Salience

Tarek Rached

Overview• David Gauthier proposes salience as a criterion for

equilibrium selection in his paper Coordination, published in in Dialogue, 14:195-221, 1975

• 13 years later, Margaret Gilbert challenges the validity of salience in her paper Rationality and Salience in Philosophical Studies, 57:61-77, 1989

• Both of these are written for a philosophical audience, although Nash and others are cited

Gauthier• Introduces the problem of coordination

– Successful coordination requires each player to maximize utility based on the other player’s actions

– Successful coordination will always result in an equilibrium

• Discusses the difference between explicit and tacit coordination– Restricts his arguments to “situations in which all [players] are

sufficiently informed about the circumstances” without allowing communication

• Discusses optimality– Notes that in games such as the Prisoner’s Dilemma, the optimal

outcome and the equilibirum outcome may not be the same.– Restricts his arguments to games in which they are the same

Gauthier (cont’d)• Omits any mention of mixed strategies, implicitly

restricting the paper to pure strategies• Proposes a Principle of Coordination

– “In a situation with one and only one outcome which is both optimal and a best equilibrium” the action leading to that outcome is the rational choice

• Seeks a Theory of Rational Coordination– By which in any situation, a single best action can be

determined

Multiple Equilibria:An Example

• Suppose you are coming from Richmond to Charlottesville on a bus, and I have agreed to meet you.

• Unfortunately, once you are on your way, I discover that there are two buses from Richmond, one of which arrives at the Corner, the other at the Downtown Mall. – Which bus do you take?– Where do I go to meet you?

Multiple Equilibria:An Example

D. Mall Corner

D. Mall 5, 5 0, 0

Corner 0, 0 5, 5

D. Mall Corner

D. Mall 5, 5 0, 0

Corner 0, 0 5, 5

• 2 identical equilibria as shown

– no way to choose between them

The Problem of Multiple Equilibria

D. Mall Corner

D. Mall 5, 5 0, 1

Corner 1, 0 5, 5

• Suppose we both know that if we don’t meet, we would both rather be at the Corner because you want to check out some records at Plan 9, and I like smoothies.

• Then going to the Corner becomes the salient choice for both of us

D. Mall Corner

D. Mall 5, 5 0, 1

Corner 1, 0 5, 5

The Problem of Multiple Equilibria

D. Mall Corner

D. Mall 5, 5 0, 0

Corner 0, 0 5, 5

• Suppose instead that we both independently check the bus schedules and find that the bus to the Downtown Mall runs more frequently

• Then going to the Downtown Mall becomes the salient choice for both of us

D. Mall Corner

D. Mall 5, 5 0, 0

Corner 0, 0 5, 5

Seek

Salience

Ignore

Salience

SS Meet 50%

IS 50% 50%

Seek

Salience

Ignore

Salience

SS 5, 5 2.5, 2.5

IS 2.5, 2.5 2.5, 2.5

Seek

Salience

Ignore

Salience

SS 5, 5 2.5, 2.5

IS 2.5, 2.5 2.5, 2.5

Expected Utility

What is Salience?

• The salient choice is the choice “which is apprehended as as standing out from the others.”

• How do we determine which choice stands out from the others?– External information (as seen) – Structure of the game

• Let’s look at some more examples

A New Game

Here, the salient solution is clear

(It’s also the best equilibrium)

A B C

A 3, 3 0, 0 0, 0

B 0, 0 2, 2 0, 0

C 0, 0 0, 0 2, 2

A B C

A 3, 3 0, 0 0, 0

B 0, 0 2, 2 0, 0

C 0, 0 0, 0 2, 2

A New Game, cont’d

A B C

A 3, 3 0, 0 0, 0

B 0, 0 2, 2 0, 0

C 0, 0 0, 0 3, 3

A B C

A 3, 3 0, 0 0, 0

B 0, 0 2, 2 0, 0

C 0, 0 0, 0 3, 3

Seek

Salience

Ignore

Salience

SS 2, 2 2/3, 2/3

IS 2/3, 2/3 8/9, 8/9

• Whereas here, it is not obvious– Gauthier claims you would “ignore salience” by

randomizing over the 3 possibilities, I would say you would randomize over the 2 best equilibra

– Seeking salience still a best equilbria in the right table

Seek

Salience

Ignore

Salience

SS 2, 2 2/3, 2/3

IS 2/3, 2/3 8/9, 8/9

Expected Utility

What’s Going On?

