2 – Decentralized Communication
F. Koessler / May 28, 2010
Strategic Information Transmission: 2 – Decentralized Communication (with Two Players) Ecole d’´ et´ e, Aussois (31 mai – 4 juin) 1/
Cheap talk = communication which is • strategic and non-binding (no contract, no commitment) • costless, without direct impact on payoffs • direct / face-to-face / unmediated • possibly several communication stages 2/
• soft information (not verifiable, not certifiable, not provable) ⇒ different, e.g., from information revelation by a price system in rational expectation general equilibrium models (Radner, 1979), from mechanism design (contract), from signaling ` a la Spence (1973),. . . In its simplest form, a cheap talk game is a specific signaling game in which messages are costless (i.e., do not enter into players’ utility functions)
2 – Decentralized Communication
F. Koessler / May 28, 2010
Example 1. (Signal of productivity in a labor market) Extremely simplified version of Spence (1973) model of education: The sender (the expert) is a worker with private information about his ability k ∈ {kL , kH } = {1, 3} The receiver (the decisionmaker) is an employer who must chose a salary j ∈ {jL , jM , jH } = {1, 2, 3} The worker’s productivity is assumed to be equal to his ability 3/
Perfect competition among employers, so the employer chooses a salary equal to the expected productivity of the worker (zero expected profits) The worker chooses a level of education e ∈ {eL , eH } = {0, 3} (which does no affect his productivity, but is costly) Ak (j) = j − c(k, e) = j − e/k B k (j) = − k − j 2
(3, −4)
(2, −1) jM
jH eL
(1, 0)
jL
jH
kH
N Employer
4/ jH
Employer
kL
Worker eH
(3, 0)
jM
jL
(0, −4) (−1, −1) (−2, 0)
jH
(2, 0)
(worker) (employer)
(2, −1) jM
(1, −4)
jL
eL Worker eH
jM (1, −1)
jL
(0, −4)
Figure 1: Fully revealing equilibrium in the labor market signaling game (example 1)
2 – Decentralized Communication
F. Koessler / May 28, 2010
What happens if we replace the level of education e by cheap talk? Then, the message “my ability is high” is not credible anymore: whatever his type, the worker always wants the employer to believe that his ability is high (in order to get a high salary)
kL
5/
kH
jH = 3
jM = 2
jL = 1
3, −4
2, −1
1, 0
jH = 3
jM = 2
jL = 1
3, 0
2, −1
1, −4
Pr(kL ) = 1/2
Pr(kH ) = 1/2
Associated one-shot cheap talk game with two possible messages (3, −4)
(2, −1) jM
jH mL
jH
(3, −4)
jL
(3, 0)
Employer
kL
Worker mH
6/
(1, 0)
jH
kH
N Employer
jM (2, −1)
jL
(1, 0)
jH
(3, 0)
(2, −1) jM
(1, −4)
jL
mL Worker mH
jM (2, −1)
jL
(1, −4)
Fully revealing equilibrium? No, because the worker of type kL deviates by sending the same message as the worker of type kH ➥
Non-revealing equilibrium (always exists in cheap talk games)
2 – Decentralized Communication
F. Koessler / May 28, 2010
Can cheap talk be credible and help to transmit relevant information? Example 2. (Credible information revelation)
j1
j2
k1
1, 1
0, 0
p
k2
0, 0
3, 3
(1 − p)
7/
Y (p) =
{j1 }
if p > 3/4,
{j2 }
if p < 3/4,
∆(J)
if p = 3/4.
The sender’s preference over the receiver’s beliefs are positively correlated with the truth
(1, 1)
(0, 0)
j1
j2 a
8/
j1
(1, 1)
N
(0, 0)
j2 a
k2
Sender b
Receiver j2
(3, 3)
j1
Receiver k1
Sender b
(0, 0)
j1
(0, 0)
j2
(3, 3)
Figure 2: Fully revealing equilibrium in Example 2.
2 – Decentralized Communication
F. Koessler / May 28, 2010
Example 3. (Revelation of information which is not credible)
j1
j2
k1
5, 2
1, 0
p
k2
3, 0
1, 4
(1 − p)
9/ Y (p) =
{j1 }
if p > 2/3,
{j2 }
if p < 2/3,
∆(J)
if p = 2/3.
