Signaling game

A signal game ( also signaling game, Eng. Signaling game) is in game theory a dynamic game with incomplete information. In the standard form of signal games are played with two players, one player, the transmitter emits signals and the receiver attempts by observing the emitted signals, to draw conclusions on the type of transmitter. It should be noted that the transmission of signals is connected to the transmitter with costs, and the cost of the transmitter type dependent. Signal Games are a form of Bayesian games.

• 3.1 Separating Equilibrium
• 3.2 Semi- Separating Equilibrium
• 3.3 Pooling Equilibrium

• 4.1 roommate wanted
• 4.2 The beer - quiche game

Course of a signal game

Basically, a signal game can be divided into three rounds:

Short example

Suppose a university professor is looking for a new academic staff. In this case, the professor wishes to have someone who is very busy, as it will help him in correcting exams. The applicant is a student who corresponds with the notes fro the ideas of the professor, but it is not possible for the professor to determine whether the candidate is hardworking ( workaholic ) or lazy ( lazy ). To verify that puts the professor in the room in which the student is waiting for the professor several magazines messy, because he thinks that a diligent student will arrange the magazines.

• In the first round is determined by using a natural turn with a probability of p, if the student is busy or not.
• In the second round, the student may decide whether he sorted the magazines or not.
• In the last round decides the professor after he has been evaluated according to the signal, whether the applicant is a workaholic or a slacker if he sets the students or not.

Formal definition

Determining the types of players

Strategy choice:

The payouts depend on:

Expiration

The payouts are dependent on:

Possible equilibria

If the signal matches distinguishes between three different equilibria between pooling equilibriums (also associative equilibria ), Separating equilibriums (also separating equilibria ) and semi- Separating equilibriums (also semi -pooling equilibrium called ). The basic concept of equilibrium that is at play signal application, is the concept of Perfect Bayesian equilibrium. The following Separating and Pooling Equilibrium are described in detail.

Separating Equilibrium

A separating equilibrium exists if each transmitter on its type, depending plays different strategies. This means that there must be an equal number of transmitter types and signals, and each transmitter type is assigned to exactly one signal (Given and must apply: and ). It is important that none of the two types has an incentive to deviate from the chosen strategy, but strictly, playing his chosen strategy.

If there are for example two different types and in a game and the possible signals and are, then there is a separating equilibrium when strictly and strictly sends. The receiver then selects on the basis of the observed signal his -maximizing strategy for him. In the above example a separating equilibrium would mean that, for example, a workaholic would strictly sort the magazines and a slacker would strictly not sort the magazines.

Semi- Separating Equilibrium

A semi- Separating equilibrium can occur if there are more types as signals (Given and must apply: ).

If there are four different types but there are, for example, only two possible signals in a game, then there is a semi- Separating equilibrium before when playing for example, and a strict and and strictly.

Pooling Equilibrium

One common equilibrium exists when the transmitter is always playing, regardless of its type, the same strategy. This means that the receiver can not distinguish the signal by the different types of transmitters. In the case of unifying the balance receiver beliefs must form, by estimating how likely a particular type. Beliefs About this receiver will then maximize its strategies. In short example of a pooling equilibrium would mean that, for example, the slacker has an incentive to sort the journals and thus to disguise themselves as workaholics. This he did but only if he has to expect the camouflage a higher payout than if he is not disguised.

Examples of signal Games

The following are illustrative of signal matches to a one example of a separating balance and on the other one for a given associative balance.

Looking for roommate

The following game is a living community seeks a new roommate. Because so far all the inhabitants of the WG very funny people, they look for the empty room also a funny man. For this purpose, the WG organized a casting to find the new roommate. To check whether an applicant (B ) is a funny or a serious type, they lay during the casting a red clown nose on the coffee table, because they hope that serious bidders would never touched down the nose.

The WG can decide whether to receive or reject a candidate. In reality, it is also true that a funny candidate happy to wear the nose, while even a personal gain of 5 achieved since he rumalbert like. A serious candidate would, however, in the event that he is the nose touches a personal loss of -5 to accept because he is ashamed with the attached nose.

If the WG find a funny roommate, they would achieve a gain of 4, in the event of a serious roommate WG would just come to 0, as a serious roommate completely abolishes the positive effect of additional rent. The candidate achieved in each case a gain of 4, when he gets the room. The likelihood that the applicant is funny, is given exogenously.

The end of the game:

Through the strategy choice of player the subgame trees for eliminate and, since both strictly dominated strategies. It follows for the Beliefs of the WG: and.

It follows as separierendes balance:

The payoff for the two players thus depends only on the probability of whether the applicant is funny or serious. In the event arising as disbursements:

The beer - quiche game

In the following a rowdy game enters a pub in the morning in order to beat with a guest. Here, the Rowdy would fight as possible with a wimp because it does not strike back and this earns him a profit of 1. However, are located in the pub and thugs. When the rowdy wars with a bat, he prefers the shorter, which means a loss of -1 for the Rowdy. Therefore, the Rowdy also has the ability to escape. Although the Rowdy can not distinguish between soft eggs and bats, but he can observe what the people eat breakfast in the pub. They can choose between a beer and breakfast quiche breakfast. It is also so that wimps prefer to consume most like a quiche breakfast and bat a beer breakfast. Both have a personal gain of 1, when they eat their favorite breakfast, and a personal loss of -1 if they would eat the other breakfast. Both soft eggs and racket want most like to avoid a fight, because both carry a personal loss of -1 if they fight with Rowdy. Should the Rowdy flee, both a wimp and a bat has a personal profit of 4, because they are pleased that the disagreeable Rowdy is gone.

The probability that the guest is a wimp, is given exogenously. Here it is equal to 0.5.

The end of the game:

The fact that the wimp has an incentive to deviate, R can not close 100 % using the signal on the type, as it may be that a sissy chooses a beer breakfast to disguise himself as a batsman and thus to escape the brawl. Therefore, the R Beliefs must form with which he estimates how likely it is that a sissy beer breakfast or selects a racket the quiche breakfast.

Consequently, there are two in this example pooling equilibriums:

And and and

However, the second example is very unrealistic because a bat hardly has an incentive to have breakfast quiche, and thus the Rowdy would never believe with a probability of about 0.5 that a bat eating a quiche breakfast. However Mathematically, the second equilibrium not invalidate using the elimination of dominant strategies, but one needs developed by In -Koo Cho and David Kreps process of the intuitive criterion.

Meaning and application of signal games

Signal games just in the practical game theory a major role. They make it very vividly depict situations in which people want to interact with each other or have to, but do not have all the relevant information about each other.

Especially when " employer-employee matches " come Signal games used. With the help of signal games but can also be a company investments in advertising, or the emergence of social norms model. Another purpose of fulfilling signal games but also in the analysis of unexpected acts of the interaction partner. Especially with Pooling Equilibrium can easily observe how people assess situations where they can not distinguish the other people. Basically, you can model with signaling games all everyday situations where you have to go out of different types of interaction partners and these guys know it, but can not assign its interaction partners.