Dynamic Games of Incomplete Information

Christopher Makler

Stanford University Department of Economics

Econ 51: Lecture 14

pollev.com/chrismakler

What song is playing right now?

Poker Night: Next Week

• I'm in. So is Scott.
• Econ 50, 51, 102A
• Details TBA...stay tuned

What's on the test?

• One question on strategy spaces
• One question on Bayes Nash Equilibrium
• One question on Perfect Bayesian Equilibrium
  with pure strategies (separating/pooling)
• One question on Perfect Bayesian Equilibrium with mixed strategies

• Conditional Beliefs and Updating
• Perfect Bayesian Equilibrium
• Separating and Pooling Equilibria
• Equilibria in Mixed Strategies

Today's Agenda

Time

Static

(Simultaneous)

Dynamic

(Sequential)

Complete

Incomplete

WEEK 5

WEEK 6

LAST TIME

TODAY

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"

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.

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.

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.

Separating and Pooling Equilibria

Steps for calculating perfect Bayesian equilibria: Guess and Check!

1. Start with a strategy for player 1 (pooling or separating).
2. 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.
3. Given the updated beliefs, calculate player 2’s optimal action.
4. 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)

Mixed Strategies

Think about Carrot in a Box

Step 1: reveal if there's a carrot to player 1.

Step 2: player 1 claims there's a carrot or not.

Step 3: player 2 decides whether to switch 

Let's play a stupid game of poker.

Player 1 has one of these three cards.

Player 2 has this card.

🂮

🂫

🃛

🂽

Player 1 has one of these three cards.

Player 2 has this card.

🂮

🂫

🃛

🂽

Will player 1 ever fold a King?

🂮

🂫

🃛

NATURE

King (Prob \({1 \over 3}\))

Jack (Prob \({2 \over 3}\))

1

1

1

1

4

0

2

2

0

4

2

2

🂮

🂫

🃛

NATURE

King (Prob \({1 \over 3}\))

Jack (Prob \({2 \over 3}\))

1

1

1

1

4

0

2

2

0

4

2

2

Bid a King, and fold a Jack

Candidate strategy:

Intuition: if you only bid a King, player 2 will fold; and if player 2 is going to fold, you would have wanted to also bid the Jack.

🂮

🂫

🃛

NATURE

King (Prob \({1 \over 3}\))

Jack (Prob \({2 \over 3}\))

1

1

1

1

4

0

2

2

0

4

2

2

Always bid

Candidate strategy:

🂮

🂫

🃛

NATURE

King (Prob \({1 \over 3}\))

Jack (Prob \({2 \over 3}\))

1

1

1

1

4

0

2

2

0

4

2

2

So: no equilibrium exists with player 1 playing a pure strategy.

🂮

🂫

🃛

NATURE

1

1

1

1

4

0

2

2

0

4

2

2

So: no equilibrium exists with player 1 playing a pure strategy.

What about a mixed strategy?

Consider the following candidate strategy: player 1 bids with the Jack of clubs, but folds with the Jack of spades.

🂮

🂫

🃛

NATURE

1

1

1

1

4

0

2

2

0

4

2

2

Two potential sources of randomness
that determine beliefs:

• Moves of nature
• Mixed strategies

What's on the test?

• One question on strategy spaces
• One question on Bayes Nash Equilibrium
• One question on Perfect Bayesian Equilibrium
  with pure strategies (separating/pooling)
• One question on Perfect Bayesian Equilibrium with mixed strategies

Review Session

Sunday, 4-6pm in Shriram 104