РЕФЕРАТ по дисциплинеМГТУ МИРЭА
Факультет информационных технологий (ИТ) Кафедра информатики и информационных систем (ИИС)
РЕФЕРАТ по дисциплине « Анализ данных, статистика и приятие решений »
Москва 2015
ECONOMIC APPLICATIONS OF THE THEORY OF GAMES The object of study of the Theory of Games are games. A game is a process in which two or more people make decisions and actions, the structure of which is registered in a set of rules (which may be formal or informal), at the end of profit. Each combination of decisions and actions determines a particular situation, and given that the decisions and actions of the players involved can be combined in numerous ways, generated numerous situations will also be equal to the magnitude of the combinations of decisions and actions agents. The total set of possible situations will be called Situational Game Table. Following the above reasoning, we find that each situation (each point of the situational picture) generates a combination of certain awards. The prize gives a player a particular situation can be compared to the prizes offered other situations. A golden rule of game analysis is as follows: "Every player can seek your highest good." Thus, when we study the behavior of a player, we know that it must qualify every situation and always pursue the particular situations that offer the greatest good. An important concept is that of payment. As noted above, each situation offers a combination of prizes, as follows: If you are two players, the situation offers a prize for the first and one for the second. If there are three players, the situation generates a prize for each player. This is the logic of the awards and situations. Each prize is called payment. Another key concept is that of the utility function. The utility function makes welfare payments. For example, if a payment of five units of money was achieved, it could generate a welfare payment of five units of welfare, and we are talking about an identity function. If the utility function were a square root, paying only five units correspond to five units being being. In this paper we will mainly utility functions identity. When required trying different utility functions (such as the square root).
IMPORTANCE OF GAME THEORY Like any theory, game theory makes use of specific terminology and complex models. Where one wonders what is the prize won by those who wish to deeply study the theory of games. Game theory has applications economic. Since we are all economic agents should study this theory, the end to understand what theoretical and practical operations could offer bigger cash prizes. Some applications of game theory to real life are: · Contracts · Military-wars · Trade-wars · Marketing For competition in the markets · Domestic Negotiation, · Commercial Negotiation, · Collective Negotiation, · Partnerships To any other situation in which two or more individuals requiring interact late financial gain must be included in the list. As the human being is a homo economicus, he can find plenty of applications to Game Theory USING GAME THEORY TO REALITY To use game theory as an application for a real situation, it is necessary to build simplified models of reality. In these models, needs to represent each player with their respective forms of behavior. When it comes to two players, usually know perfectly well what our behavior, but we know only in part of our rival or opponent. Thus it becomes easier to represent our conduct simplified to represent the behavior of the rival. In any case, it is required to adequately represent the behavior of the two (or more) players. Our behavior will be known with certainty, while the only rival as likely (in scientific language, stochastic). Sometimes you will need to make two or more representations of the probable behavior of the rival. Each performance is called scenario. Each scenario is a simple game. The set of two or more stages is a set composed. An ethical and economic implementation of situational analysis in the Prisoner's Dilemma This is a famous dilemma that focuses on the problem of a thief who has been imprisoned. This problem becomes unethical when every thief must decide whether to trust or not trust a second thief, who is his partner. And it has to decide whether or not betraying betrays his partner. Combinations of decision and action are: · Trust your partner and do not betray. · Trust your partner and gives it away. · No trust your partner and gives it away. · No trust your partner and do not betray. The first and third behaviors can be seen as "ethically consistent", while the second behavior is "opportunistic and openly immoral" behavior and the last is "purely altruistic." The member has the same number of combinations of decision and action. The situational analysis should estimate three scenarios for each partner: ethical consistency, opportunism and altruism. With this, we have set up a total of nine possible moral scenarios (since we know with certainty what is the nature of either partner). As an exercise, let the nine moral scenarios: · Partner A (first) and partner B (the second) are ethical (position 1). · A Is ethical and B is opportunistic (position 2). · A Is ethical and altruistic B (position 3). · A Is opportunistic and B is ethical (situation 4). · A And B are opportunistic (situation 5). · A Is opportunistic and B is altruistic (position 6). · A Is altruistic and B is ethical (position 7). · A Is altruistic and B is opportunistic (position 8). · A And B are altruistic (position 9). The situational analysis will look into the situation. Finally, we will know how would the distribution of the final results of this game. Altruism and opportunism reduce the possibility of varying the decision. An altruistic simply not betray, and do not want to create problems for your partner. Always betray an opportunist as it seeks its profit regardless of the damage to his partner. In the situation 9 bialtruistc, the solution is "A and B does not betray." In situation 5, biopportunistic, the solution is "A and B also betrays". In any situation where there is a single altruistic or opportunistic one, the solution will depend on the confidence level of another agent