Signal game

A signal margin (also signaling game , Eng .: signaling game ) is in game theory , a dynamic game with incomplete information . In the standard form, signal games are played with two players, with one player, the transmitter, emitting signals and the receiver trying to draw conclusions about the type of transmitter by observing the signals emitted. It should be noted that sending signals is associated with costs for the transmitter, the costs depending on the type of transmitter. Signal games are a form of Bayesian games .

Course of a signal game

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

1. In the first round, the type of transmitter is determined using a random move (also called a natural move).
2. In the second round, depending on its type, the transmitter emits a signal from which it expects the greatest possible profit.
3. In the third round, the receiver tries to assess the type of transmitter by observing the signal emitted and, depending on this, selects the answer that is best for him.

Brief example

Suppose a university professor is looking for a new research assistant. In doing so, the professor would like someone who is very hardworking because he should help him correct exams. The applicant is a student whose grades correspond to the expectations of the professor, but it is not possible for the professor to determine whether the applicant is hard-working (workaholic) or lazy (lazy). To check this, the professor lays out several magazines in a mess in the room where the student is waiting for the professor, since he thinks that a hardworking student will sort the magazines.

• In the first round, a natural move is used to determine whether the student is hardworking or not with a probability of p.
• In the second round, the student can decide whether to sort the magazines or not.
• In the last round, the professor decides, after assessing based on the signal whether the applicant is a workaholic or a lazy man, whether he will hire the student or not.

Formal definition

Definition of the player types

Player i, with i = S (sender), E (receiver)

Random move determines the type of transmitter , with ${\displaystyle \mathbf {t_{j}\in T} }$ ${\displaystyle T=\{t_{1},\dotsb ,t_{J}\}}$ ${\displaystyle j=1,\dotsc ,J}$

${\displaystyle J}$ = Number of possible types of transmitter

Strategy choice:

S chooses its strategy , with ${\displaystyle \mathbf {n_{k}\in N} }$ ${\displaystyle N=\{n_{1},\dotsb ,n_{K}\}}$ ${\displaystyle n=1,\dotsc ,K}$

${\displaystyle K}$ = Number of different signals that S can give

E chooses his response strategy , with ${\displaystyle \mathbf {a_{l}\in A} }$ ${\displaystyle A=\{a_{1},\dotsb ,a_{L}\}}$ ${\displaystyle a=1,\dotsc ,L}$

${\displaystyle L}$ = Number of possible response strategies that E can choose

The payouts depend on: and ${\displaystyle U_{S}(t_{j},n_{k},a_{l})}$ ${\displaystyle U_{E}(t_{j},n_{k},a_{l})}$

procedure

In the first round, nature randomly draws a type for the transmitter with a probability , where: and ${\displaystyle \mathbf {t_{j}\ \in T} }$ ${\displaystyle \mathbf {p(t_{j})} }$ ${\displaystyle p(t_{j})>0}$ ${\displaystyle p(t_{1})+p(t_{2})+\dotsb +p(t_{J})=1}$

In the second round, the sender S selects a signal (S knows which type it is) ${\displaystyle \mathbf {t_{j}} }$ ${\displaystyle \mathbf {n_{k}\in N} }$

In the third and last round, E observes all signals given and chooses a response strategy ${\displaystyle \mathbf {n_{k}} }$ ${\displaystyle \mathbf {a_{l}\in A} }$

The payouts depend on:

${\displaystyle \mathbf {U_{S}(t_{j},n_{k},a_{l})} }$

${\displaystyle \mathbf {U_{E}(t_{j},n_{k},a_{l})} }$

Possible equilibria

In signal games, a basic distinction is made between three different equilibriums, between pooling equilibriums (also known as unifying equilibriums), separating equilibriums (also known as separating equilibria) and semi-separating equilibriums (also called semi-pooling equilibrium). The basic concept of equilibrium used in signal games is the concept of perfect Bayesian equilibrium . Separating and Pooling Equilibrium are described in more detail below.

