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Game Theory

- Developed to explain the optimal strategy in two-person interactions.
- Initially, von Neumann and Morganstern
- Zero-sum games
- John Nash
- Nonzero-sum games
- Harsanyi, Selten
- Incomplete information

An example:Big Monkey and Little Monkey

c

w

Big monkey

c

w

c

Little monkey

w

0,0

9,1

4,4

5,3

- What should Big Monkey do?
- If BM waits, LM will climb – BM gets 9
- If BM climbs, LM will wait – BM gets 4
- BM should wait.
- What about LM?
- Opposite of BM (even though we’ll never get to the right side
- of the tree)

An example:Big Monkey and Little Monkey

- These strategies (w and cw) are called best responses.
- Given what the other guy is doing, this is the best thing to do.
- A solution where everyone is playing a best response is called a Nash equilibrium.
- No one can unilaterally change and improve things.
- This representation of a game is called extensive form.

An example:Big Monkey and Little Monkey

- What if the monkeys have to decide simultaneously?

c

w

Big monkey

c

w

c

Little monkey

w

0,0

9,1

6-2,4

7-2,3

Now Little Monkey has to choose before he sees Big Monkey move

Two Nash equilibria (c,w), (w,c)

Also a third Nash equilibrium: Big Monkey chooses between c & w

with probability 0.5 (mixed strategy)

An example:Big Monkey and Little Monkey

- It can often be easier to analyze a game through a different representation, called normal form

Little Monkey

c

v

Big Monkey

5,3

4,4

c

v

9,1

0,0

Choosing Strategies

- In the simultaneous game, it’s harder to see what each monkey should do
- Mixed strategy is optimal.
- Trick: How can a monkey maximize its payoff, given that it knows the other monkeys will play a Nash strategy?
- Oftentimes, other techniques can be used to prune the number of possible actions.

Eliminating Dominated Strategies

- The first step is to eliminate actions that are worse than another action, no matter what.

c

w

Big monkey

c

w

c

w

c

9,1

4,4

w

Little monkey

We can see that Big

Monkey will always choose

w.

So the tree reduces to:

9,1

0,0

9,1

6-2,4

7-2,3

Little Monkey will

Never choose this path.

Or this one

Eliminating Dominated Strategies

- We can also use this technique in normal-form games:

Column

a

b

9,1

4,4

a

Row

b

0,0

5,3

Eliminating Dominated Strategies

- We can also use this technique in normal-form games:

a

b

9,1

4,4

a

b

0,0

5,3

For any column action, row will prefer a.

Eliminating Dominated Strategies

- We can also use this technique in normal-form games:

a

b

9,1

4,4

a

b

0,0

5,3

Given that row will pick a, column will pick b.

(a,b) is the unique Nash equilibrium.

Prisoner’s Dilemma

- Each player can cooperate or defect

Column

cooperate

defect

cooperate

-1,-1

-10,0

Row

defect

-8,-8

0,-10

Prisoner’s Dilemma

- Each player can cooperate or defect

Column

cooperate

defect

cooperate

-1,-1

-10,0

Row

defect

-8,-8

0,-10

Defecting is a dominant strategy for row

Prisoner’s Dilemma

- Each player can cooperate or defect

Column

cooperate

defect

cooperate

-1,-1

-10,0

Row

defect

-8,-8

0,-10

Defecting is also a dominant strategy for column

Prisoner’s Dilemma

- Even though both players would be better off cooperating, mutual defection is the dominant strategy.
- What drives this?
- One-shot game
- Inability to trust your opponent
- Perfect rationality

Prisoner’s Dilemma

- Relevant to:
- Arms negotiations
- Online Payment
- Product descriptions
- Workplace relations
- How do players escape this dilemma?
- Play repeatedly
- Find a way to ‘guarantee’ cooperation
- Change payment structure

Definition of Nash Equilibrium

- A game has n players.
- Each player ihas a strategy set Si
- This is his possible actions
- Each player has a payoff function
- pI: S R
- A strategy tiin Siis a best response if there is no other strategy in Si that produces a higher payoff, given the opponent’s strategies.

Definition of Nash Equilibrium

- A strategy profile is a list (s1, s2, …, sn) of the strategies each player is using.
- If each strategy is a best response given the other strategies in the profile, the profile is a Nash equilibrium.
- Why is this important?
- If we assume players are rational, they will play Nash strategies.
- Even less-than-rational play will often converge to Nash in repeated settings.

An Example of a Nash Equilibrium

Column

a

b

a

1,2

0,1

Row

b

1,0

2,1

(b,a) is a Nash equilibrium.

To prove this:

Given that column is playing a, row’s best response is b.

Given that row is playing b, column’s best response is a.

Finding Nash Equilibria – Dominated Strategies

- What to do when it’s not obvious what the equilibrium is?
- In some cases, we can eliminate dominated strategies.
- These are strategies that are inferior for every opponent action.
- In the previous example, row = a is dominated.

Example

Column

- A 3x3 example:

a

b

c

a

73,25

57,42

66,32

Row

b

80,26

35,12

32,54

c

28,27

63,31

54,29

c dominates a for the column player

Example

Column

- A 3x3 example:

a

b

c

a

73,25

57,42

66,32

Row

b

80,26

35,12

32,54

c

28,27

63,31

54,29

b is then dominated by both a and c for the row player.

