(By Tiffiney Whitmire)
"All things throughout our universe seem to follow the same fundamental blueprint or geometric patterns. These geometrical archetypes, which reveal to us the nature of each form and its vibrational resonances. They are also symbolic of the underlying metaphysical principle of the inseparable relationship of the part to the whole. It is this principle of oneness underlying all geometry that permeates the architecture of all inseparability and union provides us with a continuous reminder of our relationship to the whole--a blueprint for the mind to the sacred foundation of all things created."
We call this blueprint "Sacred Geometry".
Showing posts with label Maths. Show all posts
Showing posts with label Maths. Show all posts
Sunday, August 16, 2009
Saturday, April 28, 2007
Game theory: Nash Equilibrium
We have applied game theory from a simpler Prisoners' Dilemma situation to a more realistic game model in a real-world example of strategic thinking, say choosing an information system in the followings.
For this example, the players will be a company considering the choice of a new information system, and a supplier who is considering producing it. The two choices are to install a technically advanced or a more proven system with less functionality. We'll assume that the more advanced system really does supply a lot more functionality.
Again, in this case, we can express all this compactly in a payoff table. Basically, the table would indicate that if both the company and supplier choose the technically advanced information system, each earns $20 million in profits from the system, but if the company chooses the advanced system and the supplier does not choose to produce it or vice versa, then both earn nil profits for the period under consideration. However, if both choose the proven information system, then both earn only $5 million of profits each.
We see that both players can be better off, on net, if an advanced system is installed. But the worst that can happen is for one player to commit to an advance system while the other player stays with the proven one. In that case there is no deal, and no payoffs for anyone. The problem is that the supplier and the user must establish a compatible standard, in order to work together, and since the choice of a standard is a strategic choice, their strategies have to mesh.
Although it looks a lot like the Prisoners' Dilemma at first glance, this is a more complicated game. We'll take several complications in turn:
1. By observing the table carefully, we would notice that there are no dominated strategies in this game. The best strategy for each participant depends on the strategy chosen by the other participant. Thus, we need a new concept of game-equilibrium that will allow for that complication. When there are no dominant strategies, we often use an equilibrium conception called the Nash Equilibrium, named after Nobel Memorial Laureate John Nash.
2. Nash Equilibrium occurs when there is a set of strategies with the property that no player can benefit by changing his / her strategy while the other players keep their strategies unchanged. In this case, this set of strategies and the corresponding payoffs constitute the Nash Equilibrium.
3. The Nash Equilibrium is a pretty simple idea: we have a Nash Equilibrium if each participant chooses the best strategy, given the strategy chosen by the other participant. In the example, if the user opts for the advanced system, then it is best for the supplier to do that too. So (Advanced, Advanced) is a Nash-equilibrium.
4. If the user chooses the proven system, it's best for the supplier to do that too. There are as such two Nash Equilibria. It may seem easy enough to opt for the advanced system which is better all around, but if each participant believes that the other will stick with the proven system, then it will be best for each player to choose the proven system. This is a danger typical of a class of games called coordination games -- and what we have learned is that the choice of compatible standards is a coordination game.
5. We have assumed that the payoffs are known and certain. In the real world, every strategic decision is risky -- and a decision for the advanced system is likely to be riskier than a decision for the proven system. Thus, we would have to take into account the players' subjective attitudes toward risk, in other words their risk aversion, to make the example fully realistic.
6. The example assumes that payoffs are measured in money. Thus, we are not only leaving risk aversion out of the picture, but also any other subjective rewards and penalties that cannot be measured in money. Economists have ways of measuring subjective rewards in money terms. To simplify the analysis, we assume that all rewards and penalties are measured in money and are transferable from the user to the supplier and vice versa.
7. Real choices of information systems are likely to involve more than two players, at least in the long run. The user may choose among several suppliers, and suppliers may have many customers. That makes the coordination problem harder to solve. Suppose, for example, that "beta" is the advanced system and "VHS" is the proven system, and suppose that about 90% of the market uses "VHS." Then "VHS" may take over the market from "beta" even though "beta" is the better system. Many economists, game theorists and others believe this is a main reason why certain technical standards gain dominance.
8. On the other hand, the user and the supplier don't have to just sit back and wait to see what the other person does. They can sit down and talk it out, and commit themselves to a contract. In fact, they have to do so, because the amount of payment from the user to the supplier also has to be agreed upon. In other words, unlike the Prisoners' Dilemma, this is a cooperative game, not a non-cooperative game. On the one hand, that will make the problem of coordinating standards easier, at least in the short run. On the other hand, Cooperative games call for a different approach to solution.
For this example, the players will be a company considering the choice of a new information system, and a supplier who is considering producing it. The two choices are to install a technically advanced or a more proven system with less functionality. We'll assume that the more advanced system really does supply a lot more functionality.
Again, in this case, we can express all this compactly in a payoff table. Basically, the table would indicate that if both the company and supplier choose the technically advanced information system, each earns $20 million in profits from the system, but if the company chooses the advanced system and the supplier does not choose to produce it or vice versa, then both earn nil profits for the period under consideration. However, if both choose the proven information system, then both earn only $5 million of profits each.
We see that both players can be better off, on net, if an advanced system is installed. But the worst that can happen is for one player to commit to an advance system while the other player stays with the proven one. In that case there is no deal, and no payoffs for anyone. The problem is that the supplier and the user must establish a compatible standard, in order to work together, and since the choice of a standard is a strategic choice, their strategies have to mesh.
Although it looks a lot like the Prisoners' Dilemma at first glance, this is a more complicated game. We'll take several complications in turn:
1. By observing the table carefully, we would notice that there are no dominated strategies in this game. The best strategy for each participant depends on the strategy chosen by the other participant. Thus, we need a new concept of game-equilibrium that will allow for that complication. When there are no dominant strategies, we often use an equilibrium conception called the Nash Equilibrium, named after Nobel Memorial Laureate John Nash.
2. Nash Equilibrium occurs when there is a set of strategies with the property that no player can benefit by changing his / her strategy while the other players keep their strategies unchanged. In this case, this set of strategies and the corresponding payoffs constitute the Nash Equilibrium.
3. The Nash Equilibrium is a pretty simple idea: we have a Nash Equilibrium if each participant chooses the best strategy, given the strategy chosen by the other participant. In the example, if the user opts for the advanced system, then it is best for the supplier to do that too. So (Advanced, Advanced) is a Nash-equilibrium.
4. If the user chooses the proven system, it's best for the supplier to do that too. There are as such two Nash Equilibria. It may seem easy enough to opt for the advanced system which is better all around, but if each participant believes that the other will stick with the proven system, then it will be best for each player to choose the proven system. This is a danger typical of a class of games called coordination games -- and what we have learned is that the choice of compatible standards is a coordination game.
5. We have assumed that the payoffs are known and certain. In the real world, every strategic decision is risky -- and a decision for the advanced system is likely to be riskier than a decision for the proven system. Thus, we would have to take into account the players' subjective attitudes toward risk, in other words their risk aversion, to make the example fully realistic.