The sender’s preference over the receiver’s beliefs is not correlated with the truth. The unique equilibrium of the cheap talk game in NR, even if when p < 2/3 communication of information would increase both players’ payoffs
Example 4. (Revelation of information which is not credible)
j1
j2
k1
3, 2
4, 0
p
k2
3, 0
1, 4
(1 − p)
10/ Y (p) =
{j1 }
if p > 2/3,
{j2 }
if p < 2/3,
∆(J)
if p = 2/3.
The sender’s preference over the receiver’s beliefs is negatively correlated with the truth. The unique equilibrium of the cheap talk game in NR
2 – Decentralized Communication
F. Koessler / May 28, 2010
Example 5. (Partial revelation of information) j1
j2
j3
j4
j5
k1
1, 10
3, 8
0, 5
3, 0
1, −8
p
k2
1, −8
3, 0
0, 5
3, 8
1, 10
1−p
11/ Y (p) =
{j5 } {j4 } {j3 } {j2 } {j } 1
if p < 1/5 if p ∈ (1/5, 3/8) if p ∈ (3/8, 5/8) if p ∈ (5/8, 4/5) if p > 4/5
Partially revealing equilibrium when p = 1/2: 3 σ(k1 ) = a + 4 1 σ(k2 ) = a + 4
12/
1 b 4 3 b 4
Pr(a | k1 ) Pr(k1 ) = 3/4 Pr(k1 | a) = Pr(a) ⇒ Pr(b | k1 ) Pr(k1 ) = 1/4 Pr(k1 | b) = Pr(b) ⇒
(
τ (a) = j2
τ (b) = j4
⇒ equilibrium, expected utility = 34 (3, 8) + 14 (3, 0) = (3, 6) (better for the sender than the NRE and FRE)
2 – Decentralized Communication
F. Koessler / May 28, 2010
Basic Decision Problem
Two players Player 1 = sender, expert (with no decision) Player 2 = receiver, decisionmaker (with no information)
13/
Two possible types for the expert (can be easily generalized): K = {k1 , k2 } = {1, 2}, Pr(k1 ) = p, Pr(k2 ) = 1 − p Action of the decisionmaker: j ∈ J Payoffs: Ak (j) and B k (j)
Silent Game
1 p
k1
A1 (1),
k2
A2 (1),
B 1 (1)
···
j
···
A1 (j),
··· B 1 (j)
···
Γ(p) 14/
1 1−p
B 2 (1)
···
j
···
A2 (j),
··· B 2 (j)
···
2 – Decentralized Communication
F. Koessler / May 28, 2010
• Mixed action of the DM: y ∈ ∆(J)
⇒ expected payoffs
X Ak (y) = y(j) Ak (j) j∈J
B k (y) =
X
y(j) B k (j)
j∈J
• Optimal mixed actions in Γ(p) (non-revealing “equilibria”): 15/ Y (p) ≡ arg max p B 1 (y) + (1 − p) B 2 (y) y∈∆(J)
1
= {y : p B (y) + (1 − p) B 2 (y) ≥ p B 1 (j) + (1 − p) B 2 (j), ∀ j ∈ J} Remark Mixed actions are used in the communication extension of the game to construct equilibria in which the expert is indifferent between several messages. They also serve as punishments off the equilibrium path in communication games with certifiable information (persuasion games)
• “Equilibrium” payoffs in Γ(p): E(p) ≡ {(a, β) : ∃ y ∈ Y (p), a = A(y), β = p B 1 (y) + (1 − p) B 2 (y)}
16/
2 – Decentralized Communication
F. Koessler / May 28, 2010
Unilateral Communication Game Γ0S (p) Unilateral information transmission from the expert to the decisionmaker Set of messages (“keyboard”) of the expert: M = {a, b, . . . , }, 17/
3 ≤ |M | < ∞
Communication phase
Information Phase
The expert sends m ∈ M
The expert learns k ∈ K
Action phase The DM chooses j ∈ J
Strategy of the expert: σ : K → ∆(M ) Strategy of the DM: τ : M → ∆(J)
Example. Two messages (M = {a, b}) 1 1 · · · A (j), B (j) · · ·
2 2 · · · A (j), B (j) · · ·
j
j
2 a
a k1
N
k2
18/ b
b 2 j
· · · A1 (j), B 1 (j) · · ·
j · · · A2 (j), B 2 (j) · · ·
ES0 (p): Equilibrium payoffs of Γ0S (p)
2 – Decentralized Communication
F. Koessler / May 28, 2010
