The latest Lifestyle | Daily Life news, tips, opinion and advice from The Sydney Morning Herald covering life and relationships, beauty, fashion, health & wellbeing The Society for Music Theory promotes the development of and engagement with music theory as a scholarly and pedagogical discipline. We construe this discipline broadly as embracing all approaches, from conceptual to practical, and all perspectives, including those of the scholar, listener, composer, performer, teacher, and student In game theory, the Nash equilibrium, named after the mathematician John Forbes Nash Jr., is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equilibrium strategies of the other players and no player has anything to gain by changing only their own strategy
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In game theorydissertation game theory, the Nash equilibriumnamed after the mathematician John Forbes Nash Jr. In a Nash equilibrium, each player is assumed to know the equilibrium strategies of the other players and no player has anything to gain by changing only their own strategy, dissertation game theory. If each player has chosen a strategy — an action plan choosing their own actions based on what has happened so far in the game — and no player can increase their own expected payoff by changing their strategy dissertation game theory the other players keep theirs unchanged, then the current set of strategy choices constitutes a Nash equilibrium, dissertation game theory.
If two players Alice and Bob choose strategies A and B, A, B is a Nash equilibrium if Alice has no other strategy available that does better than A at maximizing her payoff in response to Bob choosing B, and Bob has no other strategy available that does better than B at maximizing his payoff in response to Alice choosing A, dissertation game theory. In a game in which Carol and Dan are also players, A, B, C, D is a Nash equilibrium if A is Alice's best response to B, C, DB is Bob's best response to A, C, dissertation game theory, Dand so forth.
Nash showed that there is a Nash equilibrium for every finite game: see further the article dissertation game theory strategy. Game theorists use Nash equilibrium to analyze the outcome of the strategic interaction of several decision makers.
In a strategic interaction, the outcome for each decision-maker depends on the decisions of the others as well as their own. The simple insight underlying Nash's idea is that one cannot predict the choices of multiple decision makers if one analyzes those decisions in isolation.
Nash equilibrium requires that their choices be consistent: no player wishes to undo their decision given what the others dissertation game theory deciding, dissertation game theory. The concept has been used to analyze hostile situations such as wars and arms races [3] see prisoner's dilemmaand also how conflict may be mitigated by repeated interaction see tit-for-tat.
It has also been used to study to what extent people with different preferences can cooperate see battle of the sexesand whether they will take risks to achieve a cooperative outcome see stag hunt, dissertation game theory. It has been used to study the adoption of technical standards[ citation needed ] and also the occurrence of bank runs and currency crises see coordination game, dissertation game theory.
Other applications include traffic flow see Wardrop's principlehow to organize auctions see auction theorythe outcome of efforts exerted by multiple parties in the education process, [4] regulatory legislation such as environmental regulations see tragedy of the commons[5] natural resource management, [6] analysing strategies in marketing, [7] even penalty kicks in football see matching pennies[8] energy systems, transportation systems, dissertation game theory, evacuation problems [9] and wireless communications.
Nash equilibrium is named after American mathematician John Forbes Nash Jr. The same idea was used in a particular application in by Antoine Augustin Cournot in his theory of oligopoly. The best output for one firm depends on the outputs of the others. A Cournot equilibrium occurs when each firm's output maximizes its profits given the output of the other firms, which is a pure-strategy Nash equilibrium. Cournot also introduced the concept of best response dynamics in his analysis of the stability of equilibrium.
Cournot did not use the idea in any other applications, dissertation game theory, however, or define it generally. The concept of a mixed-strategy equilibrium was introduced by John von Neumann and Oskar Morgenstern in their book The Theory of Games and Economic Behaviorbut their analysis was restricted to the special case of zero-sum games.
They showed that a mixed-strategy Nash equilibrium will exist for any zero-sum game with a finite set of actions. The key to Nash's ability to prove existence far more generally than von Neumann lay in his definition of equilibrium.
According to Nash, "an equilibrium point is an n-tuple such that each player's mixed strategy maximizes his payoff if the strategies of the others are held fixed. Thus each player's strategy is optimal against those of the others. His paper used the simpler Brouwer fixed-point theorem for the same purpose.