in your partner. Consider the following situations: · A Is ethical and trusts B, that is opportunistic. B betrays A. · A Is ethical and does not trust B, which is opportunistic. Both reveal. · A Is ethical and trusts B, that is altruistic. Nobody betrays. · A Is ethical and does not trust B, which is altruistic. A betrays B. · A Is opportunistic and B is ethical and trust betrays A. B. · A Is opportunistic and B is ethical and does not trust A. Both reveal. · A And B are opportunists. Both reveal. · A Is opportunistic and B is altruistic. A betrays B. · A Is altruistic and B is ethical and A. Nobody betrays trust. · A Is altruistic and B is ethical and not trust betrays A. B A. · A Is altruistic and B is opportunistic. B betrays A. · A And B are altruistic. Nobody betrays. It is twelve possible situations. We have not seem the possibility that the two are ethically consistent. Let's see what would happen if our study subjects were ethically consistent: · A And B rely on their respective partner. Nobody betrays. · A trust B.B do not trust A.B betrays A. · A does not trust B, B trust A. A betrays B. · A Does not trust B, which does not trust A. Both reveal We now have sixteen possibilities for the final status of the game. If we assign a rating from one point to each situation and relate to tabulate the possible outcomes, we have: Just A betrays B: 4 out of 16 = 25%. Just B betrays A: 4 out of 16 =) 25%. Nobody betrays: 4 out of 16 = 25%. They both reveal: 4 out of 16 = 25%. Complex reasoning has led us to find that four possible scenarios for the end of the game occur. Each has a 25% chance to occur. Of course, we have assumed that the moral characteristics are distributed uniformly (33.33% for each feature) is that trust and an ethical agent to the other was 50%. In this case, there occurs imminence or fatality. If the likelihood that each would be ethical, altruistic or opportunistic 70%, 20% and 10% were made, it would have to obtain the following distribution: Just A betrays B: 4 out of 16 = 24.75%. Just B betrays A: 4 out of 16 =) 24.75%. Nobody betrays: 4 out of 16 = 20.25%. They both reveal: 4 out of 16 = 30.25%. It is indicating that the most likely outcome of the game in this case is the double betrayal. Game Theory is important because it allows us to find the impending results of several games that we face every day in the real world. Game theory is no less important just because you can not analyze all the games we play in the real world. GAME STRATEGIES ECONOMIC DECISIONS Economics is a science that through many years of development has sought to analyze and mathematically explain the relations of production, marketing, storage and redistribution of different goods in a society. In an economic analysis the relationships between multiple agents are reduced to a couple of them to understand the theoretical explanation of the phenomenon to interpret, which sometimes causes confusion by simplifying reality. Thus, the economy sees the world in a somewhat abstract way which is criticized by many who really do not understand and thus underestimate their uselessness claiming to explain the real world of the entrepreneur. However, some reason have those who criticize the abstract character to check in the corporate world that those formulas learned in college are of little use to understand and project the reality of its macroeconomic environment. These criticisms should not be ignored by those involved in the study and research of economics and, rather, should serve to open the eyes and question the validity of some statements that do not work as stated in the book, either by being out of historical context, or they are developed for economies with their own and different from those to be analyzed characteristics. Evolutionary economy includes all these concerns and comes to question many ideas of classical economics. Try to adjust to the realities of daily life more than it does the traditional theory and get to compare economic development with biological evolution, to conceive the economy as a developing organism and not as a machine that can handle levers, as have believed the Keynesians and neo-Keynesians. Evolutionary economy takes tools of game theory, mathematics, physics and genetic biology to explain that people do not always act rationally from an economic perspective. Survival and cooperation may be complementary according to this point of view in which the interaction and the ability to adapt are the foundation that explains many behaviors of society. Evolutionary economics then comprises the so-called strategic altruism and more, but is this the one that can help businesses to plan optimal strategies in various areas such as finance, marketing and management. As Forbes states: "Biology and economics are coming to the conclusion that nature is not composed exclusively of teeth and claws, and the ability to cooperate can be one of the tools for the evolutionary survival, and also for corporate survival. " Smart business people may think that this is just common sense and do not need economists to tell them that the world is not static. They know that all self-action leads to a reaction of competitors, so they must know what will be the reaction before acting. However, some of them believe that the key to success is to finish the competition, and it turns out the real key to long term success is not to eliminate opponents but to adapt to changing situations. ECONOMIC APPLICATIONS OF GAME THEORY (IN STRATEGY) Game theory is a type of mathematical analysis oriented predict the certain result or the likely outcome of a dispute between two individuals. It was designed and developed by the mathematician John von Neumann and economist Oskar Morgenstern in 1939 in order to carry out economic analysis of certain processes of negotiation. Von Neumann and Morgenstern wrote the book The Theory of Games and Economic Behaviour (1944). The mathematician John Nash (John F. Nash and John Forbes Nash, Jr., 1928) created in 1950 the notion of "Nash equilibrium", which corresponds to a situation where two rival sides agree to certain game situation or negotiation, whose alteration has disadvantages to both sides.