6. The example assumes that payoffs are measured in money. Thus, we are not only leaving risk aversion out of the picture, but also any other subjective rewards and penalties that cannot be measured in money. Economists have ways of measuring subjective rewards in money terms. To simplify the analysis, we assume that all rewards and penalties are measured in money and are transferable from the user to the supplier and vice versa.
7. Real choices of information systems are likely to involve more than two players, at least in the long run. The user may choose among several suppliers, and suppliers may have many customers. That makes the coordination problem harder to solve. Suppose, for example, that "beta" is the advanced system and "VHS" is the proven system, and suppose that about 90% of the market uses "VHS." Then "VHS" may take over the market from "beta" even though "beta" is the better system. Many economists, game theorists and others believe this is a main reason why certain technical standards gain dominance.
8. On the other hand, the user and the supplier don't have to just sit back and wait to see what the other person does. They can sit down and talk it out, and commit themselves to a contract. In fact, they have to do so, because the amount of payment from the user to the supplier also has to be agreed upon. In other words, unlike the Prisoners' Dilemma, this is a cooperative game, not a non-cooperative game. On the one hand, that will make the problem of coordinating standards easier, at least in the short run. On the other hand, Cooperative games call for a different approach to solution.
Wednesday, March 28, 2007
Game theory: Prisoners’ Dilemma
Game theory is a branch of applied mathematics that deals with the analysis of games (i.e., situations involving parties with conflicting interests). It is a mathematical method of decision-making which involves searching for the best strategy contingent upon what another player will or will not do. Typically, a competitive situation is analyzed to determine the optimal course of action for an interested topic. It is generally taught in mathematics classes such as applied combinatorics, and in economics classes such as industrial organization. In addition to the mathematical elegance and complete "solution" which is possible for simple games, the principles of game theory also find applications to complicated games such as cards, checkers, and chess, as well as real-world problems as diverse as economics, property division, politics, and warfare.
Game theory has two distinct branches: combinatorial game theory and classical game theory.
Combinatorial game theory covers two-player games of perfect knowledge such as go, chess, or checkers. Notably, combinatorial games have no chance element, and players take turns.
In classical game theory, players move, bet, or strategize simultaneously. Both hidden information and chance elements are frequent features in this branch of game theory, which is also a branch of economics.
The Prisoners’ Dilemma is a non-zero sum problem founded in game theory initially discussed by Albert W. Tucker. Tucker's invention of the Prisoners' Dilemma example did not come out via a research paper, but in a classroom. In 1950, while addressing an audience of psychologists at Stanford University in his capacity of visiting professor, Tucker created the Prisoners' Dilemma to illustrate the difficulty of analyzing certain kinds of games.
Tucker’s actual Prisoners' Dilemma example is as follows:
Two burglars, Bob and Al, are captured near the scene of a burglary and are given the "third degree" separately by the police. Each has to choose whether or not to confess and implicate the other. If neither man confesses, then both will serve one year on a charge of carrying a concealed weapon. If each confesses and implicates the other, both will go to prison for 10 years. However, if one burglar confesses and implicates the other, and the other burglar does not confess, the one who has collaborated with the police will go free, while the other burglar will go to prison for 20 years on the maximum charge.
The strategies in this case are those of whether to confess or don't confess. The payoffs or penalties in this case, are the sentences served. We can express all this compactly in a payoff table which has become quite standard in game theory. Basically, the table would indicate that if they both confess, each gets 10 years, but if Al confesses and Bob does not, Bob gets 20 and Al goes free, and vice versa. However, if both do not confess, then both get 1 year each.
A dilemma arises in deciding the best course of action in the absence of knowledge of the other prisoner's decision, as in what strategies are "rational" if both men want to minimize the time they spend in jail. Each prisoner's best strategy would appear to be to turn the other in. Al might reason as follows: "Two things can happen: Bob can confess or Bob can keep quiet. Suppose Bob confesses. Then I get 20 years if I don't confess, 10 years if I do, so in that case it's best to confess. On the other hand, if Bob doesn't confess, and I don't either, I get a year; but in that case, if I confess I can go free. Either way, it's best if I confess. Therefore, I'll confess."
But Bob will presumably reason in the same manner. Therefore, given that both of them confess, both will go to prison for 10 years each. Yet, if they had acted "irrationally" and kept quiet, they each could have gotten off with one year each.
What has happened here is that the two prisoners have fallen into something known as "dominant strategy equilibrium".
A dominant strategy is defined as follows:
If we were to allow an individual player in a game to evaluate separately each of the strategy combinations he may face, and, for each combination, choose from his own strategies the one that gives the best payoff. If the same strategy is chosen for each of the different combinations of strategies the player might face, that strategy is called a "dominant strategy" for that player in that game.
Therefore, dominant strategy equilibrium occurs if, in a game, each player has a dominant strategy, and each player plays the dominant strategy, then that combination of the dominant strategies and the corresponding payoffs are said to constitute the dominant strategy equilibrium for that game.
In the Prisoners' Dilemma game, to confess is a dominant strategy, and when both prisoners confess, dominant strategy equilibrium occurs. In this case, the individually rational action results in both persons being made worse off in terms of their own self-interested purposes. This revelation has wide implications in modern social science. This is because there are many interactions in the modern world that seem very much like this, from arms races through road congestion and pollution to the depletion of fisheries and the overexploitation of some subsurface water resources. These are all quite different interactions in detail, but are interactions in which individually rational action leads to inferior results for each person, and the Prisoners' Dilemma suggests something of what is going on in each of them.
A number of critical issues can be raised with the Prisoners' Dilemma in view of its simplified and abstract conception of many real life interactions, and each of these issues has been the basis of a large scholarly literature:
1. The Prisoners' Dilemma is a two-person game, but many of the applications of the idea are really many-person interactions;
2. We have assumed that there is no communication between the two prisoners. If they could communicate and commit themselves to coordinated strategies, we would expect a quite different outcome;
3. In the Prisoners' Dilemma, the two prisoners interact only once. Repetition of the interactions might lead to quite different results;
4. Compelling as the reasoning that leads to the dominant strategy equilibrium may be, it is not the only way this problem might be reasoned out. Perhaps it is not really the most rational answer after all.
The Prisoners' Dilemma has wide applications to economics and business. Let’s take an example of two firms, say A and B, selling similar products. Each must decide on a pricing strategy. They best exploit their joint market power when both charge a high price; each makes a profit of $10 million per month. If one sets a competitive low price, it wins a lot of customers away from the rival. Suppose its profit rises to $12 million, and that of the rival falls to $7 million. If both set low prices, the profit of each is $9 million. In this case, the low-price strategy is akin to the prisoner's confession, and the high-price akin to keeping silent. Let’s term the former cheating, and the latter cooperation. In this case, cheating is each firm's dominant strategy, but the result when both cheat is worse for each than that of both cooperating.
On a superficial level the Prisoners' Dilemma appears to run counter to Adam Smith's idea of the invisible hand. When each person in the game pursues his private interest, he does not promote the collective interest of the group. But often a group's cooperation is not in the interests of society as a whole. Collusion to keep prices high, for example, is not in society's interest because the cost to consumers from collusion is generally more than the increased profit of the firms. Therefore companies that pursue their own self-interest by cheating on collusive agreements often help the rest of society. Similarly cooperation among prisoners under interrogation makes convictions more difficult for the police to obtain. One must understand the mechanism of cooperation before one can either promote or defeat it in the pursuit of larger policy interests.
