Lesson overview
Chapter 9 Uncertainty and Information
Lesson II.3 Cheap Talk and Adverse Selection Theory
Each Example Game Introduces some Game Theory
• Example 1: Cheap Talk with Perfectly Aligned Interests
• Example 2: Cheap Talk with Perfectly Conflicting Interests
• Example 3: Cheap Talk with Partially Aligned Interests
• Example 4: Cheap Talk Review
• Example 5: Adverse Selection and Market Failure
Lesson II.4 Signaling and Screening Theory
Lesson II.5 Signaling and Screening Applications
BA 592 Lesson II.3 Cheap Talk and Adverse Selection Theory
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Example 1: Cheap Talk with Perfectly Aligned Interests
The simplest way to give information to others seems to be to tell
them, and the simplest way to get information is to ask. But in a
game of strategy, players are aware that others may not tell the
truth, and there own talk may not be believed by others. It is said
that talk is cheap; that is, talk has zero or negligible direct cost.
However, it can indirectly affect the outcome and payoffs of a
game by changing one player’s beliefs about another player’s
actions, and so selecting one equilibrium out of multiple
equilibria. Call talk that has no direct cost cheap talk. For for a
game with cheap talk, call an equilibrium that is affected by talk a
cheap talk equilibrium, and call an equilibrium that is not affected
by talk a babbling equilibrium. It turns out there is always a
babbling equilibrium. We now find out when there is also a
cheap talk equilibrium.
BA 592 Lesson II.3 Cheap Talk and Adverse Selection Theory
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Example 1: Cheap Talk with Perfectly Aligned Interests
The normal form for the Stag Hunt assurance game between
Hunter 1 and Hunter 2 identifies two Nash equilibria in pure
strategies, one where they are both certain the other is hunting the
stag, and the other where they are both certain about the hare.
The problem is they are playing the game noncooperatively,
making choices independently.
Hunter 2
Hunter 1
Stag
Hare
Stag
2,2
1,0
Hare
0,1
1,1
BA 592 Lesson II.3 Cheap Talk and Adverse Selection Theory
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Example 1: Cheap Talk with Perfectly Aligned Interests
Hunter 2
Hunter 1
Stag
Hare
Stag
2,2
1,0
Hare
0,1
1,1
But suppose Hunter 1 is given an opportunity to send the message
“I am going to hunt the stag” or “I am going to hunt a hare” to
Hunter 2 (or Hunter 2 is given an opportunity to ask a question
and Hunter 1 replies) before the choices are made. In game
theory terms, we created a first stage to the game where Hunter 1
selects one of two messages.
Define the game tree of this Cheap-Talk-Stag-Hunt Assurance
Game, and compute all the rollback equilibria
BA 592 Lesson II.3 Cheap Talk and Adverse Selection Theory
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Example 1: Subgames
The two-stage game tree:
First stage: Message Game
Hunter 1
Say Stag
Say Hare
Second stage: Assurance Game Second stage: Assurance Game
Hunter 2
Hunter 1
Stag
Hare
Stag
2,2
1,0
Hare
0,1
1,1
Hunter 2
Hunter 1
Stag
Hare
Stag
2,2
1,0
BA 592 Lesson II.3 Cheap Talk and Adverse Selection Theory
Hare
0,1
1,1
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Example 1: Subgames
A cheap talk equilibrium: Say Stag (honest), and both hunt Stag.
Most likely if Hunter 1 has a reputation for honest.
Say Stag
2,2
First stage: Message Game Hunter 1 Say Hare 1,1
Second stage: Assurance Game Second stage: Assurance Game
Hunter 2
Hunter 1
Stag
Hare
Stag
2,2
1,0
Hare
0,1
1,1
Second stage equilibrium,
Hunter 1 Stag and Hunter 2
Stag, generating payoffs 2,2
Hunter 2
Hunter 1
Stag
Hare
Stag
2,2
1,0
Hare
0,1
1,1
Second stage equilibrium,
Hunter 1 Hare and Hunter 2
Hare, generating payoffs 1,1
BA 592 Lesson II.3 Cheap Talk and Adverse Selection Theory
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Example 1: Subgames
Another cheap talk equilibrium: Say Hare (lie), and both hunt
Stag. Most likely if Hunter 1 has a reputation for dishonesty.