Game theorists have discovered that in some circumstances Nash equilibrium makes invalid predictions or fails to make a unique prediction. They have proposed many solution concepts 'refinements' of Nash equilibria designed to rule out implausible Nash equilibria.
One particularly important issue is that some Nash equilibria may be based on threats that dissertation game theory not ' credible '. In Reinhard Selten proposed subgame perfect equilibrium as a refinement that eliminates equilibria which depend on non-credible threats. Other extensions of the Nash equilibrium concept have addressed what happens if a game is repeatedor what happens if a game is played in the absence of complete information.
However, subsequent refinements and extensions of Nash equilibrium share the main insight on which Nash's concept rests: the equilibrium is a set of strategies such that each player's strategy is optimal given the choices of the others. A strategy profile is a set of strategies, one for each player. Informally, a strategy profile is a Nash equilibrium if no player can do better by unilaterally changing his strategy.
To see what this means, imagine that each player is told the strategies of the others. Dissertation game theory then that each player asks himself: "Knowing the strategies of the other players, and treating the strategies of the other players as set in stone, dissertation game theory, can I benefit by changing my strategy? If any player could answer "Yes", then that set of strategies is not a Nash equilibrium. But if every player prefers not to switch or is indifferent between switching and not then the strategy profile is a Nash equilibrium.
Thus, dissertation game theory, each strategy in a Nash equilibrium is a best response to the other players' strategies in that equilibrium. A game can have more than one Nash equilibrium. Even if the equilibrium is unique, it might be weak : a player might be indifferent among several strategies given the other players' choices. It is unique and called a strict Nash equilibrium if the inequality is strict so one strategy is the unique best response:, dissertation game theory.
Most simply, a player might choose between two strategies, e. Nash's existence proofs assume a finite strategy set, but the concept of Nash equilibrium does not require it. The Nash equilibrium may sometimes appear non-rational in a third-person perspective. This is because a Nash equilibrium is not necessarily Pareto optimal.
Nash equilibrium may also have non-rational consequences in sequential games because players may "threaten" each other with threats they would not actually carry out.
For such games the subgame perfect Nash equilibrium may be more meaningful as a tool of analysis. Suppose that in the Nash equilibrium, each player asks themselves: "Knowing the strategies of the other players, and treating the strategies of the other players as set in stone, would I suffer a loss by changing my strategy?
If every player's answer is "Yes", then the equilibrium is classified as a strict Nash equilibrium. If instead, for some player, there is exact equality between the strategy in Nash equilibrium and some other strategy that gives exactly the same payout i. this player is indifferent between switching and notthen the equilibrium is classified as a weak Nash dissertation game theory. A game can have a pure-strategy or a mixed-strategy Nash equilibrium.
In the latter a pure strategy is chosen dissertation game theory with a fixed probability. Nash proved that if mixed strategies where a player chooses probabilities of using various pure strategies are allowed, then every game with a finite number of players in which each player dissertation game theory choose from finitely many pure strategies has dissertation game theory least one Nash equilibrium, which might be a pure strategy for each player or might be a probability distribution over strategies for each dissertation game theory. Nash equilibria need not exist if the set of choices is infinite and non-compact, dissertation game theory.
An example is a game where two players simultaneously name a number and the player naming the larger number wins, dissertation game theory. Another example is where each of two players chooses a real number strictly less than 5 and the winner is whoever has the biggest number; no biggest number strictly less than 5 exists if the number could equal 5, the Nash equilibrium would have both players choosing 5 and tying the game, dissertation game theory.
However, a Nash equilibrium exists if the set of choices is compact with each player's payoff continuous in the strategies of all the players.
The coordination game is a classic two-player, two- strategy game, as shown in the example payoff matrix to the right. There are two pure-strategy equilibria, A,A with payoff 4 for each player and B,B with payoff 2 for each. The combination B,B is a Nash equilibrium because if either player unilaterally changes his strategy from B to A, his payoff will fall from 2 to 1.
A famous example of a coordination game is the stag hunt. Two players may choose to hunt a stag or a rabbit, the stag providing more meat 4 utility units, 2 for each player than the rabbit 1 utility unit. The caveat is that the stag must be cooperatively hunted, so if one player attempts to hunt the stag, while the dissertation game theory hunts the rabbit, the stag hunter will totally fail, for a payoff of 0, whereas the rabbit-hunter will succeed, for a payoff of 1.
