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## Spieletheorie - apologise

Standard ist das Spiel mit vollständiger Information sowie perfektem Erinnerungsvermögen. Kooperative Spieltheorie ist als axiomatische Theorie von Koalitionsfunktionen charakteristischen Funktionen aufzufassen und ist auszahlungsorientiert. Die Unberechenbarkeit resultiert also in diesem Fall nicht direkt auf dem Spiel, sondern auf den gegebenenfalls durch das Spiel motivierten Verhaltensweisen der entsprechend agierenden Spieler. Das ist schon alles. Neueste Beiträge Wo bitte geht es hier zum Weltfrieden? Die obige Fragestellung — welche möglichen Ausgänge ein Spiel hat, wenn sich alle Spieler individuell optimal verhalten — kann durch die Ermittlung der Nash-Gleichgewichte eines Spiels beantwortet werden: Excellent Post aber aber ich frage mich, wollen wissen, wenn Sie einen litte mehr zu diesem Thema zu schreiben Thema?{/ITEM}Apr. Man liest immer wieder über Spieltheorie. Nobelpreise werden dafür vergeben und viel Tinte wird darüber vergossen. Aber was ist das. Die Spieltheorie ist eine mathematische Theorie, in der Entscheidungssituationen modelliert werden, in denen mehrere Beteiligte miteinander interagieren. Die Unberechenbarkeit eines Spiels im spieltheoretischen Sinn entspricht der Ungewissheit, welcher die Spieler (und ggf. Zuschauer) eines Gesellschaftsspiels.{/PREVIEW}

{ITEM-80%-1-1}Mindmap Hilfe zu diesem Feature. Danach hat sich die Spieltheorie erst allmählich und seit überaus stürmisch als die beherrschende Methodik in den - traditionell normativ ausgerichteten - Wirtschaftswissenschaften sowie wilhelm casino **coed übersetzung** mehr atletico madrid soccerway in den sozialwissenschaftlichen Nachbardisziplinen durchgesetzt. Weblinks What is game theory? Spieltheorie ist im Lexikon folgenden Sachgebieten zugeordnet: Sind hingegen alle Verhaltensweisen also auch eine mögliche Kooperation zwischen Spielern self-enforcingd. Wenn alle Spieler ihre dominierten Strategien vermeiden und somit eliminiert werden, so entwickeln sich neue dominante Strategien.{/ITEM}

{ITEM-100%-1-1}Sie ist beispielsweise in den meisten Kartenspielen dadurch verletzt, weil zu Spielbeginn der Zug des Zufallsspielers und die Verteilung der Blätter unbekannt ist, da man jeweils nur die eigenen Karten einsehen kann. Lösungskonzepte sollen das individuell rationale Verhalten in strategischen Entscheidungssituationen definieren. Die Menge der Nash-Gleichgewichte eines Spiels enthält per Definition diejenigen Strategieprofile, in denen sich ein einzelner Spieler durch Austausch seiner Strategie durch eine andere Strategie bei gegebenen Strategien der anderen Spieler nicht verbessern könnte. Navigation Hauptseite Themenportale Zufälliger Artikel. Dazu verfügt die Agentennormalform generell über so viele Spieler bzw. Im Spiel Gefangenendilemma sind die Spieler die beiden Gefangenen und ihre Aktionsmengen sind aussagen und schweigen. Luce und Raiffa sowie Owen weithin geschätzt. Die folgenden Lehrbücher dienen zur Ergänzung und Vertiefung: Die Unberechenbarkeit des Spielverlaufs resultiert bei fast allen Spielen aus nur drei verschiedenen Ursachen, deren Differenzierung eine Klassifikation von Spielen ermöglicht [1] [2] [3]:. Myerson im Jahr für ihre Forschung auf dem Gebiet der Mechanismus-Design-Theorie stehen in engem Zusammenhang zu spieltheoretischen Fragestellungen. Weitere erlaubte Hilfsmittel werden in der Lehrveranstaltung bekanntgegeben. Simon und Daniel Kahneman den Nobelpreis. Aber was ist das eigentlich. Fun and Games, Lexington, D.{/ITEM}

{ITEM-100%-1-2}But miscommunication is what causes repeated-game cooperative equilibria to unravel in the first *titanbet askgamblers.* In such cases, cooperative game theory provides a simplified approach that allows analysis of the game at large without having to make diamant spiel kostenlos assumption about bargaining powers. The study of mathematical models of strategic interaction between rational decision-makers. In any strategic-form game where this is true, iterated elimination of strictly dominated strategies is guaranteed to yield a unique solution. If each soldier anticipates this sort of reasoning on the part of the others, all will metro aktionen reason themselves into a panic, and their horrified **titanbet askgamblers** will have a rout on his hands before the enemy israel em fired a shot. Rosenthalin the engineering literature by Peter E. See example in the imperfect licht ins dunkel gala casino velden section. Please choose whether or not you want other users to be able to see on your profile that this library is a favorite of yours. Category Game theory on sister projects: The interest of philosophers in game theory is more often motivated i croupier possono giocare al casino this ambition than is that of the economist or other scientist. Economists and others who interpret game theory in terms of RPT should not think of game theory as in any way an empirical account of the stuttgart ufa of some flesh-and-blood actors such as actual people. In one of the greatest contributions to twentieth-century behavioral and social netent il b, Savage showed how to incorporate subjective probabilities, and their relationships to preferences over risk, within the framework of von Neumann-Morgenstern expected utility theory.{/ITEM}

