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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

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slide1

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.

BA 592 Lesson II.3 Cheap Talk and Adverse Selection Theory

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Example 1: Cheap Talk with Perfectly Aligned Interests

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

Second stage: Assurance Game

Second stage: Assurance Game

BA 592 Lesson II.3 Cheap Talk and Adverse Selection Theory

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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.

First stage: Message Game

Second stage: Assurance Game

Second stage: Assurance Game

Second stage equilibrium, Hunter 1 Stag and Hunter 2 Stag, generating payoffs 2,2

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.

First stage: Message Game

Second stage: Assurance Game

Second stage: Assurance Game

Second stage equilibrium, Hunter 1 Hare and Hunter 2 Hare, generating payoffs 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.

First stage: Message Game

Second stage: Assurance Game

Second stage: Assurance Game

Second stage equilibrium, Hunter 1 Stag and Hunter 2 Stag, generating payoffs 2,2

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.

First stage: Message Game

Second stage: Assurance Game

Second stage: Assurance Game

Second stage equilibrium, Hunter 1 Hare and Hunter 2 Hare, generating payoffs 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

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.

BA 592 Lesson II.3 Cheap Talk and Adverse Selection Theory

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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.

BA 592 Lesson II.3 Cheap Talk and Adverse Selection Theory

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Example 2: Cheap Talk with Perfectly Conflicting Interests

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

Second stage: Zero-Sum Game

Second stage: Zero-Sum Game

BA 592 Lesson II.3 Cheap Talk and Adverse Selection Theory

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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.

First stage: Message Game

Second stage: Zero-Sum Game

Second stage: Zero-Sum Game

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

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 thusverifies 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.

BA 592 Lesson II.3 Cheap Talk and Adverse Selection Theory

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Example 3: Cheap Talk with Partially Aligned Interests

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

BA 592 Lesson II.3 Cheap Talk and Adverse Selection Theory

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Example 3: Cheap Talk with Partially Aligned Interests

The two-stage game tree:

First stage: Message Game

Second stage: Coordination Game

Second stage: Coordination Game

BA 592 Lesson II.3 Cheap Talk and Adverse Selection Theory

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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.

First stage: Message Game

Second stage: Coordination Game

Second stage: Coordination Game

Second stage equilibrium, Husband and Wife each choose Football, generating payoffs 3,2

Second stage equilibrium, Husband and Wife each choose 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

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 long-run 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

in this stock buying game?

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.

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? This is actually a three stage game.

Second stage: Message Game

Third stage: Follow advice to buy?

Third stage: Follow advice to sell?

Second stage equilibrium: If you believe he says the truth that the stock is good, You choose Buy.

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

Third stage: Message Game

Second stage equilibrium: If the Broker knows the stock is Good, he says Buy.

Second stage equilibrium: If the Broker knows the stock is Bad, 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

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 fof 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 xf + $16,000 x (1-f ) = $6,000 + $10,000 xf .

All cars are thus demanded if $6,000 + $10,000 xf > p

BA 592 Lesson II.3 Cheap Talk and Adverse Selection Theory

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Example 5: Adverse Selection and Market Failure

All cars will be supplied if p > $6,000. Putting it all together, all cars will be traded if

$6,000 + $10,000 xf > p > $6,000,

which requires f > 0.65. And in that case, the price p is bid up near $6,000 + $10,000 xf .

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

end of lesson ii 1

BA 592 Game Theory

End of Lesson II.1

BA 592 Lesson II.3 Cheap Talk and Adverse Selection Theory