The game has two equilibria, dissertation game theory, stag, stag and rabbit, rabbitbecause a player's optimal strategy depend on his expectation on what the other player will do. If one hunter trusts that the other will hunt the stag, he should hunt the stag; however if he thinks the other will hunt the rabbit, he too will hunt the rabbit.
This game is used as an analogy for social cooperation, since much of the benefit that people gain in society depends upon people cooperating and implicitly trusting one another to act in a manner corresponding with cooperation.
Driving on a road against an oncoming car, and dissertation game theory to choose either to swerve on the left or to swerve on the right of the road, is also a coordination game. For example, with payoffs 10 meaning no crash and 0 meaning a crash, the coordination game can be defined with the following payoff matrix:.
In this case there are two pure-strategy Nash equilibria, when both choose to either drive on the left or on the right. An application of Nash dissertation game theory is in determining the expected flow of traffic in a dissertation game theory. Consider the graph dissertation game theory the right.
If we assume that there are x "cars" traveling from A to D, what is the expected distribution of traffic in the network? This situation can be modeled as a "game" where every traveler has a choice of 3 strategies, where each strategy is a route from A to D one of ABDABCDor ACD. The "payoff" of each strategy is the travel time of each route. Thus, payoffs for any given strategy depend on the choices of the other players, as is usual.
However, the goal, dissertation game theory, in this case, is to minimize travel time, not maximize it. Equilibrium will occur when the time on all paths is exactly the same.
When that happens, no single driver has any incentive to switch routes, since it can only add to their travel time. For the graph on the right, if, for example, cars are travelling from A to D, then equilibrium will occur when 25 drivers travel via ABD50 via ABCDand 25 via ACD, dissertation game theory.
Every driver now has a total travel time of 3. Notice that this distribution is not, actually, socially optimal. If the cars agreed that 50 travel via ABD and the other 50 through ACDthen travel time for any single car would dissertation game theory be 3. This is also the Nash equilibrium if the path between B and C is removed, which means that adding another possible route can decrease the efficiency of dissertation game theory system, a phenomenon known as Braess's paradox.
This can be illustrated by a two-player game in which both players simultaneously choose an integer from 0 to 3 and they both win the smaller of the two numbers in points. In addition, dissertation game theory, if one player chooses a larger number than the other, then they have to give up two points to the other. This game has a unique pure-strategy Nash equilibrium: both players choosing 0 highlighted in light red.
Any other strategy can be improved by a player switching their number to one less than that of the other player. In the adjacent table, if the game begins at the green square, it is in player 1's interest to move to the purple square and it is in player 2's interest to move to the blue square. Although it would not fit the definition of a competition game, if the game is modified so that the two players win the named amount if they both choose the same number, and otherwise win nothing, then there are 4 Nash equilibria: 0,01,12,2and 3,3.
There is an easy numerical way to identify Nash equilibria on a payoff matrix. It is especially helpful in two-person games where players have more than two strategies. In this case formal analysis may become too dissertation game theory. This rule does not apply to the case where mixed stochastic strategies are of interest. The rule goes as follows: if the first payoff number, in the payoff pair of the cell, is the maximum of the column of the cell and if the second number is the maximum of the row of the cell - then the cell represents a Nash equilibrium.
Using the rule, we can very quickly much faster than with formal analysis see that the Nash equilibria cells are B,AA,Band C,C. Indeed, for cell B,A 40 is the maximum of the first column and 25 is the maximum of the second row. For A,B 25 is the maximum of the second column and 40 is the maximum of the first row, dissertation game theory.
Game Theory: Nash Equilibria
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Dissertation definition is - an extended usually written treatment of a subject; specifically: one submitted for a doctorate. How to use dissertation in a sentence We would like to show you a description here but the site won’t allow blogger.com more The book (biography) A Beautiful Mind. Written by journalism professor Sylvia Nasar, this book is an unauthorized biography of the award-winning mathematician John Forbes Nash, Jr. In , he was one of three recipients who shared the Nobel Memorial Prize in Economic Sciences for their work with game theory
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