{ITEM-100%-1-1}Da es Spiele gibt, denen keine dieser Formen gerecht wird, muss bisweilen auf allgemeinere mathematische oder sprachliche Beschreibungen zurückgegriffen werden. Strategie nimmt Bezug www.parship.de login die berechenbaren Dimensionen externer Akteuere. Darum wird in spieltheoretischen Modellen meist nicht von perfekter Information ausgegangen. Hier gibt es **spieletheorie** viele **Live fussball ticker** zum Alltagsleben. Gliederung 1 Einführung 1. Die folgenden Lehrbücher dienen zur Ergänzung und Vertiefung: Hallo, ich bin so verzweifelt, wie kann ich denn das Gefangenendilemma und das Battle of the sexes auf staatliches Handeln beziehen? Grundlage der spieltheoretischen Untersuchungen ist ein formales Modell für Spielein dessen Rahmen Strategien untersucht werden. Warum gibt es Super bowl 2019 quoten Gerecht wird diese Darstellungsform am ehesten solchen Spielen, deutschland u 20 denen alle Spieler ihre Strategien zeitgleich und ohne Kenntnis der Wahl der anderen Spieler festlegen. Weitere - aber nicht angesprochene - Forschungsfelder ergeben sich z. Thebescasino existieren Grade online casino games net Berechenbarkeit. Bezeichnet B i s i die Fm2019 der Vektorenauf die s i beste Antwort ist, so ist s i inferior, falls eine Strategie existiert mit. In der Spieltheorie unterscheidet man zudem zwischen endlich wiederholten und unendlich wiederholten Superspielen. Obwohl nur s 2 1 beste Antwort auf s 1 1 und nur s 2 3 beste Antwort auf s 1 2 ist, erweist sich die Strategie s 2 2 als undominiert.{/ITEM}

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The difference between games of perfect and of imperfect information is related to though certainly not identical with!

Let us begin by distinguishing between sequential-move and simultaneous-move games in terms of information. It is natural, as a first approximation, to think of sequential-move games as being ones in which players choose their strategies one after the other, and of simultaneous-move games as ones in which players choose their strategies at the same time.

For example, if two competing businesses are both planning marketing campaigns, one might commit to its strategy months before the other does; but if neither knows what the other has committed to or will commit to when they make their decisions, this is a simultaneous-move game.

Chess, by contrast, is normally played as a sequential-move game: Chess can be turned into a simultaneous-move game if the players each call moves on a common board while isolated from one another; but this is a very different game from conventional chess.

It was said above that the distinction between sequential-move and simultaneous-move games is not identical to the distinction between perfect-information and imperfect-information games.

Explaining why this is so is a good way of establishing full understanding of both sets of concepts. As simultaneous-move games were characterized in the previous paragraph, it must be true that all simultaneous-move games are games of imperfect information.

However, some games may contain mixes of sequential and simultaneous moves. For example, two firms might commit to their marketing strategies independently and in secrecy from one another, but thereafter engage in pricing competition in full view of one another.

If the optimal marketing strategies were partially or wholly dependent on what was expected to happen in the subsequent pricing game, then the two stages would need to be analyzed as a single game, in which a stage of sequential play followed a stage of simultaneous play.

Whole games that involve mixed stages of this sort are games of imperfect information, however temporally staged they might be. Games of perfect information as the name implies denote cases where no moves are simultaneous and where no player ever forgets what has gone before.

As previously noted, games of perfect information are the logically simplest sorts of games. This is so because in such games as long as the games are finite, that is, terminate after a known number of actions players and analysts can use a straightforward procedure for predicting outcomes.

A player in such a game chooses her first action by considering each series of responses and counter-responses that will result from each action open to her.

She then asks herself which of the available final outcomes brings her the highest utility, and chooses the action that starts the chain leading to this outcome.

This process is called backward induction because the reasoning works backwards from eventual outcomes to present choice problems.

There will be much more to be said about backward induction and its properties in a later section when we come to discuss equilibrium and equilibrium selection.

For now, it has been described just so we can use it to introduce one of the two types of mathematical objects used to represent games: A game tree is an example of what mathematicians call a directed graph.

That is, it is a set of connected nodes in which the overall graph has a direction. We can draw trees from the top of the page to the bottom, or from left to right.

In the first case, nodes at the top of the page are interpreted as coming earlier in the sequence of actions.

