Game Theory Mechanism Design

Game Theory and Mechanism Design

Jeff Jacobs

Stanford University

jjacobs3@cs.stanford.edu

November 2, 2013

1 / 26

Game Theory Mechanism Design

Sad (and happy!) introduction

Invented by von Neumann and Morgenstern in 1945

Not related to video games

:(

Can be used to analyze video games!

:)

2 / 26

Game Theory Mechanism Design

Sad (and happy!) introduction

Invented by von Neumann and Morgenstern in 1945

Not related to video games

:(

Can be used to analyze video games!

:)

3 / 26

Game Theory Mechanism Design

Sad (and happy!) introduction

Invented by von Neumann and Morgenstern in 1945

Not related to video games

:(

Can be used to analyze video games!

:)

4 / 26

Game Theory Mechanism Design

Sad (and happy!) introduction

Invented by von Neumann and Morgenstern in 1945

Not related to video games

:(

Can be used to analyze video games!

:)

5 / 26

Game Theory Mechanism Design

Sad (and happy!) introduction

Invented by von Neumann and Morgenstern in 1945

Not related to video games

:(

Can be used to analyze video games!

:)

6 / 26

Game Theory Mechanism Design

Your First Game: Rock, Paper, Scissors

Rock Paper Scissors

Rock 0, 0 −1, 1 1,−1

Paper 1,−1 0, 0 −1, 1

Scissors −1, 1 1,−1 0, 0

7 / 26

Game Theory Mechanism Design

Nash Equilibrium: Prisoners’ Dilemma

Stay Silent Confess

Stay Silent −1,−1 −10, 0

Confess 0,−10 −3,−3

8 / 26

Game Theory Mechanism Design

Nash Equilibrium: Prisoners’ Dilemma

Stay Silent Confess

Stay Silent −1,−1 −10, 0

Confess 0,−10 −3,−3

9 / 26

Game Theory Mechanism Design

Nash Equilibrium: Prisoners’ Dilemma

Stay Silent Confess

Stay Silent −1,−1 −10, 0

Confess 0,−10 −3,−3

10 / 26

Game Theory Mechanism Design

Nash Equilibrium: Prisoners’ Dilemma

Stay Silent ConfessStay Silent −1,−1 −10, 0

Confess 0,−10 −3,−3

11 / 26

Game Theory Mechanism Design

Nash Equilibrium: Prisoners’ Dilemma

Stay Silent ConfessStay Silent −1,−1 −10, 0

Confess 0,−10 −3,−3

12 / 26

Game Theory Mechanism Design

Nash Equilibrium: Prisoners’ Dilemma

Stay Silent Confess

Stay Silent −1,−1 −10, 0

Confess 0,−10 −3,−3

13 / 26

Game Theory Mechanism Design

Nash Equilibrium: Prisoners’ Dilemma

Stay Silent Confess

Stay Silent −1,−1 −10, 0Confess 0,−10 −3,−3

14 / 26

Game Theory Mechanism Design

Nash Equilibrium: Prisoners’ Dilemma

Stay Silent Confess

Stay Silent −1,−1 −10, 0Confess 0,−10 −3,−3

15 / 26

Game Theory Mechanism Design

Nash Equilibrium: Prisoners’ Dilemma

Stay Silent Confess

Stay Silent −1,−1 −10, 0Confess 0,−10 −3,−3

16 / 26

Game Theory Mechanism Design

Nash Equilibrium: Prisoners’ Dilemma

Stay Silent Confess

Stay Silent −1,−1 −10, 0Confess 0,−10 −3,−3

17 / 26

Game Theory Mechanism Design

Nash Equilibrium: Prisoners’ Dilemma

Stay Silent ConfessStay Silent −1,−1 −10, 0Confess 0,−10 −3,−3

18 / 26

Game Theory Mechanism Design

Why is Nash Equilibria Important?

Nash’s Theorem (Nash, 1951)

Every game with a finite number of players and actions has at leastone Nash equilibrium.

19 / 26

Game Theory Mechanism Design

What is Mechanism Design?

We get to design a game

We want the game to have a certain outcome

Let’s design it so that the outcome we want is a Nashequilibrium!

20 / 26

Game Theory Mechanism Design

Auctions

We have a good g that we want to sell

We have n bidders, each of whom has a valuation vi for thegood

They submit their bids bi in sealed envelopes

Who do we give the item to?

21 / 26

Game Theory Mechanism Design

The Vickrey Auction

Instead of giving to the highest bidder for their bid, give tothe highest bidder for the second highest bid.

Claim: Vickrey auction induces an honest bidding Nashequilibrium.

22 / 26

Game Theory Mechanism Design

The Vickrey Auction

Thought Experiment 1: What happens if I bid lower than myvalue?

Thought Experiment 2: What happens if I bid higher than myvalue?

Conclusion: Honest bidding is a Nash equilibrium!

23 / 26

Game Theory Mechanism Design

The Vickrey Auction

Thought Experiment 1: What happens if I bid lower than myvalue?

Thought Experiment 2: What happens if I bid higher than myvalue?

Conclusion: Honest bidding is a Nash equilibrium!

24 / 26

Game Theory Mechanism Design

The Vickrey Auction

Thought Experiment 1: What happens if I bid lower than myvalue?

Thought Experiment 2: What happens if I bid higher than myvalue?

Conclusion: Honest bidding is a Nash equilibrium!

25 / 26

Game Theory Mechanism Design

The End

26 / 26