Lying and Deception in Games Joel Sobel August 2, 2016 Lying and - - PowerPoint PPT Presentation

Lying and Deception in Games

Joel Sobel August 2, 2016

Lying and Deception Sobel

What is the Talk About?

Definitions:

• 1. Lying
• 2. Deception
• 3. Bluff

and simple properties.

Lying and Deception Sobel

Why do this?

• 1. Coherence
• 2. To facilitate scholarly communication
• 3. To identify and separate characteristics of strategic

communication

3.1 Common Language 3.2 Theory of Mind 3.3 Who Gains from Behavior

Lying and Deception Sobel

Today’s Talk

• 1. The basic model
• 2. A definition of Lying
• 3. A quick, self-serving look at an enormous literature
• 4. Properties of Lying
• 5. A definition of beliefs
• 6. A definition of deception
• 7. Many small results

Lying and Deception Sobel

But First . . .

In mathematics, names are free. It is perfectly allowable to call a self-adjoint operator an elephant, and a spectral resolution a trunk. One can then prove a theorem, whereby all elephants have trunks. What is not allowable is to pretend that this result has anything to do with certain large gray animals.

Lying and Deception Sobel

But First . . .

In mathematics, names are free. It is perfectly allowable to call a self-adjoint operator an elephant, and a spectral resolution a trunk. One can then prove a theorem, whereby all elephants have trunks. What is not allowable is to pretend that this result has anything to do with certain large gray animals. Hector Sussman, quoted in Ivar Ekeland, “Mathematics and the Unexpected”

Lying and Deception Sobel

But First . . .

In mathematics, names are free. It is perfectly allowable to call a self-adjoint operator an elephant, and a spectral resolution a trunk. One can then prove a theorem, whereby all elephants have trunks. What is not allowable is to pretend that this result has anything to do with certain large gray animals. Hector Sussman, quoted in Ivar Ekeland, “Mathematics and the Unexpected” So conclusions of the form: “Deception is possible in equilibrium” are only as insightful if my definition is appropriate.

Lying and Deception Sobel

. . . and

What I shall have to say here is neither difficult nor contentious; the only merit I should like to claim for it is that of being true, at least in parts. The phenomenon to be discussed is very widespread and obvious, and it cannot fail to have been already noticed, at least here and there, by others. Yet I have not found attention paid to it specifically.

Lying and Deception Sobel

. . . and

What I shall have to say here is neither difficult nor contentious; the only merit I should like to claim for it is that of being true, at least in parts. The phenomenon to be discussed is very widespread and obvious, and it cannot fail to have been already noticed, at least here and there, by others. Yet I have not found attention paid to it specifically.

• J. L. Austin, “How to Do Things with Words”

Lying and Deception Sobel

. . . and

What I shall have to say here is neither difficult nor contentious; the only merit I should like to claim for it is that of being true, at least in parts. The phenomenon to be discussed is very widespread and obvious, and it cannot fail to have been already noticed, at least here and there, by others. Yet I have not found attention paid to it specifically.

• J. L. Austin, “How to Do Things with Words”

sometimes is it useful to state the obvious

Lying and Deception Sobel

. . . and

What I shall have to say here is neither difficult nor contentious; the only merit I should like to claim for it is that of being true, at least in parts. The phenomenon to be discussed is very widespread and obvious, and it cannot fail to have been already noticed, at least here and there, by others. Yet I have not found attention paid to it specifically.

• J. L. Austin, “How to Do Things with Words”

sometimes is it useful to state the obvious but sometimes it isn’t.

Lying and Deception Sobel

A Basic Model

• 1. Two players: Sender and Receiver.
• 2. Sender observes θ ∈ Θ
• 3. Sender sends message m ∈ M.
• 4. Receiver hears m.
• 5. Receiver takes action y ∈ Y .
• 6. Preferences Ui(θ, m, y); UR(·) typically independent of m.
• 7. Prior P(θ).
• 8. All sets are finite (unless I want them to be infinite).

Lying and Deception Sobel

Model Includes . . .

• 1. Standard (Spence) signaling.
• 2. Cheap talk.
• 3. Some sequential Games with Incomplete Information.

