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Game Theory: Summary and Analysis

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Game theory began as a small branch in the financial industry with a great book written by John von Neumann and Oskar Morgenstern “Game Theory and Economic Behavior” on zero-sum games. The main focus is the analysis of decisions in strategic situations (games) and interactions in which the loss of one player will be equal to the win of player two.

In addition, game theory is fully integrated in the field of applied mathematics and its applications are many and interesting. Many other scholars have contributed to the development of this theory, such as John Forbes Nash, whose life was turned into a movie called “A Beautiful Mind,” who generalized the problem to non-zero-sum games and offered as a solution, the Nash Equilibrium. Reinhard Selten is also very important when it comes to game theory, as he paved the way for a satisfactory solution of the problem in dynamic games, with the concept of Subgame Perfect Nash Equilibrium and trembling hand perfect equilibrium. Lastly, the other scholar that played a huge role in the development of game theory is John Harsanyi, who dealt with games under partial information. For their work Nash, Selten and Harsanyi were awarded the 1994 Nobel Prize.

The last 30 years, game theory has found wide applications in economics, where whole sectors based on methods such as industrial organization, planning mechanisms, the most important sub-sector planning auctions and many more. Also, game theory has been used in political economies and especially in the theory of collective action, which explains any cooperation between different players. However, it is also widely used in other sciences such as evolutionary biology, psychology, sociology and others. In some games there is no cooperation, but it may occur spontaneously.

A famous example is the "prisoner's dilemma" which is the cooperation between two criminals who are suspected of being involved in a robbery and are arrested and interrogated separately. The investigator gives them 4 different choices and they have to pick one. The choices are as follows: If one player confesses and player two does not confess, player one will be free and layer two will serve a four year prison sentence. If they both confess, they will share the sentence which is three years each. If no one confesses, they will get the minimum, which is a one year sentence each. Both players have all the information, but are separated and cannot communicate.

For such games, Nash proved the existence of equilibrium. Equilibrium is a combination of the "best" strategies. In the prisoner's dilemma game, the Nash equilibrium is when both criminals confess. Indeed, the risk of imprisonment for four years is higher than the potential benefit from imprisonment for one year. The results of this kind of game may seem obvious, but the same calculation techniques can be applied to situations more complex, which provide results that are less obvious. However, the so-called "cooperative games" are extremely complex. For example, it is difficult to determine which of the many shareholders of a company has control, because the possible alliances make the situations unpredictable.

Suppose that the United States of America decided to privatize a company and it has to determine the percentage that can be sold in order to continue to have control. At first reading this problem seems that, holding 51% of shares means that the state remains the in control. Despite the obvious, is this decision economically smart? The answer is no. The country may continue to be at the helm of the company by holding 35% or even less. Of course this needs a lot of attention, because if the US keeps 35% and sells the remaining 65% to a tycoon, the company no longer belongs to the state, but the tycoon. If it wants to maintain control, then it must ensure that the remaining shares fall into the hands of thousands of small shareholders instead of one big company.