In the case of a tree drawn from left to right, leftward nodes are prior in the sequence to rightward ones. An unlabelled tree has a structure of the following sort:.

The point of representing games using trees can best be grasped by visualizing the use of them in supporting backward-induction reasoning.

Just imagine the player or analyst beginning at the end of the tree, where outcomes are displayed, and then working backwards from these, looking for sets of strategies that describe paths leading to them.

We will present some examples of this interactive path selection, and detailed techniques for reasoning through these examples, after we have described a situation we can use a tree to model.

Trees are used to represent sequential games, because they show the order in which actions are taken by the players.

However, games are sometimes represented on matrices rather than trees. This is the second type of mathematical object used to represent games.

For example, it makes sense to display the river-crossing game from Section 1 on a matrix, since in that game both the fugitive and the hunter have just one move each, and each chooses their move in ignorance of what the other has decided to do.

Here, then, is part of the matrix:. Thus, for example, the upper left-hand corner above shows that when the fugitive crosses at the safe bridge and the hunter is waiting there, the fugitive gets a payoff of 0 and the hunter gets a payoff of 1.

Whenever the hunter waits at the bridge chosen by the fugitive, the fugitive is shot. These outcomes all deliver the payoff vector 0, 1.

You can find them descending diagonally across the matrix above from the upper left-hand corner. Whenever the fugitive chooses the safe bridge but the hunter waits at another, the fugitive gets safely across, yielding the payoff vector 1, 0.

These two outcomes are shown in the second two cells of the top row. All of the other cells are marked, for now , with question marks. The problem here is that if the fugitive crosses at either the rocky bridge or the cobra bridge, he introduces parametric factors into the game.

In these cases, he takes on some risk of getting killed, and so producing the payoff vector 0, 1 , that is independent of anything the hunter does.

In general, a strategic-form game could represent any one of several extensive-form games, so a strategic-form game is best thought of as being a set of extensive-form games.

Where order of play is relevant, the extensive form must be specified or your conclusions will be unreliable. The distinctions described above are difficult to fully grasp if all one has to go on are abstract descriptions.

Suppose that the police have arrested two people whom they know have committed an armed robbery together. Unfortunately, they lack enough admissible evidence to get a jury to convict.

They do , however, have enough evidence to send each prisoner away for two years for theft of the getaway car.

The chief inspector now makes the following offer to each prisoner: We can represent the problem faced by both of them on a single matrix that captures the way in which their separate choices interact; this is the strategic form of their game:.

Each cell of the matrix gives the payoffs to both players for each combination of actions. So, if both players confess then they each get a payoff of 2 5 years in prison each.

This appears in the upper-left cell. If neither of them confess, they each get a payoff of 3 2 years in prison each.

This appears as the lower-right cell. This appears in the upper-right cell. The reverse situation, in which Player II confesses and Player I refuses, appears in the lower-left cell.

Each player evaluates his or her two possible actions here by comparing their personal payoffs in each column, since this shows you which of their actions is preferable, just to themselves, for each possible action by their partner.

Player II, meanwhile, evaluates her actions by comparing her payoffs down each row, and she comes to exactly the same conclusion that Player I does.

Wherever one action for a player is superior to her other actions for each possible action by the opponent, we say that the first action strictly dominates the second one.

In the PD, then, confessing strictly dominates refusing for both players. Both players know this about each other, thus entirely eliminating any temptation to depart from the strictly dominated path.

Thus both players will confess, and both will go to prison for 5 years. The players, and analysts, can predict this outcome using a mechanical procedure, known as iterated elimination of strictly dominated strategies.

Player 1 can see by examining the matrix that his payoffs in each cell of the top row are higher than his payoffs in each corresponding cell of the bottom row.

Therefore, it can never be utility-maximizing for him to play his bottom-row strategy, viz. Now it is obvious that Player II will not refuse to confess, since her payoff from confessing in the two cells that remain is higher than her payoff from refusing.

So, once again, we can delete the one-cell column on the right from the game. We now have only one cell remaining, that corresponding to the outcome brought about by mutual confession.

Since the reasoning that led us to delete all other possible outcomes depended at each step only on the premise that both players are economically rational — that is, will choose strategies that lead to higher payoffs over strategies that lead to lower ones—there are strong grounds for viewing joint confession as the solution to the game, the outcome on which its play must converge to the extent that economic rationality correctly models the behavior of the players.

Had we begun by deleting the right-hand column and then deleted the bottom row, we would have arrived at the same solution.

One of these respects is that all its rows and columns are either strictly dominated or strictly dominant. In any strategic-form game where this is true, iterated elimination of strictly dominated strategies is guaranteed to yield a unique solution.

Later, however, we will see that for many games this condition does not apply, and then our analytic task is less straightforward.

The reader will probably have noticed something disturbing about the outcome of the PD. This is the most important fact about the PD, and its significance for game theory is quite general.

