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Game Theory homework help by top economics problem solvers. We guarantee accurate solutions, affordable charges, 24 hours support online & assured on-time delivery. Key definitions, applications with Nash Equilibrium and Prisoner's Dilemma with examples.
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Game theory is an area of applied mathematics which provides tools to predict and analyze potential situations where participants, called players, are required to make interdependent decisions. Because of this interdependence, players must consider the possible decision or strategy of the other player/s in order to formulate their own strategy. Applied across several fields of study, game theory provides the players with a theoretical framework to conceive situations, best possible decisions and the outcome of the “game” that may affect them. It is often used to simulate real-world scenarios for situations such as product release, pricing and market competition to predict possible outcomes. As there is a great emphasis on optimal strategic decision-making of competing and independent players, game theory is also referred to as a science of strategy by scholars. Emphasizing on the decision-making aspect and those controlled by the participants rather than chance, game theory not just supplements the classic mathematical theory of probability but also goes beyond it.
Game theory was pioneered during the 1940s by Hungary born American mathematician John von Neumann and his Princeton fellow, economist Oskar Morgenstern and developed especially for application in economics. They asserted that mathematics used in the physical sciences provided a poor model to study economics. They also made the observation that economics was similar to a game in the sense that like in games, players in economics are required to anticipate the moves of others, make moves to counter them, and vie for the reward. An interesting fact about the theory is that it was initially applied to the analysis of parlor games to study the decisions, behavior and resulting outcome of players. Renowned mathematician John Nash also played a crucial role in the development and extension of the theory to the form in which it is known today.
Proponents of the theory proposed that at the center of it is the game or the interactive scenario which exists among the players. Game theory is built on the basic understanding that the payoff received by any one of the players depends on the strategy implemented by the other party and not just that player. Identities, possible strategies, preferences, and how the implementation of those strategies will affect the outcome are decided by the game itself. The decisions and actions of any one of the players will affect the outcome for all the others involved in the game. A key assumption of the theory is that players participating in the game will make rational decisions and want to maximize their payoffs. A number of other assumptions and requirements may also be necessary depending on the given model scenario. Apart from economics, the range of applications of game theory includes business, psychology, politics, war and evolutionary biology. Despite its immense scope and utility in these fields, game theory is still considered a relatively new and developing science.
The game theory can be applied to predict the most probable outcomes any time a situation between two or more players arises in which there are measurable consequences and known or quantifiable payoffs. To understand how it actually works, it is important to first define the key terms of the theory. These are:
Game: A game is defined as any set of circumstances in which two or more players take actions to arrive at a given result or outcome.
Players: They are the strategic decision makers who participate in the game and want a favorable result for themselves.
Strategy: It is the complete and most rational set of actions that players implement according to the situations within the game.
Payoff: It is the payout that a player will receive at the end of the game. The payoff must be in a quantifiable form and can be anything from money to utility.
Information Set: It is the amount of information available to the players at any given point in the game. Information is generally applicable in games which have sequential elements or components.
Equilibrium: It is the point at the end of the game where all the players have made their decisions and a valid outcome has been reached.
The Nash equilibrium developed by mathematician John Nash is a result or outcome of the game which when achieved, none of the players can increase their payoffs further by unilaterally changing their decision. This equilibrium is also understood as the point of no regrets because it is assumed that when a final decision is made, players will not have any regrets with regard to the impact of the decision on the resulting payoff. Nash demonstrated that in any given game having a set of possible outcomes and related utilities for the players, there is at least one unique outcome which satisfies the following conditions:
The outcome does not depend on the choice of utility function.
The outcome is not affected by irrelevant or unrelated alternatives even if they are added to or removed from the game.
Both or all players cannot achieve a better result than the others simultaneously. This condition is also known as Pareto-optimality.
The unique outcome is always symmetrical which means that even if the roles of the players were interchanged, the outcome will remain unchanged. The exception here is that in such a case the payoff will also be reversed with the role.
In most scenarios, the Nash equilibrium is reached gradually over a period of time. Once it is achieved after the period of trying and testing, the Nash equilibrium stays unchanged. Therefore, for economists and other professionals, it does not make any sense to study how a unilateral decision would have affected the situation. This is why the equilibrium is called the point of no regrets.
There can be several equilibriums within a single game. However, multiple equilibriums are generally seen in games that have more complex components and outcomes rather than those in which there are just two players who can make just one decision each. In simultaneous, complex and repeatable games that are sustained over a longer period of time, these equilibriums can be determined after conducting some trial and error. The presence of multiple equilibriums creates the scenario of multiple choices for players. Such a scenario can be seen most commonly in the business world where two competing businesses often have to determine the price of strong substitute products.
The prisoner’s dilemma is a classic game theory scenario developed by American mathematician Albert Tucker. It demonstrates the difficulties in making a decision in a two-person game with multiple outcomes where the players are assumed to be non-cooperative. To understand the prisoner’s dilemma, consider this example: suppose two persons—A and B have been arrested on the suspicion of committing a robbery together. Assume that the prosecutors could not find admissible evidence to prove the crime in court. Therefore, in order to make the suspects confess to their crimes, officials question both of them separately. The suspects have no way to communicate with each other. They both are presented with four deals by the officials:
1. If both confess to the crime, they will get 5 years of imprisonment each.
2. If person A confesses and person B does not, A will get 3 years and B will get 10 years of imprisonment.
3. If person B confesses and person A does not, B will get 3 years and A will get 10 years of imprisonment.
4. If neither of them confesses to the crime, both will get 2 years of imprisonment.
It is clear that the best strategy would be not to confess. However, because neither of them knows the strategy of the other person, they cannot be certain that the other will also not confess. In this situation the safest bet would be to confess to get the best possible deal. Thus both A and B will confess and get sentenced for 5. Therefore, when faced with prisoner’s dilemma, both players will make the decision which is best for them individually but not so collectively.
The application of game theory has revolutionized the field of economics by providing solutions to critical problems associated with older mathematical models. For example, economists using neoclassical economic theory found it very difficult to understand imperfect competition and entrepreneurial anticipation. Game theory provided a solution by turning their attention to the market process from the older concept of steady-state equilibrium.
In the business world, game theory is credited for improving the modeling of the competing behaviors of different agents in the economy. Businesses often have to choose from multiple strategic alternatives that can affect their ability to make economic gains. For example, they may be faced with situations of dilemma such as whether to develop new products and discontinue existing products; lower prices to challenge the competition; and adopt a new marketing strategy. Game theory provides solutions to most such situations. It is also a key theory that helps economists understand firm behavior in an oligopolistic market (a market where a small number of suppliers dominate the market) by predicting the most likely outcome when firms behave in a certain way and adopt strategies such as collusion and price fixing.
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