Lexikon Online ᐅSpieltheorie: Die Spieltheorie ist eine mathematische Methode, die das rationale Entscheidungsverhalten in sozialen Konfliktsituationen. Apr. Man liest immer wieder über Spieltheorie. Nobelpreise werden dafür vergeben und viel Tinte wird darüber vergossen. Aber was ist das. Bei einem Spiel im Sinne der Spieltheorie handelt es sich um ein mathematisches Modell zur Beschreibung von Vorgängen, in denen mehrere Akteure.
Few contemporary political theorists think that the particular steps by which Hobbes reasons his way to this conclusion are both sound and valid. Working through these issues here, however, would carry us away from our topic into details of contractarian political philosophy.
What is important in the present context is that these details, as they are in fact pursued in the contemporary debates, all involve sophisticated interpretation of the issues using the resources of modern game theory.
Notice that Hobbes has not argued that tyranny is a desirable thing in itself. The structure of his argument is that the logic of strategic interaction leaves only two general political outcomes possible: Sensible agents then choose tyranny as the lesser of two evils.
The distinction between acting parametrically on a passive world and acting non-parametrically on a world that tries to act in anticipation of these actions is fundamental.
The values of all of these variables are independent of your plans and intentions, since the rock has no interests of its own and takes no actions to attempt to assist or thwart you.
Furthermore, his probable responses should be expected to visit costs upon you, which you would be wise to consider.
Finally, the relative probabilities of his responses will depend on his expectations about your probable responses to his responses. The logical issues associated with the second sort of situation kicking the person as opposed to the rock are typically much more complicated, as a simple hypothetical example will illustrate.
Suppose first that you wish to cross a river that is spanned by three bridges. Assume that swimming, wading or boating across are impossible. The first bridge is known to be safe and free of obstacles; if you try to cross there, you will succeed.
The second bridge lies beneath a cliff from which large rocks sometimes fall. The third is inhabited by deadly cobras.
Now suppose you wish to rank-order the three bridges with respect to their preferability as crossing-points. The first bridge is obviously best, since it is safest.
To rank-order the other two bridges, you require information about their relative levels of danger. Your reasoning here is strictly parametric because neither the rocks nor the cobras are trying to influence your actions, by, for example, concealing their typical patterns of behaviour because they know you are studying them.
It is obvious what you should do here: Now let us complicate the situation a bit. Your decision-making situation here is slightly more complicated, but it is still strictly parametric.
However, this is all you must decide, and your probability of a successful crossing is entirely up to you; the environment is not interested in your plans.
However, if we now complicate the situation by adding a non-parametric element, it becomes more challenging.
Suppose that you are a fugitive of some sort, and waiting on the other side of the river with a gun is your pursuer. She will catch and shoot you, let us suppose, only if she waits at the bridge you try to cross; otherwise, you will escape.
As you reason through your choice of bridge, it occurs to you that she is over there trying to anticipate your reasoning.
It will seem that, surely, choosing the safe bridge straight away would be a mistake, since that is just where she will expect you, and your chances of death rise to certainty.
So perhaps you should risk the rocks, since these odds are much better. But wait … if you can reach this conclusion, your pursuer, who is just as rational and well-informed as you are, can anticipate that you will reach it, and will be waiting for you if you evade the rocks.
So perhaps you must take your chances with the cobras; that is what she must least expect. But, then, no … if she expects that you will expect that she will least expect this, then she will most expect it.
This dilemma, you realize with dread, is general: You appear to be trapped in indecision. All that might console you a bit here is that, on the other side of the river, your pursuer is trapped in exactly the same quandary, unable to decide which bridge to wait at because as soon as she imagines committing to one, she will notice that if she can find a best reason to pick a bridge, you can anticipate that same reason and then avoid her.
We know from experience that, in situations such as this, people do not usually stand and dither in circles forever. However, until the s neither philosophers nor economists knew how to find it mathematically.
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.
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. 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 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 ,  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: 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: 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.