Lying and Deception Sobel

Variations

• 1. Many Players
• 2. Noisy Messages
• 3. Incompletely informed Sender
• 4. (*) Sender takes payoff relevant action

(*) No time today.

Lying and Deception Sobel

Common Language

• 1. For each Θ0 ⊂ Θ, there exists a message mΘ0 ∈ M and there

is a common understanding that mΘ0 means “θ ∈ Θ0.’

• 2. Exactly one such message for each subset.
• 3. m has accepted meaning if m = mΘ0.
• 4. Some messages may have no accepted meaning.

Lying and Deception Sobel

Comments

• 1. S need not tell the truth.
• 2. There may be costs associated with lying (or telling the truth).
• 3. All definitions can be interpreted from S’s perspective.

Lying and Deception Sobel

Lying: Definitions

Lying arises when the Sender says something that she believes to be false.

Definition (Lying)

• 1. The message m is a lie given θ if m = mΘ0 and θ /

∈ Θ0.

• 2. The message m is a lie of omission given θ if m = mΘ0 and

{θ} Θ0.

• 3. The message m is true given θ if m = mθ.

Lying and Deception Sobel

Observations

• 1. Lying does not depend on preferences.
• 2. Lying does not depend on R’s behavior (or S’s beliefs about

R’s behavior)

• 3. Definition depends on only on support of the distribution. (So

there is not word for “A and B are equally likely.”)

Lying and Deception Sobel

Other Approaches

Philosophy, theology, legal studies, evolutionary biology, computer science, . . . I’ll only say:

• St. Augustine provides taxonomy (eight types of lie).

He asks: Is it ever moral to lie? Dishonest statements told in jest are not lies according to St. Augustine.

Lying and Deception Sobel

Linguistic Anthropology

Coleman and Kay argue that one evaluates whether a statement is a lie by assessing the extent to which it satisfies three criteria:

• 1. the statement is false
• 2. the speaker believes the statement to be false
• 3. the intention of the speaker is to deceive

Experimentally people classify statements as lies using the second criterion.

Lying and Deception Sobel

Another Voice

It takes two to speak truth – one to speak and another to hear.

Lying and Deception Sobel

Another Voice

It takes two to speak truth – one to speak and another to hear. Henry David Thoreau, “A Week on the Concord and Merrimack Rivers: Wednesday”

Lying and Deception Sobel

Lies in Cheap-Talk Games

Simple cheap-talk game:

• 1. Θ = [0, 1]; Y = R
• 2. for i = S, R, Ui(θ, y) is continuous, strictly concave in a, and

satisfies Ui

12 > 0;

• 3. yi(θ) ≡ arg max Ui(θ, y) is well defined;
• 4. yS(θ) > yR(θ).

Lying and Deception Sobel

Results

Proposition

In a cheap-talk game with a common language, there always exist equilibria involving lies.

Proposition

In a cheap-talk game with a common language, any non-trivial equilibrium type-action distribution can be induced by an equilibrium in which each agent’s message is a lie.

Proposition

In a simple cheap talk game, there is a positive probability of lying every equilibrium.

Proposition

In a simple cheap-talk game with a common language, every equilibrium type-action distribution can be supported as an equilibrium with only lies of omission.

Lying and Deception Sobel

Observations

• 1. Expect exaggeration.
• 2. Banning lies benefit both players.

Lying and Deception Sobel

Honest Equilibria

Honesty is typically incompatible with strategic behavior, but if there is no conflict of interest between the players, there is a possibility that the Sender will report honestly in equilibrium.

• 1. yR(θ, m) be a solution to max UR(θ, y, m);
• 2. let m(θ) solve:

max US(θ, yR(θ, m), m). (1)

• 3. When m(·) is single valued and one-to-one, the game is

potentially revealing.

Proposition

In any potentially revealing game in which US = UR, there exists a specification of language under which an honest equilibrium exists.

Lying and Deception Sobel

Beliefs

Each m′ induces a posterior distribution µ(θ | m′).

Definition (Properties of Beliefs)

• 1. The belief µ(· | m′) is completely inaccurate given m′ and θ

if P(θ) = 0 for all θ such that µR(θ | m′) > 0.