Would the Prisoners be able to extricate themselves from the Dilemma and sustain cooperation when each has a powerful incentive to cheat? The most common path to cooperation arises from repetitions of the game. In the above example, one month's cheating gets the cheater an extra $2 million. But a switch from mutual cooperation to mutual cheating loses $1 million. If one month's cheating is followed by two months' retaliation, therefore, the result is a wash for the cheater. Any stronger punishment of a cheater would be a clear deterrent.
This idea needs some comment and elaboration:
1. The cheater's reward comes at once, while the loss from punishment lies in the future. If players heavily discount future payoffs, then the loss may be insufficient to deter cheating. Thus, cooperation is harder to sustain among very impatient players.
2. Punishment won't work unless cheating can be detected and punished. Therefore, companies cooperate more when their actions are more easily detected (setting prices, for example) and less when actions are less easily detected (deciding on non-price attributes of goods, such as repair warranties). Punishment is usually easier to arrange in smaller and closed groups. Thus, industries with few firms and less threat of new entry are more likely to be collusive.
3. Punishment can be made automatic by following strategies like "tit for tat," which was popularized by University of Michigan political scientist Robert Axelrod. In this case, you cheat if and only if your rival cheated in the previous round. But if rivals' innocent actions can be misinterpreted as cheating, then tit for tat runs the risk of setting off successive rounds of unwarranted retaliation.
4. A fixed, finite number of repetitions are logically inadequate to yield cooperation. Both or all players know that cheating is the dominant strategy in the last play. Given this, the same goes for the second-last play, then the third-last, and so on. But in practice we see some cooperation in the early rounds of a fixed set of repetitions. The reason may be either that players don't know the number of rounds for sure, or that they can exploit the possibility of "irrational niceness" to their mutual advantage.
5. Cooperation can also arise if the group has a large leader, who personally stands to lose a lot from outright competition and therefore exercises restraint, even though he knows that other small players will cheat. For example, Saudi Arabia's role of "swing producer" in the OPEC cartel is an instance of this.
Game theory has two distinct branches: combinatorial game theory and classical game theory.
Combinatorial game theory covers two-player games of perfect knowledge such as go, chess, or checkers. Notably, combinatorial games have no chance element, and players take turns.
In classical game theory, players move, bet, or strategize simultaneously. Both hidden information and chance elements are frequent features in this branch of game theory, which is also a branch of economics.
The Prisoners’ Dilemma is a non-zero sum problem founded in game theory initially discussed by Albert W. Tucker. Tucker's invention of the Prisoners' Dilemma example did not come out via a research paper, but in a classroom. In 1950, while addressing an audience of psychologists at Stanford University in his capacity of visiting professor, Tucker created the Prisoners' Dilemma to illustrate the difficulty of analyzing certain kinds of games.
Tucker’s actual Prisoners' Dilemma example is as follows:
Two burglars, Bob and Al, are captured near the scene of a burglary and are given the "third degree" separately by the police. Each has to choose whether or not to confess and implicate the other. If neither man confesses, then both will serve one year on a charge of carrying a concealed weapon. If each confesses and implicates the other, both will go to prison for 10 years. However, if one burglar confesses and implicates the other, and the other burglar does not confess, the one who has collaborated with the police will go free, while the other burglar will go to prison for 20 years on the maximum charge.
The strategies in this case are those of whether to confess or don't confess. The payoffs or penalties in this case, are the sentences served. We can express all this compactly in a payoff table which has become quite standard in game theory. Basically, the table would indicate that if they both confess, each gets 10 years, but if Al confesses and Bob does not, Bob gets 20 and Al goes free, and vice versa. However, if both do not confess, then both get 1 year each.
A dilemma arises in deciding the best course of action in the absence of knowledge of the other prisoner's decision, as in what strategies are "rational" if both men want to minimize the time they spend in jail. Each prisoner's best strategy would appear to be to turn the other in. Al might reason as follows: "Two things can happen: Bob can confess or Bob can keep quiet. Suppose Bob confesses. Then I get 20 years if I don't confess, 10 years if I do, so in that case it's best to confess. On the other hand, if Bob doesn't confess, and I don't either, I get a year; but in that case, if I confess I can go free. Either way, it's best if I confess. Therefore, I'll confess."
But Bob will presumably reason in the same manner. Therefore, given that both of them confess, both will go to prison for 10 years each. Yet, if they had acted "irrationally" and kept quiet, they each could have gotten off with one year each.
What has happened here is that the two prisoners have fallen into something known as "dominant strategy equilibrium".
A dominant strategy is defined as follows:
If we were to allow an individual player in a game to evaluate separately each of the strategy combinations he may face, and, for each combination, choose from his own strategies the one that gives the best payoff. If the same strategy is chosen for each of the different combinations of strategies the player might face, that strategy is called a "dominant strategy" for that player in that game.
Therefore, dominant strategy equilibrium occurs if, in a game, each player has a dominant strategy, and each player plays the dominant strategy, then that combination of the dominant strategies and the corresponding payoffs are said to constitute the dominant strategy equilibrium for that game.
In the Prisoners' Dilemma game, to confess is a dominant strategy, and when both prisoners confess, dominant strategy equilibrium occurs. In this case, the individually rational action results in both persons being made worse off in terms of their own self-interested purposes. This revelation has wide implications in modern social science. This is because there are many interactions in the modern world that seem very much like this, from arms races through road congestion and pollution to the depletion of fisheries and the overexploitation of some subsurface water resources. These are all quite different interactions in detail, but are interactions in which individually rational action leads to inferior results for each person, and the Prisoners' Dilemma suggests something of what is going on in each of them.
A number of critical issues can be raised with the Prisoners' Dilemma in view of its simplified and abstract conception of many real life interactions, and each of these issues has been the basis of a large scholarly literature:
1. The Prisoners' Dilemma is a two-person game, but many of the applications of the idea are really many-person interactions;
2. We have assumed that there is no communication between the two prisoners. If they could communicate and commit themselves to coordinated strategies, we would expect a quite different outcome;
3. In the Prisoners' Dilemma, the two prisoners interact only once. Repetition of the interactions might lead to quite different results;
4. Compelling as the reasoning that leads to the dominant strategy equilibrium may be, it is not the only way this problem might be reasoned out. Perhaps it is not really the most rational answer after all.
The Prisoners' Dilemma has wide applications to economics and business. Let’s take an example of two firms, say A and B, selling similar products. Each must decide on a pricing strategy. They best exploit their joint market power when both charge a high price; each makes a profit of $10 million per month. If one sets a competitive low price, it wins a lot of customers away from the rival. Suppose its profit rises to $12 million, and that of the rival falls to $7 million. If both set low prices, the profit of each is $9 million. In this case, the low-price strategy is akin to the prisoner's confession, and the high-price akin to keeping silent. Let’s term the former cheating, and the latter cooperation. In this case, cheating is each firm's dominant strategy, but the result when both cheat is worse for each than that of both cooperating.
