Partially specified Probabilities: decisions and games May 2007 - - PowerPoint PPT Presentation

Partially specified probabilities

Partially specified Probabilities: decisions and games

May 2007

Ehud Lehrer

Partially specified probabilities

• Nash equilibrium implicitly assumes

that each player knows what other players do.

• But what if they don’t?

The problem

Partially specified probabilities

• A driver needs to go from A to B
• There are two ways
• Traffic conditions on the blue and the

red roads are known, but not on the black ones

• Based on this partial information the

driver needs to make a decision

Traffic Games

A B

Partially specified probabilities

• There are three states of nature: x, y and z
• A decision maker has two actions: a and b
• The payoffs stochastically depend on the action

and the state:

Noisy signals

(The entries are probability distributions over the payoffs, say 7 and 9)

• So far only a has been played and 50% of the time the payoff was 7
• Based on this information the decision maker needs to choose: a or b
• We can deduce: empirical frequency of x = empirical frequency of z
• Differently, the expectation of (1,x; 0,y; -1,z) = 0

Partially specified probabilities

An urn contains 30 red balls and 60 balls that are either white or black. A ball is randomly drawn from the urn and a decision maker is given a choice between two gambles:

Ellsberg urn

X: receive $100 if a red ball is drawn Y: receive $100 if a white ball is drawn The information available: P(R)=1/3 The decision maker is also given the choice between the following two gambles: Z: receive $100 if a red or black ball is drawn T: receive $100 if a white or black ball is drawn If you prefer X to Y and T to Z, you violate the sure thing principle

Partially specified probabilities

• First day: 30 red balls and 60 balls that are either white
• r black.
• White balls are actually Amebas (one-celled organisms)
• They multiply once a day
• Second day: 30 red balls and an unknown number of others
• The probability of drawing a red ball is unknown
• What do we know?

Dynamic Ellsberg urn

Partially specified probabilities

• First day: 30 red balls and 60 balls that are either white
• r black.
• White balls are actually Amebas (one-celled organisms)
• They multiply once a day
• Second day: 30 red balls and an unknown number of others
• The probability of drawing a red ball is unknown
• What do we know?

Dynamic Ellsberg urn

At the second day we know: the expectation of the variable (1,R; 1/6,W; 0,B) is 1/3.

The probability of no (non-trivial) event is known.

Partially specified probabilities

Definition: A partially-specified probability over S is a pair

Partially-specified probability (PSP)

where is a set of random variables defined over S that contains the indicator of S, is a probability distribution over S and The decision maker gets to know for every

Partially specified probabilities

PSP – Ellsberg urn

where is what we earlier denoted by

The decision maker gets to know that Recall: First day: 30 red balls and 60 balls that are either white

• r black. White balls multiply once a day.On the second day the PSP

is given by,

Partially specified probabilities

Recall Ellsberg urn in the first day. The PSP is given by

Decision making with PSP

Based on this information we need to choose between We know that What about Y? We use the best approximation using the variables in

R

.

Partially specified probabilities

Decision making with PSP

In general, define Decision problem: choose among Z and T Solution: choose the one with a greater integral This is the evaluation of R using the best approximation with known variables

R R

Partially specified probabilities

The dual method

R R R (R);

Partially specified probabilities

The dual method

Note,

R R R (R);

Partially specified probabilities

Decision making with PSP

Implication: evaluating R according to is equivalent to evaluating R according to the worst distribution consistent with the information available

R R

Partially specified probabilities

Decision making with PSP

• Gilboa and Schmeidler (1989)’s model refers to

the minimum over a general set of priors.

• Here, the emphasis is on the information available.

The set of prior consists of the distributions that are consistent with the available information.

• This is the first step out of vN-M expected utility

model.

• The current model maintains enough structure to

enable an information-based equilibrium analysis.

Partially specified probabilities

Decision making with PSP

The decision making with PSP is axiomatized in the setting of Anscombe and Aumann (1963).

Partially specified probabilities

Partially specified equilibrium

In a partially-specified equilibrium

• each player plays a pure or a mixed strategy
• each player obtains partial information about other players’ strategies
• each player maximizes her payoff against the worst strategy consistent

with her information.

Partially specified probabilities

Partially specified equilibrium

2/3 1/3

PSE eq.

Partially specified probabilities

Partially specified equilibrium

2/3

½ ½

1/3

PSE eq.

Partially specified probabilities

Partially specified equilibrium

2/3

½ ½

1/3

Notice: (2/3,1/3) is the unique best response for player 1. Neither T nor B are best responses. PSE eq.

Partially specified probabilities

Partially specified equilibrium

2/3

½ ½

1/3 2/3 1/3 2/3 1/3

Nash eq. PSE eq.

Partially specified probabilities

Partially specified equilibrium

1

1 0

Unique partially-specified equilibrium

Partially specified probabilities

Partially specified equilibrium (PSE)

Partially specified probabilities

Consistency with information

1/2 1/2 1/2 1/2

Partially specified probabilities

Best response

Partially specified probabilities

PSE - definition

Partially specified probabilities

Partially specified eq. - properties

• Always exists (for every

)

• Resistant to duplication of strategies
• Coincides with Nash eq. when strategies

are fully specified

Partially specified probabilities

Information-based

The notion of partially-specified equilibrium is information- based. The information structure, namely, the information available to each player, determines the set of equilibria. No player has a prior belief about other players' strategies. The belief a player has about the actual strategies played is determined by the actual strategies, as well as by the partial information a player obtains about them.

Partially specified probabilities

Other possible approaches

Responding to a distribution that maximizes uncertainty (e.g, entropy or another concave function).

Partially specified probabilities

Other possible approaches

Cons:

• Sensitive to duplications
• The set of independent distributions is not convex – might

create existence problems

Responding to a distribution that maximizes uncertainty (e.g, entropy or another concave function).

Partially specified probabilities

Other possible approaches

Optimistic approach: maximizing against the best distribution consistent with information

Cons:

• Lack of existence

Partially specified probabilities

Question

• Is the above the only possible definition,

which could guarantee existence, while not being sensitive to strategy duplications?

Partially specified probabilities

Partially-specified correlated equilibrium

Partially specified probabilities

Partially-specified correlated equilibrium This is a partially-specified correlated equilibrium.

Partially specified probabilities

Partially-specified correlated equilibrium

Partially specified probabilities

Learning to play PSCE

• What is it good for?
• Adaptive learning with imperfect monitoring
• The game is played repeatedly
• Each player observes a noisy signal, which depends on her
• wn strategy and on that of the others
• Each player plays a regret-free strategy (“I have no regret for

playing the mixed strategy p because this is the best I could do in response to the worst strategy of the others, consistent with the signals I received while playing p.”)

• The empirical frequency of the mixed strategies played

converges to PSCE.

Partially specified probabilities

Possible extensions

• Games with incomplete information: a state is randomly chosen,

and the resulting game is played. The probability according to which the state is selected might be partially specified

• The probability could be specified more vaguely, for instance, as an

interval that contains the expectation of each

• But then, how would the information in a game be defined? In other

words, how would you specify the information that one player has about the strategy actually been played by another?

• Hybrid equilibrium: players play independently but each player

reacts as if they are coordinated

Partially specified probabilities

• It is unrealistic to expect that each player knows all
• ther players’ strategies
• Players may obtain and utilize partial information about
• ther players’ strategies
• An interactive model with PSP is called for
• Taking an extreme pessimistic approach, I defined

two notions: partially-specified equilibrium and partially-specified correlated equilibrium

• I assume that there are other plausible approaches to the

problem of interactive decision making with a Partially-Specified Probability

Summary