Characterization of NE Payoffs of Γ0S (p) Recall. E(p) ⊆ R2 × R: NE payoffs in the silent game Γ(p) Modified equilibrium payoffs of Γ(p): E + (p): the expert can have a (virtual) payoff which is higher than his equilibrium payoff when his type has zero probability ➥ (a, β) ∈ R2 × R such that there exists an optimal action y ∈ Y (p) in the silent game Γ(p) satisfying 19/
(i) ak ≥ Ak (y), for all k ∈ K (ii) a1 = A1 (y) if p 6= 0 and a2 = A2 (y) if p 6= 1 (iii) β = p B 1 (y) + (1 − p) B 2 (y) (Thus, E + (p) = E(p) if p ∈ (0, 1)) Graph of the modified equilibrium payoff correspondence: gr E + ≡ {(a, β, p) ∈ R2 × R × [0, 1] : (a, β) ∈ E + (p)}
Hart (1985, MOR), Aumann and Hart (2003, Ecta): Without any assumption on the utility functions, all equilibrium payoffs of the unilateral communication game Γ0S (p) can be geometrically characterized only from the graph of the equilibrium payoff correspondence of the silent game
20/
Theorem (Characterization of ES0 (p)) Let p ∈ (0, 1). A payoff profile (a, β) is a Nash equilibrium payoff of the unilateral communication game Γ0S (p) if and only if (a, β, p) belongs to conva (gr E + ), the set points obtained by convexification of the set gr E + in (β, p) by keeping the expert’s payoff, a, constant: ES0 (p) = {(a, β) ∈ R2 × R : (a, β, p) ∈ conva (gr E + )}
2 – Decentralized Communication
F. Koessler / May 28, 2010
Illustrations Unique equilibrium, non revealing (Example 1) Optimal decisions in the silent game:
21/ Y (p) =
{jH } ∆({jH , jM })
if p < 1/4 if p = 1/4
{jM } ∆({jM , jL }) {j }
if p ∈ (1/4, 3/4) if p = 3/4 if p > 3/4
L
aH
p=1
=
jM
p
2
p=0
1/ 4
jH
3
22/ 1
jL
p
=
4 3/ aL
0 0
1
2
3
Figure 3: Modified equilibrium payoffs in Example 1
2 – Decentralized Communication
F. Koessler / May 28, 2010
Full revelation of information (Example 2)
k1
a2
j1
j2
1, 1
0, 0
p j2
3 0, 0
3, 3
p=0
(1 − p)
p=1
k2
FRE
2 23/ if p > 3/4
p=
{j1 } Y (p) = {j2 } ∆(J)
1
3/4
if p < 3/4 if p = 3/4
j1
0 0
a1
1
Unique equilibrium, non-revealing (Example 3)
a2 j2
5, 2
1, 0
5 p
p=1
k1
j1
4 k2
3, 0
1, 4
(1 − p)
3
p=
2
24/
Y (p) =
{j1 }
1
j2
if p > 2/3,
{j2 }
if p < 2/3,
∆(J)
if p = 2/3
j1
2/3
p=0
a1
0 0
1
2
3
4
5
6
2 – Decentralized Communication
F. Koessler / May 28, 2010
Unique equilibrium, non-revealing (Example 4) a2 j2
3, 2
4, 0
p
p=1
k1
5
j1
4 k2
3, 0
1, 4
(1 − p)
j1
3
p=
25/
2/3
2 {j1 } Y (p) = {j2 } ∆(J)
if p > 2/3, 1
if p < 2/3,
j2
p=0
if p = 2/3
a1
0 0
1
2
3
Partial revelation of information: Example 6
j1
j2
j3
j4
k1
4, 0
2, 7
5, 9
1, 10
p
k2
1, 10
4, 7
4, 4
2, 0
1−p
26/ {j1 } ∆({j1 , j2 }) {j2 } Y (p) = ∆({j2 , j3 }) {j3 } ∆({j3 , j4 }) {j4 }
if p < 3/10 if p = 3/10 if p ∈ (3/10, 3/5) if p = 3/5 if p ∈ (3/5, 4/5) if p = 4/5 if p > 4/5
4
5
2 – Decentralized Communication
F. Koessler / May 28, 2010
a2 5
27/
p = 3/5
j2 p=1
4 3
PRE p=
p=
4/5
10 3/
2
j4
1
j3
p=0 j1 a1
0 0
1
2
3
4
5
The Art of Conversation: Multistage Communication and Compromises Aumann et al. (1968): Allowing more than one communication stage can extend and Pareto improve the set of Nash equilibria, even if only one player is privately informed 28/ Aumann and Hart (2003, Ecta): Full characterization of equilibrium payoffs induced by multistage cheap talk communication in finite two-player games with incomplete information on one side Multistage communication also extends the equilibrium outcomes in the classical model of Crawford and Sobel (1982)
