Bayesian Nash Equilibrium

Carlos Hurtado

Department of EconomicsUniversity of Illinois at Urbana-Champaign

hrtdmrt2@illinois.edu

Jun 19th, 2015

C. Hurtado (UIUC - Economics) Game Theory

On the Agenda

1 Introduction

2 Private vs. Public Information

3 Bayesian game

4 How do we model Bayesian games?

5 Bayesian Nash equilibrium

6 Exercises

C. Hurtado (UIUC - Economics) Game Theory

Introduction

On the Agenda

1 Introduction

2 Private vs. Public Information

3 Bayesian game

4 How do we model Bayesian games?

5 Bayesian Nash equilibrium

6 Exercises

C. Hurtado (UIUC - Economics) Game Theory 1 / 21

Introduction

Introduction

We would like to understand what a game of incomplete information (Bayesiangame) is.First, we would like to differentiate private vs. public informationExample: Batle of Sex (BoS)In Sequential BoS, all information is public, meaning everyone can see all the sameinformation:

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Private vs. Public Information

On the Agenda

1 Introduction

2 Private vs. Public Information

3 Bayesian game

4 How do we model Bayesian games?

5 Bayesian Nash equilibrium

6 Exercises

C. Hurtado (UIUC - Economics) Game Theory 3 / 21

Private vs. Public Information

Private vs. Public Information

In this extensive-form representation of regular BoS, Player 2 cannot observe theaction chosen by Player 1.

When one player knows something that others do not, we call this private orasymmetric information.Complete and Imperfect Information.

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Private vs. Public Information

Private vs. Public Information

Example: BoS variantPlayer 1 is unsure whether Player 2 wants to go out with her or avoid her, andthinks that these two possibilities are equally likely. Player 2 knows Player 1’spreferences. So Player 1 thinks that with probability 1/2 she is playing the gameon the left and with probability 1/2 she is playing the game on the right.

This is an example of a game in which one player does not know the payoffs of theother.

C. Hurtado (UIUC - Economics) Game Theory 5 / 21

Bayesian game

On the Agenda

1 Introduction

2 Private vs. Public Information

3 Bayesian game

4 How do we model Bayesian games?

5 Bayesian Nash equilibrium

6 Exercises

C. Hurtado (UIUC - Economics) Game Theory 6 / 21

Bayesian game

Bayesian game

More examples:- Bargaining over a surplus and you aren’t sure of the size- Buying a car of unsure quality- Job market: candidate is of unsure quality- Juries: unsure whether defendant is guilty- Auctions: sellers, buyers unsure of other buyers’ valuations

When some players do not know the payoffs of the others, a game is said to haveincomplete information. It’s also known as a Bayesian game.

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Bayesian game

Bayesian game

Example: First-price auction (game with incomplete information)1. I have a copy of the Mona Lisa that I want to sell for cash2. Each of you has a private valuation for the painting, only known to you3. I will auction it off to the highest bidder4. Everyone submits a bid (sealed → simultaneous)5. Highest bidder wins the painting, pays their bid6. If tie, I will flip a coin

C. Hurtado (UIUC - Economics) Game Theory 8 / 21

Bayesian game

Bayesian game

Example: Second-price auction (game with incomplete information)1. I have a copy of the Mona Lisa that I want to sell for cash2. Each of you has a private valuation for the painting, only known to you3. I will auction it off to the highest bidder4. Everyone submits a bid (sealed → simultaneous)5. Highest bidder wins the painting, pays the second-highest bid6. If tie, I will flip a coin

C. Hurtado (UIUC - Economics) Game Theory 9 / 21

How do we model Bayesian games?

On the Agenda

1 Introduction

2 Private vs. Public Information

3 Bayesian game

4 How do we model Bayesian games?

5 Bayesian Nash equilibrium

6 Exercises

C. Hurtado (UIUC - Economics) Game Theory 10 / 21

How do we model Bayesian games?

How do we model Bayesian games?

One approach:- Random events are considered an act of nature (that determine game structure)- Treat nature as another (non-strategic) player- Draw nature’s decision nodes in extensive form

Treat game as extensive form game with imperfect info: players may/may notobserve nature’s action

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How do we model Bayesian games?

How do we model Bayesian games?

Recall: BoS variantPlayer 1 is unsure whether Player 2 wants to go out with her or avoid her, andthinks that these two possibilities are equally likely. Player 2 knows Player 1’spreferences. So Player 1 thinks that with probability 1/2 she is playing the gameon the left and with probability 1/2 she is playing the game on the right.

Let’s put this into extensive form.

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How do we model Bayesian games?

How do we model Bayesian games?

BoS variant in extensive form:

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How do we model Bayesian games?

How do we model Bayesian games?

How do we make predictions? Think back to basics of extensive form:1. Convert to (Bayesian) normal form.2. Nature’s choice determines what type of person Player 2 is, then, 2’s strategy

specifies action of both.3. We have to write down the expected payoffs if there is uncertainty.

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How do we model Bayesian games?

How do we model Bayesian games?

The Bayesian normal form of our variant of BoS with incomplete information:Payoffs for palyer 1

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How do we model Bayesian games?

How do we model Bayesian games?

How do we make predictions? Think back to basics of extensive form:1. Convert to (Bayesian) normal form.2. Nature’s choice determines what type of person Player 2 is, then, 2’s strategy

specifies action of both.3. We have to write down the expected payoffs if there is uncertainty.4. Now do the same for Player 2, but Fill in payoff for each type, put in ordered pair

in each cell

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How do we model Bayesian games?

How do we model Bayesian games?

The Bayesian normal form of our variant of BoS with incomplete information:

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Bayesian Nash equilibrium

On the Agenda

1 Introduction

2 Private vs. Public Information

3 Bayesian game

4 How do we model Bayesian games?

5 Bayesian Nash equilibrium

6 Exercises

C. Hurtado (UIUC - Economics) Game Theory 18 / 21

Bayesian Nash equilibrium

Bayesian Nash equilibrium

Bayesian Nash equilibrium is a straightforward extension of NE:Each type of player chooses a strategy that maximizes expected utility given theactions of all types of other players and that player’s beliefs about others’ typesIn our BoS variant:

- Action of Player 1 is optimal (maximizes expected utility) given the actionsof the two types of Player 2 (and Player 1s beliefs about Player 2s type)

- Action of each type of Player 2 is optimal, given the action of Player 1- Claim: (O;OB) is a Bayesian Nash equilibrium

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Exercises

On the Agenda

1 Introduction

2 Private vs. Public Information

3 Bayesian game

4 How do we model Bayesian games?

5 Bayesian Nash equilibrium

6 Exercises

C. Hurtado (UIUC - Economics) Game Theory 20 / 21

Exercises

ExercisesYou and a friend are playing a 2 ÃŮ 2 matrix game, but you’re not sure if it’s BoSor PD. Both are equally likely.

Can you put this game into Bayesian normal form?

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