Say Stag
1,1
First stage: Message Game Hunter 1 Say Hare 2,2
Second stage: Assurance Game Second stage: Assurance Game
Hunter 2
Hunter 1
Stag
Hare
Stag
2,2
1,0
Hare
0,1
1,1
Second stage equilibrium,
Hunter 1 Hare and Hunter 2
Hare, generating payoffs 1,1
Hunter 2
Hunter 1
Stag
Hare
Stag
2,2
1,0
Hare
0,1
1,1
Second stage equilibrium,
Hunter 1 Stag and Hunter 2
Stag, generating payoffs 2,2
BA 592 Lesson II.3 Cheap Talk and Adverse Selection Theory
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Example 1: Subgames
One babbling equilibrium: Say Stag (lie), and both hunt Stag.
Likely if Hunter 1 is often ignored.
Say Stag
2,2
First stage: Message Game Hunter 1 Say Hare 2,2
Second stage: Assurance Game Second stage: Assurance Game
Hunter 2
Hunter 1
Stag
Hare
Stag
2,2
1,0
Hare
0,1
1,1
Second stage equilibrium,
Hunter 1 Stag and Hunter 2
Stag, generating payoffs 2,2
Hunter 2
Hunter 1
Stag
Hare
Stag
2,2
1,0
Hare
0,1
1,1
Second stage equilibrium,
Hunter 2 Stag and Hunter 2
Stag, generating payoffs 2,2
BA 592 Lesson II.3 Cheap Talk and Adverse Selection Theory
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Example 1: Subgames
Another babbling equilibrium: Say Stag (lie), and both hunt Hare.
Likely if Hunter 1 is often ignored.
Say Stag
1,1
First stage: Message Game Hunter 1 Say Hare 1,1
Second stage: Assurance Game Second stage: Assurance Game
Hunter 2
Hunter 1
Stag
Hare
Stag
2,2
1,0
Hare
0,1
1,1
Second stage equilibrium,
Hunter 1 Hare and Hunter 2
Hare, generating payoffs 1,1
Hunter 2
Hunter 1
Stag
Hare
Stag
2,2
1,0
Hare
0,1
1,1
Second stage equilibrium,
Hunter 2 Hare and Hunter 2
Hare, generating payoffs 1,1
BA 592 Lesson II.3 Cheap Talk and Adverse Selection Theory
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Example 1: Cheap Talk with Perfectly Aligned Interests
Hunter 2
Hunter 1
Stag
Hare
Stag
2,2
1,0
Hare
0,1
1,1
There are also two other babbling equilibrium where Hunter 1
says Hare.
Summing up, there are cheap talk equilibria. One of them is most
likely if Hunter 1 has a reputation for honesty, and the other if
Hunter 1 has a reputation for dishonesty. (Perfectly dishonest
people are worth listening to for the same reasons as listening to
perfectly honest people.) The cheap talk equilibrium with Hunter
1 honest is also a natural focal point.
There are also babbling equilibria, just like every other cheap talk
game.
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Example 2: Cheap Talk with Perfectly Conflicting Interests
The normal form for the zero-sum game between Kicker and
Goalie identifies a single Nash equilibria in mixed strategies: The
Kicker kicks Left or Right with probabilities p = .36 and (1-p) =
.64, and the Goalie guards Left or Right with probabilities q = .45
and (1-q) = .55. The Nash equilibrium generates expected
payoffs of .1q + .8(1-q) = .7q + .3(1-q) = .48 for the Kicker and
.9p + .3(1-p) = .2p + .7(1-p) = .52 for the Goalie.
Goalie
Kicker
Left
Right
Left
.1,.9
.7,.3
Right
.8,.2
.3,.7
BA 592 Lesson II.3 Cheap Talk and Adverse Selection Theory
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Example 2: Cheap Talk with Perfectly Conflicting Interests
Goalie
Kicker
Left
Right
Left
.1,.9
.7,.3
Right
.8,.2
.3,.7
Now suppose the Kicker is given an opportunity to send the
message “I am going to kick Left” or “I am going to kick Right”
to the Goalie (or the Goalie is given an opportunity to ask a
question and the Kicker replies) before the choices are made. We
created a first stage to the game where the Kicker selects one of
two messages.