2 – Decentralized Communication
F. Koessler / May 28, 2010
Examples Example. (Compromising)
L
R
T
6, 2
0, 0
B
0, 0
2, 6
Jointly controlled lottery (JCL): 1 a
29/
b 2
a
b
a
b
L
R
L
R
L
R
L
R
T
6, 2
0, 0
6, 2
0, 0
6, 2
0, 0
6, 2
0, 0
B
0, 0
2, 6
0, 0
2, 6
0, 0
2, 6
0, 0
2, 6
1 1 1 1 a + b ⇒ (T, L) + (B, R) → (4, 4) 2 2 2 2
Example. (Signalling, and then compromising)
30/
k1 T B
L (6, 2) (0, 0)
M (0, 0) (2, 6)
R (3, 0) (3, 0)
k2 T B
L (0, 0) (0, 0)
M (0, 0) (0, 0)
R (4, 4) (4, 4)
Interim equilibrium payoffs ((4, 4), 4) The two communication stages cannot be reversed (compromising should come after signalling)
2 – Decentralized Communication
F. Koessler / May 28, 2010
Example. (Compromising, and then signaling) (Example 5)
j1
j2
j3
j4
j5
k1
1, 10
3, 8
0, 5
3, 0
1, −8
p
k2
1, −8
3, 0
0, 5
3, 8
1, 10
1−p
31/
Interim equilibrium payoffs ((2, 2), 8) = 12 ((3, 3), 6) + 21 ((1, 1), 10) Of course, the two communication stages cannot be reversed (the compromise determines the type of signalling)
Example. (Signalling, then compromising, and then signalling)
32/
j1
j2
j3
j4
j5
j6
k1
1, 10
3, 8
0, 5
3, 0
1, −8
2, 0
1/3
k2
1, −8
3, 0
0, 5
3, 8
1, 10
2, 0
1/3
k3
0, 0
0, 0
0, 0
0, 0
0, 0
2, 8
1/3
Interim equilibrium payoffs ((2, 2, 2), 8)
2 – Decentralized Communication
F. Koessler / May 28, 2010
Multistage and Bilateral Cheap Talk Game Γ0n (p) Bilateral communication: the uninformed player can also send messages Player 1: informed, expert Player 2: uninformed, decision maker K: set of information states (i.e., types) of P1, probability distribution p 33/
J: set of actions of P2 P1’s payoff is Ak (j) and P2’s payoff is B k (j) M 1 : set of messages of the expert (independent of his type) M 2 : set of message of the decisionmaker
At every stage t = 1, . . . , n, P1 sends a message m1t ∈ M 1 to P2 and, simultaneously, P2 sends a message m2t ∈ M 2 to P1 At stage n + 1, P2 chooses j in J
34/
Information Phase
Communication Phase
Action Phase
Expert learns k ∈ K
Expert and DM send
DM chooses j ∈ J
(m1t , m2t ) ∈ M 1 × M 2 (t = 1, . . . n)
2 – Decentralized Communication
F. Koessler / May 28, 2010
Characterization of the Nash equilibria of Γ0n (p), n = 1, 2, . . . Hart (1985), Aumann and Hart (2003): finite case (K and J are finite sets) All Nash equilibrium payoffs of the multistage, bilateral communication games Γ0n (p), n = 1, 2, . . ., are characterized geometrically from the graph of the equilibrium correspondence of the silent game 35/ Additional stages of cheap talk can Pareto-improve the equilibria of the communication game (Aumann et al., 1968) Imposing no deadline to cheap talk can Pareto-improve the equilibria of any n-stage communication game (Forges, 1990b, QJE, Simon, 2002, GEB)
Example. (Forges, 1990a, QJE) An employer (the DM) chooses to offer a job j1 , j2 , j3 or j4 , or no job (action j0 ) to a candidate (the expert) The candidate has two possible types k1 et k2 , which determine his competence and preference for the different jobs j1
j2
j0
j3
j4
k1
6, 10
10, 9
0, 7
4, 4
3, 0
Pr[k1 ] = p
k2
3, 0
4, 4
0, 7
10, 9
6, 10
Pr[k2 ] = 1 − p
36/
{j1 } {j2 } Y (p) = {j0 } {j3 } {j } 4
if p > 4/5, if p ∈ (3/5, 4/5), if p ∈ (2/5, 3/5), if p ∈ (1/5, 2/5), if p < 1/5.
