Signaling Games and Forward Induction Songzi Du UCSD Econ 200C April 30, 2021 Songzi Du (UCSD Econ 200C) Signaling Games April 30, 2021 1 / 24 Receiver observes s (but not t), selects action a. Ui (a; t; s) payoff function. Signaling Game Two players, sender (player 1) and receiver (player 2) Nature picks t 2 T type of sender. p(t) is probability that type is t. Sender observes t, selects signal s 2 S. Songzi Du (UCSD Econ 200C) Signaling Games April 30, 2021 2 / 24 Signaling Game Two players, sender (player 1) and receiver (player 2) Nature picks t 2 T type of sender. p(t) is probability that type is t. Sender observes t, selects signal s 2 S. Receiver observes s (but not t), selects action a. Ui (a; t; s) payoff function. Songzi Du (UCSD Econ 200C) Signaling Games April 30, 2021 2 / 24 Beer-quiche game Songzi Du (UCSD Econ 200C) Signaling Games April 30, 2021 3 / 24 2 R preferences: U2(a; t; s) = −(a − t) I R should be thought of as firms who are competing (in a Bertrand fashion) to hire the worker. I Each firm gets a payoff t − a if it hires the worker at wage a. Labor market signaling (Spence) S is worker, t 2 R+ is ability, s 2 R+ is education R is market, a 2 R+ is market wage S preferences: U1(a; t; s) = a − c(s; t) Songzi Du (UCSD Econ 200C) Signaling Games April 30, 2021 4 / 24 Labor market signaling (Spence) S is worker, t 2 R+ is ability, s 2 R+ is education R is market, a 2 R+ is market wage S preferences: U1(a; t; s) = a − c(s; t) 2 R preferences: U2(a; t; s) = −(a − t) I R should be thought of as firms who are competing (in a Bertrand fashion) to hire the worker. I Each firm gets a payoff t − a if it hires the worker at wage a. Songzi Du (UCSD Econ 200C) Signaling Games April 30, 2021 4 / 24 Other signaling models 1 Verifiable information: S = P(T ), U1(a; t; s) is unimaginably small if t 62 s. 2 Cheap-talk: Ui independent of s. Songzi Du (UCSD Econ 200C) Signaling Games April 30, 2021 5 / 24 (σ∗; α∗; µ∗) is a perfect Bayesian equilibrium (PBE) if: 1 σ∗(t) solves ∗ max U1(α (s); t; s) s2S for all t, 2 α∗(s) solves X ∗ max µ (t j s)U2(a; t; s) a2A t2T for all s, 3 µ∗ derives from prior and σ∗ using Bayes's Rule whenever possible, ∗ ∗ p(t) σ (t)(s) µ (t j s) = P 0 ∗ 0 : t02T p(t ) σ (t )(s) Perfect Bayesian Equilibrium (for finite signaling game) Sender strategy: σ : T ! ∆(S). Receiver strategy: α : S ! ∆(A). Receiver belief: µ( · j s) 2 ∆(T ), 8s 2 S. Songzi Du (UCSD Econ 200C) Signaling Games April 30, 2021 6 / 24 Perfect Bayesian Equilibrium (for finite signaling game) Sender strategy: σ : T ! ∆(S). Receiver strategy: α : S ! ∆(A). Receiver belief: µ( · j s) 2 ∆(T ), 8s 2 S. (σ∗; α∗; µ∗) is a perfect Bayesian equilibrium (PBE) if: 1 σ∗(t) solves ∗ max U1(α (s); t; s) s2S for all t, 2 α∗(s) solves X ∗ max µ (t j s)U2(a; t; s) a2A t2T for all s, 3 µ∗ derives from prior and σ∗ using Bayes's Rule whenever possible, ∗ ∗ p(t) σ (t)(s) µ (t j s) = P 0 ∗ 0 : t02T p(t ) σ (t )(s) Songzi Du (UCSD Econ 200C) Signaling Games April 30, 2021 6 / 24 (σ∗; α∗; µ~∗) is a perfect Bayesian equilibrium (PBE) if: 1 σ∗(t) solves ∗ max U1(α (s); t; s) s2S for all t, 2 α∗(s) solves P X p(t) z(t)(s) U2(a; t; s) max µ~∗(z j s) t2T a2A P p(t) z(t)(s) z2Z(s) t2T for all s, 3 µ~∗ derives from prior and σ∗ using Bayes's Rule whenever possible, Q ∗ P ∗ t2T σ (t)(z(t)) t2T p(t) z(t)(s) µ~ (z j s) = P ∗ : t2T p(t) σ (t)(s) Perfect Bayesian Equilibrium based on appraisal Receiver's appraisal:µ ~( · j s) 2 ∆(Z(s)), where Z(s) is the set of pure strategies that sends s for some type (z(t) = s for some t). Songzi Du (UCSD Econ 200C) Signaling Games April 30, 2021 7 / 24 3 µ~∗ derives from prior and σ∗ using Bayes's Rule whenever possible, Q ∗ P ∗ t2T σ (t)(z(t)) t2T p(t) z(t)(s) µ~ (z j s) = P ∗ : t2T p(t) σ (t)(s) Perfect Bayesian Equilibrium based on appraisal Receiver's appraisal:µ ~( · j s) 2 ∆(Z(s)), where Z(s) is the set of pure strategies that sends s for some type (z(t) = s for some t). (σ∗; α∗; µ~∗) is a perfect Bayesian equilibrium (PBE) if: 1 σ∗(t) solves ∗ max U1(α (s); t; s) s2S for all t, 2 α∗(s) solves P X p(t) z(t)(s) U2(a; t; s) max µ~∗(z j s) t2T a2A P p(t) z(t)(s) z2Z(s) t2T for all s, Songzi Du (UCSD Econ 200C) Signaling Games April 30, 2021 7 / 24 Perfect Bayesian Equilibrium based on appraisal Receiver's appraisal:µ ~( · j s) 2 ∆(Z(s)), where Z(s) is the set of pure strategies that sends s for some type (z(t) = s for some t). (σ∗; α∗; µ~∗) is a perfect Bayesian equilibrium (PBE) if: 1 σ∗(t) solves ∗ max U1(α (s); t; s) s2S for all t, 2 α∗(s) solves P X p(t) z(t)(s) U2(a; t; s) max µ~∗(z j s) t2T a2A P p(t) z(t)(s) z2Z(s) t2T for all s, 3 µ~∗ derives from prior and σ∗ using Bayes's Rule whenever possible, Q ∗ P ∗ t2T σ (t)(z(t)) t2T p(t) z(t)(s) µ~ (z j s) = P ∗ : t2T p(t) σ (t)(s) Songzi Du (UCSD Econ 200C) Signaling Games April 30, 2021 7 / 24 Sequential Equilibrium (for finite signaling game) (σ∗; α∗; µ∗) is a sequential equilibrium (SE) if: Conditions 1 and 2 of PBE are satisfied. n n There is a sequence of trembles (σ ; µ )n2N s.t. ∗ n ∗ n I σ = limn!1 σ and µ = limn!1 µ , n I σ (t)(s) > 0 for all t, s and n, n n I µ derives from prior and σ for every n. Fact: Every SE is a PBE, and every PBE is a SE (in signaling game). Songzi Du (UCSD Econ 200C) Signaling Games April 30, 2021 8 / 24 Pooling equilibrium: σ∗(t) constant in t ∗ I µ ( · j s) = p( · ) whenever s is on the equilibrium path Separating equilibrium: the support of σ∗(t) does not intersect the support of σ∗(t0) if t 6= t0 ∗ ∗ I µ (t j s) = 1 if σ (t)(s) > 0 There are also \hybrid" equilibria in which some types pool and some types separate. Terminology s is on the equilibrium path if σ∗(t)(s) > 0 for some t. Equilibrium outcome π 2 ∆(T × S × A), induced by some PBE. Songzi Du (UCSD Econ 200C) Signaling Games April 30, 2021 9 / 24 Terminology s is on the equilibrium path if σ∗(t)(s) > 0 for some t. Equilibrium outcome π 2 ∆(T × S × A), induced by some PBE. Pooling equilibrium: σ∗(t) constant in t ∗ I µ ( · j s) = p( · ) whenever s is on the equilibrium path Separating equilibrium: the support of σ∗(t) does not intersect the support of σ∗(t0) if t 6= t0 ∗ ∗ I µ (t j s) = 1 if σ (t)(s) > 0 There are also \hybrid" equilibria in which some types pool and some types separate. Songzi Du (UCSD Econ 200C) Signaling Games April 30, 2021 9 / 24 PBE in beer-quiche game Songzi Du (UCSD Econ 200C) Signaling Games April 30, 2021 10 / 24 PBE in labor market signaling (two types) @c @2c Suppose @s > 0 and @t@s < 0. Songzi Du (UCSD Econ 200C) Signaling Games April 30, 2021 11 / 24 Definition An equilibrium outcome π satisfies forward induction if it is induced by a PBE (σ∗; α∗; µ~∗) such that at all π-relevant signals s,µ ~∗( · j s) places probability one on π-relevant strategies. Forward induction implies intuitive criterion, D1, D2, etc. Forward Induction (Govindan and Wilson, 2009) Fix a (PBE) equilibrium outcome π 2 ∆(T × S × A). A pure strategy z : T ! S is relevant for π if it is a best response to α∗ of a PBE (σ∗; α∗; µ~∗) that induces π. A signal s 2 S is relevant for π if z(t) = s for some relevant z and type t. Songzi Du (UCSD Econ 200C) Signaling Games April 30, 2021 12 / 24 Forward Induction (Govindan and Wilson, 2009) Fix a (PBE) equilibrium outcome π 2 ∆(T × S × A). A pure strategy z : T ! S is relevant for π if it is a best response to α∗ of a PBE (σ∗; α∗; µ~∗) that induces π. A signal s 2 S is relevant for π if z(t) = s for some relevant z and type t. Definition An equilibrium outcome π satisfies forward induction if it is induced by a PBE (σ∗; α∗; µ~∗) such that at all π-relevant signals s,µ ~∗( · j s) places probability one on π-relevant strategies.