Other important representatives of game theory were nationalized Hungarian American John Harsanyi (1920) and the German Reinhard Selten. Nash, Harsanyi and Selten won the Nobel Prize for Economics in 1994 for contributions to game theory. There are different tools to analyze a game, including: · -The Matrix Payments or Pay-Off Matrix · -The Reaction curves · -The Trees successive results MATRIX PAYMENTS A payoff matrix is a two-way table. The top entries indicate the options that you can take B, and entries on the left show the options you can take A. The matrix points defined by combinations of possible decisions represent RESULTS TO the game, including gains (or losses) will get each player. See the following payoff matrix:
The first component of each pair of numbers is the payment received by A if the result lies in the combination that defines each point matrix determined. The second component is the payment received by B in the same situation. ZERO SUM GAME A zero sum game is one in which everything he earns a player A player B loses and vice versa. Thus, if A wins $ 10 on a business, for example, B earns $ -10, ie losing $ 10. 10 + (-10) gives zero. A zero-sum game can have matrix notation, or not having it. Game theory, meanwhile, devoted much effort to the analysis zero sum problems likely to be noticed in matrix. NON-ZERO-SUM GAME It represents a situation in which what A win will not always be lost by B, and vice versa. Some non-zero sum games are susceptible to matrix notation. MATRIX OF A ZERO-SUM GAME The matrix of zero-sum game always sees payments from the point of view of player whose decisions are represented horizontally. See, for example, the payoff following:
The exposed matrix tells us that A and B can make different choices. If A chooses the decision or path A1 and B choose B1 decision, the game will pay 10 points to A and charge those same ten points to B. If the decision to opt for A2 and B chooses the decision B1, A loses B earns 12 points and the same 12 points. If A is decided by the way A1 and B on the road B2, A and B loses the 12 points wins. If A makes the decision making B2 A2 and B, A wins 11 points, which are lost to B. MATRIX NON-ZERO-SUM GAME Furthermore, a matrix that results usually nonzero presents several game outcomes that can be favorable for A and B results in a matrix of non-zero game (or amount "nonzero") exposed sum using commas in each cell to show what the player gets represented in the first horizontal decisions (our player A) and you get the player vertically below (our player B). If we wanted to represent a zero-sum game in juna matrix sum game non-zero, the result is as follows (based on our example):
In a zero-sum matrix, a "fair" outcome is one in which neither of the two rival individuals get positive benefit. Since any gain from A to B is a loss, just as it is when no one wins or loses. The result of a zero-sum game is not always fair. For example, in a game of chess or checkers, sometimes the result is a tie ("tables" in chess). This is a "fair" outcome from the point of view. But most of the time, the result is a draw, but appears a single winner. Chess, being a zero-sum game, shows that the optimal result is not always what is socially just. The next game is a fair result in the decisional A2-B1 combination, namely the combination 2-1
In an array of non-zero sum, the result just be one that benefits both simultaneously, or one that benefits as much as possible, affecting other as possible. An array of different amount two positive zero values assigned to each matrix point. REACTION CURVES In game theory, the reaction curves shown in a Cartesian graph, combinations of decisions (may be on the abscissa) and payments (may be on the ordinate). A simple example of reaction curves can be seen in the curves of supply and demand. Suppose that demand and supply are built by trial, as price proposals receivable and payable made by a bidder and a plaintiff in relation to a specific amount to be traded on the market. Combinations (X *, pd) offered and combinations (X *, pd) proposed by the applicant determined that there is a price difference (pd - ps) higher, lower or equal to zero. If the difference is greater than zero, the applicant must decide whether it should propose a different level of business combined with a lower price to pay. The bidder must also decide whether to propose a different level of business combined with a price to pay later. The procedure is similar when the difference is less than zero: the bidder may propose a lower price and the applicant may propose a higher price. In the case described by those who follow the law of supply and demand law of markets, it shows that there is a combination solution (X, p) having convergence and stability. In this market model a sequence of moves, on the "signal" to the price difference is made. The shifts are double, ie both players act simultaneously. Another notable model is sequential Cournot duopoly. Trees successive results