2 – Decentralized Communication
F. Koessler / May 28, 2010
p=1 /5
9 8 7
37/
4
p=0 y
j4
p=
5
x
FRE
2/5
6
p=1
Graph of modified equilibrium payoffs gr E + : a2 j3 10
j2
/5 p=4
3
j1
2
p=
1 0 j0 0
1
2
3
4
3/5 a1
5
6
7
8
9 10
Γ0S (p),
From the equilibrium characterization theorem for there is only two types of equilibria in the single-stage cheap talk game: NRE and FRE
But in the 3-stage cheap talk game, when p = 3/10, the interim payoff (3, 6) can be obtained as follows, where z = (2/5)j0 + (3/5)j3 N k1 k2 1
3 10
a
b 1 3
1
7 10
a
2 3
b 4 7
3 7
JCL
38/
JCL
H 1 6
1
H
T 5 6
2
1 6
1
a
T 5 6
b 2
j4
j1
(3, 0)
(6, 10)
2
2 z
j4
2 j4
, 26 ) (6, 10) (6, 10) ( 12 5 5
2 z (30/5, 41/5)
F. Koessler / May 28, 2010
2 – Decentralized Communication
Geometrically, this equilibrium payoff can be constructed as follows Adding a JCL before the one-stage cheap talk game at p = 1/5 yields [j3 , j4 , FRE] Adding a JCL before the one-stage cheap talk game at p = 2/5 yields [j0 , j3 , FRE] Adding a signalling stage before the JCL allows a second convexification at p fixed Hence, for all p ∈ [1/5, 2/5] (in particular, p = 3/10) we get [j3 , j4 , FRE] (in particular, a = (3, 6)) with three communication stages 39/
A subset of R2 × R × [0, 1] is diconvex if it is convex in (β, p) when a is fixed, and convex in (a, β) when p is fixed. di-co (E) is the smallest diconvex set containing E
Theorem. (Hart, 1985, Forges, 1994, Aumann and Hart, 2003) Let p ∈ (0, 1). A payoff (a, β) is an equilibrium payoff of some bilateral communication game Γ0n (p), for some length n, if and only if (a, β, p) belongs to di-co (gr E + ), the set of all points obtained by diconvexifying the set gr E + 40/
2 – Decentralized Communication
F. Koessler / May 28, 2010
Communication with No Deadline When the number of communication stages, n, is not fixed in advance, the job candidate can even achieve the expected payoff (7, 7) when p = 1/2
41/
N k1 H
1
JCL
T
1 2
1 2
a
1 2
1 2
1 a
a
t1
h1
2 j2
2 H
T 1 a 3 4
2
1 4
H
T
2
1 j3
t2 JCL
H 2 j2
H
T
b
2
2 j3
T j2
a
a 2
→ h1 j1
1 3 4
1 4
b → t1
H
JCL
T
2
1 a → h2
1 4
3 4
j2
b 2 j4
1
a
JCL
H
T
2 j3
1
JCL
JCL
j1
1 b
2
JCL
T 1 a
j3
b
T
1 2
h2
JCL
H
3 4
1 2
j4
JCL
42/
b
1 4
1 4
JCL
H
1 b
3 4
2 j1
k2
H
b 3 4
1 4
2 j4
T
1
b → t2
2 – Decentralized Communication
F. Koessler / May 28, 2010
When the number of communication rounds is unbounded or even infinite, the previous theorem applies by replacing di-co (gr E + ) by di-span (gr E + ), the set of all expectations of di-martingales whose limits are almost surely in gr E + a j3
x 43/
(7, 7) y
j2 p 1/5
2/5 1/2 3/5
4/5
1
Certifiable Information in Games Unilateral persuasion game ΓS (p): defined as the unilateral cheap talk game Γ0S (p), but the set of messages of the sender, M (k), depends on his type k ···
A1 (j), B 1 (j) · · ·
···
A2 (j), B 2 (j) · · ·
j
j 2
44/
.. .
A1 (j), B 1 (j)
a j
1
a 1
c 2
k1
N
k2
b
.. .