• 2. The belief µ(· | m′) is inaccurate given m′ and θ if P(θ) = 0

for some θ such that µR(θ | m′) > 0.

• 3. The belief µ(· | m′) is accurate given m′ and θ if

µR(θ | m′) = 1.

• 4. Given the mixed strategy σ(·) of S, the belief µ(· | m′) is

rational given m′, θ, and σ(·) if it is derived from the prior and Sender’s strategy whenever possible.

Lying and Deception Sobel

Beliefs and Honesty

Definition

The Receiver is credulous if µR(· | mΘ0) is equal to the prior distribution conditional on θ ∈ Θ0. That is µR(θ | mΘ0) =

• if θ /

∈ Θ0

P(θ)

• θ′∈Θ0 P(θ′)

if θ ∈ Θ0.

Proposition

Given a communication game with a common noise-free language,

• 1. If m is a lie and R is credulous, then µR(· | m) is completely

inaccurate given m and θ.

• 2. If m is a lie of omission and R is credulous, then µR(· | m) is

inaccurate given m and θ.

• 3. If m is true and R is credulous, then µR(· | m) is accurate

given m and θ.

Lying and Deception Sobel

Deception

¯ uR(θ, m) is R’s expected utility.

Definition (Deception)

The message m is deceptive given θ and x if there exists a message n such that ¯ uR(θ, m) < ¯ uR(θ, n).

Proposition

There is no deception if the Receiver’s actions are independent of the message received.

Proposition

Suppose that the Receiver hears the Sender’s message perfectly. There is no deception in a separating equilibrium.

Lying and Deception Sobel

Deception in Equilibrium

Proposition

Suppose that the Sender and Receiver have identical preferences, there is no deception in equilibrium.

• 1. Deception is not inducing Non-Rational Beliefs.
• 2. Cheap talk. Two actions induced. S slightly above cutoff is

deceptive.

Lying and Deception Sobel

Deception and Beliefs

I(θ′ | θ) =

• if θ′ = θ

1 if θ′ = θ. .

Definition (Misleading Messages)

The message m is misleading given θ if there exists n and a p ∈ [0, 1) such that µR(· | n) = pµR(· | m) + (1 − p)I(· | θ). The message m is truly misleading given θ if there exists n and a p ∈ [0, 1) such that µR(· | n) = pµR(· | m) + (1 − p)I(· | θ) and the Receiver’s best response to m is not a best response to n.

Proposition

If m is truly misleading, then it is deceptive. If a message is deceptive independent of the specification of R’s preferences, then it is misleading.

Lying and Deception Sobel

Bluffs

Definition (Bluff)

Given a communication game, the message m is a bluff given θ if there exists m′ = m such that US(θ, yR(θ, m), m) < US(θ, yR(θ, m′), m′).

Lying and Deception Sobel

Properties

Proposition

There are no bluffs in cheap-talk games or in perfect information games.

• 1. Bluff: S’s preferences. Deceipt: R’s preferences.
• 2. Costly communication: Bluffs with Common Preferences. Not

needed with cheap talk.

• 3. Related: Separating Equilibria in Spence models are Bluffs.
• 4. No bluffs in zero-sum games with pure equilibria.
• 5. Bluffs in mixed equilibria.

Lying and Deception Sobel

Examples of Deception

• 1. Hendricks and McAfee feints.
• 2. Crawford level-k lying.
• 3. Ettinger and Jehiel, deception with analogy based equilibrium.
• 4. Kartik, Ottaviani, and Squintani costly lying.

Lying and Deception Sobel

How to Order Your Coffee

Americans mispronounce my co-author’s first name. He tells the Starbuck’s barista that his name is Steve. This is a lie. This is deceptive and a misrepresentation. This is not effectively deceptive.

Lying and Deception Sobel

Lying Not Deception

“Thank you, Mrs. Edna Mosh,” he wrapped up, “for your eyewitness account of this dramatic siege at the Hilarius Psychiatric Clinic. Thus is KCUF Mobile Two, sending it back now to ‘Rabbit’ Warren, at the studio.” He cut his power. Something was not quite right. “Edna Mosh?” Oedipa said. “It’ll come out the right way,” Mucho said. “I was allowing for the distortion on these rigs, and then when they put it on tape.”

