Signaling Games In incomplete information games one player knows more information than the other player So far we have focused on the case where the type of the more informed player was known to that player but unknown to the less informed player Signaling games are incomplete information games where the more informed player has to decide whether to signal in some way their true type and the less informed player has to decide how to respond to both the uncertainty about his opponent s type and the signal his opponent has sent recognizing that signals may be strategically chosen What are Signals Signals are actions that more informed players use to convey information to less informed players about the unobservable type of the more informed player Example A player who wants the trust of less informed player may signal past instances of trust may provide verbal assurances of trustworthiness the names of character references former employees on a resume discuss church attendance charity work etc Signals may or may not be credible Why Because individuals will use signals strategically when it suits them Less qualified applicants may pad their resumes lie about their past work history qualifications embezzle from their church charity Talk is cheap Yeah right whatever I could care less are common The more credible signals involve costly actions e g a college diploma an artistic portfolio a published book a successful business Examples of Strategic Signaling Insurance contracts Accident prone types will want greater coverage lower deductibles while those unlikely to be in accidents will require minimal coverage higher deductibles Insurance companies respond to these signals by charging higher prices for greater coverage lower deductible Used cars The dealer has to decide whether to offer a warranty on a used car or offer the car as is Pittsburgh left turn game The left turner can attempt to signal whether he is a Pittsburgher or an Out of Towner Letter grade or pass fail grades Letter grade signals more commitment risk taking pass grade signals lowest possible passing letter grade C Example 1 Prisoner s Dilemma Again Recall the Prisoner s Dilemma game from last week where player 1 s preferences depend on whether player 2 is nice or selfish C 1 C D 2 D 4 4 0 6 6 0 2 2 Player 2 selfish C 1 C D 2 D 6 6 2 4 4 2 0 0 Player 2 nice Suppose the player 2 can costlessly signal to player 1 her action choice before Player 1 gets to choose The signal is nonbinding cheap talk Player 1 observes this signal before making his own move but still does not know what type of player 2 he is facing selfish or nice For example if player 2 signals C player 1 wants to play C if player 2 is nice but D if player 2 is selfish Example 1 in Extensive Form Note the two information sets for player 1 P1 Given a signal C or D P1 does not know if the player 2 P2 is selfish or nice Analysis of Example 1 The signal is player 2 s cheap talk message of C or D I intend to play C or I intend to play D Both player 2 types have an incentive to signal C A selfish player 2 wants player 1 to play C so she can play D and get the highest possible payoff for herself A nice player 2 wants player 1 to play C so she can play C and get the highest possible payoff for herself If the two types sent different signals player 1 would be able to differentiate between the two types of player 2 and the game would then be like a game of complete information Therefore both player 2 types signal C the signal is perfectly uninformative this is called a pooling equilibrium outcome Player 1 will play C if the prior probability that player 2 is selfish p and will play D if p In this example since p 1 3 player 1 should play C Example 2 Signaling in Coordination Games 1 Consider the battle of the sexes game Kaylee Luke Basketball Game Party Basketball Game 2 1 0 0 Party 0 0 1 2 Now suppose that prior to play of the game Kaylee says Luke I m going to the party Kaylees message is self committing if she thinks Luke believes her message that she will go to the party then her best response is to go to the party It is also self signaling Kaylee wants to say she is going to the party if and only if that is where she is planning to go Messages that are both self committing and self signaling are highly credible Signaling in Coordination Games 2 Consider this Stag Hunt game Amy and Pete are working on their Econ 1200 term project Either can choose to supply high or low effort Pete Amy High effort Low effort High effort 5 5 1 3 Low effort 3 1 2 2 Suppose prior to the game Amy says Pete I plan to put in high effort on the term project Then the game is played Amy s message is self committing if she thinks Pete will believe her message then it is a best response for her to supply high effort But Amy s message is not self signaling Amy would like Pete to believe she will supply high effort even if she plans to supply low effort So perhaps the message is less credible in this case Credibility of Signaling Continued Note that in a standard PD where it is known that there are no type distinctions i e both players are known to be selfish then a message by one player that he intends to cooperate not confess is neither self committing nor self signaling In other words it is incredible What about two way communication Generally this would lead to some kind of negotiation game Alternatively if communication is expensive time consuming one can appeal to social norms of behavior e g the bride s family pays for the wedding Example 3 Market Entry Game with Signaling Two firms incumbent is Oldstar the new firm is Nova Oldstar is a known set in its ways company but Nova is a relatively unknown startup Oldstar reasons that Nova is one of two types strong or weak Nova knows its type In a fight Oldstar can beat a weak Nova but a strong Nova can beat Oldstar The winner has the market all to itself If Oldstar has the market to itself it makes a profit of 3 and if Nova has the market to itself it makes a profit of 4 The cost of a fight is 2 to both firms These facts are reflected in the payoff matrix given to the right The Equilibrium Without Signaling Let w be the probability that Nova is weak and so 1 w is the probability that Nova is strong In the absence of any signals from Nova Oldstar will calculate the expected payoff from fighting which is w 1 1 w 2 …