.. . c2 2
1
j
b 2 j
···
A1 (j), B 1 (j) · · ·
A2 (j), B 2 (j) .. .
j ···
A2 (j), B 2 (j) · · ·
Figure 4: Extensive form of the unilateral persuasion game ΓS (p) with two types, two cheap talk messages and one certificate for each type
2 – Decentralized Communication
F. Koessler / May 28, 2010
Examples In example 3 recalled below the unique NE of the cheap talk game is NR (j2 → (a, β) = ((1, 1), 2)): j1
j2
k1
5, 2
1, 0
p = 1/2
k2
3, 0
1, 4
(1 − p) = 1/2
45/
However, if type k1 is able to prove his type, by sending a message (certificate) m = c1 which is not available to type k2 , then there is a FRE
(3, 2)
(1, 0)
j1
j2
m Expert
(3, 0)
j1
DM k1
N
j1
(3, 2)
j2 m Expert
k2
c1 46/
(1, 4)
c2 j2
(1, 0)
j1
(3, 0)
j2
(1, 4)
With certifiable information, there is also a (pure strategy) FRE in the monotonic games 1, 7 and 8, as well as in examples 2 and 5 where there already exists a FRE under cheap talk On the contrary, examples 4 and 6 don’t admit a FRE
2 – Decentralized Communication
F. Koessler / May 28, 2010
Example. j1
j2
j3
j4
j5
k1
5, 0
3, 4
0, 7
4, 9
2, 10
Pr[k1 ] = 1/2
k2
1, 10
3, 9
0, 7
5, 4
6, 0
Pr[k2 ] = 1/2
47/ Unique communication equilibrium: non-revealing (j3 → ((0, 0), 7))
10
j1
j5
9 j2
j4 j3
7
48/
4
0
1 5
2 5
3 5
4 5
1
p
Figure 5: Expected payoffs (fine lines) and best reply expected payoffs (bold lines) for the DM
2 – Decentralized Communication
F. Koessler / May 28, 2010
Fully Revealing Equilibrium (5, 0) (3, 4) (0, 7) (4, 9) (2, 10)
(1, 10) (3, 9) (0, 7) (5, 4) (6, 0)
Receiver m 49/
Sender c1
k1
N
Receiver
k2 Receiver
(5, 0) (3, 4) (0, 7) (4, 9) (2, 10)
m Sender c2
(1, 10) (3, 9) (0, 7) (5, 4) (6, 0)
Interim expected payoffs: (a, β) = ((2, 1), 10)
Non-revealing Equilibrium (5, 0) (3, 4) (0, 7) (4, 9) (2, 10)
m 50/
k1
(1, 10) (3, 9) (0, 7) (5, 4) (6, 0)
N
k2
c1
m c2
(5, 0) (3, 4) (0, 7) (4, 9) (2, 10)
(1, 10) (3, 9) (0, 7) (5, 4) (6, 0)
Interim expected payoffs: (a, β) = ((0, 0), 7) (Note: this NE is not subgame perfect)
2 – Decentralized Communication
F. Koessler / May 28, 2010
Partially Revealing Equilibrium: PRE1 (5, 0) (3, 4) (0, 7) (4, 9) (2, 10)
(1, 10) (3, 9) (0, 7) (5, 4) (6, 0)
2/3 1/3
2/3 1/3
m 2/3 51/
k1
N
m
k2
c1 1/3
c2
(5, 0) (3, 4) (0, 7) (4, 9) (2, 10)
(1, 10) (3, 9) (0, 7) (5, 4) (6, 0)
Interim expected payoffs: (a, β) = ((2, 2), 7.5)
Partially Revealing Equilibrium: PRE2 (5, 0) (3, 4) (0, 7) (4, 9) (2, 10) 4/5
m 52/
(1, 10) (3, 9) (0, 7) (5, 4) (6, 0)
1/5
k1
c1
(5, 0) (3, 4) (0, 7) (4, 9) (2, 10)
4/5
N
k2
1/5
2/3 m 1/3 c2
(1, 10) (3, 9) (0, 7) (5, 4) (6, 0)
Interim Expected Payoffs: (a, β) = ((4/5, 1), 7.5) (Note: This NE is not subgame perfect)
2 – Decentralized Communication
F. Koessler / May 28, 2010
Geometric Characterization of NE payoffs of ΓS (p) Recall: Modified equilibrium payoffs E + (p) of Γ(p): the expert can get a payoff higher than his equilibrium when his type has zero probability ➥ (a, β) ∈ R2 × R such that there exists an optimal mixed action y ∈ Y (p) of the silent game Γ(p) satisfying (i) ak ≥ Ak (y), for every k ∈ K; 53/
(ii) a1 = A1 (y) if p 6= 0 and a2 = A2 (y) if p 6= 1; (iii) β = p B 1 (y) + (1 − p) B 2 (y). Extended equilibrium payoffs E ++ (p) of Γ(p): the expert can have any payoff when his type has zero probability ➥ (a, β) ∈ R2 × R such that there exists y ∈ Y (p) satisfying (ii) and (iii)
Graph of the extended equilibrium payoff correspondence: gr E ++ ≡ {(a, β, p) ∈ R2 × R × [0, 1] : (a, β) ∈ E ++ (p)}