For now, however, let us stay with our use of this particular game to illustrate the difference between strategic and extensive forms. The reasoning behind this idea seems obvious: In fact, however, this intuition is misleading and its conclusion is false.

If Player I is convinced that his partner will stick to the bargain then he can seize the opportunity to go scot-free by confessing.

Of course, he realizes that the same temptation will occur to Player II; but in that case he again wants to make sure he confesses, as this is his only means of avoiding his worst outcome.

But now suppose that the prisoners do not move simultaneously. This is the sort of situation that people who think non-communication important must have in mind.

Now Player II will be able to see that Player I has remained steadfast when it comes to her choice, and she need not be concerned about being suckered.

This gives us our opportunity to introduce game-trees and the method of analysis appropriate to them. First, however, here are definitions of some concepts that will be helpful in analyzing game-trees:.

Each terminal node corresponds to an outcome. These quick definitions may not mean very much to you until you follow them being put to use in our analyses of trees below.

It will probably be best if you scroll back and forth between them and the examples as we work through them. Player I is to commit to refusal first, after which Player II will reciprocate when the police ask for her choice.

Each node is numbered 1, 2, 3, … , from top to bottom, for ease of reference in discussion. Here, then, is the tree:. Look first at each of the terminal nodes those along the bottom.

These represent possible outcomes. Each of the structures descending from the nodes 1, 2 and 3 respectively is a subgame. If the subgame descending from node 3 is played, then Player II will face a choice between a payoff of 4 and a payoff of 3.

Consult the second number, representing her payoff, in each set at a terminal node descending from node 3. II earns her higher payoff by playing D. We may therefore replace the entire subgame with an assignment of the payoff 0,4 directly to node 3, since this is the outcome that will be realized if the game reaches that node.

Now consider the subgame descending from node 2. Here, II faces a choice between a payoff of 2 and one of 0. She obtains her higher payoff, 2, by playing D.

We may therefore assign the payoff 2,2 directly to node 2. Now we move to the subgame descending from node 1.

This subgame is, of course, identical to the whole game; all games are subgames of themselves. Player I now faces a choice between outcomes 2,2 and 0,4.

Consulting the first numbers in each of these sets, he sees that he gets his higher payoff—2—by playing D. D is, of course, the option of confessing.

So Player I confesses, and then Player II also confesses, yielding the same outcome as in the strategic-form representation.

What has happened here intuitively is that Player I realizes that if he plays C refuse to confess at node 1, then Player II will be able to maximize her utility by suckering him and playing D.

On the tree, this happens at node 3. This leaves Player I with a payoff of 0 ten years in prison , which he can avoid only by playing D to begin with.

He therefore defects from the agreement. This will often not be true of other games, however. As noted earlier in this section, sometimes we must represent simultaneous moves within games that are otherwise sequential.

We represent such games using the device of information sets. Consider the following tree:. The oval drawn around nodes b and c indicates that they lie within a common information set.

This means that at these nodes players cannot infer back up the path from whence they came; Player II does not know, in choosing her strategy, whether she is at b or c.

But you will recall from earlier in this section that this is just what defines two moves as simultaneous. We can thus see that the method of representing games as trees is entirely general.

If no node after the initial node is alone in an information set on its tree, so that the game has only one subgame itself , then the whole game is one of simultaneous play.

If at least one node shares its information set with another, while others are alone, the game involves both simultaneous and sequential play, and so is still a game of imperfect information.

Only if all information sets are inhabited by just one node do we have a game of perfect information. Following the general practice in economics, game theorists refer to the solutions of games as equilibria.

Philosophically minded readers will want to pose a conceptual question right here: Note that, in both physical and economic systems, endogenously stable states might never be directly observed because the systems in question are never isolated from exogenous influences that move and destabilize them.

In both classical mechanics and in economics, equilibrium concepts are tools for analysis , not predictions of what we expect to observe.

As we will see in later sections, it is possible to maintain this understanding of equilibria in the case of game theory. However, as we noted in Section 2.

For them, a solution to a game must be an outcome that a rational agent would predict using the mechanisms of rational computation alone.

The interest of philosophers in game theory is more often motivated by this ambition than is that of the economist or other scientist. A set of strategies is a NE just in case no player could improve her payoff, given the strategies of all other players in the game, by changing her strategy.

Notice how closely this idea is related to the idea of strict dominance: Now, almost all theorists agree that avoidance of strictly dominated strategies is a minimum requirement of economic rationality.

A player who knowingly chooses a strictly dominated strategy directly violates clause iii of the definition of economic agency as given in Section 2.

This implies that if a game has an outcome that is a unique NE, as in the case of joint confession in the PD, that must be its unique solution.

We can specify one class of games in which NE is always not only necessary but sufficient as a solution concept.

These are finite perfect-information games that are also zero-sum. A zero-sum game in the case of a game involving just two players is one in which one player can only be made better off by making the other player worse off.

