Cheap Talk
Abdul Quadir
XLRI
9 January, 2020
Introduction
I Cheap Talk
I Screening and signaling
Reading: Dixit and Skeath, Chapter 8.
Partially Aligned Interests
I More often we observe both conflict and common interest in
games.
I Therefore the credibility of direct communication depends
which side the weight is tilted.
I Let us consider a example of a person who would like to invest
his money.
I He seeks advice from a financial adviser.
I Financial adviser could be trustworthy or not.
I Credibility of adviser’s recommendations depends on what
type relationship you establish with him.
Partially Aligned Interests
I Suppose the investor has 1 lac rupees and invest in an asset
recommended by the adviser.
I States of the world and return on asset are summarized as:
State Return
Bad (B) 50% loss
Mediocre (M) 1%
Good (G) 55%
I Adviser charges 2% up front fee.
I Adviser also earn 20% of any gain that the investor makes.
I Adviser will not share any loss.
I Suppose investor is not in a position to know which state
could occur.
I Thus, he treats them equally likely, i.e., each state occurs with
probability 13 .
Partially Aligned Interests
I Without any more information, will the investor invest his
money in the recommended asset?
I The expected payoff of the investment in the recommended
asset is:
1 1 1
× (−50, 000) + × 0.8 × 1000 + × 0.8 × 55, 000 − 2000 =
3 3 3
− 1730 − 2000 = −3730
I The investor will invest his money.
I Note also that the investor will not invest if the state is B or
M for sure or any probability mix-up of B and M.
I However, adviser knows which of three possibilities is the
truth.
I What will he do with his information?
Partially Aligned Interests
I Without any more information, will the investor invest his
money in the recommended asset?
I The expected payoff of the investment in the recommended
asset is:
1 1 1
× (−50, 000) + × 0.8 × 1000 + × 0.8 × 55, 000 − 2000 =
3 3 3
− 1730 − 2000 = −3730
I The investor will invest his money.
I Note also that the investor will not invest if the state is B or
M for sure or any probability mix-up of B and M.
I However, adviser knows which of three possibilities is the
truth.
I What will he do with his information?
Partially Aligned Interests
I The investor will update his information as he gets more
information from the adviser.
I We assume for this example that the investor will believe
whatever the adviser told to him.
I This means the investor will assign probability 1 to the asset
being the type stated by the adviser.
Short-Term Relationship
I If the adviser tells the investor that the state is B, then he will
not invest because he suffers loss amounting to
−50, 000 − 2000 = −52, 000.
I If the adviser tells the investor that the state is M, then he
will not invest because he suffers loss amounting to
0.8 × 1000 − 2000 = −1200.
I If the adviser tells the investor that the state is G , then he will
invest because he realizes return amounting to
0.8 × 55, 000 − 2000 = 42, 000.
I The adviser will tell the state is G if it is actually G .
I The adviser will also be tempted to tell the investor that the
state is G while it is either B or M in truth if the relationship
is short.
I Therefore, there is a possibility of getting wrong advice, the
investor will ignore the adviser’s recommendation. (babbling
equilibrium only).
Short-Term Relationship
I If the adviser tells the investor that the state is B, then he will
not invest because he suffers loss amounting to
−50, 000 − 2000 = −52, 000.
I If the adviser tells the investor that the state is M, then he
will not invest because he suffers loss amounting to
0.8 × 1000 − 2000 = −1200.
I If the adviser tells the investor that the state is G , then he will
invest because he realizes return amounting to
0.8 × 55, 000 − 2000 = 42, 000.
I The adviser will tell the state is G if it is actually G .
I The adviser will also be tempted to tell the investor that the
state is G while it is either B or M in truth if the relationship
is short.
I Therefore, there is a possibility of getting wrong advice, the
investor will ignore the adviser’s recommendation. (babbling
equilibrium only).
Long-Term Relationship: Full Revelation
I Now suppose that there is some cost of reputation that the
adviser has to suffer in case of wrong advice.
I Could you have relationship with a firm where the adviser
works.
I The investor is comparing the actual performance of the asset
vis-a-vis the adviser’s forecast.