On a superficial level the Prisoners' Dilemma appears to run counter to Adam Smith's idea of the invisible hand. When each person in the game pursues his private interest, he does not promote the collective interest of the group. But often a group's cooperation is not in the interests of society as a whole. Collusion to keep prices high, for example, is not in society's interest because the cost to consumers from collusion is generally more than the increased profit of the firms. Therefore companies that pursue their own self-interest by cheating on collusive agreements often help the rest of society. Similarly cooperation among prisoners under interrogation makes convictions more difficult for the police to obtain. One must understand the mechanism of cooperation before one can either promote or defeat it in the pursuit of larger policy interests.
Would the Prisoners be able to extricate themselves from the Dilemma and sustain cooperation when each has a powerful incentive to cheat? The most common path to cooperation arises from repetitions of the game. In the above example, one month's cheating gets the cheater an extra $2 million. But a switch from mutual cooperation to mutual cheating loses $1 million. If one month's cheating is followed by two months' retaliation, therefore, the result is a wash for the cheater. Any stronger punishment of a cheater would be a clear deterrent.
This idea needs some comment and elaboration:
1. The cheater's reward comes at once, while the loss from punishment lies in the future. If players heavily discount future payoffs, then the loss may be insufficient to deter cheating. Thus, cooperation is harder to sustain among very impatient players.
2. Punishment won't work unless cheating can be detected and punished. Therefore, companies cooperate more when their actions are more easily detected (setting prices, for example) and less when actions are less easily detected (deciding on non-price attributes of goods, such as repair warranties). Punishment is usually easier to arrange in smaller and closed groups. Thus, industries with few firms and less threat of new entry are more likely to be collusive.
3. Punishment can be made automatic by following strategies like "tit for tat," which was popularized by University of Michigan political scientist Robert Axelrod. In this case, you cheat if and only if your rival cheated in the previous round. But if rivals' innocent actions can be misinterpreted as cheating, then tit for tat runs the risk of setting off successive rounds of unwarranted retaliation.
4. A fixed, finite number of repetitions are logically inadequate to yield cooperation. Both or all players know that cheating is the dominant strategy in the last play. Given this, the same goes for the second-last play, then the third-last, and so on. But in practice we see some cooperation in the early rounds of a fixed set of repetitions. The reason may be either that players don't know the number of rounds for sure, or that they can exploit the possibility of "irrational niceness" to their mutual advantage.
5. Cooperation can also arise if the group has a large leader, who personally stands to lose a lot from outright competition and therefore exercises restraint, even though he knows that other small players will cheat. For example, Saudi Arabia's role of "swing producer" in the OPEC cartel is an instance of this.
Wednesday, March 01, 2006
Game theory: Prisoners’ Dilemma
Game theory is a branch of applied mathematics that deals with the analysis of games (i.e., situations involving parties with conflicting interests). It is a mathematical method of decision-making which involves searching for the best strategy contingent upon what another player will or will not do. Typically, a competitive situation is analyzed to determine the optimal course of action for an interested topic. It is generally taught in mathematics classes such as applied combinatorics, and in economics classes such as industrial organization. In addition to the mathematical elegance and complete "solution" which is possible for simple games, the principles of game theory also find applications to complicated games such as cards, checkers, and chess, as well as real-world problems as diverse as economics, property division, politics, and warfare.
Game theory has two distinct branches: combinatorial game theory and classical game theory.
Combinatorial game theory covers two-player games of perfect knowledge such as go, chess, or checkers. Notably, combinatorial games have no chance element, and players take turns.
In classical game theory, players move, bet, or strategize simultaneously. Both hidden information and chance elements are frequent features in this branch of game theory, which is also a branch of economics.
The Prisoners’ Dilemma is a non-zero sum problem founded in game theory initially discussed by Albert W. Tucker. Tucker's invention of the Prisoners' Dilemma example did not come out via a research paper, but in a classroom. In 1950, while addressing an audience of psychologists at Stanford University in his capacity of visiting professor, Tucker created the Prisoners' Dilemma to illustrate the difficulty of analyzing certain kinds of games.
Tucker’s actual Prisoners' Dilemma example is as follows:
Two burglars, Bob and Al, are captured near the scene of a burglary and are given the "third degree" separately by the police. Each has to choose whether or not to confess and implicate the other. If neither man confesses, then both will serve one year on a charge of carrying a concealed weapon. If each confesses and implicates the other, both will go to prison for 10 years. However, if one burglar confesses and implicates the other, and the other burglar does not confess, the one who has collaborated with the police will go free, while the other burglar will go to prison for 20 years on the maximum charge.
The strategies in this case are those of whether to confess or don't confess. The payoffs or penalties in this case, are the sentences served. We can express all this compactly in a payoff table as follows which has become quite standard in game theory. Here is the payoff table for the Prisoners' Dilemma game:
The above table is interpreted as follows:
Each prisoner chooses one of the two strategies. In effect, Al chooses a column and Bob chooses a row. The two numbers in each cell tell the outcomes for the two prisoners when the corresponding pair of strategies is chosen. The number to the left of the comma tells the payoff to the person who chooses the rows (Bob) while the number to the right of the column tells the payoff to the person who chooses the columns (Al). Thus (reading down the first column) if they both confess, each gets 10 years, but if Al confesses and Bob does not, Bob gets 20 and Al goes free.
A dilemma arises in deciding the best course of action in the absence of knowledge of the other prisoner's decision, as in what strategies are "rational" if both men want to minimize the time they spend in jail. Each prisoner's best strategy would appear to be to turn the other in. Al might reason as follows: "Two things can happen: Bob can confess or Bob can keep quiet. Suppose Bob confesses. Then I get 20 years if I don't confess, 10 years if I do, so in that case it's best to confess. On the other hand, if Bob doesn't confess, and I don't either, I get a year; but in that case, if I confess I can go free. Either way, it's best if I confess. Therefore, I'll confess."
But Bob will presumably reason in the same manner. Therefore, given that both of them confess, both will go to prison for 10 years each. Yet, if they had acted "irrationally" and kept quiet, they each could have gotten off with one year each.
What has happened here is that the two prisoners have fallen into something known as "dominant strategy equilibrium".
A dominant strategy is defined as follows:
If we were to allow an individual player in a game to evaluate separately each of the strategy combinations he may face, and, for each combination, choose from his own strategies the one that gives the best payoff. If the same strategy is chosen for each of the different combinations of strategies the player might face, that strategy is called a "dominant strategy" for that player in that game.
Therefore, dominant strategy equilibrium occurs if, in a game, each player has a dominant strategy, and each player plays the dominant strategy, then that combination of the dominant strategies and the corresponding payoffs are said to constitute the dominant strategy equilibrium for that game.
In the Prisoners' Dilemma game, to confess is a dominant strategy, and when both prisoners confess, dominant strategy equilibrium occurs. In this case, the individually rational action results in both persons being made worse off in terms of their own self-interested purposes. This revelation has wide implications in modern social science. This is because there are many interactions in the modern world that seem very much like this, from arms races through road congestion and pollution to the depletion of fisheries and the overexploitation of some subsurface water resources. These are all quite different interactions in detail, but are interactions in which individually rational action leads to inferior results for each person, and the Prisoners' Dilemma suggests something of what is going on in each of them.