• Gauthier seeks to use salience as the basis for his Theory of Rational Coordination

• Unfortunately…• “The apprehension of salience is

itself not, or at least not only, a rational apprehension.” - Gauthier

Gauthier (cont’d)• Seeks to apply his theory of salience to the

philisophical realm of act-utilitarianism– All persons in society are rational maximizers

of individual utility– D.H. Hodgson writes that promise keeping and

truth selling cannot be enforced in such a society

– Gauthier applies salience to argue that they are, in fact, encouraged

On to Gilbert’s Counterexample: Kidnapped

• 2 players are kidnapped and separated

• Kidnapper gives them each a box with 4 buttons, each of a different color.

• Tells the captives that in 10 minutes, each will have to press a button. If they both press the same color button, they are free.

• Otherwise, they die.

Kidnapped (cont’d)

• Clearly, 4 best equilibria

R G B Y

R 1 0 0 0

G 0 1 0 0

B 0 0 1 0

Y 0 0 0 1

R G B Y

R 1 0 0 0

G 0 1 0 0

B 0 0 1 0

Y 0 0 0 1

Kidnapped (cont’d)

• Suppose that after the kidnapper told them about the buttons and their impending decision, he then tell them that he will turn on the radio.

• The radio comes on, and it’s a sports broadcast about the Cincinnati Reds. The announcer comments on their nice new red uniforms, the red bleachers, and the red ketchup he just spilled on himself.

• What then?

Kidnapped (cont’d)

• Using salience, we are led to R-R

• Fair enough

R G B Y

R 1 0 0 0

G 0 1 0 0

B 0 0 1 0

Y 0 0 0 1

Seek

Salience

Ignore

Salience

SS 1, 1 1/4, 1/4

IS 1/4, 1/4 1/4, 1/4

Seek

Salience

Ignore

Salience

SS 1, 1 1/4, 1/4

IS 1/4, 1/4 1/4, 1/4

Expected UtilityR G B Y

R 1 0 0 0

G 0 1 0 0

B 0 0 1 0

Y 0 0 0 1

Kidnapped (cont’d)

• Suppose that after the bit on the Cincinnati Reds, the announcer breaks in with a quick news flash about the kidnapping, noting that one of the captives is notoriously salience-shy.

• Red is still the salient choice, but it would now be irrational for the captives to choose it.

Kidnapped (cont’d)

• The extreme counter-example– Instead of the sports show, all the captives hear

is a news bulletin about the kidnapping, which prominently mentions the fact that both captives hate the color red.

• Red is still the salient choice, as it is distinct from the others, but it doesn’t make any sense to choose it.

• Salience fails because it provides no rational criteria for “standing out”

Our previous example

• We can claim that the Downtown Mall is salient by virtue of its “standing out” as the inferior choice.

• Suppose we both know that if we don’t meet, we would both rather be at the Corner because you want to check out some records at Plan 9, and I like smoothies.

D. Mall Corner

D. Mall 5, 5 0, 1

Corner 1, 0 5, 5

D. Mall Corner

D. Mall 5, 5 0, 1

Corner 1, 0 5, 5

The Bottom Line

• The perception of salience itself is a non-rational process

• Even having successfully perceived salience, it is not at all clear that it is the rational choice

• However, it can aid real (non-rational) players in games where knowledge of the world around them may be a factor

Discuss