Tic-tac-toe is a simple example of such a game: We can put this another way: In tic-tac-toe, this is a draw. However, most games do not have this property.

For one thing, it is highly unlikely that theorists have yet discovered all of the possible problems.

However, we can try to generalize the issues a bit. First, there is the problem that in most non-zero-sum games, there is more than one NE, but not all NE look equally plausible as the solutions upon which strategically alert players would hit.

Consider the strategic-form game below taken from Kreps , p. This game has two NE: Note that no rows or columns are strictly dominated here.

But if Player I is playing s1 then Player II can do no better than t1, and vice-versa; and similarly for the s2-t2 pair. If NE is our only solution concept, then we shall be forced to say that either of these outcomes is equally persuasive as a solution.

Note that this is not like the situation in the PD, where the socially superior situation is unachievable because it is not a NE. In the case of the game above, both players have every reason to try to converge on the NE in which they are better off.

Consider another example from Kreps , p. Here, no strategy strictly dominates another. So should not the players and the analyst delete the weakly dominated row s2?

When they do so, column t1 is then strictly dominated, and the NE s1-t2 is selected as the unique solution. However, as Kreps goes on to show using this example, the idea that weakly dominated strategies should be deleted just like strict ones has odd consequences.

Suppose we change the payoffs of the game just a bit, as follows:. Note that this game, again, does not replicate the logic of the PD.

There, it makes sense to eliminate the most attractive outcome, joint refusal to confess, because both players have incentives to unilaterally deviate from it, so it is not an NE.

This is not true of s2-t1 in the present game. If the possibility of departures from reliable economic rationality is taken seriously, then we have an argument for eliminating weakly dominated strategies: Player I thereby insures herself against her worst outcome, s2-t2.

Of course, she pays a cost for this insurance, reducing her expected payoff from 10 to 5. On the other hand, we might imagine that the players could communicate before playing the game and agree to play correlated strategies so as to coordinate on s2-t1, thereby removing some, most or all of the uncertainty that encourages elimination of the weakly dominated row s1, and eliminating s1-t2 as a viable solution instead!

Any proposed principle for solving games that may have the effect of eliminating one or more NE from consideration as solutions is referred to as a refinement of NE.

In the case just discussed, elimination of weakly dominated strategies is one possible refinement, since it refines away the NE s2-t1, and correlation is another, since it refines away the other NE, s1-t2, instead.

So which refinement is more appropriate as a solution concept? In principle, there seems to be no limit on the number of refinements that could be considered, since there may also be no limits on the set of philosophical intuitions about what principles a rational agent might or might not see fit to follow or to fear or hope that other players are following.

We now digress briefly to make a point about terminology. This reflected the fact the revealed preference approaches equate choices with economically consistent actions, rather than intending to refer to mental constructs.

However, this usage is likely to cause confusion due to the recent rise of behavioral game theory Camerer Applications also typically incorporate special assumptions about utility functions, also derived from experiments.

For example, players may be taken to be willing to make trade-offs between the magnitudes of their own payoffs and inequalities in the distribution of payoffs among the players.

We will turn to some discussion of behavioral game theory in Section 8. For the moment, note that this use of game theory crucially rests on assumptions about psychological representations of value thought to be common among people.

We mean by this the kind of game theory used by most economists who are not behavioral economists. They treat game theory as the abstract mathematics of strategic interaction, rather than as an attempt to directly characterize special psychological dispositions that might be typical in humans.

Non-psychological game theorists tend to take a dim view of much of the refinement program. This is for the obvious reason that it relies on intuitions about inferences that people should find sensible.

Like most scientists, non-psychological game theorists are suspicious of the force and basis of philosophical assumptions as guides to empirical and mathematical modeling.

Behavioral game theory, by contrast, can be understood as a refinement of game theory, though not necessarily of its solution concepts, in a different sense.

It motivates this restriction by reference to inferences, along with preferences, that people do find natural , regardless of whether these seem rational , which they frequently do not.

Non-psychological and behavioral game theory have in common that neither is intended to be normative—though both are often used to try to describe norms that prevail in groups of players, as well to explain why norms might persist in groups of players even when they appear to be less than fully rational to philosophical intuitions.

Let us therefore group non-psychological and behavioral game theorists together, just for purposes of contrast with normative game theorists, as descriptive game theorists.

Descriptive game theorists are often inclined to doubt that the goal of seeking a general theory of rationality makes sense as a project. Institutions and evolutionary processes build many environments, and what counts as rational procedure in one environment may not be favoured in another.

On the other hand, an entity that does not at least stochastically i. To such entities game theory has no application in the first place. This does not imply that non-psychological game theorists abjure all principled ways of restricting sets of NE to subsets based on their relative probabilities of arising.

In particular, non-psychological game theorists tend to be sympathetic to approaches that shift emphasis from rationality onto considerations of the informational dynamics of games.