I The forecast could be wrong in
I small way (actual B, forecasted M or actual M, forecasted G )
I long way (actual B, forecasted G ).
I If the adviser attaches some cost if ho looses his reputation,
then he will also care about the investor’s losses too (partially
aligned interest).
I Suppose loss to reputation are 2000 rupees or 4000 rupees if
he misreports in small way or a big way, respectively.
Long-Term Relationship: Full Revelation
I It is obvious that if the state is G , he will reveal that the state
is G .
I What is the incentive to reveal G when the state is not G ?
I First consider that the state is B and the adviser reports it to
be M.
I The investor will not invest if the adviser truthfully reveals
that it is B.
I Therefore, the payoffs for both of them is zero.
I Suppose the adviser reports that the state is M.
I The investor will not invest even in this case too. Again
payoffs for both are zero.
I Thus, the adviser does not have any incentive to misreport
that a B-type is a M-type.
Long-Term Relationship: Full Revelation
I It is obvious that if the state is G , he will reveal that the state
is G .
I What is the incentive to reveal G when the state is not G ?
I First consider that the state is B and the adviser reports it to
be M.
I The investor will not invest if the adviser truthfully reveals
that it is B.
I Therefore, the payoffs for both of them is zero.
I Suppose the adviser reports that the state is M.
I The investor will not invest even in this case too. Again
payoffs for both are zero.
I Thus, the adviser does not have any incentive to misreport
that a B-type is a M-type.
Long-Term Relationship: Full Revelation
I What is the adviser reports G ?
I The investor will believe him and invest.
I Thus, the adviser will get up front fee of 2000 rupees but he
will also suffer reputational loss of large error, i.e. 4000 rupees.
I The payoff of the adviser is negative if he reports G while the
actual state is B.
I However if he reveals B truthfully, it is better for him.
I Therefore, the adviser will reveal truthfully if the state is G or
B.
I What if the state is M?
I Note that if the adviser reveals that the state is M, the the
investor will not invest and the adviser’s payoff is zero from
reporting M.
I If he reports G , then his payoff is
2000 + 0.2 × 1000 − 2000 = 200 > 0
Long-Term Relationship: Full Revelation
I What is the adviser reports G ?
I The investor will believe him and invest.
I Thus, the adviser will get up front fee of 2000 rupees but he
will also suffer reputational loss of large error, i.e. 4000 rupees.
I The payoff of the adviser is negative if he reports G while the
actual state is B.
I However if he reveals B truthfully, it is better for him.
I Therefore, the adviser will reveal truthfully if the state is G or
B.
I What if the state is M?
I Note that if the adviser reveals that the state is M, the the
investor will not invest and the adviser’s payoff is zero from
reporting M.
I If he reports G , then his payoff is
2000 + 0.2 × 1000 − 2000 = 200 > 0
Long-Term Relationship: Full Revelation
I Thus, the adviser gets benefited by reporting G while the true
state is M.
I Knowing this the investor will disregard any advice given by
the adviser.
I Thus, full information cannot be revealed credibly.
I Only babbling equilibrium exists.
I Note that the failure to achieve full revelation occurs because
the adviser misreport M as G .
I What if the adviser club M and G together and reveal only B
or not-B?
Long-Term Relationship: Partial Revelation
I What is the posterior probability now when the adviser is
restricting in reporting B and not-B?
1
I Your prior probability is 3 for each state.
I If the investor is told not-B, then he is left with two
possibilities M and G (both are equally likely).
I 1 and 12 are the posterior probabilities conditional the
2
information that the investor receives from the adviser.
I Now the investor expected payoff if adviser reports not-B
1 1
×0.8×1000+ ×0.8×55, 000−2000 = 400+22000−2000 = 20400
2 2
I The investor has the positive incentive to invest if adviser
report not-B.
I Will adviser has any incentive to lie?
Long-Term Relationship: Partial Revelation
I What is the posterior probability now when the adviser is
restricting in reporting B and not-B?
1
I Your prior probability is 3 for each state.
I If the investor is told not-B, then he is left with two
possibilities M and G (both are equally likely).
