Incomplete Information Econ 400 University of Notre Dame Econ 400 - - PowerPoint PPT Presentation

Incomplete Information

Econ 400

University of Notre Dame

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Games of Incomplete Information

In game theory, there are two sources of uncertainty related to information:

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Games of Incomplete Information

In game theory, there are two sources of uncertainty related to information: Uncertainty about the preferences or capabilities of an opponent (Incomplete Information)

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Games of Incomplete Information

In game theory, there are two sources of uncertainty related to information: Uncertainty about the preferences or capabilities of an opponent (Incomplete Information) Uncertainty about the previous actions of other players (Imperfect Information)

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Games of Incomplete Information

In game theory, there are two sources of uncertainty related to information: Uncertainty about the preferences or capabilities of an opponent (Incomplete Information) Uncertainty about the previous actions of other players (Imperfect Information) The first kind of uncertainty can actually be converted into the second, actually.

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A Simple Game with Incomplete Information

There are two players with two strategies each, S or C. However, the payoff matrix is c S C r S 1,1

• 1,xc

C xr, −1 0,0 where xr is known only to player r, and xc is known only to player c. The value xr is player r’s type, and the value xc is player c’s type.

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A Simple Game with Incomplete Information: Types and Beliefs

The players’ types follow this distribution: With probability p, xr = 0, and with probability 1 − p, xr = 2 With probability p, xc = 0, and with probability 1 − p, xc = 2

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A Simple Game with Incomplete Information: Types and Beliefs

The players’ types follow this distribution: With probability p, xr = 0, and with probability 1 − p, xr = 2 With probability p, xc = 0, and with probability 1 − p, xc = 2 But because the players’ types are random and unknown to each other, they are uncertain about “who” their opponent is. However, they have the same beliefs about frequencies of opponent types, like knowing the composition of the deck of cards from which hands are dealt.

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A Simple Game with Incomplete Information: Payoffs

There are actually four games that might be going on: c S C r S 1,1

• 1,0

C 0,-1 0,0

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A Simple Game with Incomplete Information: Payoffs

There are actually four games that might be going on: c S C r S 1,1

• 1,0

C 0,-1 0,0 c S C r S 1,1

• 1,2

C 2,-1 0,0

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A Simple Game with Incomplete Information: Payoffs

There are actually four games that might be going on: c S C r S 1,1

• 1,0

C 0,-1 0,0 c S C r S 1,1

• 1,2

C 2,-1 0,0 c S C r S 1,1

• 1,2

C 0,-1 0,0

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A Simple Game with Incomplete Information: Payoffs

There are actually four games that might be going on: c S C r S 1,1

• 1,0

C 0,-1 0,0 c S C r S 1,1

• 1,2

C 2,-1 0,0 c S C r S 1,1

• 1,2

C 0,-1 0,0 c S C r S 1,1

• 1,0

C 2,-1 0,0 The players just aren’t sure which one they’re actually in.

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A Simple Game with Incomplete Information: Strategies

What are strategies? A strategy for the row player is a rule saying what the row player should do — S or C — for each type xr = 0 or xr = 2. There are four possibilities: (0,2) → (S, S) (0,2) → (S, C) (0,2) → (C, S) (0,2) → (C, C)

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A Simple Game with Incomplete Information: Strategies

What are strategies? A strategy for the row player is a rule saying what the row player should do — S or C — for each type xr = 0 or xr = 2. There are four possibilities: (0,2) → (S, S) (0,2) → (S, C) (0,2) → (C, S) (0,2) → (C, C) Similarly, the column player has four potential strategies: (0,2) → (S, S) (0,2) → (S, C) (0,2) → (C, S) (0,2) → (C, C)

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Expected Payoffs

Note that if the column player’s strategy is, say, (0, 2) → (C, S), then the probability that column uses C is p, and the probability column uses S is 1 − p. This means that we can compute expected payoffs by using the rule that corresponds to each type.

