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 ∈ T type of sender. p(t) is probability that type is t. Sender observes t, selects signal s ∈ 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 ∈ T type of sender. p(t) is probability that type is t. Sender observes t, selects signal s ∈ 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 ∈ R+ is ability, s ∈ R+ is education R is market, a ∈ 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 ∈ R+ is ability, s ∈ R+ is education R is market, a ∈ 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 6∈ 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) s∈S for all t, 2 α∗(s) solves X ∗ max µ (t | s)U2(a, t, s) a∈A t∈T for all s, 3 µ∗ derives from prior and σ∗ using Bayes’s Rule whenever possible, ∗ ∗ p(t) σ (t)(s) µ (t | s) = P 0 ∗ 0 . t0∈T 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( · | s) ∈ ∆(T ), ∀s ∈ 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( · | s) ∈ ∆(T ), ∀s ∈ S.

(σ∗, α∗, µ∗) is a perfect Bayesian equilibrium (PBE) if: 1 σ∗(t) solves ∗ max U1(α (s), t, s) s∈S for all t, 2 α∗(s) solves X ∗ max µ (t | s)U2(a, t, s) a∈A t∈T for all s, 3 µ∗ derives from prior and σ∗ using Bayes’s Rule whenever possible, ∗ ∗ p(t) σ (t)(s) µ (t | s) = P 0 ∗ 0 . t0∈T 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) s∈S for all t, 2 α∗(s) solves P X p(t) z(t)(s) U2(a, t, s) max µ˜∗(z | s) t∈T a∈A P p(t) z(t)(s) z∈Z(s) t∈T for all s, 3 µ˜∗ derives from prior and σ∗ using Bayes’s Rule whenever possible, Q ∗ P ∗ t∈T σ (t)(z(t)) t∈T p(t) z(t)(s) µ˜ (z | s) = P ∗ . t∈T p(t) σ (t)(s)

Perfect Bayesian Equilibrium based on appraisal Receiver’s appraisal:µ ˜( · | s) ∈ ∆(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 ∗ t∈T σ (t)(z(t)) t∈T p(t) z(t)(s) µ˜ (z | s) = P ∗ . t∈T p(t) σ (t)(s)

Perfect Bayesian Equilibrium based on appraisal Receiver’s appraisal:µ ˜( · | s) ∈ ∆(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 Perfect Bayesian Equilibrium based on appraisal Receiver’s appraisal:µ ˜( · | s) ∈ ∆(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) s∈S for all t, 2 α∗(s) solves P X p(t) z(t)(s) U2(a, t, s) max µ˜∗(z | s) t∈T a∈A P p(t) z(t)(s) z∈Z(s) t∈T for all s, 3 µ˜∗ derives from prior and σ∗ using Bayes’s Rule whenever possible, Q ∗ P ∗ t∈T σ (t)(z(t)) t∈T p(t) z(t)(s) µ˜ (z | s) = P ∗ . t∈T 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 )n∈N s.t. ∗ n ∗ n I σ = limn→∞ σ and µ = limn→∞ 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 ( · | 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 | 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 ∈ ∆(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 ∈ ∆(T × S × A), induced by some PBE.

Pooling equilibrium: σ∗(t) constant in t ∗ I µ ( · | 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 | 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 ˜∗( · | 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 ∈ ∆(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 ∈ 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 ∈ ∆(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 ∈ 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 ˜∗( · | s) places probability one on π-relevant strategies.

Forward induction implies intuitive criterion, D1, D2, etc.

Songzi Du (UCSD Econ 200C) Signaling Games April 30, 2021 12 / 24 Intuition of forward induction

At a PBE, receiver tries to rationalize an oﬀ-equilibrium s, believing that s is sender’s best response to another PBE with the same outcome.

Receiver thinks that:

I Sender is confused about which PBE is in eﬀect, which is understandable since the two PBE’s result in the same outcome. I In other words, Sender is confused about receiver’s oﬀ-equilibrium play.

