Die Spieltheorie ist ein Teilgebiet der Mathematik, das sich mit der Ein Spiel im Sinne der Spieltheorie ist eine Entscheidungssituation mit mehreren. Die Spieltheorie ist eine wirtschaftstheoretische Methodenlehre, welche das Ziel hat Denkfehler bei der strategischen Planung mithilfe mathematischer Fehler. Einführung in die. Spieltheorie von. Prof. Dr. Wolfgang Leininger und. PD Dr. Erwin Amann. Lehrstuhl Wirtschaftstheorie. Universität Dortmund. Postfach

Gametree Odd Even 1. Gametree Odd Even 2. Gametree Odd Even 3. Gametree Odd Even 4. Hare and Hounds board. Illustration of the Minimax Search.

Krzywa reakcji Kraju Bertrand. MJE - Figura 1B. MJE - Figura 1C. Moivre - Doctrine of chances, - Monty Hall problem game theory.

Normal Form Game 1. Normal Form Game 2. PD with outside option. Retrieved from " https: Applied mathematics Discrete mathematics Economics Games Decision theory Subdivisions of mathematics.

Views View Edit History. In other projects Wikimedia Commons Wikipedia Wikiversity. As a result, economists were forced to treat non-parametric influences as if they were complications on parametric ones.

This is likely to strike the reader as odd, since, as our example of the bridge-crossing problem was meant to show, non-parametric features are often fundamental features of decision-making problems.

Classical economists, such as Adam Smith and David Ricardo, were mainly interested in the question of how agents in very large markets—whole nations—could interact so as to bring about maximum monetary wealth for themselves.

Economists always recognized that this set of assumptions is purely an idealization for purposes of analysis, not a possible state of affairs anyone could try or should want to try to attain.

But until the mathematics of game theory matured near the end of the s, economists had to hope that the more closely a market approximates perfect competition, the more efficient it will be.

No such hope, however, can be mathematically or logically justified in general; indeed, as a strict generalization the assumption was shown to be false as far back as the s.

This article is not about the foundations of economics, but it is important for understanding the origins and scope of game theory to know that perfectly competitive markets have built into them a feature that renders them susceptible to parametric analysis.

Because agents face no entry costs to markets, they will open shop in any given market until competition drives all profits to zero.

This implies that if production costs are fixed and demand is exogenous, then agents have no options about how much to produce if they are trying to maximize the differences between their costs and their revenues.

These production levels can be determined separately for each agent, so none need pay attention to what the others are doing; each agent treats her counterparts as passive features of the environment.

The other kind of situation to which classical economic analysis can be applied without recourse to game theory is that of a monopoly facing many customers.

However, both perfect and monopolistic competition are very special and unusual market arrangements. Prior to the advent of game theory, therefore, economists were severely limited in the class of circumstances to which they could neatly apply their models.

Philosophers share with economists a professional interest in the conditions and techniques for the maximization of human welfare.

In addition, philosophers have a special concern with the logical justification of actions, and often actions must be justified by reference to their expected outcomes.

One tradition in philosophy, utilitarianism, is based on the idea that all justifiable actions must be justified in this way. Without game theory, both of these problems resist analysis wherever non-parametric aspects are relevant.

In doing this, we will need to introduce, define and illustrate the basic elements and techniques of game theory. To this job we therefore now turn.

An economic agent is, by definition, an entity with preferences. Game theorists, like economists and philosophers studying rational decision-making, describe these by means of an abstract concept called utility.

This refers to some ranking, on some specified scale, of the subjective welfare or change in subjective welfare that an agent derives from an object or an event.

For example, we might evaluate the relative welfare of countries which we might model as agents for some purposes by reference to their per capita incomes, and we might evaluate the relative welfare of an animal, in the context of predicting and explaining its behavioral dispositions, by reference to its expected evolutionary fitness.

In the case of people, it is most typical in economics and applications of game theory to evaluate their relative welfare by reference to their own implicit or explicit judgments of it.

This is why we referred above to subjective welfare. Consider a person who adores the taste of pickles but dislikes onions.

She might be said to associate higher utility with states of the world in which, all else being equal, she consumes more pickles and fewer onions than with states in which she consumes more onions and fewer pickles.

However, economists in the early 20th century recognized increasingly clearly that their main interest was in the market property of decreasing marginal demand, regardless of whether that was produced by satiated individual consumers or by some other factors.

In the s this motivation of economists fit comfortably with the dominance of behaviourism and radical empiricism in psychology and in the philosophy of science respectively.

If this looks circular to you, it should: Like other tautologies occurring in the foundations of scientific theories, this interlocking recursive system of definitions is useful not in itself, but because it helps to fix our contexts of inquiry.

When such theorists say that agents act so as to maximize their utility, they want this to be part of the definition of what it is to be an agent, not an empirical claim about possible inner states and motivations.

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 motivations of some flesh-and-blood actors such as actual people.

Rather, they should regard game theory as part of the body of mathematics that is used to model those entities which might or might not literally exist who consistently select elements from mutually exclusive action sets, resulting in patterns of choices, which, allowing for some stochasticity and noise, can be statistically modeled as maximization of utility functions.

On this interpretation, game theory could not be refuted by any empirical observations, since it is not an empirical theory in the first place. Of course, observation and experience could lead someone favoring this interpretation to conclude that game theory is of little help in describing actual human behavior.

Some other theorists understand the point of game theory differently. They view game theory as providing an explanatory account of strategic reasoning.

These two very general ways of thinking about the possible uses of game theory are compatible with the tautological interpretation of utility maximization.

The philosophical difference is not idle from the perspective of the working game theorist, however. As we will see in a later section, those who hope to use game theory to explain strategic reasoning , as opposed to merely strategic behavior , face some special philosophical and practical problems.

Since game theory is a technology for formal modeling, we must have a device for thinking of utility maximization in mathematical terms.

Such a device is called a utility function. We will introduce the general idea of a utility function through the special case of an ordinal utility function.

Later, we will encounter utility functions that incorporate more information. Suppose that agent x prefers bundle a to bundle b and bundle b to bundle c.

We then map these onto a list of numbers, where the function maps the highest-ranked bundle onto the largest number in the list, the second-highest-ranked bundle onto the next-largest number in the list, and so on, thus:.

The only property mapped by this function is order. The magnitudes of the numbers are irrelevant; that is, it must not be inferred that x gets 3 times as much utility from bundle a as she gets from bundle c.

Thus we could represent exactly the same utility function as that above by. The numbers featuring in an ordinal utility function are thus not measuring any quantity of anything.

For the moment, however, we will need only ordinal functions. All situations in which at least one agent can only act to maximize his utility through anticipating either consciously, or just implicitly in his behavior the responses to his actions by one or more other agents is called a game.

Agents involved in games are referred to as players. If all agents have optimal actions regardless of what the others do, as in purely parametric situations or conditions of monopoly or perfect competition see Section 1 above we can model this without appeal to game theory; otherwise, we need it.

In literature critical of economics in general, or of the importation of game theory into humanistic disciplines, this kind of rhetoric has increasingly become a magnet for attack.

The reader should note that these two uses of one word within the same discipline are technically unconnected. Furthermore, original RPT has been specified over the years by several different sets of axioms for different modeling purposes.

Once we decide to treat rationality as a technical concept, each time we adjust the axioms we effectively modify the concept.

Consequently, in any discussion involving economists and philosophers together, we can find ourselves in a situation where everyone uses the same word to refer to something different.

For readers new to economics, game theory, decision theory and the philosophy of action, this situation naturally presents a challenge.

We might summarize the intuition behind all this as follows: Economic rationality might in some cases be satisfied by internal computations performed by an agent, and she might or might not be aware of computing or having computed its conditions and implications.

In other cases, economic rationality might simply be embodied in behavioral dispositions built by natural, cultural or market selection.

Each player in a game faces a choice among two or more possible strategies. The significance of the italicized phrase here will become clear when we take up some sample games below.

A crucial aspect of the specification of a game involves the information that players have when they choose strategies.

A board-game of sequential moves in which both players watch all the action and know the rules in common , such as chess, is an instance of such a game.

By contrast, the example of the bridge-crossing game from Section 1 above illustrates a game of imperfect information , since the fugitive must choose a bridge to cross without knowing the bridge at which the pursuer has chosen to wait, and the pursuer similarly makes her decision in ignorance of the choices of her quarry.

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.

Darum wird in spieltheoretischen Modellen meist nicht von perfekter Information ausgegangen. Spiele werden meist entweder in strategischer Normal- Form oder in extensiver Form beschrieben.

Weiterhin ist noch die Agentennormalform zu nennen. Gerecht wird diese Darstellungsform am ehesten solchen Spielen, bei denen alle Spieler ihre Strategien zeitgleich und ohne Kenntnis der Wahl der anderen Spieler festlegen.

Zur Veranschaulichung verwendet man meist eine Bimatrixform. Wer oder was ist eigentlich ein Spieler in einer gegebenen Situation?

