Stanford MS&E 246 - Lecture 16 - Signaling games

Unformatted text preview:

MS&E246:Lecture16SignalinggamesRamesh JohariSignalinggamesSignaling games are two-stage games where:• Player 1 (with private information)moves first.His move is observed by Player 2.• Player 2 (with no knowledge of Player 1’s private information) moves second.• Then payoffs are realized.DynamicgamesSignaling games are a key example of dynamic games ofincomplete information.(i.e., a dynamic game where the entire structure is not common knowledge)SignalinggamesThe formal description:Stage 0:Nature chooses a random variable t1,observable only to Player 1,from a distribution P(t1).SignalinggamesThe formal description:Stage 1:Player 1 chooses an actiona1from the set A1.Player 2 observes this choice of action.(The action of Player 1 is also called a “message.”)SignalinggamesThe formal description:Stage 2:Player 2 chooses an actiona2from the set A2.Following Stage 2, payoffs are realized:Π1(a1, a2; t1) ; Π2(a1, a2; t1).SignalinggamesObservations:• The modeling approach follows Harsanyi’smethod for static Bayesian games.• Note that Player 2’s payoff dependson the type of player 1!• When Player 2 moves first,and Player 1 moves second,it is called a screening game.Application1:LabormarketsA key application due to Spence (1973):Player 1: workert1: intrinsic abilitya1: education decisionPlayer 2: firm(s)a2: wage offeredPayoffs: Π1= net benefitΠ2= productivityApplication2:OnlineauctionsPlayer 1: sellert1: true quality of the gooda1: advertised qualityPlayer 2: buyer(s)a2:bid offeredPayoffs: Π1= profitΠ2= net benefitApplication3:ContractingA model of Cachon and Lariviere (2001):Player 1: manufacturert1: demand forecasta1: declared demand forecast,contract offerPlayer 2: suppliera2: capacity builtPayoffs: Π1= profit of manufacturerΠ2= profit of supplierAsimplesignalinggameSuppose there are two types for Player 1,and two actions for each player:• t1= H or t1= LLet p = P(t1= H)• A1= { a, b }• A2= { A, B }AsimplesignalinggameNature moves first:NatureAsimplesignalinggameNature moves first:NatureHLAsimplesignalinggamePlayer 1 moves second:Nature1.11.2HLAsimplesignalinggamePlayer 1 moves second:Nature1.11.2HLaabbAsimplesignalinggamePlayer 2 observes Player 1’s action:Nature1.11.2HLaabb2.22.1AsimplesignalinggamePlayer 2 moves:Nature1.11.2HLaabb2.22.1ABABABABAsimplesignalinggamePayoffs are realized: Πi(a1, a2; t1)Nature1.11.2HLaabb2.22.1ABABABABPerfectBayesianequilibriumEach player has 2 information sets,and 2 actions in each, so 4 strategies.A PBE is a pair of strategies and beliefssuch that:-each players’ beliefs are derived from strategies using Bayes’ rule (if possible)-each players’ strategies maximize expected payoff given beliefsPoolingvs.separatingequilibriaWhen player 1 plays the same action,regardless of his type,it is called a pooling strategy.When player 1 plays different actions,depending on his type,it is called a separating strategy.Poolingvs.separatingequilibriaIn a pooling equilibrium,Player 2 gains no information about t1from Player 1’s message⇒ P2(t1= H | a1) = P(t1= H) = pIn a separating equilibrium,Player 2 knows Player 1’s type exactlyfrom Player 1’s message⇒ P2(t1= H | a1) = 0 or 1AneBay‐likemodelSuppose seller has an item with quality either H (prob. p) or L (prob. 1 - p).Seller can advertise either H or L.Assume there are two bidders.Suppose that bidders always bid truthfully,given their beliefs.(This would be the case if the seller used a second price auction.)AneBay‐likemodelSuppose seller always advertises high.Then: buyers will never “trust” the seller,and always bid expected valuation.This is the pooling equilibrium:s1(H) = s1(L) = H.sB(H) = sB(L) = p H + (1 - p) L.AneBay‐likemodelIs there any equilibrium where s1(t1) = t1? (In this case the seller is truthful.)In this case the buyers bid:sB(H) = H, sB(L) = L.But if the buyers use this strategy,the seller prefers to always advertise H!AneBay‐likemodelNow suppose that if the seller lies when the true value is L,there is a cost c (in the form of lower reputation in future transactions).If H - c < L,then the seller prefers to tell the truth⇒ separating equilibrium.AneBay‐likemodelThis example highlights the importance ofsignaling costs:To achieve a separating equilibrium, there must be a difference in the costsof different messages.(When there is no cost, the resulting message is called “cheap


View Full Document

Stanford MS&E 246 - Lecture 16 - Signaling games

Download Lecture 16 - Signaling games
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Lecture 16 - Signaling games and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Lecture 16 - Signaling games 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?