A number of critical issues can be raised with the Prisoners' Dilemma in view of its simplified and abstract conception of many real life interactions, and each of these issues has been the basis of a large scholarly literature:
1. The Prisoners' Dilemma is a two-person game, but many of the applications of the idea are really many-person interactions;
2. We have assumed that there is no communication between the two prisoners. If they could communicate and commit themselves to coordinated strategies, we would expect a quite different outcome;
3. In the Prisoners' Dilemma, the two prisoners interact only once. Repetition of the interactions might lead to quite different results;
4. Compelling as the reasoning that leads to the dominant strategy equilibrium may be, it is not the only way this problem might be reasoned out. Perhaps it is not really the most rational answer after all.
The Prisoners' Dilemma has wide applications to economics and business. Let’s take an example of two firms, say A and B, selling similar products. Each must decide on a pricing strategy. They best exploit their joint market power when both charge a high price; each makes a profit of $10 million per month. If one sets a competitive low price, it wins a lot of customers away from the rival. Suppose its profit rises to $12 million, and that of the rival falls to $7 million. If both set low prices, the profit of each is $9 million. In this case, the low-price strategy is akin to the prisoner's confession, and the high-price akin to keeping silent. Let’s term the former cheating, and the latter cooperation. In this case, cheating is each firm's dominant strategy, but the result when both cheat is worse for each than that of both cooperating.
On a superficial level the Prisoners' Dilemma appears to run counter to Adam Smith's idea of the invisible hand. When each person in the game pursues his private interest, he does not promote the collective interest of the group. But often a group's cooperation is not in the interests of society as a whole. Collusion to keep prices high, for example, is not in society's interest because the cost to consumers from collusion is generally more than the increased profit of the firms. Therefore companies that pursue their own self-interest by cheating on collusive agreements often help the rest of society. Similarly cooperation among prisoners under interrogation makes convictions more difficult for the police to obtain. One must understand the mechanism of cooperation before one can either promote or defeat it in the pursuit of larger policy interests.
Would the Prisoners be able to extricate themselves from the Dilemma and sustain cooperation when each has a powerful incentive to cheat? The most common path to cooperation arises from repetitions of the game. In the above example, one month's cheating gets the cheater an extra $2 million. But a switch from mutual cooperation to mutual cheating loses $1 million. If one month's cheating is followed by two months' retaliation, therefore, the result is a wash for the cheater. Any stronger punishment of a cheater would be a clear deterrent.
This idea needs some comment and elaboration:
1. The cheater's reward comes at once, while the loss from punishment lies in the future. If players heavily discount future payoffs, then the loss may be insufficient to deter cheating. Thus, cooperation is harder to sustain among very impatient players.
2. Punishment won't work unless cheating can be detected and punished. Therefore, companies cooperate more when their actions are more easily detected (setting prices, for example) and less when actions are less easily detected (deciding on non-price attributes of goods, such as repair warranties). Punishment is usually easier to arrange in smaller and closed groups. Thus, industries with few firms and less threat of new entry are more likely to be collusive.
3. Punishment can be made automatic by following strategies like "tit for tat," which was popularized by University of Michigan political scientist Robert Axelrod. In this case, you cheat if and only if your rival cheated in the previous round. But if rivals' innocent actions can be misinterpreted as cheating, then tit for tat runs the risk of setting off successive rounds of unwarranted retaliation.
4. A fixed, finite number of repetitions are logically inadequate to yield cooperation. Both or all players know that cheating is the dominant strategy in the last play. Given this, the same goes for the second-last play, then the third-last, and so on. But in practice we see some cooperation in the early rounds of a fixed set of repetitions. The reason may be either that players don't know the number of rounds for sure, or that they can exploit the possibility of "irrational niceness" to their mutual advantage.
5. Cooperation can also arise if the group has a large leader, who personally stands to lose a lot from outright competition and therefore exercises restraint, even though he knows that other small players will cheat. For example, Saudi Arabia's role of "swing producer" in the OPEC cartel is an instance of this.
Game theory has two distinct branches: combinatorial game theory and classical game theory.
Combinatorial game theory covers two-player games of perfect knowledge such as go, chess, or checkers. Notably, combinatorial games have no chance element, and players take turns.
In classical game theory, players move, bet, or strategize simultaneously. Both hidden information and chance elements are frequent features in this branch of game theory, which is also a branch of economics.
The Prisoners’ Dilemma is a non-zero sum problem founded in game theory initially discussed by Albert W. Tucker. Tucker's invention of the Prisoners' Dilemma example did not come out via a research paper, but in a classroom. In 1950, while addressing an audience of psychologists at Stanford University in his capacity of visiting professor, Tucker created the Prisoners' Dilemma to illustrate the difficulty of analyzing certain kinds of games.
Tucker’s actual Prisoners' Dilemma example is as follows:
Two burglars, Bob and Al, are captured near the scene of a burglary and are given the "third degree" separately by the police. Each has to choose whether or not to confess and implicate the other. If neither man confesses, then both will serve one year on a charge of carrying a concealed weapon. If each confesses and implicates the other, both will go to prison for 10 years. However, if one burglar confesses and implicates the other, and the other burglar does not confess, the one who has collaborated with the police will go free, while the other burglar will go to prison for 20 years on the maximum charge.
The strategies in this case are those of whether to confess or don't confess. The payoffs or penalties in this case, are the sentences served. We can express all this compactly in a payoff table as follows which has become quite standard in game theory. Here is the payoff table for the Prisoners' Dilemma game:
The above table is interpreted as follows:
Each prisoner chooses one of the two strategies. In effect, Al chooses a column and Bob chooses a row. The two numbers in each cell tell the outcomes for the two prisoners when the corresponding pair of strategies is chosen. The number to the left of the comma tells the payoff to the person who chooses the rows (Bob) while the number to the right of the column tells the payoff to the person who chooses the columns (Al). Thus (reading down the first column) if they both confess, each gets 10 years, but if Al confesses and Bob does not, Bob gets 20 and Al goes free.
A dilemma arises in deciding the best course of action in the absence of knowledge of the other prisoner's decision, as in what strategies are "rational" if both men want to minimize the time they spend in jail. Each prisoner's best strategy would appear to be to turn the other in. Al might reason as follows: "Two things can happen: Bob can confess or Bob can keep quiet. Suppose Bob confesses. Then I get 20 years if I don't confess, 10 years if I do, so in that case it's best to confess. On the other hand, if Bob doesn't confess, and I don't either, I get a year; but in that case, if I confess I can go free. Either way, it's best if I confess. Therefore, I'll confess."
But Bob will presumably reason in the same manner. Therefore, given that both of them confess, both will go to prison for 10 years each. Yet, if they had acted "irrationally" and kept quiet, they each could have gotten off with one year each.
What has happened here is that the two prisoners have fallen into something known as "dominant strategy equilibrium".
