1.
Game theory
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Game theory is the study of mathematical models of conflict and cooperation between intelligent rational decision-makers. Game theory is used in economics, political science, and psychology, as well as logic, computer science. Originally, it addressed zero-sum games, in one persons gains result in losses for the other participants. Today, game theory applies to a range of behavioral relations, and is now an umbrella term for the science of logical decision making in humans, animals. Modern game theory began with the idea regarding the existence of equilibria in two-person zero-sum games. Von Neumanns original proof used Brouwer fixed-point theorem on continuous mappings into compact convex sets and his paper was followed by the 1944 book Theory of Games and Economic Behavior, co-written with Oskar Morgenstern, which considered cooperative games of several players. The second edition of this provided an axiomatic theory of expected utility. This theory was developed extensively in the 1950s by many scholars, Game theory was later explicitly applied to biology in the 1970s, although similar developments go back at least as far as the 1930s. Game theory has been recognized as an important tool in many fields. With the Nobel Memorial Prize in Economic Sciences going to game theorist Jean Tirole in 2014, John Maynard Smith was awarded the Crafoord Prize for his application of game theory to biology. Early discussions of examples of two-person games occurred long before the rise of modern, the first known discussion of game theory occurred in a letter written by Charles Waldegrave, an active Jacobite, and uncle to James Waldegrave, a British diplomat, in 1713. In this letter, Waldegrave provides a mixed strategy solution to a two-person version of the card game le Her. James Madison made what we now recognize as an analysis of the ways states can be expected to behave under different systems of taxation. In 1913 Ernst Zermelo published Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels and it proved that the optimal chess strategy is strictly determined. This paved the way for more general theorems, the Danish mathematician Zeuthen proved that the mathematical model had a winning strategy by using Brouwers fixed point theorem. In his 1938 book Applications aux Jeux de Hasard and earlier notes, Borel conjectured that non-existence of mixed-strategy equilibria in two-person zero-sum games would occur, a conjecture that was proved false. Game theory did not really exist as a field until John von Neumann published a paper in 1928. Von Neumanns original proof used Brouwers fixed-point theorem on continuous mappings into compact convex sets and his paper was followed by his 1944 book Theory of Games and Economic Behavior co-authored with Oskar Morgenstern
2.
Coordination game
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In game theory, coordination games are a class of games with multiple pure strategy Nash equilibria in which players choose the same or corresponding strategies. If this game is a game, then the following inequalities in payoffs hold for player 1, A > B, D > C, and for player 2. In this game the strategy profiles and are pure Nash equilibria and this setup can be extended for more than two strategies, as well as for a game with more than two players. A typical case for a game is choosing the sides of the road upon which to drive. In a simplified example, assume that two drivers meet on a dirt road. Both have to swerve in order to avoid a head-on collision, if both execute the same swerving maneuver they will manage to pass each other, but if they choose differing maneuvers they will collide. In the payoff matrix in Fig.2, successful passing is represented by a payoff of 10, in this case there are two pure Nash equilibria, either both swerve to the left, or both swerve to the right. In this example, it doesnt matter which side both players pick, as long as they pick the same. This is not true for all games, as the pure coordination game in Fig.3 shows. Pure coordination is the game where the players prefer the same Nash equilibrium outcome, here both players prefer partying over both staying at home to watch TV. The outcome Pareto dominates the outcome, just as both Pareto dominate the two outcomes, and. This is different in type of coordination game commonly called battle of the sexes. In this game both players prefer engaging in the activity over going alone, but their preferences differ over which activity they should engage in. Player 1 prefers that they both party while player 2 prefers that they stay at home. Finally, the stag hunt game in Fig.5 shows a situation in both players can benefit if they cooperate. However, cooperation might fail, because each hunter has an alternative which is safer because it not require cooperation to succeed. This example of the conflict between safety and social cooperation is originally due to Jean-Jacques Rousseau. Coordination games also have mixed strategy Nash equilibria, since d > b and d-b < a+d-b-c, p is always between zero and one, so existence is assured
3.
Normal-form game
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In game theory, normal form is a description of a game. Unlike extensive form, normal-form representations are not graphical per se, while this approach can be of greater use in identifying strictly dominated strategies and Nash equilibria, some information is lost as compared to extensive-form representations. The normal-form representation of a game includes all perceptible and conceivable strategies, in static games of complete, perfect information, a normal-form representation of a game is a specification of players strategy spaces and payoff functions. The matrix to the right is a representation of a game in which players move simultaneously. For example, if player 1 plays top and player 2 plays left, player 1 receives 4, in each cell, the first number represents the payoff to the row player, and the second number represents the payoff to the column player. Often, symmetric games are represented only one payoff. This is the payoff for the row player, for example, the payoff matrices on the right and left below represent the same game. The payoff matrix facilitates elimination of dominated strategies, and it is used to illustrate this concept. For example, in the dilemma, we can see that each prisoner can either cooperate or defect. If exactly one prisoner defects, he gets off easily and the prisoner is locked up for a long time. However, if they both defect, they both be locked up for a shorter time. One can determine that Cooperate is strictly dominated by Defect, one must compare the first numbers in each column, in this case 0 > −1 and −2 > −5. This shows that no matter what the player chooses, the row player does better by choosing Defect. Similarly, one compares the second payoff in each row, again 0 > −1 and this shows that no matter what row does, column does better by choosing Defect. This demonstrates the unique Nash equilibrium of this game is and these matrices only represent games in which moves are simultaneous. The above matrix does not represent the game in which player 1 moves first, observed by player 2, in order to represent this sequential game we must specify all of player 2s actions, even in contingencies that can never arise in the course of the game. In this game, player 2 has actions, as before, Left, unlike before he has four strategies, contingent on player 1s actions. Accordingly, to specify a game, the payoff function has to be specified for each player in the player set P=. D. Fudenberg and J. Tirole, Game Theory
4.