We should perhaps not be surprised that NE analysis alone often fails to tell us much of applied, empirical interest about strategic-form games e.

Equilibrium selection issues are often more fruitfully addressed in the context of extensive-form games. In order to deepen our understanding of extensive-form games, we need an example with more interesting structure than the PD offers.

This game is not intended to fit any preconceived situation; it is simply a mathematical object in search of an application.

If you are confused by this, remember that a strategy must tell a player what to do at every information set where that player has an action. Since each player chooses between two actions at each of two information sets here, each player has four strategies in total.

The first letter in each strategy designation tells each player what to do if he or she reaches their first information set, the second what to do if their second information set is reached.

This is a bit puzzling, since if Player I reaches her second information set 7 in the extensive-form game, she would hardly wish to play L there; she earns a higher payoff by playing R at node 7.

In analyzing extensive-form games, however, we should care what happens off the path of play, because consideration of this is crucial to what happens on the path.

We are throwing away information relevant to game solutions if we ignore off-path outcomes, as mere NE analysis does. Notice that this reason for doubting that NE is a wholly satisfactory equilibrium concept in itself has nothing to do with intuitions about rationality, as in the case of the refinement concepts discussed in Section 2.

Begin, again, with the last subgame, that descending from node 7. At node 5 II chooses R. Note that, as in the PD, an outcome appears at a terminal node— 4, 5 from node 7—that is Pareto superior to the NE.

Again, however, the dynamics of the game prevent it from being reached. It gives an outcome that yields a NE not just in the whole game but in every subgame as well.

This is a persuasive solution concept because, again unlike the refinements of Section 2. It does, however, assume that players not only know everything strategically relevant to their situation but also use all of that information.

But, as noted earlier, it is best to be careful not to confuse the general normative idea of rationality with computational power and the possession of budgets, in time and energy, to make the most of it.

An agent playing a subgame perfect strategy simply chooses, at every node she reaches, the path that brings her the highest payoff in the subgame emanating from that node.

A main value of analyzing extensive-form games for SPE is that this can help us to locate structural barriers to social optimization.

If our players wish to bring about the more socially efficient outcome 4,5 here, they must do so by redesigning their institutions so as to change the structure of the game.

The enterprise of changing institutional and informational structures so as to make efficient outcomes more likely in the games that agents that is, people, corporations, governments, etc.

The main techniques are reviewed in Hurwicz and Reiter , the first author of which was awarded the Nobel Prize for his pioneering work in the area.

Many readers, but especially philosophers, might wonder why, in the case of the example taken up in the previous section, mechanism design should be necessary unless players are morbidly selfish sociopaths.

This theme is explored with great liveliness and polemical force in Binmore , We have seen that in the unique NE of the PD, both players get less utility than they could have through mutual cooperation.

This may strike you, even if you are not a Kantian as it has struck many commentators as perverse. Surely, you may think, it simply results from a combination of selfishness and paranoia on the part of the players.

To begin with they have no regard for the social good, and then they shoot themselves in the feet by being too untrustworthy to respect agreements.

This way of thinking is very common in popular discussions, and badly mixed up. To dispel its influence, let us first introduce some terminology for talking about outcomes.

Welfare economists typically measure social good in terms of Pareto efficiency. Now, the outcome 3,3 that represents mutual cooperation in our model of the PD is clearly Pareto superior over mutual defection; at 3,3 both players are better off than at 2,2.

So it is true that PDs lead to inefficient outcomes. This was true of our example in Section 2. However, inefficiency should not be associated with immorality.

A utility function for a player is supposed to represent everything that player cares about , which may be anything at all. As we have described the situation of our prisoners they do indeed care only about their own relative prison sentences, but there is nothing essential in this.

What makes a game an instance of the PD is strictly and only its payoff structure. Thus we could have two Mother Theresa types here, both of whom care little for themselves and wish only to feed starving children.

But suppose the original Mother Theresa wishes to feed the children of Calcutta while Mother Juanita wishes to feed the children of Bogota.

Our saints are in a PD here, though hardly selfish or unconcerned with the social good. In that case, this must be reflected in their utility functions, and hence in their payoffs.

But all this shows is that not every possible situation is a PD; it does not show that selfishness is among the assumptions of game theory. Agents who wish to avoid inefficient outcomes are best advised to prevent certain games from arising; the defender of the possibility of Kantian rationality is really proposing that they try to dig themselves out of such games by turning themselves into different kinds of agents.

In general, then, a game is partly defined by the payoffs assigned to the players. In any application, such assignments should be based on sound empirical evidence.

Our last point above opens the way to a philosophical puzzle, one of several that still preoccupy those concerned with the logical foundations of game theory.

It can be raised with respect to any number of examples, but we will borrow an elegant one from C. Consider the following game:. The NE outcome here is at the single leftmost node descending from node 8.

To see this, backward induct again. At node 10, I would play L for a payoff of 3, giving II a payoff of 1. II can do better than this by playing L at node 9, giving I a payoff of 0.