A dominant strategy is defined as follows:
If we were to allow an individual player in a game to evaluate separately each of the strategy combinations he may face, and, for each combination, choose from his own strategies the one that gives the best payoff. If the same strategy is chosen for each of the different combinations of strategies the player might face, that strategy is called a "dominant strategy" for that player in that game.
Therefore, dominant strategy equilibrium occurs if, in a game, each player has a dominant strategy, and each player plays the dominant strategy, then that combination of the dominant strategies and the corresponding payoffs are said to constitute the dominant strategy equilibrium for that game.
In the Prisoners' Dilemma game, to confess is a dominant strategy, and when both prisoners confess, dominant strategy equilibrium occurs. In this case, the individually rational action results in both persons being made worse off in terms of their own self-interested purposes. This revelation has wide implications in modern social science. This is because there are many interactions in the modern world that seem very much like this, from arms races through road congestion and pollution to the depletion of fisheries and the overexploitation of some subsurface water resources. These are all quite different interactions in detail, but are interactions in which individually rational action leads to inferior results for each person, and the Prisoners' Dilemma suggests something of what is going on in each of them.
A number of critical issues can be raised with the Prisoners' Dilemma in view of its simplified and abstract conception of many real life interactions, and each of these issues has been the basis of a large scholarly literature:
1. The Prisoners' Dilemma is a two-person game, but many of the applications of the idea are really many-person interactions;
2. We have assumed that there is no communication between the two prisoners. If they could communicate and commit themselves to coordinated strategies, we would expect a quite different outcome;
3. In the Prisoners' Dilemma, the two prisoners interact only once. Repetition of the interactions might lead to quite different results;
4. Compelling as the reasoning that leads to the dominant strategy equilibrium may be, it is not the only way this problem might be reasoned out. Perhaps it is not really the most rational answer after all.
The Prisoners' Dilemma has wide applications to economics and business. Let’s take an example of two firms, say A and B, selling similar products. Each must decide on a pricing strategy. They best exploit their joint market power when both charge a high price; each makes a profit of $10 million per month. If one sets a competitive low price, it wins a lot of customers away from the rival. Suppose its profit rises to $12 million, and that of the rival falls to $7 million. If both set low prices, the profit of each is $9 million. In this case, the low-price strategy is akin to the prisoner's confession, and the high-price akin to keeping silent. Let’s term the former cheating, and the latter cooperation. In this case, cheating is each firm's dominant strategy, but the result when both cheat is worse for each than that of both cooperating.
On a superficial level the Prisoners' Dilemma appears to run counter to Adam Smith's idea of the invisible hand. When each person in the game pursues his private interest, he does not promote the collective interest of the group. But often a group's cooperation is not in the interests of society as a whole. Collusion to keep prices high, for example, is not in society's interest because the cost to consumers from collusion is generally more than the increased profit of the firms. Therefore companies that pursue their own self-interest by cheating on collusive agreements often help the rest of society. Similarly cooperation among prisoners under interrogation makes convictions more difficult for the police to obtain. One must understand the mechanism of cooperation before one can either promote or defeat it in the pursuit of larger policy interests.
Would the Prisoners be able to extricate themselves from the Dilemma and sustain cooperation when each has a powerful incentive to cheat? The most common path to cooperation arises from repetitions of the game. In the above example, one month's cheating gets the cheater an extra $2 million. But a switch from mutual cooperation to mutual cheating loses $1 million. If one month's cheating is followed by two months' retaliation, therefore, the result is a wash for the cheater. Any stronger punishment of a cheater would be a clear deterrent.
This idea needs some comment and elaboration:
1. The cheater's reward comes at once, while the loss from punishment lies in the future. If players heavily discount future payoffs, then the loss may be insufficient to deter cheating. Thus, cooperation is harder to sustain among very impatient players.
2. Punishment won't work unless cheating can be detected and punished. Therefore, companies cooperate more when their actions are more easily detected (setting prices, for example) and less when actions are less easily detected (deciding on non-price attributes of goods, such as repair warranties). Punishment is usually easier to arrange in smaller and closed groups. Thus, industries with few firms and less threat of new entry are more likely to be collusive.
3. Punishment can be made automatic by following strategies like "tit for tat," which was popularized by University of Michigan political scientist Robert Axelrod. In this case, you cheat if and only if your rival cheated in the previous round. But if rivals' innocent actions can be misinterpreted as cheating, then tit for tat runs the risk of setting off successive rounds of unwarranted retaliation.
4. A fixed, finite number of repetitions are logically inadequate to yield cooperation. Both or all players know that cheating is the dominant strategy in the last play. Given this, the same goes for the second-last play, then the third-last, and so on. But in practice we see some cooperation in the early rounds of a fixed set of repetitions. The reason may be either that players don't know the number of rounds for sure, or that they can exploit the possibility of "irrational niceness" to their mutual advantage.
5. Cooperation can also arise if the group has a large leader, who personally stands to lose a lot from outright competition and therefore exercises restraint, even though he knows that other small players will cheat. For example, Saudi Arabia's role of "swing producer" in the OPEC cartel is an instance of this.
Monday, January 30, 2006
Pareto's Principle (80/20 Rule)
The Pareto principle, also known as the 80/20 rule, the law of the vital few and the principle of factor sparsity, states that for many phenomena, 80% of the consequences stem from only 20% of the causes.
The principle was suggested by the Quality Management pioneer and thinker, Dr. Joseph M. Juran, in the late 1940s. Juran inaccurately attributed the 80/20 Rule to a phenomenon who was first proposed by the Italian economist Vilfredo Pareto, who in 1906, created a mathematical formula to describe the unequal distribution of wealth in his country. He observed that 80% of all properties in Italy were owned by only 20% of the Italian population. In other words, 20% of the Italians owned 80% of the country’s wealth.
After Pareto made his observation and created his formula, many others, like Juran, observed similar phenomena in their own areas of expertise. Juran, recognized a universal principle which he called the "vital few and trivial many" and reduced it to writing. In an early work, a lack of precision on Juran's part made it appear that he was applying Pareto's observations about economics to a broader body of work. As such, the name Pareto's Principle stuck on.
As a result, Dr. Juran's observation of the "vital few and trivial many", the principle that 20% of something always are responsible for 80% of the results, became known as Pareto's Principle or the 80/20 Rule.
The 80/20 Rule means that in anything a few (20%) are vital and many (80%) are trivial. In Pareto's case it meant 20% of the people owned 80% of the wealth. In Juran's initial work, he identified 20% of the defects causing 80% of the problems. Experienced project managers will know that 20% of the work (the first 10% and the last 10%) consume 80% of the time and resources available. One can apply the 80/20 Rule to almost anything, from the science of management to the physical world.
The value of the Pareto Principle for a manager is that it reminds the manager to focus on the 20% that matters. Of the things the manager do during your day, only 20% really matter. Those 20% produce 80% of the results. It is critical that the manager identifies and focuses on those things. When it comes to the crunch, the manager needs to remind himself of the 20% he needs to focus on. So if something in the schedule has to slip, if something isn't going to get done, the manager must make sure that it's not part of that 20%.