Nash equilibrium
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The Nash equilibrium is one of the foundational concepts in game theory. The reality of the Nash equilibrium of a game can be tested using experimental economics methods, Game theorists use the Nash equilibrium concept to analyze the outcome of the strategic interaction of several decision makers. The simple insight underlying John Nashs idea is that one cannot predict the result of the choices of multiple decision makers if one analyzes those decisions in isolation, instead, one must ask what each player would do, taking into account the decision-making of the others. Nash equilibrium has been used to analyze hostile situations like war and arms races and it has also been used to study to what extent people with different preferences can cooperate, and whether they will take risks to achieve a cooperative outcome. It has been used to study the adoption of technical standards, the Nash equilibrium was named after John Forbes Nash, Jr. A version of the Nash equilibrium concept was first known to be used in 1838 by Antoine Augustin Cournot in his theory of oligopoly, in Cournots theory, firms choose how much output to produce to maximize their own profit. However, the best output for one firm depends on the outputs of others, a Cournot equilibrium occurs when each firms 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, however, Nashs definition of equilibrium is broader than Cournots. It is also broader than the definition of a Pareto-efficient equilibrium, the modern game-theoretic concept of Nash equilibrium is instead defined in terms of mixed strategies, where players choose a probability distribution over possible actions. The concept of the mixed-strategy Nash equilibrium was introduced by John von Neumann and Oskar Morgenstern in their 1944 book The Theory of Games, however, their analysis was restricted to the special case of zero-sum games. They showed that a mixed-strategy Nash equilibrium will exist for any game with a finite set of actions. The key to Nashs ability to prove far more generally than von Neumann lay in his definition of equilibrium. According to Nash, a point is an n-tuple such that each players mixed strategy maximizes his payoff if the strategies of the others are held fixed. Thus each players strategy is optimal against those of the others, since the development of the Nash equilibrium concept, game theorists have discovered that it makes misleading predictions in certain circumstances. They have proposed many related solution concepts designed to overcome perceived flaws in the Nash concept, one particularly important issue is that some Nash equilibria may be based on threats that are not credible. In 1965 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 repeated, or what happens if a game is played in the absence of complete information. Informally, a profile is a Nash equilibrium if no player can do better by unilaterally changing his or her strategy. To see what this means, imagine that each player is told the strategies of the others
5.
Money burning
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Money burning or burning money is the purposeful act of destroying money. In the prototypical example, banknotes are destroyed by literally setting them on fire, burning money decreases the wealth of the owner without directly enriching any particular party. However, according to the quantity theory of money, because it reduces the supply of money it increases by the amount the collective wealth of everyone else who holds money. Money is usually burned to communicate a message, either for artistic effect, as a form of protest, in some games, a player can sometimes benefit from the ability to burn money. Burning money is illegal in some jurisdictions, for the purposes of macroeconomics, burning money is equivalent to removing the money from circulation, and locking it away forever, the salient feature is that no one may ever use the money again. Burning money shrinks the money supply, and is therefore a special case of monetary policy that can be implemented by anyone. In the usual case, the central bank money from circulation by selling government bonds or foreign currency. The difference with money burning is that the bank does not have to exchange any assets of value for the money burnt. Money burning is thus equivalent to gifting the money back to the central bank, if the economy is at full employment equilibrium, shrinking the money supply causes deflation, increasing the real value of the money left in circulation. Assuming that the money is paper money with negligible intrinsic value, no real goods are destroyed. Instead, all surviving money slightly increases in value, everyone gains wealth in proportion to the amount of money they already hold, Economist Steven Landsburg proposes in The Armchair Economist that burning ones fortune is a form of philanthropy more egalitarian than deeding it to the United States Treasury. In 1920, Thomas Nixon Carver wrote that money into the sea is better for society than spending it wastefully. Central banks routinely collect and destroy worn-out coins and banknotes in exchange for new ones and this does not affect the money supply, and is done to maintain a healthy population of usable currency. The practice raises an interesting possibility, if an individual can steal the money before it is incinerated, the effect is the opposite of burning money, the thief is enriched at the expense of the rest of society. One such incident at the Bank of England inspired the 2001 TV movie Hot Money, another, more common near-opposite is the creation of counterfeit money. Another way to analyze the cost of forgery is to consider the effects of a banks monetary policy. The interest earnings on those bonds is turned over to the US Treasury, so any lost interest must be made up by U. S. taxpayers, behaviorally speaking, burning money is usually seen as a purely negative act. The cognitive impact of burning money can even be a motivational tool