Example

Column

- A 3x3 example:

a

b

c

a

73,25

57,42

66,32

Row

b

80,26

35,12

32,54

c

28,27

63,31

54,29

Given this, b dominates c for the column player –

the column player will always play b.

Example

Column

- A 3x3 example:

a

b

c

a

73,25

57,42

66,32

Row

b

80,26

35,12

32,54

c

28,27

63,31

54,29

Since column is playing b, row will prefer c.

Example

Column

a

b

c

a

73,25

57,42

66,32

Row

b

80,26

35,12

32,54

c

28,27

63,31

54,29

We verify that (c,b) is a Nash Equilibrium by observation:

If row plays c, b is the best response for column.

If column plays b, c is the best response by row.

Coordination Games

- Consider the following problem:
- A supplier and a buyer need to decide whether to adopt a new purchasing system.

Buyer

new

old

new

20,20

0,0

Supplier

old

5,5

0,0

No dominated strategies!

new

old

new

20,20

0,0

Supplier

old

5,5

0,0

Coordination Games- This game has two Nash equilibria (new,new) and (old,old)
- Real-life examples: Beta vs VHS, Mac vs Windows vs Linux, others?
- Each player wants to do what the other does
- which may be different than what they say they’ll do
- How to choose a strategy? Nothing is dominated.

Solving Coordination Games

- Coordination games turn out to be an important real-life problem
- Technology/policy/strategy adoption, delegation of authority, synchronization
- Human agents tend to use “focal points”
- Solutions that seem to make “natural sense”
- e.g. pick a number between 1 and 10
- Social norms/rules are also used
- Driving on the right/left side of the road
- These strategies change the structure of the game

Price-matching Example

- Two sellers are offering the same book for sale.
- This book costs each seller $25.
- The lowest price gets all the customers; if they match, profits are split.
- What is the Nash Equilibrium strategy?

Mixed strategies

- Unfortunately, not every game has a pure strategy equilibrium.
- Rock-paper-scissors
- However, every game has a mixed strategy Nash equilibrium.
- Each action is assigned a probability of play.
- Player is indifferent between actions, given these probabilities.

Mixed Strategies

- In many games (such as coordination games) a player might not have a pure strategy.
- Instead, optimizing payoff might require a randomized strategy (also called a mixed strategy)

Wife

football

shopping

football

2,1

0,0

Husband

shopping

1,2

0,0

football

shopping

football

2,1

0,0

Husband

shopping

1,2

0,0

Strategy SelectionIf we limit to pure strategies:

Husband: U(football) = 0.5 * 2 + 0.5 * 0 = 1

U(shopping) = 0.5 * 0 + 0.5 * 1 = ½

Wife: U(shopping) = 1, U(football) = ½

Problem: this won’t lead to coordination!

Mixed strategy

- Instead, each player selects a probability associated with each action
- Goal: utility of each action is equal
- Players are indifferent to choices at this probability
- a=probability husband chooses football
- b=probability wife chooses shopping
- Since payoffs must be equal, for husband:
- b*1=(1-b)*2 b=2/3
- For wife:
- a*1=(1-a)*2 = 2/3
- In each case, expected payoff is 2/3
- 2/9 of time go to football, 2/9 shopping, 5/9 miscoordinate
- If they could synchronize ahead of time they could do better.

Example: Rock paper scissors

Column

rock

paper

scissors

0,0

-1,1

1,-1

rock

Row

paper

1,-1

0,0

-1,1

scissors

-1,1

1,-1

0,0

Setup

- Player 1 plays rock with probability pr, scissors with probability ps, paper with probability 1-pr –ps
- P2: Utility(rock) = 0*pr + 1*ps – 1(1-pr –ps) = 2 ps + pr -1
- P2: Utility(scissors) = 0*ps + 1*(1 – pr – ps) – 1pr = 1 – 2pr –ps
- P2: Utility(paper) = 0*(1-pr –ps)+ 1*pr – 1ps = pr –ps

Player 2 wants to choose a probability for each strategy

so that the expected payoff for each strategy is the same.

Repeated games

- Many games get played repeatedly
- A common strategy for the husband-wife problem is to alternate
- This leads to a payoff of 1, 2,1,2,…
- 1.5 per week.
- Requires initial synchronization, plus trust that partner will go along.
- Difference in formulation: we are now thinking of the game as a repeated set of interactions, rather than as a one-shot exchange.

Repeated vs Stage Games

- There are two types of multiple-action games:
- Stage games: players take a number of actions and then receive a payoff.
- Checkers, chess, bidding in an ascending auction
- Repeated games: Players repeatedly play a shorter game, receiving payoffs along the way.
- Poker, blackjack, rock-paper-scissors, etc

Analyzing Stage Games

- Analyzing stage games requires backward induction
- We start at the last action, determine what should happen there, and work backwards.
- Just like a game tree with extensive form.
- Strange things can happen here:
- Centipede game
- Players alternate – can either cooperate and get $1 from nature or defect and steal $2 from your opponent
- Game ends when one player has $100 or one player defects.

Analyzing Repeated Games

- Analyzing repeated games requires us to examine the expected utility of different actions.
- Assumption: game is played “infinitely often”
- Weird endgame effects go away.
- Prisoner’s Dilemma again:
- In this case, tit-for-tat outperforms defection.
- Collusion can also be explained this way.
- Short-term cost of undercutting is less than long-run gains from avoiding competition.

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