Die Agentennormalform beantwortet diese Frage so: Wichtige sind das Minimax-Gleichgewicht , das wiederholte Streichen dominierter Strategien sowie Teilspielperfektheit und in der kooperativen Spieltheorie der Core, der Nucleolus , die Verhandlungsmenge und die Imputationsmenge.

Damit ist eine reine Strategie der Spezialfall einer gemischten Strategie, in der immer dann, wenn die Aktionsmenge eines Spielers nichtleer ist, die gesamte Wahrscheinlichkeitsmasse auf eine einzige Aktion der Aktionsmenge gelegt wird.

Man kann leicht zeigen, dass jedes Spiel, dessen Aktionsmengen endlich sind, ein Nash-Gleichgewicht in gemischten Strategien haben muss. In der Spieltheorie unterscheidet man zudem zwischen endlich wiederholten und unendlich wiederholten Superspielen.

Die Analyse wiederholter Spiele wurde wesentlich von Robert J. Man unterstellt also allgemein bekannte Spielregeln, bzw. Evolutionstheoretisch besagt diese Spieltheorie, dass jeweils nur die am besten angepasste Strategie bzw.

Der Begriff Spieltheorie engl. Generell wird die nichtkooperative von der kooperativen Spieltheorie so unterschieden: Kooperative Spieltheorie ist als axiomatische Theorie von Koalitionsfunktionen charakteristischen Funktionen aufzufassen und ist auszahlungsorientiert.

Nichtkooperative Spieltheorie ist dagegen aktions- bzw. Aumann und John Forbes Nash Jr. Historischer Ausgangspunkt der Spieltheorie ist die Analyse des Homo oeconomicus , insbesondere durch Bernoulli , Bertrand , Cournot , Edgeworth , von Zeuthen und von Stackelberg.

Dieses Buch gilt auch heute noch als wegweisender Meilenstein. Simon und Daniel Kahneman den Nobelpreis. Maskin und Roger B.

Die Spieltheorie modelliert die verschiedensten Situationen als ein Spiel. Im Spiel Gefangenendilemma sind die Spieler die beiden Gefangenen und ihre Aktionsmengen sind aussagen und schweigen.

In der Informatik versucht man, mit Hilfe von Suchstrategien und Heuristiken allgemein: Man spricht in diesem Zusammenhang vom first movers advantage bzw.

Unterschieden werden hierbei drei Begriffe: Game theory author Mitsuo Suzuki. Game Theory by Morton Davis - Flickr - brewbooks.

Game theory schwimmbad kino. Myerson - Flickr - brewbooks. Game with no value. Gametheory Example Extensive Game. Gametree Odd Even 1. Gametree Odd Even 2.

Gametree Odd Even 3. Gametree Odd Even 4. Hare and Hounds board. Illustration of the Minimax Search. Krzywa reakcji Kraju Bertrand. MJE - Figura 1B.

MJE - Figura 1C. Moivre - Doctrine of chances, - Monty Hall problem game theory. Normal Form Game 1.

This need not be perfect information about every action of earlier players; it might be very little knowledge. The difference between simultaneous and sequential games is captured in the different representations discussed above.

Often, normal form is used to represent simultaneous games, while extensive form is used to represent sequential ones. The transformation of extensive to normal form is one way, meaning that multiple extensive form games correspond to the same normal form.

Consequently, notions of equilibrium for simultaneous games are insufficient for reasoning about sequential games; see subgame perfection.

An important subset of sequential games consists of games of perfect information. A game is one of perfect information if all players know the moves previously made by all other players.

Most games studied in game theory are imperfect-information games. Many card games are games of imperfect information, such as poker and bridge.

Games of incomplete information can be reduced, however, to games of imperfect information by introducing " moves by nature ".

Games in which the difficulty of finding an optimal strategy stems from the multiplicity of possible moves are called combinatorial games.

Examples include chess and go. Games that involve imperfect information may also have a strong combinatorial character, for instance backgammon.

There is no unified theory addressing combinatorial elements in games. There are, however, mathematical tools that can solve particular problems and answer general questions.

Games of perfect information have been studied in combinatorial game theory , which has developed novel representations, e. These methods address games with higher combinatorial complexity than those usually considered in traditional or "economic" game theory.

A related field of study, drawing from computational complexity theory , is game complexity , which is concerned with estimating the computational difficulty of finding optimal strategies.

Research in artificial intelligence has addressed both perfect and imperfect information games that have very complex combinatorial structures like chess, go, or backgammon for which no provable optimal strategies have been found.