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Expected Payoffs

Note that if the column player’s strategy is, say, (0, 2) → (C, S), then the probability that column uses C is p, and the probability column uses S is 1 − p. This means that we can compute expected payoffs by using the rule that corresponds to each type. For example, if column uses (0, 2) → (C, S), then row’s payoff is purow(σrow, C, trow, 0) + (1 − p)urow(σrow, S, trow, 2)

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Expected Payoffs

Note that if the column player’s strategy is, say, (0, 2) → (C, S), then the probability that column uses C is p, and the probability column uses S is 1 − p. This means that we can compute expected payoffs by using the rule that corresponds to each type. For example, if column uses (0, 2) → (C, S), then row’s payoff is purow(σrow, C, trow, 0) + (1 − p)urow(σrow, S, trow, 2)

• r if column uses (0, 2) → (C, C), then row’s payoff is

purow(σrow, C, trow, 0) + (1 − p)urow(σrow, C, trow, 2)

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Equilibrium

A set of strategies for each player-type is a Bayesian Nash equilibrium if no player-type can deviate and get a higher expected payoff. Or, if we make a table, Type Strategy t11 s11 . . . . . . t22 s21 . . . . . . tNK sNK So player i’s k-th type, tik, row ik is assigned a strategy sik. A table like the one above is an equilibrium if no player-type tik can choose a strategy s′ that gives a strictly higher payoff than sik.

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Equilibrium 1: Always confess

Suppose Type Strategy xr = 0 C xr = 2 C xc = 0 C xc = 2 C so everyone confesses no matter what. Is this a Bayesian Nash equilibrium?

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Equilibrium 1: Always confess

We check that no player-type wants to deviate (for each row in the table, playing as suggested is better than switching to something else): Does the 0-type row player want to deviate?

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Equilibrium 1: Always confess

We check that no player-type wants to deviate (for each row in the table, playing as suggested is better than switching to something else): Does the 0-type row player want to deviate? The expected payoff from confessing (assuming all other types do what it says in the table) is p(0) + (1 − p)(0) = 0 The expected payoff from deviating and remaining silent (assuming all the other types do as the table says) is p(−1) + (1 − p)(−1) = −1

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Equilibrium 1: Always confess

We check that no player-type wants to deviate (for each row in the table, playing as suggested is better than switching to something else): Does the 0-type row player want to deviate? The expected payoff from confessing (assuming all other types do what it says in the table) is p(0) + (1 − p)(0) = 0 The expected payoff from deviating and remaining silent (assuming all the other types do as the table says) is p(−1) + (1 − p)(−1) = −1 So the 0-type row player doesn’t want to deviate.

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Equilibrium 1: Always confess

Does the 2-type row player want to deviate? The expected payoff from confessing (assuming all the other types do as the table says) is pur(C, C, 2) + (1 − p)ur(C, C, 2) = 0 The expected payoff from deviating to remaining silent (assuming all the other types do as the table says) is p(−1) + (1 − p)(−1) = −1

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Equilibrium 1: Always confess

Does the 2-type row player want to deviate? The expected payoff from confessing (assuming all the other types do as the table says) is pur(C, C, 2) + (1 − p)ur(C, C, 2) = 0 The expected payoff from deviating to remaining silent (assuming all the other types do as the table says) is p(−1) + (1 − p)(−1) = −1 So the 2-type row player doesn’t want to deviate either.

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Equilibrium 1: Always confess

Does the 2-type row player want to deviate? The expected payoff from confessing (assuming all the other types do as the table says) is pur(C, C, 2) + (1 − p)ur(C, C, 2) = 0 The expected payoff from deviating to remaining silent (assuming all the other types do as the table says) is p(−1) + (1 − p)(−1) = −1 So the 2-type row player doesn’t want to deviate either. These calculations are the same for the column player, so “always confess” is an equilibrium: If all your opponent’s player-types are confessing, you can’t do any better than confess yourself, no matter what your player-type is.

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Equilibrium 2: Take a chance on cooperation

Suppose Type Strategy xr = 0 C xr = 2 S xc = 0 C xc = 2 S so that players try to cooperate when they get a low payoff to confessing (0), but confess when they get a high payoff to confessing (2).