Songzi Du (UCSD Econ 200C) Signaling Games April 30, 2021 13 / 24 Forward induction in beer-quiche game

Songzi Du (UCSD Econ 200C) Signaling Games April 30, 2021 14 / 24 Forward induction in job market signaling (two types)

Songzi Du (UCSD Econ 200C) Signaling Games April 30, 2021 15 / 24 2 For any off-equilibrium s0, it suffices to consider relevant strategy z such that at every t, z(t) is either s0 or σ∗(t). 0 0 ∗ ∗ I Make up α (s ) (and given α (σ (t))) so that every t weakly prefers σ∗(t) to s0; and if z(t) = s0 then t is indifferent between σ∗(t) and s0. 3 If s0 is π-relevant (there exists a relevant z of the form above s.t. z(t) = s0 for some t), then chooseµ ˜∗( · | s0) so that it puts probability 1 on the relevant z. Specify α∗(s0) as a best response to µ˜∗( · | s0). ∗ 0 ∗ 0 I If for all your choices ofµ ˜ ( · | s ) and α (s ) some t wants to deviate from σ∗(t) to s0, then π does not satisfy forward induction. 4 If you do (3) at every π-relevant s0, and no t wants to deviate, then π satisfies forward induction.

A recipe for forward induction in signaling game

Songzi Du (UCSD Econ 200C) Signaling Games April 30, 2021 16 / 24 3 If s0 is π-relevant (there exists a relevant z of the form above s.t. z(t) = s0 for some t), then chooseµ ˜∗( · | s0) so that it puts probability 1 on the relevant z. Specify α∗(s0) as a best response to µ˜∗( · | s0). ∗ 0 ∗ 0 I If for all your choices ofµ ˜ ( · | s ) and α (s ) some t wants to deviate from σ∗(t) to s0, then π does not satisfy forward induction. 4 If you do (3) at every π-relevant s0, and no t wants to deviate, then π satisﬁes forward induction.

A recipe for forward induction in signaling game

Songzi Du (UCSD Econ 200C) Signaling Games April 30, 2021 16 / 24 4 If you do (3) at every π-relevant s0, and no t wants to deviate, then π satisﬁes forward induction.

A recipe for forward induction in signaling game

0 For simplicity, suppose the sender is playing a pure strategy. 1 Fix the equilibrium outcome π. (This tells you the sender’s equilibrium strategy σ∗ as well as the receiver’s equilibrium action α∗(s) for s ∈ {σ∗(t): t ∈ T }.) 2 For any off-equilibrium s0, it suffices to consider relevant strategy z such that at every t, z(t) is either s0 or σ∗(t). 0 0 ∗ ∗ I Make up α (s ) (and given α (σ (t))) so that every t weakly prefers σ∗(t) to s0; and if z(t) = s0 then t is indifferent between σ∗(t) and s0. 3 If s0 is π-relevant (there exists a relevant z of the form above s.t. z(t) = s0 for some t), then chooseµ ˜∗( · | s0) so that it puts probability 1 on the relevant z. Specify α∗(s0) as a best response to µ˜∗( · | s0). ∗ 0 ∗ 0 I If for all your choices ofµ ˜ ( · | s ) and α (s ) some t wants to deviate from σ∗(t) to s0, then π does not satisfy forward induction.

Songzi Du (UCSD Econ 200C) Signaling Games April 30, 2021 16 / 24 A recipe for forward induction in signaling game

Songzi Du (UCSD Econ 200C) Signaling Games April 30, 2021 16 / 24 Extensive form and normal form

Two extensive-form games may have the same reduced normal form (i.e., the same normal form up to duplicated strategies).

Songzi Du (UCSD Econ 200C) Signaling Games April 30, 2021 17 / 24 Theorem (Kohlberg and Mertens) In a generic extensive-form game, there exists an invariant sequential equilibrium outcome.