I 1 and 12 are the posterior probabilities conditional the
2
information that the investor receives from the adviser.
I Now the investor expected payoff if adviser reports not-B
1 1
×0.8×1000+ ×0.8×55, 000−2000 = 400+22000−2000 = 20400
2 2
I The investor has the positive incentive to invest if adviser
report not-B.
I Will adviser has any incentive to lie?
Long-Term Relationship: Partial Revelation
I Suppose the true type is B, will the adviser report not-B?
I If the adviser reports B, then the investor will not invest and
the adviser payoff is zero.
I If he report not-B, then the investor will invest and the
adviser will get up front fee of 2000 rupees.
I The adviser also incurs cost of loosing reputation. The
expected loss of reputation is 21 × 2000 + 12 × 4000 = 3000.
I Now if the true state is B and the adviser report not-B, his
net payoff is 2000 − 3000 = −1000.
I Thus, it is not in the best interest of the adviser to misreport.
I Thus, there is a cheap talk equilibrium with credible partial
revelation of information.
I Payoffs of the adviser and the investor are calculated by the
rules of the game.
Long-Term Relationship: Partial Revelation
I Suppose the true type is B, will the adviser report not-B?
I If the adviser reports B, then the investor will not invest and
the adviser payoff is zero.
I If he report not-B, then the investor will invest and the
adviser will get up front fee of 2000 rupees.
I The adviser also incurs cost of loosing reputation. The
expected loss of reputation is 12 × 2000 + 12 × 4000 = 3000.
I Now if the true state is B and the adviser report not-B, his
net payoff is 2000 − 3000 = −1000.
I Thus, it is not in the best interest of the adviser to misreport.
I Thus, there is a cheap talk equilibrium with credible partial
revelation of information.
I Payoffs of the adviser and the investor are calculated by the
rules of the game.
Formal Analysis of Cheap Talk
I We can analyze the cheap talk by drawing a tree.
I There is a new neutral player ‘Nature’ here without any its
payoff.
I Observe that by under-reporting the adviser cannot benefit.
I Therefore, we ignore this information to simplify the tree.
I The reputational cost is denoting by S or L when the error is
small or big, respectively.
2,-50
I
Inv.
N 0, 0
B
Adv. Inv. I 2 − S, −52
M N
Inv. I 0,0
B G 2 − L, −52
N
Inv.
M I 0,0
M Adv. 2.2, -1.2
Nature N
Inv. 0,0
G I 2.2 − S, −1.2
G Adv. N 0,0
I 13, 42
Inv.
G N 0,0
Formal Analysis of Cheap Talk
I We can further ignore those strategies which cannot be played
in equilibrium.
I Ignore the following strategies:
1. ‘Choose I when report is B’. This is dominated by ‘Choose N
when report is B’
2. ‘Choose I when report is M’. This is dominated by ‘Choose N
when report is M’
3. ‘Choose I when report is B’.
I Similarly ignore those strategies which will not make no
difference in the search of cheap talk equilibrium.
I Thus, for the adviser ‘report B’ and ‘report M’ leads to no
investment by the investor. Ignore them.
Formal Analysis of Cheap Talk
I We can further ignore those strategies which cannot be played
in equilibrium.
I Ignore the following strategies:
1. ‘Choose I when report is B’. This is dominated by ‘Choose N
when report is B’
2. ‘Choose I when report is M’. This is dominated by ‘Choose N
when report is M’
3. ‘Choose I when report is B’.
I Similarly ignore those strategies which will not make no
difference in the search of cheap talk equilibrium.
I Thus, for the adviser ‘report B’ and ‘report M’ leads to no
investment by the investor. Ignore them.
2,-50
I
Inv.
N 0, 0
B
Adv. Inv. I 2 − S, −52
M N
Inv. I 0,0
B G 2 − L, −52
N
Inv.
M I 0,0
M Adv. 2.2, -1.2
Nature N
Inv. 0,0
G I 2.2 − S, −1.2
G Adv. N 0,0
I 13, 42
Inv.
G N 0,0
2,-50
I
Inv.