The practical solutions involve computational heuristics, like alpha-beta pruning or use of artificial neural networks trained by reinforcement learning , which make games more tractable in computing practice.

Games, as studied by economists and real-world game players, are generally finished in finitely many moves. Pure mathematicians are not so constrained, and set theorists in particular study games that last for infinitely many moves, with the winner or other payoff not known until after all those moves are completed.

The focus of attention is usually not so much on the best way to play such a game, but whether one player has a winning strategy.

The existence of such strategies, for cleverly designed games, has important consequences in descriptive set theory. Much of game theory is concerned with finite, discrete games, that have a finite number of players, moves, events, outcomes, etc.

Many concepts can be extended, however. Continuous games allow players to choose a strategy from a continuous strategy set.

The problem of finding an optimal strategy in a differential game is closely related to the optimal control theory.

In particular, there are two types of strategies: A particular case of differential games are the games with a random time horizon.

Therefore, the players maximize the mathematical expectation of the cost function. It was shown that the modified optimization problem can be reformulated as a discounted differential game over an infinite time interval.

Evolutionary game theory studies players who adjust their strategies over time according to rules that are not necessarily rational or farsighted.

Such rules may feature imitation, optimization or survival of the fittest. In the social sciences, such models typically represent strategic adjustment by players who play a game many times within their lifetime and, consciously or unconsciously, occasionally adjust their strategies.

Individual decision problems with stochastic outcomes are sometimes considered "one-player games". These situations are not considered game theoretical by some authors.

Although these fields may have different motivators, the mathematics involved are substantially the same, e. Stochastic outcomes can also be modeled in terms of game theory by adding a randomly acting player who makes "chance moves" " moves by nature ".

For some problems, different approaches to modeling stochastic outcomes may lead to different solutions. For example, the difference in approach between MDPs and the minimax solution is that the latter considers the worst-case over a set of adversarial moves, rather than reasoning in expectation about these moves given a fixed probability distribution.

The minimax approach may be advantageous where stochastic models of uncertainty are not available, but may also be overestimating extremely unlikely but costly events, dramatically swaying the strategy in such scenarios if it is assumed that an adversary can force such an event to happen.

General models that include all elements of stochastic outcomes, adversaries, and partial or noisy observability of moves by other players have also been studied.

The " gold standard " is considered to be partially observable stochastic game POSG , but few realistic problems are computationally feasible in POSG representation.

These are games the play of which is the development of the rules for another game, the target or subject game. Metagames seek to maximize the utility value of the rule set developed.

The theory of metagames is related to mechanism design theory. The term metagame analysis is also used to refer to a practical approach developed by Nigel Howard.

Subsequent developments have led to the formulation of confrontation analysis. 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.

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Nash-Gleichgewicht (in reinen Strategien) einfach erklärt ● Gehe auf proyectocesar.eu Überprüfung von Steuerzahlern durch das Finanzamt, ob genügend Steuern gezahlt wurden, während der zu Überprüfende dieser lieber entgehen würde. Sie sollten daher als Lösungsstrategien ausscheiden und - ähnlich wie dominierte Strategien - wiederholt eliminiert werden. Eine Einführung in die Spieltheorie, Berlin, Springer Diese Funktion ordnet jedem möglichen Spielausgang einen Auszahlungsvektor zu, d. Perfekte Information , also die Kenntnis sämtlicher Spieler über sämtliche Züge sämtlicher Spieler, ist eine rigorose Forderung, die in vielen klassischen Spielen nicht erfüllt ist: Leider existieren nicht immer strikte Gleichgewichte vgl. Dies ist eine Aufgabe der experimentellen Wirtschaftsforschung, die in der Tat eine Vielzahl robuster Phänomene identifizieren konnte, die im Widerspruch zur spieltheoretischen Analyse stehen. Gerecht wird diese Darstellungsform am ehesten solchen Spielen, bei denen alle Spieler ihre Strategien zeitgleich und ohne Kenntnis der Wahl der anderen Spieler festlegen. In derartigen Spielen sollte eine Lösungskonzeption diejenigen Strategien der Spieler auszeichnen, die den intuitiven Anforderungen an rationales Entscheiden genügen. In solchen erweiterten Rahmen kann z. Perfekte Gleichgewichte sind immer auch sequenzielle Gleichgewichte, wobei die Umkehrung nicht immer, aber fast immer zutrifft. Kooperative Spieltheorie ist als axiomatische Theorie von Koalitionsfunktionen charakteristischen Funktionen aufzufassen und ist auszahlungsorientiert.

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