Common knowledge (logic)

Common Knowledge, in German referred to as common knowledge or common knowledge, is a game- theoretical concept of the information structure of players. Accordingly, there is the knowledge of the players in addition to the pure knowledge of a fact or an event and from knowledge of the individual players on their knowledge with each other. For the analysis of games within the framework of game theory, it is important to know what is present to the players as common knowledge and what is not.

• 2.1 Formal representation
• 2.2 Example

• 3.1 In the case of complete information
• 3.2 In the case of incomplete information
• 3.3 Bayesian Games

Term

Definition

Common knowledge is information or events that everyone knows the players and each of them knows that they are all known to others, and also that all in turn know that everyone knows that they are known to all, etc. The knowledge of the players infinitely interleaved with each other. Accordingly, an information common knowledge, if

• All players know the information
• All players know that all players know the information
• All players know that all players know that all players know the information

Over an infinite number of levels of knowledge of time.

If the process has only a finite number of levels of knowledge across before, it is called Bounded Knowledge. The difference is mainly in subgame perfect games play a significant role. The concept of information Bounded Knowledge, however, has in game theory a far less important than common knowledge.

Example

This example illustrates the required infinite regress, which is required for formation of common knowledge. It is designed in accordance with the example in Rieck (2007).

In the context of this example, the information that both prisoners want to break the next morning never become common knowledge. To make the information common knowledge, would have the confirmation of the receipt of the news of an infinite number of levels available. The number is limited here to a finite number of iterations, so that it can at any time come to an outbreak here. A finite number of steps as it is in the example, only results in the Bounded Knowledge assumption. Suppose A prisoner would not receive the confirmation of the confirmation here, so although both would know the planned breakout the next day, but the information would not be common knowledge. There would be no escape of the two prisoners.

History

Thomas Schelling first time in 1960 found that common knowledge arises from an infinite regress, the actual concept of common knowledge can be found in 1969 when the philosopher David Kellogg Lewis. However, it measures the concept a more holistic meaning to be a purely game-theoretic. The first formal presentation delivers Robert Aumann in Agreeing to disagree (1976). The mathematician develops while the theorem that players who trust each other, in their knowledge about an event in retrospect can not agree to be in disagreement. 2005 Aumann and Schelling were honored for their achievements in the field of game theory with the price of Economics at the Swedish Riksbank in Memory of Alfred Nobel.

Formal representation

In its mathematical formalization Aumann uses the laws of set theory. The following formalization is analogous to his.

Given a set of states S. The event E is a subset of S. For each player i put pi is a partition of S. In this partition should be the knowledge of player i in a state. In the state s i player knows that, in one of the sets of the partition P ( s ) is included, but not which one. Pi ( s ) stands for the single-element Pi, which contains s.

One can now define a knowledge function K as follows:

In the event that "no e" can define the following formula:

With the assumptions for the e, , e function can then define a common knowledge function:

Example

The concept of Common Knowledge is often illustrated with a version of the following story.

It raises the question of implications attracts the notice of acquaintances by itself.

K = 1

K = 2

K = 3

Conclusion

Significance for game theory

The concept of common knowledge is necessary for the development of solution concepts of games, especially in the area of non-cooperative game theory, the most important part of game theory, it is of utmost importance. Non- cooperative games are games in which the player, as opposed to coalitions based on cooperative game theory, can not make binding agreements. The non-cooperative game theory is a field of microeconomics and is mainly concerned with actions and strategies of interacting players trying their utility in (partial ) to maximize knowledge of their environment. It is of great importance what knowledge everyone has here. Common knowledge is in the non-cooperative game theory in almost all cases, a basic assumption or prior knowledge into the analysis a. It is generally assumed that the rules are always common knowledge. The rules specify the exact course of a game. In addition, they also contain information about the payoffs, the information districts, the probabilities of random trains, etc. Usually, it is also assumed that all players behave rationally and also know that all know that all behave rationally, etc. The rationality of the players is also common knowledge. It is relevant for the description of a game, what information in the players at what point of the game. Thus, different information can lead to a decision point to different decisions and payoffs of the players. Common knowledge makes it possible to draw conclusions about the information stands of the players.

In the full information

Games with complete information are easy to analyze in general. If in addition to the common knowledge assumption, the strategy sets Si and payoff function ui ( s ) of all the players are common knowledge, is by definition before a game with complete information. The game with complete information contains information on the amount of players, the strategy and the utility function and can be expressed in the form = (N, S, u). The players always choose the train that generates them the maximum benefit, and this is known to all other players. Due to the complete transparency and common knowledge of rationality assumption regarding players can guess the behavior of the other players in the various game situations and adapt their own strategy accordingly. In the analysis of games with complete information, the transparency will be taken advantage of, which creates the common knowledge assumption. The solution concepts equilibrium in dominant strategies and Nash equilibrium implicitly assume. Disadvantage of games with complete information, however, is that no games can be mapped, in which some more players, or other information available as others.

When developed by Reinhard Selten subgame perfection is a solution concept in which an analysis of the identification of the Nash equilibria of all existing games part of a game is determined by backward induction. The principle of the Nash equilibrium, in the subgame perfection refined with the aim of ruling out implausible equilibria ( equilibria with implausible threats ). Basically, the solution of games with subgame perfection with the assumption of common knowledge compatible. In some games but can be justified node that can not be attained by subgame perfection, by assuming that either

• At least one of the players is not rational or
• The rationality of the players is not common knowledge and there is only a finite number of steps ( Bounded Knowledge ).

In the area of ​​incomplete information

In games with incomplete information can be captured with complete information, aspects of game situations that arise from information asymmetries in contrast to games. Incomplete information means that a player i certain properties are not known to the other players. These properties can, for example, preferences, act conjectures about other players or Initial ownership. One speaks in this context of private information of the players. In games with incomplete information, the common knowledge assumption is violated, as not all players fully informed of the rules of the game. For this reason, an alternative solution concept is developed for games of this type. It is possible to convert a game with incomplete information into a game with complete but imperfect information. By the transformation of the game is well defined and there is no more uncertainty with respect to the rules. To Imperfect information is when certain actions of players other are not observable. If in the game but no private information before, we speak of perfect information.

Bayesian games

A game with incomplete information can also be called a Bayesian game, as the recourse here applied solution concept to the Bayesian formula. It was developed by Harsanyi. The existing uncertainty in these games is with respect to the unequal information bypassed by a trick: At the beginning of the game Zufallszug nature (referred to as player 0 ) is introduced, which defines the type ti of player i. This specific type is only the concerned player i known. The other players have only beliefs about the probability with which a player belongs to a certain type ti. These ideas are called beliefs. The transformation of the games with incomplete information in games with imperfect information, the common knowledge assumption in the framework of Bayesian games can be maintained. This makes it possible to solve games of this type.