Graph of interim individually rational payoffs: INTIR ≡ {(a, β, p) ∈ R2 × R × [0, 1] : ∃ y ∈ ∆(J), ak ≥ Ak (y) ∀ k ∈ K} 54/ Forges and Koessler (2008, JET): If every event is certifiable, all Nash equilibrium payoffs of the unilateral persuasion game ΓS (p) can be geometrically characterized from the graph of the equilibrium payoff correspondence of the silent game
2 – Decentralized Communication
F. Koessler / May 28, 2010
Assumptions: • For every k there exists ck ∈ M 1 such that M −1 (ck ) = {k} • |M (k1 ) ∩ M (k2 )| ≥ 3
55/
Theorem (Characterization of ES (p)) Let p ∈ (0, 1). A payoff (a, β) is an equilibrium payoff of the unilateral persuasion game ΓS (p) if and only if (a, β, p) belongs to conva (gr E ++ ) ∩ INTIR, the set of all points obtained by convexifying the set gr E ++ in (β, p) while keeping constant and individually rational the expert’s payoff, a: ES (p) = {(a, β) ∈ R2 × R : (a, β, p) ∈ conva (gr E ++ ) ∩ INTIR}.
p=1
a2
j5
6
p=
4/5
j4
=
3/ 5
5
j2
3
p
2
PRE1 PRE2
1/ 5
j1
FRE
p=0
=
2/ 5
1
=
a1
p
56/
p
4
0j 3 NRE 0 1
2
3
4
5
2 – Decentralized Communication
F. Koessler / May 28, 2010
Equilibrium Refinement in Persuasion Games
Contrary to the cheap talk case, a Nash equilibrium in a persuasion game may rely on irrational choices off the equilibrium path For instance, in the previous example, the NRE and the PRE2 are not subgame perfect 57/ Similarly, the NRE is not subgame perfect in the persuasion games associated with example 1 when p > 1/4, example 2 for every p, example 3 when p < 2/3, example 5 when p ∈ (3/8, 5/8), example 7 when p ∈ (1/3, 2/3)
The example below, which is a modified version of example 4 by adding the strictly dominated action j3 , has a subgame perfect FRE when x ≤ 3 et y ≤ 1, but it is not a perfect Bayesian equilibrium
58/
j1
j2
j3
k1
3, 2
4, 0
x, −1
p
k2
3, 0
1, 4
y, −1
(1 − p)
2 – Decentralized Communication
F. Koessler / May 28, 2010
4
p=1
a2
3
j1
p=
59/
2/3
2
j2
FRE
1
p=0
y j3 0 0
1
a1 x
2
3
4
5
Formally, in the geometric characterization of the theorem, the payoff a = (a1 , a2 ) of the expert should also satisfy ∃ y 1 ∈ Y (1) t.q. a1 ≥ A1 (y 1 ) ∃ y 2 ∈ Y (0) t.q. a2 ≥ A2 (y 2 ) for a subgame perfect NE (⇒ north-east of FRE)
60/
and ∃ p ∈ ∆(K), y ∈ Y (p) t.q. ak ≥ Ak (y) ∀ k ∈ K for a perfect Bayesian equilibrium (⇒ north-east of [j1 , j2 ])
2 – Decentralized Communication
F. Koessler / May 28, 2010
Long Persuasion Games In the unilateral persuasion game associated with the game
61/
j1
j2
j3
j4
j5
k1
5, 0
3, 4
0, 7
4, 9
2, 10
Pr[k1 ] = 1/2
k2
1, 10
3, 9
0, 7
5, 4
6, 0
Pr[k2 ] = 1/2
the highest payoff for the expert is (2, 2) at the partially revealing equilibrium PRE1 However, in the 3-stage bilateral persuasion game, there is an equilibrium in which the expert can get (3, 3) by delaying information certification
Stage 1: Signaling The expert sends message a or b with a type dependent positive probability Equilibrium condition: he must be indifferent between sending a or b, whatever his type Stage 2: Jointly controlled lottery (JCL) Both players decide jointly on how to continue the game 62/ Stage 3: Possible certification According to the outcome of the JCL, either P2 makes his decision immediately or P1 first fully certifies his type
2 – Decentralized Communication
F. Koessler / May 28, 2010
Signaling Info.