Separating Equilibrium

A separating equilibrium exists when each broadcaster plays different strategies depending on its type. This means that there must be the same number of transmitter types and signals and that exactly one signal must be assigned to each transmitter type (given and must apply: and ). It is important that none of the types have an incentive to deviate from the chosen strategy, but strictly play their chosen strategy. ${\displaystyle T\{t_{1},\dots ,t_{J}\}}$ ${\displaystyle N\{n_{1},\dotsb ,n_{K}\}}$ ${\displaystyle J=K}$ ${\displaystyle \{(t_{1},n_{1}),\dotsb ,(t_{J},n_{K})\}}$

For example, if two different types and exist in a game and the possible signals are and , then there is a separating equilibrium when sending strictly and strictly . The receiver then chooses its maximizing strategy based on the observed signal . In the above example, a separating equilibrium would mean that, for example, a workaholic would strictly sort the magazines and a lazy would strictly not sort the magazines. ${\displaystyle \mathbf {t_{1}} }$ ${\displaystyle \mathbf {t_{2}} }$ ${\displaystyle \mathbf {n_{1}} }$ ${\displaystyle \mathbf {n_{2}} }$ ${\displaystyle \mathbf {t_{1}} }$ ${\displaystyle \mathbf {n_{1}} }$ ${\displaystyle \mathbf {t_{2}} }$ ${\displaystyle \mathbf {n_{2}} }$ ${\displaystyle \mathbf {n_{k}} }$

Semi-separating equilibrium

A semi-separating equilibrium can occur if there are more types than signals (given and must apply:) . ${\displaystyle T\{t_{1},\dots ,t_{J}\}}$ ${\displaystyle N\{n_{1},\dotsb ,n_{K}\}}$ ${\displaystyle J>K}$

If, for example, there are four different types but only two possible signals in a game, then a semi-separating equilibrium is present if, for example, and play strictly and and strictly . ${\displaystyle \mathbf {t_{1},\dotsb ,t_{4}} }$ ${\displaystyle \mathbf {n_{1},n_{2}} }$ ${\displaystyle \mathbf {t_{1}} }$ ${\displaystyle \mathbf {t_{2}} }$ ${\displaystyle \mathbf {n_{1}} }$ ${\displaystyle \mathbf {t_{3}} }$ ${\displaystyle \mathbf {t_{4}} }$ ${\displaystyle \mathbf {n_{2}} }$

Pooling Equilibrium

A unifying equilibrium exists when the broadcaster always plays the same strategy regardless of its type. This means that the receiver cannot distinguish the different types of transmitters through the signal. In the case of a unifying balance, the recipient must form beliefs by estimating how likely a particular type is. The recipient then uses these beliefs to maximize their strategies. In the short example, a pooling equilibrium would mean that, for example, the slacker has an incentive to sort the magazines and thus disguise himself as a workaholic. But he only has this if he can expect a higher payout from the camouflage than if he does not camouflage himself. ${\displaystyle \mathbf {t_{j}} }$

Examples of signal games

In the following, an example for a separating equilibrium and an example for a uniting equilibrium are given to illustrate signal games.

roommate wanted

In the following game, a shared apartment is looking for a new roommate. Since all the residents of the flat share are very funny people so far, they are also looking for a funny person for the empty room. In addition, the shared flat is holding a casting to find the new roommate . To check if a candidate (B) is a funny or a serious guy, they put a red clown nose on the living room table during the casting in the hope that serious applicants would never put their noses on . ${\displaystyle \mathbf {(WG)} }$ ${\displaystyle \mathbf {(B)} }$ ${\displaystyle \mathbf {(t_{L})} }$ ${\displaystyle \mathbf {(t_{E})} }$ ${\displaystyle \mathbf {(n_{ab})} }$

The flat share can decide whether to accept or reject an applicant . In reality it is also the case that a funny applicant likes to put on his nose and even make a personal profit of 5 because he likes to fool around. A serious applicant, on the other hand, would accept a personal loss of −5 in the event that he put his nose on, as he would be ashamed of having his nose on. ${\displaystyle \mathbf {(a_{ja})} }$ ${\displaystyle \mathbf {(a_{\text{nein}})} }$ ${\displaystyle \mathbf {(n_{\text{auf}})} }$

If the flat share found a funny flatmate, it would generate a profit of 4, in the case of a serious flatmate the flat share would come to exactly 0, since a serious flatmate completely negates the positive effect of the additional rent. The applicant earns a profit of 4 in each case if they receive the room. The probability that the applicant is funny is exogenous. ${\displaystyle \mathbf {p(t_{L})} }$

Signal game tree wanted for the example of roommate

Course of the game:

1st round: chooses with a probability of whether ${\displaystyle \mathbf {Z} }$ ${\displaystyle \mathbf {p} }$ ${\displaystyle B(t_{j})=t_{L}}$

