the hills and Cortez took over without a fight. Another example is in the Novel Henry V, the Count of Agincourt kills the French prisoners in full view of the enemy. This allows his men to see that the enemy sees that their men have been killed therefore if they do not win the battle and are taken prisoner they will most likely be killed as well, this makes them want to fight better and win the war. This also allows the enemy to know that their opponents know that if they are taken prisoner they will be killed, so they know that they are going to fight even harder, therefore their resolve might weaken. At the same time however, they are enraged that their fellow countrymen have been killed so they will fight harder as well to avenge the deaths of their friends. And this cycle goes round and round. It is a classic example of the mirror within the mirror; it goes on and on and on.

While doing a search on game theory I came across an interesting article in Fortune magazine, in it John McDonald described the results of a one and a half year study of the way top executives in large companies make decisions. In his report he described game theory as "uniquely qualified to make sense of the forces of work and how it related to the strategies of some actual corporations caught up in conglomerate warfare". He went on to talk about airline competition, plant location, and product diversification as fertile area for the use of game theory.

Now while there are many different applications of game theory, such as: one-person games, two person games, the utility theory, the n-person game etc. What most people deal with, and what we have been talking about so far and will continue to talk about is the Two-Person, Zero-Sum Games of Perfect Information, or in layman’s terms, a game that involves two people, parties, or corporations etc.

When dealing with two or more parties, the main element that makes it a game is the amount of information available to the players, and these complicate matters considerably. The final outcome of a game depends on the actions of both players, while trying to think in advance of what your opponent will do in case of your move. "If I do this, will my opponent do this or that, or if I do that move will my opponent do that or this", these are the classic questions that come up when applying this theory. For example the stone-paper-scissors game, in this game the players much choose their strategies simultaneously, neither knowing what the other is going to do. Now if you know what your opponent is going to do, then the answer is simple, but that would be a case of a one person game. In this game one would try to create a theory of what his opponent will do based on what he has done previously, and what he is going to do based upon the moves of his opponent. This information is of course kept in his head, as well as the possible outcomes, and the circle once again goes round and round, and we are back to the mirror in the mirror.

On this topic, in Von Neumann and Morgenstern’s book, they put the problem this way: "Let us imagine that there exists a complete theory of the zero-sum two-person game which tells a player what to do and which is absolutely convincing. If the players knew such a theory then each player would have to assume that his strategy has been "found out" by his opponent. The opponent knows the theory, and he knows that a player would be unwise not to follow it. Thus the hypothesis of the existence of a satisfactory theory legitimatize our investigation of the situation when a player’s strategy is found out by his opponent". (The Theory of Games and Economic Behavior: John Von Neumann, and Oskar Morgenstern in 1944.)

Now we come to the paradox of this as well, if we are successful in constructing a theory of stone-paper-scissors that indicates which of the three strategies is best, an intelligent opponent with access to all the information can use the same logic and deduce our strategy. He then second guesses and wins, so if the original player uses the "best" strategy it would end up being fatal, however if the first player knows that the second player knows the theory and the first player knows that the second player knows that the first player knows that he knows, then he is back to the beginning of the paradox. Once again we are back with the mirrors.

This is where the mathematical equations come into effect and help us to figure out what needs to be done to ensure us of our victory. I am not going to get into all of that here, as it is too hard to explain and will take too long. However, when I give my presentation I will be showing some different mathematical equations that show how this theory works and I will be talking about it then.

Another theory we must look at that is closely related to game theory is the Nash Equilibrium, where each player’s strategy is optimal given the strategies of the other players. Yes this is confusing, in better terms: If there is a set of strategies with the property that no player can benefit by changing her strategy while the other players keep their strategies unchanged, then that set of strategies and the corresponding payoffs constitute the Nash Equilibrium.

It actually is not that major of an issue while trying to explain game theory, and it can lead to more confusion, so that is all I will be talking about on the Nash Equilibrium theory.

**In Summary:**

In constructing a theory of a 2 or more person’s game, the game theorist has a choice, he can focus on one aspect of the game and disregard others, or he can try to capture all the relevant features of the game in a single model. Yet in either of these choices he/she still will end up paying a price.

If player/players one chooses the first course, and assumed that they were dealing with other competitive players who could communicate and have simultaneous and instantaneous access to each other, then this theory is convincing and may be used to predict what the range of punishment or reward will be.

Game theory describes the situations involving conflict in which the payoff is effected by the actions and counter-actions of intelligent opponents.

It has been our intention to make this report as clear as precise as possible, to do this we have dappled in or discussed application that are excessively technical. Not only would it take too long, but also it is difficult for us to understand as well, to do this we would have to mention applications that have been made to international trade, the use of national resources, collective bargaining. These are way too in depth and detailed and would take several hundred pages and months of studying.

However, hopefully my readers have some idea of game theory after reading this, and some idea of what the problems are and what to do about them.

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