# Dynamic Games of Incomplete Information

Christopher Makler

Stanford University Department of Economics

Econ 51: Lecture 15

- Beliefs and Updating
- Perfect Bayesian Equilibrium
- Job Market Signaling

# Today's Agenda

- Complete information: everyone knows everything about the game
- Incomplete information: players have private information

(e.g., they know what their valuation of a good is)

- Incomplete information: players have private information
- Perfect information: everyone can perfectly observe everyone else's moves
- Imperfect information: players' moves are hidden from other players

(e.g. I cannot observe how much time you spend studying)

- Imperfect information: players' moves are hidden from other players

# Information

# Time

# Information

Static

(Simultaneous)

Dynamic

(Sequential)

Complete

Incomplete

WEEKS 5 & 6

WEEK 7

LAST TIME

TODAY

Prisoners' Dilemma

Cournot

Entry Deterrence

Stackelberg

Auctions

Job Market Signaling

Collusion

Cournot with Private Information

Poker

# Time

# Information

Static

(Simultaneous)

Dynamic

(Sequential)

Complete

Incomplete

**Strategy**: an **action**

**Equilbirium**: Nash Equilibrium

**Strategy**: a mapping from the history of the game onto an action.

**Equilibrium**: Subgame Perfect NE

Strategy: a **plan of action **that

specifies what to do after every possible history of the game, based on one's own private information and (updating) beliefs about other players' private information.

Equilibrium: Perfect Bayesian Equilibrium

**Strategy**: a mapping from one's private information onto an action.

**Equilibrium**: Bayesian NE

WEEKS 5 & 6

WEEK 7

LAST TIME

TODAY

Prisoners' Dilemma

Cournot

Entry Deterrence

Stackelberg

Auctions

Job Market Signaling

Collusion

Cournot with Private Information

Poker

# Bayes' Rule and Conditional Beliefs

Suppose you don't know whether it's raining out,

but you can observe whether

I'm carrying an umbrella or not.

Ex ante, you believe the **joint probabilities**

of these events are given by this table:

Bayes' Rule:

Before you see whether I'm carrying an umbrella, with what probability do you believe it's raining?

# Bayes' Rule and Conditional Beliefs

Suppose you don't know whether it's raining out,

but you can observe whether

I'm carrying an umbrella or not.

Ex ante, you believe the **joint probabilities**

of these events are given by this table:

Bayes' Rule:

Suppose you see me with an umbrella. Now with what probability do you think it's raining?

## Conditional Beliefs and Strategies

- Suppose you're at an information set; you don't know which of several nodes you might be at.
- How can you use what you know about other players' strategies to form beliefs about the true state of the world?

# Perfect Bayesian Equilibrium

Consider a **strategy profile **for the players, as well as **beliefs **over the nodes at all information sets.

These are called a **perfect Bayesian Equilibrium (PBE) **if:

- Each player’s strategy is optimal to them at each infoset, given beliefs at this infoset and opponents’ strategies (“Sequential Rationality”)
- The beliefs are obtained from strategies using Bayes’ Rule wherever possible (i.e. at each infoset that is reached with a positive probability) (“consistency of beliefs”)

"Gift Giving Game"

Nature determines whether player 1 is a "friend" or "enemy" to player 2.

Player 1, knowing their type, can decide to give a gift to player 2 or not.

If player 1 gives a gift, player 2 can choose to accept it or not. Player 2 wants to accept a gift from a friend, but not from an enemy.

Whenever a player reaches an information set, they have *some* updated beliefs over which node they are.

Based on these *beliefs*, they should choose the action that maximizes their expected payoff.

"Gift Giving Game"

Nature determines whether player 1 is a "friend" or "enemy" to player 2.

Player 1, knowing their type, can decide to give a gift to player 2 or not.

If player 1 gives a gift, player 2 can choose to accept it or not. Player 2 wants to accept a gift from a friend, but not from an enemy.

"Gift Giving Game"

Nature determines whether player 1 is a "friend" or "enemy" to player 2.

Player 1, knowing their type, can decide to give a gift to player 2 or not.

If player 1 gives a gift, player 2 can choose to accept it or not. Player 2 wants to accept a gift from a friend, but not from an enemy.

In equilibrium, players' beliefs should be **consistent** with the strategies being played.

What is \(q\) if player 1 plays \(G^FN^E\)?

What is \(q\) if player 1 plays \(N^FG^E\)?

What is \(q\) if player 1 plays \(G^FG^E\)?

What is \(q\) if player 1 plays \(N^FN^E\)?

# Separating and Pooling Equilibria

**Separating Equilibrium**: Each type of informed player chooses differently,

thereby conveying information about their type to the uninformed player

**Pooling Equilibrium**: Each type of informed player chooses the same,

thereby leaving the uninformed player with their *prior* belief.

**Steps for calculating perfect Bayesian equilibria: Guess and Check!**

- Start with a strategy for player 1 (pooling or separating).
- If possible, calculate updated beliefs (
*q*in the example) by using Bayes’ rule.

*In the event that Bayes’ rule cannot be used, you must arbitrarily select an updated belief; here you will generally have to check different potential**values for the updated belief with the next steps of the procedure*. - Given the updated beliefs, calculate player 2’s optimal action.
- Check whether player 1’s strategy is a best response to player 2’s strategy.

If so, you have found a PBE.

Guided Exercise from Watson (p. 385)

There are two types of workers: "high-ability" and "low-ability."

High-ability workers

are worth \(y_H\) to a firm

Low-ability workers

are worth \(y_L\) to a firm

Assume both firms and high-ability workers would be better off if firms could observe their ability.

Need some mechanism to create a **separating equilibrium**.

# Job Market Signaling

#### Econ 51 | 15 | Dynamic Games of Incomplete Information

By Chris Makler

# Econ 51 | 15 | Dynamic Games of Incomplete Information

Perfect Bayesian Equilibrium and Signaling Models

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