Define the game tree of this Cheap-Talk-Penalty-Kick Zero-Sum
Game, and compute all the rollback equilibria
BA 592 Lesson II.3 Cheap Talk and Adverse Selection Theory
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Example 2: Cheap Talk with Perfectly Conflicting Interests
The two-stage game tree:
First stage: Message Game
Kicker
Second stage: Zero-Sum Game
Say Left
Say Right
Second stage: Zero-Sum Game
Goalie
Kicker
Left
Right
Left
.1,.9
.7,.3
Right
.8,.2
.3,.7
Goalie
Kicker
Left
Right
Left
.1,.9
.7,.3
BA 592 Lesson II.3 Cheap Talk and Adverse Selection Theory
Right
.8,.2
.3,.7
13
Example 2: Cheap Talk with Perfectly Conflicting Interests
Two babbling equilibrium: 1) Say Left (lie), and both play
mixed-strategy Nash equilibrium. 2) Say Right (lie), and Nash.
Say Left .48,.52
First stage: Message Game Kicker Say Right .48,.52
Second stage: Zero-Sum Game
Second stage: Zero-Sum Game
Goalie
Kicker
Left
Right
Left
.1,.9
.7,.3
Right
.8,.2
.3,.7
Goalie
Kicker
Left
Right
Left
.1,.9
.7,.3
Right
.8,.2
.3,.7
Second stage equilibrium is the unique mixed-strategy Nash
equilibrium: Kicker kicks Left or Right with probabilities
(.36,.64) and Goalie guards Left or Right with probabilities
(.45,.55), generating expected payoffs .48,.52
BA 592 Lesson II.3 Cheap Talk and Adverse Selection Theory
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Example 2: Cheap Talk with Perfectly Conflicting Interests
Goalie
Kicker
Left
Right
Left
.1,.9
.7,.3
Right
.8,.2
.3,.7
Summing up, there is no cheap talk equilibrium. And that is true
for any Cheap-Talk-Zero-Sum Game. Do not believe your
opponent will do what he says (he’ll use that against you), do not
believe he will do the opposite (he’ll also use that against you),
but disregard what he says.
Rollback analysis thus verifies what Juan Sanchez Villa-Lobos
Ramirez (Sean Connery) said in the movie Highlander, “Never
take what your opponent offers you”.
BA 592 Lesson II.3 Cheap Talk and Adverse Selection Theory
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Example 3: Cheap Talk with Partially Aligned Interests
The normal form for the Battle of the Sexes game between
Husband and Wife identifies two Nash equilibria in pure
strategies, one where they are both certain the other is at the
football game and the other where they are both certain about the
opera.
Wife
Football
Football
3,2
Husband
Opera
0,0
Opera
0,0
2,3
BA 592 Lesson II.3 Cheap Talk and Adverse Selection Theory
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Example 3: Cheap Talk with Partially Aligned Interests
Wife
Football
Football
3,2
Husband
Opera
0,0
Opera
0,0
2,3
Now suppose Husband is given an opportunity to send the
message “I am going to Football” or “I am going to Opera” to
Wife (or Wife is given an opportunity to ask a question and
Husband replies) before the choices are made. We created a first
stage to the game where Husband selects one of two messages.
Define the game tree of this Cheap-Talk-Battle-of-the-Sexes
Game, and compute all the rollback equilibria
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Example 3: Cheap Talk with Partially Aligned Interests
The two-stage game tree:
First stage: Message Game
Husband
Say Football
Say Opera
Second stage: Coordination
Game
Wife
Second stage: Coordination
Game
Wife
Football
Football
3,2
Husband
Opera
0,0
Football
Football
3,2
Husband
Opera
0,0
Opera
0,0
2,3
BA 592 Lesson II.3 Cheap Talk and Adverse Selection Theory
Opera
0,0
2,3
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Example 3: Cheap Talk with Partially Aligned Interests
One of the Cheap Talk equilibria: Say Football, and both
coordinate on the Football Nash equilibrium.