A measure of the ability to control a shareholder in the company, is the so-called "power indicator", which can be calculated in many ways. The most famous, is the index of Saplei, which is the name of the initiator, Lloyd Stogouel Saplei, who was also Nash’s classmate at Princeton. This index can be used for the sharing of profits, which is not necessarily based on the number of shares held by each shareholder. Here is a concrete example: If 100% of the shares have been split into four partners holding respectively 10%, 20%, 30% and 40%, the index of Saplei shows that the profits will be distributed as follows: 8% 3%, 25%, 25% and 41.6%. Calculation of "utilities" In games where you can set the "benefit" of each player can be found very specific solutions. A good example is the deal, which was studied by Nash in 1950: two people have to share a sum of money, one rich and the other not. While the poor, of necessity, will be satisfied even with a few, the rich, because power will be enjoyed only by a lot of money. This model leads to an unfair result. If Nas solved, if the amount is EUR 500, the 310 will get rich while the poor just 190. The Nash takes into account one crucial factor: things, even money, have a different value for every person and this affects the game. "If you play both, it pays to bluff only when you have the worst cards, not when you have moderate '. Samaras - Papandreou We come then to the notorious question. What should Antonis Samaras? Saying YES vote in the medium or not the NO vote. Before you answer one slightly heart must take into account that to win Antonis Samaras should be able to "play" with both data. And with the YES and NO. Here, however, care is needed to have someone on hand because as I said the words "cooperative games" is extremely complex. For example, it is difficult to determine which of the participants 'players' in the game in control, because the possible alliances make unpredictable situation. If we want to implement the "prisoner's dilemma" to give an answer to what should be done by the ND leader then leads therefore necessarily prudent solution below equilibrium. "Gone Papandreou in passing with 150? - Anthony chooses NOT because there is nothing to lose, nor be held responsible. Gone Papandreou in passing with 180? - Anthony chooses YES to contribute to the preservation of the state since the position of this forced movement emerges as the salvation of the state. " These movements Antonis Samaras are the unique "best solution" for the New Democracy Antonis Samaras and sent to the Prime Minister that margin should ultimately publicly thank Anthony for the position of responsibility that will get (say now ... ) The paradox of blackmailer The Reuben and Simon are in a small room with a suitcase containing $ 100,000 in cash. The owner of the suitcase makes them the following sentence: "I will give you all the money in the suitcase, but only on condition that they will negotiate and come to a friendly agreement on sharing. This is the only way to give you the money. " The Roumpen are reasonable man, believes the golden opportunity presented to him and send for Simon with the following obvious proposition: "Come on, you'll get half the amount and I the other half, so everyone will walk away with $ 50,000" . To his surprise, Simon with a serious face and resolute voice says: "Listen, I do not know what your intentions for the money but I'm leaving this room with less than $ 90,000. Take it or get out. I'm fully prepared to go home with nothing. " The Roumpen can not believe his ears. What happened to Simon? he asks. Why should he get 90% and I only 10%? He decides to talk to Simon. "Come on, be reasonable" begs. "Both of us are in and we both want the money. Let us divide the amount in straight and let's move forward. " But the rationale of his friend, does not appear to have resulted in Simon. Listens carefully the words of Reuben but then states with even greater emphasis: "There is nothing to discuss. 90-10 or anything, this is my final offer! ". Reuben's face turns red with anger. Wants to skampilisei Simon but soon rethink. Realizes that Simon is determined to walk away with the most money and that the only way to leave the room with some money is to succumb to blackmail Simon. He straightens his clothes, pulls out the suitcase the amount of $ 10,000, gesture exchange with Simon and goes from seedy room. This event is called "the paradox of the blackmailer." The paradox that appears in this case is that the logical Reuben is eventually forced to act irrationally conspicuously order to win the maximum that is available to him. The logic behind this strange result is that Simon emits a total faith and confidence in excessive claims and that is able to convince Reuben to succumb to blackmail in order to ensure a minimum profit.

H application in Biology

Finally, we must stand in the application of game theory to biology, namely the implementation of the "best strategy" in the (co) competition or cooperation between players at species or individual animals. This has been done in those cases where it is difficult to predict the effects of natural selection, because the best that can be done depends on what do the other members of a population. The techniques of game theory were actually used in simple models "evolutionary games" to offer an explanation for the evolution of certain characteristics. Thus the British biologist John Maynard Smith made an evolutionary game theory, leading to the concept of "evolutionarily stable strategy" which, if almost all members of a population adopt no other mutant strategy can perform better with regard to him and to "threaten" within the population. On the other hand, some had already guessed that probably do not need a perfectly rational being to identify the best strategy and tried to apply the theory in models of fundamental microbiological structures. Interestingly, it was discovered that small RNA molecules can indeed engage in simple two-player games.


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