N 0, 0
B
Adv. Inv. I 2 − S, −52
M N
Inv. I 0,0
B G 2 − L, −52
N
Inv.
M I 0,0
M Adv. 2.2, -1.2
Nature N
Inv. 0,0
G I 2.2 − S, −1.2
G Adv. N 0,0
I 13, 42
Inv.
G N 0,0
Inv. I
B G 2 − L, −52
N
0,0
M
Nature
Inv.
G I 2.2 − S, −1.2
G Adv. N 0,0
I 13, 42
Inv.
G N 0,0
Formal Analysis of Cheap Talk
I This elimination leaves us with only six terminal nodes.
I These nodes arise from the strategies which includes adviser
reporting G .
I Note that there are three interesting strategies for the adviser.
I ‘Report G always.
I ‘Report G when true state is B or M’.
I ‘Report G when true state is G ’.
I Two interesting strategies for the investor:
I ‘Choose I if report is G ’.
I ‘Choose N even if report is G ’
I This is summarized in the following game matrix.
Game Matrix
I/A I if G N if G
Always G (17.2-S-L)/3, -11.2/3 0,0
G only if M or G (15.2-S)/3,40.8/3 0,0
G if and only if G 13/3,42/3 0,0
I The best response of the investor is ‘N if G ’ if the adviser
plays ‘always G ’.
I The best response of the investor is ‘I if G ’ if the adviser
plays ‘G if M or G ’ or ‘G if and only if G ’.
I The best response of the adviser any thing if the investor
plays is ‘N if G ’.
I Thus, (Always G , N if G ) is a babbling equilibrium.
I This is the earlier case that we have seen.
Game Matrix
I/A I if G N if G
Always G (17.2-S-L)/3, -11.2/3 0,0
G only if M or G (15.2-S)/3,40.8/3 0,0
G if and only if G 13/3,42/3 0,0
I The best response of the investor is ‘N if G ’ if the adviser
plays ‘always G ’.
I The best response of the investor is ‘I if G ’ if the adviser
plays ‘G if M or G ’ or ‘G if and only if G ’.
I The best response of the adviser any thing if the investor
plays is ‘N if G ’.
I Thus, (Always G , N if G ) is a babbling equilibrium.
I This is the earlier case that we have seen.
Cheap Talk Equilibrium
I The best response of the adviser if the investor plays‘I if G ’
could be G if M or G ’ or ‘G if and only if G ’.
I The values of S and L matter.
I (‘G if M or G ’, ‘I if G ’) is a Nash equilibrium if
15.2 − S > 17.2 − S − L and 15.2 − S > 13
I The first equation implies that L > 2 and the second equation
implies that S < 2.2.
I If S and L meet these requirements, then (‘G if M or G ’, ‘I if
G ’) is a Nash equilibrium.
I In this equilibrium the report G does not allow you to know
whether the true state is M or G but you get positive
expected payoff.
I Thus, the partial revelation equilibrium emerges.
Cheap Talk Equilibrium
I When (‘G if and only if G ’, ‘I if G ’) is a Nash equilibrium?
I Again we have to check that
13 > 17.2 − S − L and 13 > 15.2 − S
I From the second equation we get S > 2.2.
I As we have assumed that L > S, any vale of L greater than S
will work.
I Thus, (‘G if and only if G ’, ‘I if G ’) is Nash equilibrium if
S > 2.2 as long as L > S.
I Note that in either case Babbling equilibrium exists.
Another Example of Cheap Talk
I Consider two friends, Friend 1 and Friend 2.
I Friend 1 is from Delhi and now resides there.
I Friend 2 is from Jamshedpur and she recently has got a job at
a company in Gurgaon.
I She would like to relocate herself to Delhi.
I Friend 2 really wants to live closer to Friend 1 with having any
easy commute to her office.
I She would like to live closer to her office in Gurgoan if traffic
is worse and closer to me in Delhi if the traffic is good.
I She would prefer to live in between Gurgaon and Delhi if
traffic is moderate.
I Friend 1 cares more about her living close to him than she
does.