k1
N
k2
1 2
1 a
1 2
b 3 4
a
1 4
JCL Action Certification
H
T 1 2
T 1 2
1
1 2
c1 2
3 4
JCL
H
63/
b 1 4
JCL
1 2
1
2
1 c2
2
2
2
2
j4
j5
j2
j4
j1
j2
(4, 9)
(2, 10)
(3, 4)
(5, 4)
(1, 10)
(3, 9)
Γn (p): Information and actions phases as in the signalling game ΓS (p) but • Bilateral communication. Player 2’s message set M 2 , |M 2 | ≥ 2 • n ≥ 1 communication rounds, perfect monitoring Information phase
Talking phase (n ≥ 1 rounds)
Action phase
Expert learns k ∈ K
Both send (m1t , m2t ) ∈ M (k) × M 2
DM chooses j ∈ J
(t = 1, . . . n) 64/ En (p): Nash equilibrium payoffs of Γn (p) EB (p) =
S
n≥1
En (p): NE payoffs of all multistage, bilateral persuasion games
2 – Decentralized Communication
F. Koessler / May 28, 2010
Theorem (Characterization of EB (p)) Let p ∈ (0, 1). A payoff (a, β) is an equilibrium payoff of a multistage bilateral persuasion game Γn (p), for some length n, if and only if (a, β, p) belongs to di-co (gr E ++ ) ∩ INTIR, the set of all points obtained by diconvexifying the set of all payoffs in gr E ++ that are interim individually rational for the expert: EB (p) = {(a, β) ∈ R2 × R : (a, β, p) ∈ di-co (gr E ++ ) ∩ INTIR}. 65/ Remarks. The characterization is more complicate with more than two types; INTIR should be applied w.r.t. the posteriors at each step of the diconvexification process To get PBE, INTIR should be defined for sequentially rational decisions only
p=1
a2 j5
6
p=
4/5
j4
=
3/ 5
5
j2
3
p = 5 1/
2
j1
1
=
2/ 5
p=0 a1
p
66/
p
4
0j 3 0
1
2
3
4
5
F. Koessler / May 28, 2010
2 – Decentralized Communication
References Aumann, R. J. and S. Hart (2003): “Long Cheap Talk,” Econometrica, 71, 1619–1660. Aumann, R. J., M. Maschler, and R. Stearns (1968): “Repeated Games with Incomplete Information: An Approach to the Nonzero Sum Case,” Report of the U.S. Arms Control and Disarmament Agency, ST-143, Chapter IV, pp. 117–216. Crawford, V. P. and J. Sobel (1982): “Strategic Information Transmission,” Econometrica, 50, 1431–1451. Forges, F. (1990a): “Equilibria with Communication in a Job Market Example,” Quarterly Journal of Economics, 105, 375–398. ——— (1990b): “Universal Mechanisms,” Econometrica, 58, 1341–1364.
67/
——— (1994): “Non-Zero Sum Repeated Games and Information Transmission,” in Essays in Game Theory: In Honor of Michael Maschler, ed. by N. Megiddo, Springer-Verlag. Forges, F. and F. Koessler (2008): “Long Persuasion Games,” Journal of Economic Theory, 143, 1–35. Hart, S. (1985): “Nonzero-Sum Two-Person Repeated Games with Incomplete Information,” Mathematics of Operations Research, 10, 117–153. Radner, R. (1979): “Rational Expectations Equilibrium: Generic Existence and the Information Revealed by Prices,” Econometrica, 47, 655–678. Simon, R. S. (2002): “Separation of Joint Plan Equilibrium Payoffs from the Min-Max Functions,” Games and Economic Behavior, 41, 79–102. Spence, A. M. (1973): “Job Market Signaling,” Quarterly Journal of Economics, 87, 355–374.