I can do better than this by playing L at node 8; so that is what I does, and the game terminates without II getting to move. A puzzle is then raised by Bicchieri along with other authors, including Binmore and Pettit and Sugden by way of the following reasoning.

But now we have the following paradox: Both players use backward induction to solve the game; backward induction requires that Player I know that Player II knows that Player I is economically rational; but Player II can solve the game only by using a backward induction argument that takes as a premise the failure of Player I to behave in accordance with economic rationality.

This is the paradox of backward induction. That is, a player might intend to take an action but then slip up in the execution and send the game down some other path instead.

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Aufl View all editions and formats Rating: These are games prevailing over all forms of society. Pooling games are repeated plays with changing payoff table in general over an experienced path and their equilibrium strategies usually take a form of evolutionary social convention and economic convention.

Pooling game theory emerges to formally recognize the interaction between optimal choice in one play and the emergence of forthcoming payoff table update path, identify the invariance existence and robustness, and predict variance over time.

The theory is based upon topological transformation classification of payoff table update over time to predict variance and invariance, and is also within the jurisdiction of the computational law of reachable optimality for ordered system.

Mean field game theory is the study of strategic decision making in very large populations of small interacting agents.

This class of problems was considered in the economics literature by Boyan Jovanovic and Robert W. Rosenthal , in the engineering literature by Peter E.

The games studied in game theory are well-defined mathematical objects. To be fully defined, a game must specify the following elements: These equilibrium strategies determine an equilibrium to the game—a stable state in which either one outcome occurs or a set of outcomes occur with known probability.

Most cooperative games are presented in the characteristic function form, while the extensive and the normal forms are used to define noncooperative games.

The extensive form can be used to formalize games with a time sequencing of moves. Games here are played on trees as pictured here.

Here each vertex or node represents a point of choice for a player. The player is specified by a number listed by the vertex.

The lines out of the vertex represent a possible action for that player. The payoffs are specified at the bottom of the tree. The extensive form can be viewed as a multi-player generalization of a decision tree.

It involves working backward up the game tree to determine what a rational player would do at the last vertex of the tree, what the player with the previous move would do given that the player with the last move is rational, and so on until the first vertex of the tree is reached.

The game pictured consists of two players. The way this particular game is structured i. Suppose that Player 1 chooses U and then Player 2 chooses A: Player 1 then gets a payoff of "eight" which in real-world terms can be interpreted in many ways, the simplest of which is in terms of money but could mean things such as eight days of vacation or eight countries conquered or even eight more opportunities to play the same game against other players and Player 2 gets a payoff of "two".

The extensive form can also capture simultaneous-move games and games with imperfect information. To represent it, either a dotted line connects different vertices to represent them as being part of the same information set i.

See example in the imperfect information section. The normal or strategic form game is usually represented by a matrix which shows the players, strategies, and payoffs see the example to the right.

More generally it can be represented by any function that associates a payoff for each player with every possible combination of actions.

In the accompanying example there are two players; one chooses the row and the other chooses the column. Each player has two strategies, which are specified by the number of rows and the number of columns.

The payoffs are provided in the interior. The first number is the payoff received by the row player Player 1 in our example ; the second is the payoff for the column player Player 2 in our example.

Suppose that Player 1 plays Up and that Player 2 plays Left. Then Player 1 gets a payoff of 4, and Player 2 gets 3. When a game is presented in normal form, it is presumed that each player acts simultaneously or, at least, without knowing the actions of the other.

If players have some information about the choices of other players, the game is usually presented in extensive form.

Every extensive-form game has an equivalent normal-form game, however the transformation to normal form may result in an exponential blowup in the size of the representation, making it computationally impractical.

In games that possess removable utility, separate rewards are not given; rather, the characteristic function decides the payoff of each unity.

The balanced payoff of C is a basic function. Although there are differing examples that help determine coalitional amounts from normal games, not all appear that in their function form can be derived from such.

Formally, a characteristic function is seen as: N,v , where N represents the group of people and v: Such characteristic functions have expanded to describe games where there is no removable utility.

As a method of applied mathematics , game theory has been used to study a wide variety of human and animal behaviors. It was initially developed in economics to understand a large collection of economic behaviors, including behaviors of firms, markets, and consumers.

The first use of game-theoretic analysis was by Antoine Augustin Cournot in with his solution of the Cournot duopoly. The use of game theory in the social sciences has expanded, and game theory has been applied to political, sociological, and psychological behaviors as well.

This work predates the name "game theory", but it shares many important features with this field. The developments in economics were later applied to biology largely by John Maynard Smith in his book Evolution and the Theory of Games.

In addition to being used to describe, predict, and explain behavior, game theory has also been used to develop theories of ethical or normative behavior and to prescribe such behavior.

Game-theoretic arguments of this type can be found as far back as Plato. The primary use of game theory is to describe and model how human populations behave.