I Thus, Friend 1 cares less how bad a commute she has.
Formal Representation
I Consider Friend 1 as Player 1 who is the sender of
information.
I Consider Friend 2 as Player 2 who is the receiver of the
information.
I The expected traffic condition is θ ∈ {1, 3, 5}, where 1 is bad,
3 moderate and 5 good.
I Player 1 knows the true value of θ but Player 2 knows only
the prior distribution of θ.
I Player 1 sends a message (his action) to Player 2 about the
traffic condition.
I Player 2 then chooses an action where to live
a2 ∈ A2 = {1, 2, 3, 4, 5}, where 1 is Gurgaon, 5 is Delhi and
2,3 and 4 are places in between the two cities in that order.
Formal Representation
I The payoff function of Player 2 is 5 minus the distance she
ends up living.
u2 (a2 , θ) = 5 − (θ − a2 )2
I Observe that the best response of Player 2: given any level of
traffic, she wants to choose her residence location equal to the
traffic level.
I Thus, her optimal choose is a∗ (θ) = θ.
I We know that Player 1 is biased that she lives closer to him.
I Thus, his payoff is
u1 (a2 , θ) = 5 − (θ + b − a2 )2
where b > 1 is the bias of Player 1.
I Note that the action of Player affects the payoffs of both
Player 1 and 2.
I This is known as common-values game.
Formal Representation
I The payoff function of Player 2 is 5 minus the distance she
ends up living.
u2 (a2 , θ) = 5 − (θ − a2 )2
I Observe that the best response of Player 2: given any level of
traffic, she wants to choose her residence location equal to the
traffic level.
I Thus, her optimal choose is a∗ (θ) = θ.
I We know that Player 1 is biased that she lives closer to him.
I Thus, his payoff is
u1 (a2 , θ) = 5 − (θ + b − a2 )2
where b > 1 is the bias of Player 1.
I Note that the action of Player affects the payoffs of both
Player 1 and 2.
I This is known as common-values game.
Formal Representation
I The payoff function of Player 2 is 5 minus the distance she
ends up living.
u2 (a2 , θ) = 5 − (θ − a2 )2
I Observe that the best response of Player 2: given any level of
traffic, she wants to choose her residence location equal to the
traffic level.
I Thus, her optimal choose is a∗ (θ) = θ.
I We know that Player 1 is biased that she lives closer to him.
I Thus, his payoff is
u1 (a2 , θ) = 5 − (θ + b − a2 )2
where b > 1 is the bias of Player 1.
I Note that the action of Player affects the payoffs of both
Player 1 and 2.
I This is known as common-values game.
Equilibrium Analysis
I Player 1 can sends any message from a1 ∈ A1 = {1, 3, 5}.
I Take b = 1.1.
I What is the babbling equilibrium?
I Player 1 has a incentive to over-report his message.
I Suppose θ = 3, then Player 2 will choose a2 = 3.
I However, Player 1’s optimal is a1 = 4 or evern he prefers to
report a1 = 5 than a1 = 3.
I Suppose Player 1 sends any message with equal probability of
1
3 regardless of θ.
1
I Thus, Player knows that Pr (θ) = 3 for any θ ∈ {1, 3, 5}.
I Player 2 maximizes her expected payoff:
1 1 1
max Ev2 (a2 , θ) = 5−[ (1−a2 )2 + (3−a2 )2 + (5−a2 )2 ]
a2 ∈{1,2,3,4,5} 3 3 3
Equilibrium Analysis
I Player 1 can sends any message from a1 ∈ A1 = {1, 3, 5}.
I Take b = 1.1.
I What is the babbling equilibrium?
I Player 1 has a incentive to over-report his message.
I Suppose θ = 3, then Player 2 will choose a2 = 3.
I However, Player 1’s optimal is a1 = 4 or evern he prefers to
report a1 = 5 than a1 = 3.
I Suppose Player 1 sends any message with equal probability of
1
3 regardless of θ.
1
I Thus, Player knows that Pr (θ) = 3 for any θ ∈ {1, 3, 5}.