Theorem (Govindan and Wilson) For a generic two-player extensive-form game, any invariant sequential equilibrium outcome satisﬁes forward induction.

Invariant backward induction implies forward induction

Two extensive form games are equivalent if they have the same reduced normal form.

A sequential equilibrium (SE) outcome of an extensive-form game is invariant if for every equivalent extensive-form game, there is a SE with the equivalent outcome.

I Robust to diﬀerent presentations of the extensive form.

Songzi Du (UCSD Econ 200C) Signaling Games April 30, 2021 18 / 24 Theorem (Govindan and Wilson) For a generic two-player extensive-form game, any invariant sequential equilibrium outcome satisﬁes forward induction.

Invariant backward induction implies forward induction

Two extensive form games are equivalent if they have the same reduced normal form.

A sequential equilibrium (SE) outcome of an extensive-form game is invariant if for every equivalent extensive-form game, there is a SE with the equivalent outcome.

I Robust to diﬀerent presentations of the extensive form.

Theorem (Kohlberg and Mertens) In a generic extensive-form game, there exists an invariant sequential equilibrium outcome.

Songzi Du (UCSD Econ 200C) Signaling Games April 30, 2021 18 / 24 Invariant backward induction implies forward induction

Two extensive form games are equivalent if they have the same reduced normal form.

A sequential equilibrium (SE) outcome of an extensive-form game is invariant if for every equivalent extensive-form game, there is a SE with the equivalent outcome.

I Robust to diﬀerent presentations of the extensive form.

Theorem (Kohlberg and Mertens) In a generic extensive-form game, there exists an invariant sequential equilibrium outcome.

Theorem (Govindan and Wilson) For a generic two-player extensive-form game, any invariant sequential equilibrium outcome satisﬁes forward induction.

Songzi Du (UCSD Econ 200C) Signaling Games April 30, 2021 18 / 24 ∗ ∗ ∗ ∗ Let (σ , α , µ˜ ) be a SE with outcome π (σ (t) = s1 for all t). Suppose π is an invariant sequential equilibrium outcome.

Suppose s2 is a π-relevant signal.

Let z be a π-relevant strategy that sends s2. Note: all of above are w.r.t. the original signaling game Γ.

A simple case

Proposition In a signaling game where |S| = 2, any invariant sequential equilibrium outcome satisﬁes forward induction. Proof: It suﬃces to consider pooling equilibrium outcome.

Suppose S = {s1, s2}, and consider an equilibrium outcome π where all types send s1.

Songzi Du (UCSD Econ 200C) Signaling Games April 30, 2021 19 / 24 Suppose s2 is a π-relevant signal.

Let z be a π-relevant strategy that sends s2. Note: all of above are w.r.t. the original signaling game Γ.

A simple case

Proposition In a signaling game where |S| = 2, any invariant sequential equilibrium outcome satisﬁes forward induction. Proof: It suﬃces to consider pooling equilibrium outcome.

Suppose S = {s1, s2}, and consider an equilibrium outcome π where all types send s1. ∗ ∗ ∗ ∗ Let (σ , α , µ˜ ) be a SE with outcome π (σ (t) = s1 for all t). Suppose π is an invariant sequential equilibrium outcome.

Songzi Du (UCSD Econ 200C) Signaling Games April 30, 2021 19 / 24 A simple case

Proposition In a signaling game where |S| = 2, any invariant sequential equilibrium outcome satisﬁes forward induction. Proof: It suﬃces to consider pooling equilibrium outcome.

Suppose s2 is a π-relevant signal.

Let z be a π-relevant strategy that sends s2. Note: all of above are w.r.t. the original signaling game Γ.