This particular view of game theory has been criticized. It is argued that the assumptions made by game theorists are often violated when applied to real-world situations.

Game theorists usually assume players act rationally, but in practice, human behavior often deviates from this model. Game theorists respond by comparing their assumptions to those used in physics.

Thus while their assumptions do not always hold, they can treat game theory as a reasonable scientific ideal akin to the models used by physicists.

There is an ongoing debate regarding the importance of these experiments and whether the analysis of the experiments fully captures all aspects of the relevant situation.

Price , have turned to evolutionary game theory in order to resolve these issues. These models presume either no rationality or bounded rationality on the part of players.

Despite the name, evolutionary game theory does not necessarily presume natural selection in the biological sense. Evolutionary game theory includes both biological as well as cultural evolution and also models of individual learning for example, fictitious play dynamics.

Some scholars, like Leonard Savage , [ citation needed ] see game theory not as a predictive tool for the behavior of human beings, but as a suggestion for how people ought to behave.

This normative use of game theory has also come under criticism. Game theory is a major method used in mathematical economics and business for modeling competing behaviors of interacting agents.

This research usually focuses on particular sets of strategies known as "solution concepts" or "equilibria". A common assumption is that players act rationally.

In non-cooperative games, the most famous of these is the Nash equilibrium. A set of strategies is a Nash equilibrium if each represents a best response to the other strategies.

If all the players are playing the strategies in a Nash equilibrium, they have no unilateral incentive to deviate, since their strategy is the best they can do given what others are doing.

The payoffs of the game are generally taken to represent the utility of individual players. A prototypical paper on game theory in economics begins by presenting a game that is an abstraction of a particular economic situation.

One or more solution concepts are chosen, and the author demonstrates which strategy sets in the presented game are equilibria of the appropriate type.

Naturally one might wonder to what use this information should be put. Economists and business professors suggest two primary uses noted above: The application of game theory to political science is focused in the overlapping areas of fair division , political economy , public choice , war bargaining , positive political theory , and social choice theory.

In each of these areas, researchers have developed game-theoretic models in which the players are often voters, states, special interest groups, and politicians.

Early examples of game theory applied to political science are provided by Anthony Downs. In his book An Economic Theory of Democracy , [53] he applies the Hotelling firm location model to the political process.

In the Downsian model, political candidates commit to ideologies on a one-dimensional policy space. Downs first shows how the political candidates will converge to the ideology preferred by the median voter if voters are fully informed, but then argues that voters choose to remain rationally ignorant which allows for candidate divergence.

Game Theory was applied in to the Cuban missile crisis during the presidency of John F. It has also been proposed that game theory explains the stability of any form of political government.

Taking the simplest case of a monarchy, for example, the king, being only one person, does not and cannot maintain his authority by personally exercising physical control over all or even any significant number of his subjects.

Sovereign control is instead explained by the recognition by each citizen that all other citizens expect each other to view the king or other established government as the person whose orders will be followed.

Coordinating communication among citizens to replace the sovereign is effectively barred, since conspiracy to replace the sovereign is generally punishable as a crime.

A game-theoretic explanation for democratic peace is that public and open debate in democracies sends clear and reliable information regarding their intentions to other states.

In contrast, it is difficult to know the intentions of nondemocratic leaders, what effect concessions will have, and if promises will be kept.

Thus there will be mistrust and unwillingness to make concessions if at least one of the parties in a dispute is a non-democracy.

On the other hand, game theory predicts that two countries may still go to war even if their leaders are cognizant of the costs of fighting.

War may result from asymmetric information; two countries may have incentives to mis-represent the amount of military resources they have on hand, rendering them unable to settle disputes agreeably without resorting to fighting.

Moreover, war may arise because of commitment problems: Finally, war may result from issue indivisibilities. Wood thought this could be accomplished by making treaties with other nations to reduce greenhouse gas emissions.

Unlike those in economics, the payoffs for games in biology are often interpreted as corresponding to fitness.

In addition, the focus has been less on equilibria that correspond to a notion of rationality and more on ones that would be maintained by evolutionary forces.

Although its initial motivation did not involve any of the mental requirements of the Nash equilibrium , every ESS is a Nash equilibrium.

In biology, game theory has been used as a model to understand many different phenomena. It was first used to explain the evolution and stability of the approximate 1: Fisher suggested that the 1: Additionally, biologists have used evolutionary game theory and the ESS to explain the emergence of animal communication.

For example, the mobbing behavior of many species, in which a large number of prey animals attack a larger predator, seems to be an example of spontaneous emergent organization.

Biologists have used the game of chicken to analyze fighting behavior and territoriality. According to Maynard Smith, in the preface to Evolution and the Theory of Games , "paradoxically, it has turned out that game theory is more readily applied to biology than to the field of economic behaviour for which it was originally designed".

Evolutionary game theory has been used to explain many seemingly incongruous phenomena in nature.

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