Pareto's Principle, the 80/20 Rule, should serve as a daily reminder to focus 80% of the time and energy on the 20% of the work that is really important. So the wise manager should not just work smart, he should work smart on the right things.
The principle was suggested by the Quality Management pioneer and thinker, Dr. Joseph M. Juran, in the late 1940s. Juran inaccurately attributed the 80/20 Rule to a phenomenon who was first proposed by the Italian economist Vilfredo Pareto, who in 1906, created a mathematical formula to describe the unequal distribution of wealth in his country. He observed that 80% of all properties in Italy were owned by only 20% of the Italian population. In other words, 20% of the Italians owned 80% of the country’s wealth.
After Pareto made his observation and created his formula, many others, like Juran, observed similar phenomena in their own areas of expertise. Juran, recognized a universal principle which he called the "vital few and trivial many" and reduced it to writing. In an early work, a lack of precision on Juran's part made it appear that he was applying Pareto's observations about economics to a broader body of work. As such, the name Pareto's Principle stuck on.
As a result, Dr. Juran's observation of the "vital few and trivial many", the principle that 20% of something always are responsible for 80% of the results, became known as Pareto's Principle or the 80/20 Rule.
The 80/20 Rule means that in anything a few (20%) are vital and many (80%) are trivial. In Pareto's case it meant 20% of the people owned 80% of the wealth. In Juran's initial work, he identified 20% of the defects causing 80% of the problems. Experienced project managers will know that 20% of the work (the first 10% and the last 10%) consume 80% of the time and resources available. One can apply the 80/20 Rule to almost anything, from the science of management to the physical world.
The value of the Pareto Principle for a manager is that it reminds the manager to focus on the 20% that matters. Of the things the manager do during your day, only 20% really matter. Those 20% produce 80% of the results. It is critical that the manager identifies and focuses on those things. When it comes to the crunch, the manager needs to remind himself of the 20% he needs to focus on. So if something in the schedule has to slip, if something isn't going to get done, the manager must make sure that it's not part of that 20%.
Pareto's Principle, the 80/20 Rule, should serve as a daily reminder to focus 80% of the time and energy on the 20% of the work that is really important. So the wise manager should not just work smart, he should work smart on the right things.
Thursday, April 21, 2005
Syllogism - Aristotlean logic
A syllogism is an inference in which one proposition (the conclusion) follows of necessity from two others (known as premises) and this forms the foundation of traditional logic. This definition is traditional in nature, but is derived loosely from Aristotle's Prior Analytics, as such, syllogism is also known popularly as Aristotlean logic. As a matter of interest, the word “syllogism” has its roots from the Greek word "sullogismos", which means "deduction".
Syllogisms consist of three things: major, minor (the premises) and conclusion, which follows logically from the major and the minor. A major is a general principle. A minor is a specific statement. Logically, the conclusion follows from applying the major to the minor.
For example, this is the classic "Barbara" syllogism, given by Aristotle:
If all humans (B's) are mortal (A), (major)
and all Greeks (C's) are humans (B's), (minor)
then all Greeks (C's) are mortal (A). (conclusion)
That is,
Men die. (general principle)
Socrates is a man. (specific statement)
Socrates will die. (application of major to minor)
A metaphor, in contrast, resembles a form of syllogism called affirming the consequent, which is a logical fallacy:
Dogs (B) die (A).
Men (C's) die (A).
Men (C's) are dogs (B).
A Barbara syllogism involves grammar and logical types; it has a subject (e.g. Socrates) and a predicate (mortal). Affirming the Consequent, the basis of metaphor, is grammatically symmetrical: it equates two predicates. This form of syllogism is logically invalid.
Syllogisms may also be invalid if they have four terms or the middle term is not distributed.
Epagoge are weak syllogisms that rely on inductive reasoning.
The conclusion is a biconditional only when all premises are biconditionals. This statement is of great practical value. In a succession of deductions we must pay close attention to see if the transition from one proposition to the other takes place by means of a biconditional or only of a conditional. There is no equivalence between two extreme propositions unless all intermediate deductions are equivalences; in other words, if there is one single implication in the chain, the relation of the two extreme propositions is only that of implication.
Syllogisms consist of three things: major, minor (the premises) and conclusion, which follows logically from the major and the minor. A major is a general principle. A minor is a specific statement. Logically, the conclusion follows from applying the major to the minor.
For example, this is the classic "Barbara" syllogism, given by Aristotle:
If all humans (B's) are mortal (A), (major)
and all Greeks (C's) are humans (B's), (minor)
then all Greeks (C's) are mortal (A). (conclusion)
That is,
Men die. (general principle)
Socrates is a man. (specific statement)
Socrates will die. (application of major to minor)
A metaphor, in contrast, resembles a form of syllogism called affirming the consequent, which is a logical fallacy:
Dogs (B) die (A).
Men (C's) die (A).
Men (C's) are dogs (B).
A Barbara syllogism involves grammar and logical types; it has a subject (e.g. Socrates) and a predicate (mortal). Affirming the Consequent, the basis of metaphor, is grammatically symmetrical: it equates two predicates. This form of syllogism is logically invalid.
Syllogisms may also be invalid if they have four terms or the middle term is not distributed.
Epagoge are weak syllogisms that rely on inductive reasoning.
The conclusion is a biconditional only when all premises are biconditionals. This statement is of great practical value. In a succession of deductions we must pay close attention to see if the transition from one proposition to the other takes place by means of a biconditional or only of a conditional. There is no equivalence between two extreme propositions unless all intermediate deductions are equivalences; in other words, if there is one single implication in the chain, the relation of the two extreme propositions is only that of implication.
Wednesday, December 08, 2004
Six Degrees of Separation
Defination
The Six Degrees of Separation theory, also known as the Small World Phenomenon, contends that all people in the world are connected together through a chain of no more than six people. The theory is that there are only six degrees (or levels) of separation between you and everyone else in the world. This is the idea that everyone in the world can be reached through a short chain of social acquaintances. In other words, everyone in the world is separated from anyone else by no more than six degrees of separation, or simply speaking, six acquaintances or friend. A degree of separation is defined as an acquaintance or friend who separates you from someone else. As such, there is zero degree of separation between you and your immediate friends or acquaintances.
History
In 1967, the Harvard Social Psychologist Stanley Milgram sent out roughly 300 letters to randomly selected people in Omaha, Nebraska with the instruction to get the letter to a single "target" person in Boston using only personal contacts.
Milgram gave each "sender" some information about the target including name, location, and occupation, so that if the sender did not know the target, and it was extremely unlikely that they would, they could send the letter to someone whom they did know who they thought would be "closer" to the target. This began a chain of senders, with each member of the chain attempting to zero in on the target by sending the letter to someone else, may these people be a friend, a family member, a business associate, or a casual acquaintance.
Milgram's surprising finding was that for the 60 chains that eventually reached the target, the average number of steps in a chain was around six, a result that has entered folklore as the phrase "six degrees of separation."
While Milgram's first experiment suggests it is, other experiments have been less conclusive, and no experiment has been done to test the theory on a global scale.
Mathematics
We assume that a person only knows 50 other persons, and each of these 50 persons in turn know another 50 non-redundant persons (this assumption eradicates the possibility of duplication of persons with the first person). In this case, each of the 50 persons would then know another 50 non-redundant persons and so on up to six degrees.