Songzi Du (UCSD Econ 200C) Signaling Games April 30, 2021 19 / 24 Γ, where p(t1) = 0.6 and p(t2) = 0.4

Songzi Du (UCSD Econ 200C) Signaling Games April 30, 2021 20 / 24 Γ(x), where x ∈ [0, 1], p(t1) = 0.6 and p(t2) = 0.4

Songzi Du (UCSD Econ 200C) Signaling Games April 30, 2021 21 / 24 At a sequential equilibrium in Γ(x) equivalent to (σ∗, α∗):

∗ I After sender rejects σ in Γ(x), by sequential rationality he must either ∗ 0 ∗ choose xσ + (1 − x)z or something in argmaxz0∈Z U1(z , α ).

I Now, conditional on observing s2, receiver’s belief about types must be derived from π-relevant strategies, i.e., forward induction.

Proof of the proposition (continued) Let Z be the set of pure strategies for sender:

∗ 0 ∗ σ ∈ argmax U1(z , α ), z0∈Z

0 ∗ Z0 = Z \ argmax U1(z , α ). z0∈Z Choose x suﬃciently close to 1 so that ∗ ∗ 0 ∗ U1(xσ + (1 − x)z, α ) > U1(z , α ) 0 for all z ∈ Z0.

Songzi Du (UCSD Econ 200C) Signaling Games April 30, 2021 22 / 24 I Now, conditional on observing s2, receiver’s belief about types must be derived from π-relevant strategies, i.e., forward induction.

Proof of the proposition (continued) Let Z be the set of pure strategies for sender:

∗ 0 ∗ σ ∈ argmax U1(z , α ), z0∈Z

0 ∗ Z0 = Z \ argmax U1(z , α ). z0∈Z Choose x suﬃciently close to 1 so that ∗ ∗ 0 ∗ U1(xσ + (1 − x)z, α ) > U1(z , α ) 0 for all z ∈ Z0.

At a sequential equilibrium in Γ(x) equivalent to (σ∗, α∗):

∗ I After sender rejects σ in Γ(x), by sequential rationality he must either ∗ 0 ∗ choose xσ + (1 − x)z or something in argmaxz0∈Z U1(z , α ).

Songzi Du (UCSD Econ 200C) Signaling Games April 30, 2021 22 / 24 Proof of the proposition (continued) Let Z be the set of pure strategies for sender:

∗ 0 ∗ σ ∈ argmax U1(z , α ), z0∈Z

0 ∗ Z0 = Z \ argmax U1(z , α ). z0∈Z Choose x suﬃciently close to 1 so that ∗ ∗ 0 ∗ U1(xσ + (1 − x)z, α ) > U1(z , α ) 0 for all z ∈ Z0.

At a sequential equilibrium in Γ(x) equivalent to (σ∗, α∗):

∗ I After sender rejects σ in Γ(x), by sequential rationality he must either ∗ 0 ∗ choose xσ + (1 − x)z or something in argmaxz0∈Z U1(z , α ).

I Now, conditional on observing s2, receiver’s belief about types must be derived from π-relevant strategies, i.e., forward induction.

Songzi Du (UCSD Econ 200C) Signaling Games April 30, 2021 22 / 24 More invariance: endogenous signal game

The incumbent (Sender) always prefers the rival (Receiver) to stay out. The rival prefers to stay out (enter) if it thinks the incumbent has (not) invested in the project. The incumbent prefers (not) to invest in the project if it intends to choose a low (high) price. Lots of SE, with diﬀerent outcomes.

Songzi Du (UCSD Econ 200C) Signaling Games April 30, 2021 23 / 24 Unique SPE: (L, I , N; O, E), equivalent to SE (I , L, L; O, E) in the original game.

In and Wright (2017), Signaling Private Choices, Restud.

More invariance: endogenous signal game

Songzi Du (UCSD Econ 200C) Signaling Games April 30, 2021 24 / 24 More invariance: endogenous signal game

Unique SPE: (L, I , N; O, E), equivalent to SE (I , L, L; O, E) in the original game.

In and Wright (2017), Signaling Private Choices, Restud.

Songzi Du (UCSD Econ 200C) Signaling Games April 30, 2021 24 / 24