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Equilibrium 2: Take a chance on cooperation

Suppose Type Strategy xr = 0 C xr = 2 S xc = 0 C xc = 2 S so that players try to cooperate when they get a low payoff to confessing (0), but confess when they get a high payoff to confessing (2). Is this a Bayesian Nash equilibrium?

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Equilibrium 2: Take a chance on cooperation

When a player draws a 2, the payoff to confessing (the proposed strategy) is p(2) + (1 − p)0 for both players. The payoff to remaining silent is p(1) + (1 − p)(−1). Since 2p > 2p − 1, neither player wants to remain silent when their type is 2

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Equilibrium 2: Take a chance on cooperation

When a player draws a 2, the payoff to confessing (the proposed strategy) is p(2) + (1 − p)0 for both players. The payoff to remaining silent is p(1) + (1 − p)(−1). Since 2p > 2p − 1, neither player wants to remain silent when their type is 2 When a player draws a 0, the payoff to remaining silent (the proposed strategy) is p(1) + (1 − p)(−1). The payoff to confessing is p(0) + (1 − p)(0) = 0. Then remaining silent is better than confessing if p − (1 − p) ≥ 0

• r

p ≥ 1 2

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Equilibrium 2: Take a chance on cooperation

When a player draws a 2, the payoff to confessing (the proposed strategy) is p(2) + (1 − p)0 for both players. The payoff to remaining silent is p(1) + (1 − p)(−1). Since 2p > 2p − 1, neither player wants to remain silent when their type is 2 When a player draws a 0, the payoff to remaining silent (the proposed strategy) is p(1) + (1 − p)(−1). The payoff to confessing is p(0) + (1 − p)(0) = 0. Then remaining silent is better than confessing if p − (1 − p) ≥ 0

• r

p ≥ 1 2 So the proposed strategies are an equilibrium when p ≥ 1/2.

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What are “types”, really?

We use the word “type” to refer to a player’s “pocket cards”, or the private information they have. What are examples of types from games that people commonly study? The best example are pocket cards in poker: Each player is dealt some cards, but no one else gets to see them.

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What are “types”, really?

We use the word “type” to refer to a player’s “pocket cards”, or the private information they have. What are examples of types from games that people commonly study? The best example are pocket cards in poker: Each player is dealt some cards, but no one else gets to see them. In an auction, a bidder usually knows their “true” value privately, while the other players only have an estimate of another player’s value.

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What are “types”, really?

We use the word “type” to refer to a player’s “pocket cards”, or the private information they have. What are examples of types from games that people commonly study? The best example are pocket cards in poker: Each player is dealt some cards, but no one else gets to see them. In an auction, a bidder usually knows their “true” value privately, while the other players only have an estimate of another player’s value. When a customer goes to a bank to get a loan, the customer privately knows whether or not they are a good credit risk, but the bank can

• nly see things like income, outstanding debts, education, etc.

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What are “types”, really?

We use the word “type” to refer to a player’s “pocket cards”, or the private information they have. What are examples of types from games that people commonly study? The best example are pocket cards in poker: Each player is dealt some cards, but no one else gets to see them. In an auction, a bidder usually knows their “true” value privately, while the other players only have an estimate of another player’s value. When a customer goes to a bank to get a loan, the customer privately knows whether or not they are a good credit risk, but the bank can

• nly see things like income, outstanding debts, education, etc.

Insurance companies give people policies in case of accidents or bad health conditions, but people privately know whether their family has a history of cancer or whether they go rock climbing without ropes.

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What are “types”, really?

When a mechanic quotes you a price for a repair to your car, the mechanic has already examined the car, but you typically have no idea whether something is wrong or not.

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What are “types”, really?

When a mechanic quotes you a price for a repair to your car, the mechanic has already examined the car, but you typically have no idea whether something is wrong or not. Your future employer can observe where you went to school, what you studied, etc., but cannot observe whether or not you are actually a good potential employee (especially in a world where grades are all top-coded to “A”).