This works out to be:
50x 50 x 50 x 50 x 50 x 50 = 15.63 x 1010 or about 15 to 16 billion persons!
If the current entire population of the whole world is only about 6 to 7 billion persons, then this computation would show that six degrees of separation is enough to cover the world's population.
Thursday, June 24, 2004
The Butterfly Effect
Predictability: Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?
The butterfly effect technically refers to the propensity of a system to be sensitive to initial conditions. It is the sensitive dependency on its initial conditions that the system, over time, becomes wholly unpredictable.
The butterfly effect is however not quite as similiar as the domino effect. For the domino effect, there is no doubt dependency of the system on the initial conditions. But a simple linear row of dominoes would merely allow one event to initiate another similar event upon each iteration. The butterfly effect however amplifies the condition upon each iteration.
The butterfly effect has been most commonly associated with the weather system as this is where the discovery of non-linear phenomenon within a complex and dynamic system began. Due to nonlinearities in the weather processes, a butterfly flapping its wings in Brazil can, in theory, produce a tornado in Texas. This strong dependence of outcomes on very slightly differing initial conditions is a distinct characteristic of the mathematical behavior known as chaos. The idea in meteorology that the flapping of a butterfly's wing will create a disturbance that in the chaotic motion of the atmosphere will become amplified eventually to change the large scale atmospheric motion, so that the long term behavior becomes impossible to forecast.
Flowing from the above argument, the butterfly effect in fact represents the essence of chaos. A complex and dynamical system is deemed to be chaotic if it
1. Has a dense collection of members with periodic orbits,
2. Is sensitive to the initial condition of the system (so that initially nearby members can evolve quickly into very different states), and
3. Is topologically transitive.
Chaotic systems exhibit irregular, unpredictable behavior. The boundary between linear and chaotic behavior is often characterized by periodic doubling in orbits, followed by quadrupling in orbits etc., although other kinds of combinations are also possible.
Animal populations are also subjected to the same phenomenon. Empirical evidence suggests that predator-prey systems too have complex dynamics making them prone to cycles. Such a system even with two simple variables such as rabbits and foxes can create a system that is really much more complex than initially thought to be. The lack of foxes may mean that the rabbit population can increase initially. But the increasing numbers of rabbits may also mean that the foxes have more food and are therefore more likely to survive and reproduce, which then in turn decreases the number of rabbits. It is possible for such systems to find a state of equilibrium, and even though species can become extinct, there is a tendency for populations to be robust. However, they can vary dramatically under certain circumstances. Real populations, of course having more than two variables, are even more complex than that of the illustration as given above.
The effects of the butterfly effect is best demonstrated by the Lorenz Attractor. The Lorenz Attractor is a graphical representation of the time variation of three variables coupled by non-linear evolution equations. You will observe that for the two separate non-linear evolution equations that are made to run simultaneously from slightly differing initial conditions, the tiny difference in the initial conditions becomes amplified by the evolution, until such time the two trajectories evolve quite separately. The amplification is exponential, the difference grows very rapidly and after a surprisingly short time the two solutions behave quite differently.
After having explained the butterfly effect from the scientific angle, it may also be appropriate to examine it from the layman's angle. There is a clever set of lyrics that is derived from an old english rhyme which originated for the purpose of encouraging children to apply logical progression to the consequences of their actions. The rhyme is often used to gently chastise a child whilst explaining the possible events that may follow a thoughtless act. But in this case, it perfectly explains the butterfly effect.
For want of a nail
For want of a nail, the shoe was lost.
For want of a shoe, the horse was lost.
For want of a horse, the rider was lost.
For want of a rider, the battle was lost.
For want of a battle, the kingdom was lost.
And all for the want of a horseshoe nail.
The butterfly effect technically refers to the propensity of a system to be sensitive to initial conditions. It is the sensitive dependency on its initial conditions that the system, over time, becomes wholly unpredictable.
The butterfly effect is however not quite as similiar as the domino effect. For the domino effect, there is no doubt dependency of the system on the initial conditions. But a simple linear row of dominoes would merely allow one event to initiate another similar event upon each iteration. The butterfly effect however amplifies the condition upon each iteration.
The butterfly effect has been most commonly associated with the weather system as this is where the discovery of non-linear phenomenon within a complex and dynamic system began. Due to nonlinearities in the weather processes, a butterfly flapping its wings in Brazil can, in theory, produce a tornado in Texas. This strong dependence of outcomes on very slightly differing initial conditions is a distinct characteristic of the mathematical behavior known as chaos. The idea in meteorology that the flapping of a butterfly's wing will create a disturbance that in the chaotic motion of the atmosphere will become amplified eventually to change the large scale atmospheric motion, so that the long term behavior becomes impossible to forecast.
Flowing from the above argument, the butterfly effect in fact represents the essence of chaos. A complex and dynamical system is deemed to be chaotic if it
1. Has a dense collection of members with periodic orbits,
2. Is sensitive to the initial condition of the system (so that initially nearby members can evolve quickly into very different states), and
3. Is topologically transitive.
Chaotic systems exhibit irregular, unpredictable behavior. The boundary between linear and chaotic behavior is often characterized by periodic doubling in orbits, followed by quadrupling in orbits etc., although other kinds of combinations are also possible.
Animal populations are also subjected to the same phenomenon. Empirical evidence suggests that predator-prey systems too have complex dynamics making them prone to cycles. Such a system even with two simple variables such as rabbits and foxes can create a system that is really much more complex than initially thought to be. The lack of foxes may mean that the rabbit population can increase initially. But the increasing numbers of rabbits may also mean that the foxes have more food and are therefore more likely to survive and reproduce, which then in turn decreases the number of rabbits. It is possible for such systems to find a state of equilibrium, and even though species can become extinct, there is a tendency for populations to be robust. However, they can vary dramatically under certain circumstances. Real populations, of course having more than two variables, are even more complex than that of the illustration as given above.
The effects of the butterfly effect is best demonstrated by the Lorenz Attractor. The Lorenz Attractor is a graphical representation of the time variation of three variables coupled by non-linear evolution equations. You will observe that for the two separate non-linear evolution equations that are made to run simultaneously from slightly differing initial conditions, the tiny difference in the initial conditions becomes amplified by the evolution, until such time the two trajectories evolve quite separately. The amplification is exponential, the difference grows very rapidly and after a surprisingly short time the two solutions behave quite differently.
After having explained the butterfly effect from the scientific angle, it may also be appropriate to examine it from the layman's angle. There is a clever set of lyrics that is derived from an old english rhyme which originated for the purpose of encouraging children to apply logical progression to the consequences of their actions. The rhyme is often used to gently chastise a child whilst explaining the possible events that may follow a thoughtless act. But in this case, it perfectly explains the butterfly effect.
For want of a nail
For want of a nail, the shoe was lost.
For want of a shoe, the horse was lost.
For want of a horse, the rider was lost.
For want of a rider, the battle was lost.
For want of a battle, the kingdom was lost.
And all for the want of a horseshoe nail.
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