Lying and Deception Sobel

Lying Not Deception

“Thank you, Mrs. Edna Mosh,” he wrapped up, “for your eyewitness account of this dramatic siege at the Hilarius Psychiatric Clinic. Thus is KCUF Mobile Two, sending it back now to ‘Rabbit’ Warren, at the studio.” He cut his power. Something was not quite right. “Edna Mosh?” Oedipa said. “It’ll come out the right way,” Mucho said. “I was allowing for the distortion on these rigs, and then when they put it on tape.” Pynchon, “The Crying of Lot 49”

Lying and Deception Sobel

Deception Not Lying

Two Jews met in a railway carriage at a station in

• Galicia. “Where are you going?” said one. “To Cracow,”

was the answer. “What a liar you are!” broke out the

• ther. “If you say you’re going to Cracow, you want me

to believe you’re going to Lemberg. But I know that in fact you’re going to Cracow. So why are you lying to me?” . . . Is it the truth if we describe things as they are without troubling to consider how our hearer will understand what we say? Or is this only jesuitical truth, and does not genuine truth consist in taking the hearer into account and giving him a faithful picture of our own knowledge.

Lying and Deception Sobel

Deception Not Lying

Two Jews met in a railway carriage at a station in

• Galicia. “Where are you going?” said one. “To Cracow,”

was the answer. “What a liar you are!” broke out the

• ther. “If you say you’re going to Cracow, you want me

to believe you’re going to Lemberg. But I know that in fact you’re going to Cracow. So why are you lying to me?” . . . Is it the truth if we describe things as they are without troubling to consider how our hearer will understand what we say? Or is this only jesuitical truth, and does not genuine truth consist in taking the hearer into account and giving him a faithful picture of our own knowledge. Freud, “Jokes and their Relation to the Unconscious”

Lying and Deception Sobel

Another Definition

Kartik, Ottaviani, and Squintani: view deception as the act of inducing false beliefs by means of communication, and exploiting them to one’s

• wn advantage. Such false beliefs are clearly incompatible

with traditional equilibrium analysis. Deception in this interpretation is distinct from the notion that a player may choose not to disclose private information in order to exploit the imprecise – but not incorrect – belief induced in a counterpart (e.g., bluffing in poker).

Lying and Deception Sobel

Comparison

• 1. I do not limit deception to communication.
• 2. KOS beliefs are false if and only if they are consistent with

prior information and equilibrium strategies.

• 3. My definition deception is disadvantageous to the Receiver,

but need not be advantageous to the Sender.

• 4. I do not view bluffing in poker as a decision not to disclose

private information.

• 5. KOS and I agree:

5.1 that bluffing depends on an informational asymmetry between the players; 5.2 that the Sender bluffs with the intention to exploit this asymmetry; 5.3 that bluffing can be compatible with equilibrium behavior.

Lying and Deception Sobel

Informal Summary

words, words, words

Lying and Deception Sobel

Informal Summary

words, words, words but shortly before he described his reading habits, Hamlet defined honesty: To be honest, as this world goes, is to be one man picked out of ten thousand.

Lying and Deception Sobel

Informal Summary

words, words, words but shortly before he described his reading habits, Hamlet defined honesty: To be honest, as this world goes, is to be one man picked out of ten thousand.

Lying and Deception Sobel

Summary

I give you:

• 1. Definition of Lying (requires language, but no preferences, no

empathy)

• 2. Definition of Misleading/Inaccurate (requires beliefs about

beliefs, no language, no preferences directly)

• 3. Definition of Deception (requires information about Receiver’s

preferences)

• 4. Definition of Bluff (requires information about Sender’s

preferences) With these:

◮ Some basic logical connections and properties ◮ An implicit faith that the the taxonomy is useful (details

another time)

Lying and Deception Sobel

Coda

Thank you

Lying and Deception Sobel

Coda

Thank you and Many Happy Returns

Lying and Deception Sobel