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What are “types”, really?

When a mechanic quotes you a price for a repair to your car, the mechanic has already examined the car, but you typically have no idea whether something is wrong or not. Your future employer can observe where you went to school, what you studied, etc., but cannot observe whether or not you are actually a good potential employee (especially in a world where grades are all top-coded to “A”). Originally, types were meant to be more complicated... but game theorists and economists have largely forgotten that debate.

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Games with Incomplete Information

What are the important ingredients of the games we’ve looked at so far? A Bayesian Game is

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Games with Incomplete Information

What are the important ingredients of the games we’ve looked at so far? A Bayesian Game is There’s a set of players (row and column) Each player has a set of types (0 or 2, cards drawn from a deck) Each player has a set of feasible actions (S or C, raise or fold) Even though each player’s type is known only to him, the types are distributed according to a commonly known distribution (p and 1 − p, the frequency of cards of different types) The players have payoffs functions, which depend on their own, privately known type and the actions of all the players (the four payoff matrices)

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Updating Beliefs and Bayes’ Rule

This means that, just like in the poker game, knowing a player’s type might give you information about the types of the other players.

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Updating Beliefs and Bayes’ Rule

This means that, just like in the poker game, knowing a player’s type might give you information about the types of the other players. In particular, we just use Bayes’ rule on the common prior: Player-type ti’s beliefs about the types of his opponents t−i are given by pr[t−i|ti] = pr(ti ∩ t−i) pr(ti) This is like saying, “Given that I have drawn four Aces (ti), what is the probability that my opponents have draw a flush or full house (t−i)?”

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Strategies and Expected Payoffs in Bayesian Games

Definition

A strategy in Bayesian Games is a mapping from a player’s types to his strategies, si(ti), that says what strategy he should pick for each of his possible types. (For each card hand I draw, should I raise or fold?)

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Strategies and Expected Payoffs in Bayesian Games

Definition

A strategy in Bayesian Games is a mapping from a player’s types to his strategies, si(ti), that says what strategy he should pick for each of his possible types. (For each card hand I draw, should I raise or fold?) A strategy profile is a list of strategies for all of the player-types, s(t) = (s1(t1), s2(t2), ..., sN(tN)). (For each player and each card they draw, what should their strategy be? This is the “table” assigning a strategy to each type.)

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Strategies and Expected Payoffs in Bayesian Games

Definition

A strategy in Bayesian Games is a mapping from a player’s types to his strategies, si(ti), that says what strategy he should pick for each of his possible types. (For each card hand I draw, should I raise or fold?) A strategy profile is a list of strategies for all of the player-types, s(t) = (s1(t1), s2(t2), ..., sN(tN)). (For each player and each card they draw, what should their strategy be? This is the “table” assigning a strategy to each type.)

Definition

Player-type ti’s expected utility of playing s′

i, given knowledge of their type

ti and the strategy profile adopted by the other players s−i(t−i), is Ui(s′

i, s−i(t−i), ti) = Et−i[ui(s′ i , s−i(t−i), ti, t−i)|ti]

where the uncertainty is about the other players’ types t−i.

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Bayesian Nash Equilibrium

Definition

A strategy profile s∗(t) is a Bayesian Nash equilibrium if, for every player-type ti and any alternative strategy s′

i that player-type ti could use,

Ui(s∗

i (ti), s∗ −i(t−i), ti, t−i) ≥ Ui(s′ i , s∗ −i(t−i), ti, t−i)

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Bayesian Nash Equilibrium

Definition

A strategy profile s∗(t) is a Bayesian Nash equilibrium if, for every player-type ti and any alternative strategy s′

i that player-type ti could use,

Ui(s∗

i (ti), s∗ −i(t−i), ti, t−i) ≥ Ui(s′ i , s∗ −i(t−i), ti, t−i)

Theorem

In games with a finite number of player-types and a finite number of strategies, Bayesian Nash Equilibria always exist (potentially in mixed strategies).