2nd round: Applicant chooses strategy depending on : ${\displaystyle \mathbf {t_{j}} }$

B plays strictly when he is ${\displaystyle \mathbf {n_{\text{auf}}} }$ ${\displaystyle \mathbf {t_{L}} }$

B plays strictly when he is ${\displaystyle \mathbf {n_{\text{ab}}} }$ ${\displaystyle \mathbf {t_{E}} }$

3rd round: WG decides, depending on the observed signal , whether to accept or reject the applicant: ${\displaystyle \mathbf {n_{k}} }$

WG strictly plays when watching ${\displaystyle \mathbf {a_{\text{ja}}} }$ ${\displaystyle \mathbf {n_{\text{auf}}} }$

WG strictly plays when watching ${\displaystyle \mathbf {a_{\text{nein}}} }$ ${\displaystyle \mathbf {n_{\text{ab}}} }$

The player's choice of strategy eliminates the sub- trees for and , as both are strictly dominated strategies. From this it follows for the beliefs of the flat share: and . ${\displaystyle \mathbf {\{t_{E},n_{auf}\}} }$ ${\displaystyle \mathbf {\{t_{L},n_{ab}\}} }$ ${\displaystyle \alpha =1}$ ${\displaystyle \beta =0}$

This results in the separating equilibrium: ${\displaystyle \mathbf {[(n_{\text{auf}},n_{\text{ab}}),(a_{\text{ja}},a_{\text{nein}})],\alpha =1,\beta =0} }$

The payout of the two players therefore only depends on the probability of whether the applicant is funny or serious. In this case , the payouts are: ${\displaystyle p=0{,}5}$

${\displaystyle {\begin{aligned}U_{B}^{\text{Ges}}&=p\cdot U_{B}(t_{L},n_{\text{auf}},a_{ja})+(1-p)\cdot U_{B}(t_{E},n_{\text{ab}},a_{\text{nein}})\\&=0{,}5\cdot 9+0{,}5\cdot 0=4{,}5\\\end{aligned}}}$

${\displaystyle {\begin{aligned}U_{WG}^{\text{Ges}}&=p\cdot U_{W}G(t_{L},n_{\text{auf}},a_{\text{ja}})+(1-p)\cdot U_{W}G(t_{E},n_{\text{ab}},a_{\text{nein}})\\&=0{,}5\cdot 4+0{,}5\cdot 0=2\\\end{aligned}}}$

The beer quiche game

In the following game, a rowdy walks into a pub in the morning to fight a guest . The rowdy would like to fight with a wimp as much as possible , because he does not hit back and this brings him a profit of 1. However, there are also thugs in the pub . If the rowdy takes on a bat, he pulls the shorter one, which means -1 damage to the rowdy. Therefore, the rowdy also has the opportunity to escape . The rowdy cannot distinguish between wimp and thugs, but he can watch what people are having breakfast in the pub. You can choose between a beer breakfast and a quiche breakfast . It is also the case that wimps prefer a quiche breakfast and thugs prefer a beer breakfast. Both have a personal gain of 1 if they eat their favorite breakfast and a personal loss of −1 if they eat the other breakfast. Both sissies and thugs want to avoid a brawl because both of them will suffer personal damage of −1 if they fight the thug. Should the rowdy escape, both a wimp and a thug have a personal gain of 4 because they are happy that the uncomfortable rowdy is gone. ${\displaystyle \mathbf {(R)} }$ ${\displaystyle \mathbf {(G)} }$ ${\displaystyle \mathbf {(a_{D})} }$ ${\displaystyle \mathbf {(t_{W})} }$ ${\displaystyle \mathbf {(t_{S})} }$ ${\displaystyle \mathbf {(a_{F})} }$ ${\displaystyle \mathbf {(n_{B})} }$ ${\displaystyle \mathbf {(n_{Q})} }$

The probability that the guest is a wimp is exogenous. Here it is equal to 0.5. ${\displaystyle \mathbf {p(t_{W})} }$

Signal game tree for the example of a beer quiche game

Course of the game:

1st round: chooses with a probability of whether ${\displaystyle \mathbf {Z} }$ ${\displaystyle \mathbf {p} }$ ${\displaystyle B(t_{j})=\!\ t_{W}}$

2nd round: The guest chooses a strategy depending on : ${\displaystyle \mathbf {t_{j}} }$

