Partially specified probabilities Partially specified Probabilities: decisions and games May 2007 Ehud Lehrer
The problem Partially specified probabilities • Nash equilibrium implicitly assumes that each player knows what other players do. • But what if they don’t?
Traffic Games Partially specified probabilities - A driver needs to go from A to B - There are two ways B - 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 A
Noisy signals 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: (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
Ellsberg urn 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: 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
Dynamic Ellsberg urn Partially specified probabilities - First day: 30 red balls and 60 balls that are either white or 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 or 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? 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 probability (PSP) Partially specified probabilities Definition: A partially-specified probability over S is a pair where is a probability distribution over S and is a set of random variables defined over S that contains the indicator of S, The decision maker gets to know for every
PSP – Ellsberg urn Partially specified probabilities Recall: First day: 30 red balls and 60 balls that are either white or black. White balls multiply once a day.On the second day the PSP is given by, where is what we earlier denoted by The decision maker gets to know that
Decision making with PSP Partially specified probabilities Recall Ellsberg urn in the first day. The PSP is given by Based on this information we need to choose between We know that . What about Y ? We use the best approximation using R the variables in
Decision making with PSP Partially specified probabilities In general, define R R This is the evaluation of R using the best approximation with known variables Decision problem: choose among Z and T Solution: choose the one with a greater integral
Partially specified probabilities R The dual method ( R ); R R
Partially specified probabilities R The dual method ( R ); Note, R R
Decision making with PSP Partially specified probabilities Implication: evaluating R according to R R is equivalent to evaluating R according to the worst distribution consistent with the information available
Decision making with PSP Partially specified probabilities - 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.
Decision making with PSP Partially specified probabilities The decision making with PSP is axiomatized in the setting of Anscombe and Aumann (1963).
Partially specified equilibrium Partially specified probabilities 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 equilibrium Partially specified probabilities PSE eq. 2/3 1/3
Partially specified equilibrium Partially specified probabilities ½ ½ PSE eq. 2/3 1/3
Partially specified equilibrium Partially specified probabilities ½ ½ PSE eq. 2/3 1/3 Notice: (2/3,1/3) is the unique best response for player 1. Neither T nor B are best responses.
Partially specified equilibrium Partially specified probabilities ½ ½ PSE eq. 2/3 2/3 1/3 1/3 1/3 2/3 Nash eq.
Partially specified equilibrium Partially specified probabilities 1 0 1 0 Unique partially-specified equilibrium
Partially specified equilibrium (PSE) Partially specified probabilities
Consistency with information Partially specified probabilities 1/2 1/2 1/2 1/2
Best response Partially specified probabilities
PSE - definition Partially specified probabilities
Partially specified eq. - properties Partially specified probabilities - Always exists ( for every ) - Resistant to duplication of strategies - Coincides with Nash eq. when strategies are fully specified
Information-based Partially specified probabilities 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.
Other possible approaches Partially specified probabilities Responding to a distribution that maximizes uncertainty (e.g, entropy or another concave function).
Other possible approaches Partially specified probabilities Responding to a distribution that maximizes uncertainty (e.g, entropy or another concave function). Cons: - Sensitive to duplications - The set of independent distributions is not convex – might create existence problems
Other possible approaches Partially specified probabilities Optimistic approach: maximizing against the best distribution consistent with information Cons: - Lack of existence
Question Partially specified probabilities - Is the above the only possible definition, which could guarantee existence, while not being sensitive to strategy duplications?
Partially-specified correlated equilibrium Partially specified probabilities
Partially-specified correlated equilibrium Partially specified probabilities This is a partially-specified correlated equilibrium.
Partially-specified correlated equilibrium Partially specified probabilities
Learning to play PSCE Partially specified probabilities - 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 own 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.
Possible extensions Partially specified probabilities - 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
Summary Partially specified probabilities - It is unrealistic to expect that each player knows all other players’ strategies - Players may obtain and utilize partial information about other 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
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