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Cournot with Incomplete Information

There is an incumbent firm with known marginal costs c. There is an entrant, who is either high cost cH > c with probability p, or low cost cL < c with probability 1 − p. The firms compete in quantities, with the incumbent choosing qi, and the entrant choosing qe. The market price is p = A − qi − qe.

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Cournot with Incomplete Information

There is an incumbent firm with known marginal costs c. There is an entrant, who is either high cost cH > c with probability p, or low cost cL < c with probability 1 − p. The firms compete in quantities, with the incumbent choosing qi, and the entrant choosing qe. The market price is p = A − qi − qe. What is the Bayesian Nash equilibrium of the game?

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Cournot: Firm Types and Strategies

The incumbent has a single type, since he has no private information. So he has a single strategy, qi. The entrant has two player-types: H and L. Then the entrant has two player-strategies, qH and qL.

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Cournot: Expected Payoffs

The incumbent’s expected payoff is E[πi] = p(A − qi − qH)qi + (1 − p)(A − qi − qL)qi − cqi

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Cournot: Expected Payoffs

The incumbent’s expected payoff is E[πi] = p(A − qi − qH)qi + (1 − p)(A − qi − qL)qi − cqi The high-type entrant’s expected payoff is E[πe|H] = (A − qi − qH)qH − cHqH

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Cournot: Expected Payoffs

The incumbent’s expected payoff is E[πi] = p(A − qi − qH)qi + (1 − p)(A − qi − qL)qi − cqi The high-type entrant’s expected payoff is E[πe|H] = (A − qi − qH)qH − cHqH The low-type entrant’s expected payoff is E[πe|L] = (A − qi − qL)qL − cLqL

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Cournot: Bayesian Nash Equilibrium

Definition

A set of strategies (q∗

i , q∗ H, q∗ L) is a Bayesian Nash equilibrium if neither the

incumbent nor the two entrant types can deviate to some other quantity and get a higher expected payoff, given the strategies of the other players. How do we solve for this...? Well, we maximize each player-type’s expected utility, taking the other player-types’ strategies as given, and look for a solution. (Algebra in the handout version).

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Equilibrium in the Cournot Game

q∗

i = A − 2c + pcH + (1 − p)cL

3 q∗

H = 2A + 2c − 3cH − (pcH + (1 − p)cL)

6 q∗

L = 2A + 2c − 3cL − (pcH + (1 − p)cL)

6

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Equilibrium in the Cournot Game

q∗

i = A − 2c + pcH + (1 − p)cL

3 q∗

H = 2A + 2c − 3cH − (pcH + (1 − p)cL)

6 q∗

L = 2A + 2c − 3cL − (pcH + (1 − p)cL)

6 Notice how cL appears in q∗

H, and cH appears in q∗ L — each entrant-type’s

strategy depends on the other entrant-type’s marginal cost. Why?

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Solving for Bayesian Nash Equilibria

Figure out all of the player-types. For each player-type, write out the strategies they could choose.

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Solving for Bayesian Nash Equilibria

Figure out all of the player-types. For each player-type, write out the strategies they could choose. Write down the expected payoff for each player-type, with the correct beliefs about opponent player-types.

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Solving for Bayesian Nash Equilibria

Figure out all of the player-types. For each player-type, write out the strategies they could choose. Write down the expected payoff for each player-type, with the correct beliefs about opponent player-types. Look for a strategy profile in player-types from which no player-type has an incentive to deviate. (Easier said than done)

Econ 400 (ND) Incomplete Information 25 / 25

Solving for Bayesian Nash Equilibria

Figure out all of the player-types. For each player-type, write out the strategies they could choose. Write down the expected payoff for each player-type, with the correct beliefs about opponent player-types. Look for a strategy profile in player-types from which no player-type has an incentive to deviate. (Easier said than done) Since I usually tell you to check whether a particular set of strategies is a Bayesian Nash equilibrium or not, you don’t have to worry about finding them from scratch (which can be tedious if you don’t know what you’re looking for).

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