A tree diagram is used successive results in games involving sequences of movements (a movement is a binomial decision-action). In this tree, a starting point is defined (eg, the initial position of the player A). From the beginning, branches representing the different movements that can make the player who started the competition stretch. The various movements or branches define as many results or payments, which can serve as a starting point for further decisions the next player (for example, player B). The process is repeated until the number of moves that A and B can be performed. A game with a move for A and one for B has two generations of branches. A game with two movements for A and two moves to B has four generations of branches. In general, a game with movements to movements A and n B (mn absolute value can not exceed 1) has m + n generations of branches. The tips of the branches of art contain a description of the possible outcomes of the game. In the case of both A and B can take only two choices at each stage of the game the number of points of the tree will be 2m + n. Order of moves in the game A game can be simultaneous or sequential movements. The popular game "rock-paper-scissors" is a simultaneous game, while the checkers and chess games are sequential. Cournot duopoly is also a sequential play. Each of these game types has different foci of interest to game theory. The most notable simultaneous play is called "prisoner's dilemma". This "dilemma" is a non-zero sum game. In this interesting case, the dilemma of every two prisoners is not to betray your partner or give him away. Reducing the time of incarceration is paid. A negative reduction corresponds to increased prison sentence. Interrogators are close to each of the two defendants allegedly involved in a crime carried out jointly. Each inmate is told the following: "The period of preventive detention is three months, so if we can prove that your partner committed the crime and you do not, you will reduce the sentence in three months and will leave free instantly and he will increase it in three months, leaving in six months. But if we prove that you and your partner are criminals, the prison term shall be five months increasing by two months (to fall in under two months). Finally, if we can not prove that you and your partner are guilty, the penalty is reduced to two months, so both should spend a month in the shade, while a series of procedures is done " A payment in this case is the time when the penalty is reduced. The penalty increases are reductions negative. Payments corresponding matrix is:
What is the solution of this game? It is seen that if A and B cooperate, ie neither betrays the other, reduction of sentence is obtained in two months for each (result 2: 2). If A wants to leave immediately, you can try their luck with the reduction of three months, looking outcome 3, -3. To do this, A must give away. B can tempt fate with the result -3, 3, and must also betray. If A and B choose denunciation, rather than obtain instantaneous output, both creditors get a penalty increase in two months (reductions -2, -2). To solve the problem will be necessary to analyze the position of any player (for example, A). A can initially choose not to reveal. If B assumed that A is not going to betray, he will conclude that his sentence be reduced to two months or three months. Under the assumption that A did not betray, B will feel compelled to betray him, and that way the result is maximum. The analysis of the position of B leads to parallel conclusions strategy A. The imperative search for the best final position leads them to opt for the strategy of betrayal. Outcome of a game by analyzing a payoff Results are obtained by convergenca criteria and stability. Convergence occurs when decisions of A and B tend to generate a favorable outcome for both. A combination of decisions that appeals to B and not bother to allow both get benefits of play. Similarly, a decision that benefits not disturb A and B will benefit both. The analysis in the payoff matrix shows that often the paths chosen by A and B roads converge elected by a point. Arguably, then, "the paths of A and B lead to Rome". The convergence of the paths, as in the case of the true paths followed in any region, it occurs at a certain point. To this point it is called the optimum collective game. The convergence often produces a stable solution. A stable solution corresponds to a decision that will not change in the future. Suppose A decided to conduct an action A1 and B decided to take action B1. Once both have reached this optimal decision, you may feel satisfied and rested with the solution. In this case, it is said that the solution is stable point (a point solution two actions is defined by any one of a subject A and a subject B other, such that said point provides significant benefits to both subjects or players). A stable point will be considered as a sink or attractor. It is a drain that remembers where water leaves the tank, all vectors occurring water are directed towards the point of escape. It is an attractor that, considering the entire array of payments as an indoor space region of points, the stable point attracts all these points. The concept of stability often leads to the concept of Nash equilibrium. NASH EQUILIBRIUM Given any situation defined by a choice of A and B choice, if it happens that A assumes that B will not change your choice and choose not to modify their own and simultaneously assumed that B will not change his election and also chooses not to change yours, it says that such a situation is a Nash equilibrium. As shown, the Nash equilibrium is a situation that has advantages for both players, and because of such advantages, neither A nor B will change their decision. However, it may happen that A notice that you can win a few more benefits if disappoints B. Such is the case of an unstable point solution. As the