${\displaystyle \mathbf {G(t_{W})} }$would prefer to play, but since he knows that R would never avoid a duel if he is observing and strictly avoid a duel if he observes, has an incentive to deviate, since: ${\displaystyle \mathbf {n_{Q}} }$ ${\displaystyle \mathbf {n_{Q}} }$ ${\displaystyle \mathbf {n_{B}} }$ ${\displaystyle \mathbf {G(t_{W})} }$

${\displaystyle {\begin{aligned}&\ \ \ \ \ \alpha \cdot u_{G}(t_{W},n_{B},a_{F})>\beta \cdot u_{G}(t_{W},n_{Q},a_{D})\\&\Leftrightarrow \alpha *(3)>\beta *(0)\\&\Leftrightarrow 3\alpha >0\\\end{aligned}}}$

G plays strictly when he is ${\displaystyle \mathbf {n_{B}} }$ ${\displaystyle \mathbf {t_{S}} }$

3rd round: R decides, depending on the observed signal , whether to duel or flee: ${\displaystyle \mathbf {n_{k}} }$

R would, if he could separate, play strictly as soon as he observed ${\displaystyle \mathbf {a_{D}} }$ ${\displaystyle \mathbf {n_{Q}} }$

R would, if he could separate, play strictly as soon as he observed ${\displaystyle \mathbf {a_{F}} }$ ${\displaystyle \mathbf {n_{B}} }$

Because the wimp has an incentive to deviate, R can no longer infer 100% of the type from the signal, since a wimp may choose a beer breakfast in order to disguise himself as a thug and thus avoid the brawl . Therefore, the R must form beliefs with which he estimates how likely it is that a wimp will choose the beer breakfast or a thug the quiche breakfast. ${\displaystyle (\alpha \land \beta )}$

Therefore, in this example there are two pooling equilibriums:

${\displaystyle \mathbf {([(n_{B},n_{B}),(a_{F},a_{D})],\alpha \leq 0{,}5} }$and and and ${\displaystyle \mathbf {\beta \geq 0{,}5)} }$ ${\displaystyle \mathbf {([(n_{Q},n_{Q}),(a_{D},a_{F})],\alpha \geq 0{,}5} }$ ${\displaystyle \mathbf {\beta \leq 0{,}5)} }$

The second example is very unrealistic, however, since a bat has little incentive to eat quiche and thus the rowdy would never believe with a probability of more than 0.5 that a bat is eating a quiche breakfast. Mathematically, however, the second equilibrium cannot be overridden by eliminating dominant strategies, but rather the method of the intuitive criterion developed by In-Koo Cho and David Kreps .

Significance and application of signal games

Signal games play a major role in practical game theory. They make it possible to illustrate very clearly situations in which people want or have to interact with one another, but do not have all the relevant information about one another.

Signal games are particularly used in “employer-employee games”. With the help of signal games, however, a company's investments in advertising or the emergence of social norms can also be modeled. Signal games also serve a further purpose when analyzing unexpected actions of the interaction partner. With pooling equilibriums in particular, you can very well observe how people assess situations in which they cannot differentiate between other people. Basically, you can use signal games to model all everyday situations in which you have to assume different types of interaction partner and although you know these types, you cannot assign them to your interaction partner.

literature

• Robert Gibbon: A primer in game theory. Prentice Hall, Harlow 1992, ISBN 0-7450-1159-4
• Hans Peters: Game Theory-A Multi-Leveled Approach. Springer Verlag, Berlin Heidelberg 2008
• Siegfried K. Berninghau et al .: Strategic Games - An Introduction to Game Theory. Springer Verlag, Berlin Heidelberg 2002
• Andreas Diekmann: Game Theory. Rowohlt Taschenbuch Verlag, Reinbek near Hamburg 2009, ISBN 978-3-499-55701-9
• Manfred J. Holler , Gerhard Illing: Introduction to game theory. , 7th edition, Springer Verlag, Berlin Heidelberg 2009

Web links

• Encyclopedia for Game Theory - Project of the Mathematical Faculty of the Ludwig Maximilians University (LMU) Munich
• Lecture notes on game theory - archive of the economics faculty of the LMU Munich

Individual evidence

1. ^ Robert Gibbons: A primer in game theory. P. 183, see literature
2. ↑ ^{a b} Andreas Diekmann: Game theory. P. 235, see literature