Say Football
3,2
First stage: Message Game Husband Say Opera 2,3
Second stage: Coordination
Game
Wife
Second stage: Coordination
Game
Wife
Football
Football
3,2
Husband
Opera
0,0
Football
Football
3,2
Husband
Opera
0,0
Opera
0,0
2,3
Opera
0,0
2,3
Second stage equilibrium,
Second stage equilibrium,
Husband and Wife each choose Husband and Wife each choose
Football, generating payoffs 3,2 Opera, generating payoffs 3,2
BA 592 Lesson II.3 Cheap Talk and Adverse Selection Theory
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Example 3: Cheap Talk with Partially Aligned Interests
Wife
Football
Football
3,2
Husband
Opera
0,0
Opera
0,0
2,3
Summing up, there are cheap talk equilibria. One of them is most
likely if Husband has a reputation for honesty. That cheap talk
equilibrium with Husband honest is also a natural focal point.
There are also babbling equilibria, just like every other cheap talk
game.
BA 592 Lesson II.3 Cheap Talk and Adverse Selection Theory
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Example 4: Cheap Talk Review
When your stockbroker recommends that you should buy a
particular stock, he may be doing so as part of developing a longrun relationship with you for the future commissions that your
business will bring him, or he may be trying to dump a loser that
his firm wants to get rid of for a quick profit. You have to guess
the relative importance of these two possibilities to his payoffs.
BA 592 Lesson II.3 Cheap Talk and Adverse Selection Theory
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Example 4: Cheap Talk Review
Suppose there are just two possibilities: the stock has good
prospects or bad. If good, you should buy; if bad, sell, or sell
short. The table below are payoffs consistent with that data. (It
is not a payoff matrix of a game since the columns are not the
choices of a strategic player.
The broker knows whether the stock is good or bad; you don’t.
He can recommend buy or sell.
Should you follow his advice
The Stock is
in this stock buying game?
You
Buy
Sell
Good
1
-1
Bad
-1
1
BA 592 Lesson II.3 Cheap Talk and Adverse Selection Theory
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Example 4: Cheap Talk Review
Should you follow his advice? It depends on your broker’s
payoffs. Suppose his payoffs are a mixture of two
considerations. One is the long-term relationship with you: that
part of his payoffs is just a replica of yours. But he also gets an
extra kickback X from his firm if he can persuade you to buy a
bad stock that the firm wants to dump. Then his payoffs are in
the table below.
The Stock is
You
Buy
Sell
Good
1
-1
Bad
-1+X
1
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Example 4: Cheap Talk Review
Can there be a cheap talk equilibrium with truthful
communication? This is actually a three stage game.
Say Buy
Broker
Second stage: Message
Say Sell
Game
Third stage: Follow advice to
buy?
The Stock is
You
Buy
Sell
Good
1
-1
Bad
-1
1
Second stage equilibrium: If
you believe he says the truth
that the stock is good, You
choose Buy.
Third stage: Follow advice to
sell?
The Stock is
You
Buy
Sell
Good
1
-1
Bad
-1
1
Second stage equilibrium: If
you believe he says the truth
that the stock is bad, You
choose Sell.
BA 592 Lesson II.3 Cheap Talk and Adverse Selection Theory
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Example 4: Cheap Talk Review
Can there be a cheap talk equilibrium with truthful
communication?
First stage: Nature determines the stock is Good or bad.
Second stage: Message Game
Broker
The Stock is
Good
Say Buy
1
Say Sell
-1
Third stage: Message Game
Broker
The Stock is
Bad
Say Buy
-1+X
Say Sell
1
Second stage equilibrium: If the Second stage equilibrium: If the
Broker knows the stock is
Broker knows the stock is Bad,
Good, he says Buy.
he says Sell if 1 > -1+X (X < 2)
but not if X > 2.
BA 592 Lesson II.3 Cheap Talk and Adverse Selection Theory
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Example 4: Cheap Talk Review
Summing up, Should you follow your broker’s advice? It
depends on your broker’s payoffs.
If X < 2, there is a cheap talk equilibrium with truthful
communication. That equilibrium is also a natural focal point, so
you may want to follow your broker’s advice and choose the
cheap talk equilibrium over the babbling equilibria that are
present in every cheap talk game.)
But if X > 2, do not follow your broker’s advice since there are
only babbling equilibria.
BA 592 Lesson II.3 Cheap Talk and Adverse Selection Theory
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Example 5: Adverse Selection and Market Failure
Many games are like the stock pricing game above where one of
the players knows something about the outcomes that the other
players don’t know. And many games are like the stock pricing
game with X > 2 where there are only babbling equilibria, and
cheap talk does not matter.