payoff matrix is analyzed in two dimensions, convergence is what gives the attraction. It is seen that the attraction does not always stable. The attraction exerted by the decisions of A and B makes this point a solution, while the repulsion exerted by them makes it an unstable point. B means tapping defraud the social optimum position to further raise the benefits obtainable from the game. For example, if A and B decided the following: A1: A will not sell merchandise in Area C. B1: B will not sell merchandise in Area C. (C proves to be a neutral area). If finally to decide to sell merchandise in Area C that is you can achieve greater profits, you disappoint B. B, seeing it, decide that it is useless to respect the rule violated by A. A and B had as optimal social outcome an equitable distribution of the sales areas. Now, A and B lose that optimum social position as a result of having sought each individual advantage. The point of convergence and stability problems characterizes both called zero sum game (where the payoff amount is zero) as games nonzero sum. NASH EQUILIBRIUM: AN APPLICATION TO THE COMPETITION TWO COMPANIES When the problem of analysis of the games is the definition of a Nash equilibrium indicates that this balance is a set of actions such that none of the players, if it considers that the actions of your opponent are given, want to change their own action. A Nash equilibrium is a game situation in which, once each player when it considers that the actions taken by the opponent are unchanged, will resist change its own action. In a Nash equilibrium, the player will notice that, as the action of his opponent's default, he can choose his own action within a range of possibilities. If the game has resulted in a situation S1i, 2j corresponding to a Nash equilibrium in which the player performs an action i and j opponent action, the player will reject any possibility of a different action of i. A NASH EQUILIBRIUM LEVEL OF PRICE COMPETITION Particularize the following case, two companies producing different goods replaceable offer its product in two different markets. Suppose demand corresponding structures are: They are the unit costs of production of items w1 and w2. Consider that unit costs are equal to each other, and have a value w. The unit benefits are: The total benefits are: Calculating the second derivative of the total profits on the price of the company, we see that its value is negative. Calculating the second derivative of the total profits on the price of the alternative company, we see that its value is zero. This ensures that maximum benefits can be obtained by applying the first derivative test for null. The benefits we derive once and get: The criterion of the first zero-derivative generates a set of two simultaneous equations in two unknowns. In resolving such a system, prices that optimize business decisions appear: The optimal pricing decisions determine, for the following point: This price point is a Nash equilibrium: neither competitors want to alter their decision regarding the price
MORE ABOUT THE NASH EQUILIBRIUM The behavior of both players can set any of the following situations: · The Nash equilibrium corresponds to the result of applying pure strategies. · The Nash equilibrium corresponds to the result of applying mixed strategies. · There is a Nash equilibrium in the game. · Two or more Nash equilibria in the game. · There is not even a Nash equilibrium. A pure strategy is one decision that is certain. In contrast to this concept, a mixed strategy is a decision taken with a certain probability. When a problem does not reach a solution via pure strategies often they can be approached from the perspective of mixed strategies. Thus, it is said that the problems that have no solution via satellite pure strategies can have it mixed strategy. Both situations can be seen as certain solutions versus ranges of probable solutions. Pure strategy equilibria can constitute various magnitudes, as a unique balance, two or more balances (a discrete number), countless balances on a subset of the total final game situations, or infinite balances that cover the entire late game situations. In any case, an equilibrium is a pure strategy whose final situation likely to give maximum benefit (within the neighborhood of situations) both players is one. As mentioned, when no pure strategy Nash equilibrium, it is often possible to determine Nash equilibria in mixed strategies. It is usual in such contexts find multiple equilibria in mixed strategies Nash each balance associated with a couple of players' decisions, each decision in turn associated with a probability of being taken. Therefore, we can say that the analysis shows Nash resulting probability distributions produced by Nash equilibria. The method for finding the probability distributions of the mixed strategies is to assume that a subset of the final standings (which can sometimes cover the entire set of final situations) has a unique and maximum expected value. That way, you can calculate the probability distributions that allow such equivalence occurs. The technical details of the method will not be discussed here. The decisions of the players have associated distributions calculated odds in the manner described. Two players and determine an area of distribution of the probabilities associated with the final standings. Some final situations are more likely than others.
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