I Player 2 maximizes her expected payoff:
1 1 1
max Ev2 (a2 , θ) = 5−[ (1−a2 )2 + (3−a2 )2 + (5−a2 )2 ]
a2 ∈{1,2,3,4,5} 3 3 3
Equilibrium Analysis
I Player 1 can sends any message from a1 ∈ A1 = {1, 3, 5}.
I Take b = 1.1.
I What is the babbling equilibrium?
I Player 1 has a incentive to over-report his message.
I Suppose θ = 3, then Player 2 will choose a2 = 3.
I However, Player 1’s optimal is a1 = 4 or evern he prefers to
report a1 = 5 than a1 = 3.
I Suppose Player 1 sends any message with equal probability of
1
3 regardless of θ.
1
I Thus, Player knows that Pr (θ) = 3 for any θ ∈ {1, 3, 5}.
I Player 2 maximizes her expected payoff:
1 1 1
max Ev2 (a2 , θ) = 5−[ (1−a2 )2 + (3−a2 )2 + (5−a2 )2 ]
a2 ∈{1,2,3,4,5} 3 3 3
Babbling Equilibrium
I We have to compute the expected payoff for each
a2 ∈ {1, 2, 3, 4, 5} to pick which one maximizes it.
a2 Eu2 (a2 , θ)
1 5- 3 (0 + 4 + 16)=− 35
1
2 5- 31 (1 + 1 + 9)= 43
3 5- 31 (4 + 0 + 4)= 73
4 5- 31 (9 + 1 + 1)= 43
5 5- 31 (16 + 4 + 0)=− 35
I Thus a2 = 3 maximizes the expected payoff.
I Player 2 will ignore the information sent by the Player 1.
I There is no truthful equilibrium because Player 1 is biased
towards higher actions.
Cheap Talk Equilibrium
I To construct a cheap talk equilibrium, some value information
only will be sent to Player 2.
I Player 1 will reveal truthfully if θ = 1 and club θ = 3 and
θ = 5 together.
I This will lead to truthful partial equilibrium.
I Thus, Player 1’s action when θ = 1 is a1 = 1 and take action
a1 ∈ {3, 5} with probability 12 when θ ∈ {3, 5}.
I The posterior probability for Player 2:
1
Pr (θ = 1|a1 = 1) = 1 and Pr (θ = 3|a1 ) = Pr (θ = 5|a1 ) = .
2
I Thus, Player 2’s expected payoff is
1 1
max Ev2 (a2 , θ) = 5 − [ (3 − a2 )2 + (5 − a2 )2 ]
a2 ∈{1,2,3,4,5} 2 2
Cheap Talk Equilibrium
I To construct a cheap talk equilibrium, some value information
only will be sent to Player 2.
I Player 1 will reveal truthfully if θ = 1 and club θ = 3 and
θ = 5 together.
I This will lead to truthful partial equilibrium.
I Thus, Player 1’s action when θ = 1 is a1 = 1 and take action
a1 ∈ {3, 5} with probability 12 when θ ∈ {3, 5}.
I The posterior probability for Player 2:
1
Pr (θ = 1|a1 = 1) = 1 and Pr (θ = 3|a1 ) = Pr (θ = 5|a1 ) = .
2
I Thus, Player 2’s expected payoff is
1 1
max Ev2 (a2 , θ) = 5 − [ (3 − a2 )2 + (5 − a2 )2 ]
a2 ∈{1,2,3,4,5} 2 2
Cheap Talk Equilibrium
a2 Eu2 (a2 , θ)
1 5- 21 (4 + 16)=−5
2 5- 21 (1 + 9)=0
3 5- 21 (0 + 4)=3
4 5- 11 (1 + 1)=4
5 5- 31 (4 + 0)=3
I Player 2’s optimal action is a2 = 4.
I Player 1’s strategy reporting not-θ = 1 or reporting {3, 4}
makes him indifferent choosing a1 = 3 or a1 = 5 and better
than a1 = 1.
I Likewise if θ = 1, Player 2 prefers a2 = 1 over a2 = 4 and
Player 1 prefers a1 = 1 over other possible actions.