BA 592 Lesson II.3 Cheap Talk and Adverse Selection Theory
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Example 5: Adverse Selection and Market Failure
Employers knows less about the skills of potential employees
than does the employee, with other important matters like work
attitude and collegiality even harder to observe. Insurance
companies know less about the heath and driving skills of
insurance applicants than do the applicants. And buyers of used
cars know less about a car than do the sellers.
In those examples, cheap talk cannot be screened for information
by the less informed player. Unskilled, lazy, and obnoxious
potential employees will claim to have skills and will act hard
working and collegial during interviews to get higher pay.
Insurance applicants who are bad risks will claim good health and
driving habits to get lower rates. And sellers of used cars will
withhold any problems about their cars. For those same reasons,
cheap talk cannot signal information by the more informed player
since the other player knows there are incentives to lie.
BA 592 Lesson II.3 Cheap Talk and Adverse Selection Theory
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Example 5: Adverse Selection and Market Failure
If there are no other ways to screen or signal information, then
insurance companies would have to offer the same rates to all
applicants, say 5 cents per dollar of coverage. That rate would be
especially attractive to applicants who know there own risk (of
illness or car crash) exceeds 5%. Hence, the pool of applicants
for that insurance policy will have a larger proportion of poorer
risks than the proportion of those risks in the population as a
whole. The insurance company thus selectively attracts an
unfavorable, or adverse, group of customers. That phenomenon
is common in transactions involving asymmetric information and
is known as adverse selection.
Adverse selection can make markets inefficient, or fail altogether.
BA 592 Lesson II.3 Cheap Talk and Adverse Selection Theory
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Example 5: Adverse Selection and Market Failure
The Ford Pinto was Ford Motor Company's first domestic North
American subcompact automobile marketed beginning on
September 11, 1970. The Pinto was popular in sales, with
100,000 units delivered by January 1971. Its reputation suffered
as a popular small, inexpensive car from a controversy
surrounding the safety of its gas tank. Think of the market in
1974 for a 1971 Ford Pinto.
Suppose each owner knows whether his Pinto is a lemon (bad) or
a peach (good). Suppose each owner of a peach values it at
$12,500, and each owner of a lemon at $3,000. Suppose each
buyer values a peach at $16,000, and a lemon at $6,000. Since
buyers value cars more than sellers, it benefits everyone if all cars
sell. The price for an orange should be between $12,500 and
$16,000, and a lemon between $3,000 and $6,000. Suppose there
is a limited stock of cars, and a larger number of buyers.
BA 592 Lesson II.3 Cheap Talk and Adverse Selection Theory
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Example 5: Adverse Selection and Market Failure
The Car is a
Value to
Buyer
Seller
Peach
$16,000
$12,500
Lemon
$6,000
$3,000
If Peaches and Lemons can be separated with certainty, then
competition among buyers drives prices up to their full values,
peaches at $16,000 and lemons at $6,000.
But without such separation, there must be just one price, p.
Whether efficient trade is possible depends on the fraction f of
peaches and the remaining fraction (1-f ) of lemons. A purchased
car is thus a peach with probability f and a lemon with probability
(1-f ). The expected value of a purchased car is
$16,000 x f + $16,000 x (1-f ) = $6,000 + $10,000 x f .
All cars are thus demanded if $6,000 + $10,000 x f > p
BA 592 Lesson II.3 Cheap Talk and Adverse Selection Theory
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Example 5: Adverse Selection and Market Failure
The Car is a
Value to
Buyer
Seller
Peach
$16,000
$12,500
Lemon
$6,000
$3,000
All cars will be supplied if p > $6,000. Putting it all together, all
cars will be traded if
$6,000 + $10,000 x f > p > $6,000,
which requires f > 0.65. And in that case, the price p is bid up
near $6,000 + $10,000 x f .
If f < 0.65, then the market for peaches fails, and the only cars
traded are lemons. In that care, the price is bid up to $6,000.
BA 592 Lesson II.3 Cheap Talk and Adverse Selection Theory
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BA 592
Game Theory
End of Lesson II.1
BA 592 Lesson II.3 Cheap Talk and Adverse Selection Theory
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