Uncovering Others'
Private Information

Christopher Makler

Stanford University Department of Economics

Econ 51: Lecture 13

pollev.com/chrismakler

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Poker Night: Next Week

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• Wednesday, November 18th, 7–9 p.m.
• Lathrop 282
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• 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:

Conditional Probability

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?   

Text

Consider this game:

• Nature chooses U or D
• Player 1 observes Nature's choice,
  and chooses U or D.
• Player 2 can only observe if (U,U) was played; otherwise they're at an information set,
  and don't know which node of that set.
• Denote their beliefs as \(\alpha,\beta,\gamma\).
• What are their beliefs, for different P1 strategies?

Text

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”)

Text

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”)

Text

Are you James Talarico?

• Nature chooses whether player 1 is Talarico or Bot
• Player 1 observes Nature's choice,
  and chooses Send text (S) or Don't (D).
• You (Player 2) can only observe if you got a text.
• Let \(q\) be the probability with which you believe the text came from Talarico, conditional on receiving a text.

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.

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