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### Game Theory

Source: Google

GAME THEORY

- Game theory
- The tool used to analyze strategic behavior—behavior that recognizes mutual interdependence and takes account of the expected behavior of others.

Game Theory

- Def. Game theory: it is the that model formally problem of strategic interaction, e.g. problems in which the utility (payoff) of an individual (player) is affected by the choice made by other individuals (players).

GAME THEORY

- What Is a Game?
- All games involve three features:
- Rules
- Strategies
- Payoffs
- Prisoners’ dilemma
- A game between two prisoners that shows why it is hard to cooperate, even when it would be beneficial to both players to do so.

GAME THEORY

- The Prisoners’ Dilemma
- Art and Bob been caught stealing a car: sentence is 2 years in jail.
- DA wants to convict them of a big bank robbery: sentence is 10 years in jail.
- DA has no evidence and to get the conviction, he makes the prisoners play a game.

GAME THEORY

- Rules
- Players cannot communicate with one another.
- If both confess to the larger crime, each will receive a sentence of 3 years for both crimes.
- If one confesses and the accomplice does not, the one who confesses will receive a sentence of 1 year, while the accomplice receives a 10-year sentence.
- If neither confesses, both receive a 2-year sentence.

GAME THEORY

- Strategies
- The strategies of a game are all the possible outcomes of each player.
- The strategies in the prisoners’ dilemma are:
- Confess to the bank robbery
- Deny the bank robbery

GAME THEORY

- Payoffs
- Four outcomes:
- Both confess.
- Both deny.
- Art confesses and Bob denies.
- Bob confesses and Art denies.
- A payoff matrix is a table that shows the payoffs for every possible action by each player given every possible action by the other player.

Nash Equilibrium

- If there is a set of strategies with the property that no player can benefit by changing her strategy while the other players keep their strategies unchanged, then that set of strategies and the corresponding payoffs constitute the Nash Equilibrium

GAME THEORY

- The Nash equilibrium for the two prisoners is to confess.
- Not the Best Outcome
- The equilibrium of the prisoners’ dilemma is not the best outcome.

Dominance and Dominance Principle

- Definition: A strategy S dominates a strategy T if every outcome in S is at least as good as the corresponding outcome in T, and at least one outcome in S is strictly better than the corresponding outcome in T.
- Dominance Principle: A rational player would never play a dominated strategy.

Dominant Strategy Equilibrium

- If every player in the game has a dominant strategy, and each player plays the dominant strategy, then that combination of strategies and the corresponding payoffs are said to constitute the dominant strategy equilibrium for that game.

An example:Big Monkey and Little Monkey

- Monkeys usually eat ground-level fruit
- Occasionally climb a tree to get a coconut (1 per tree)
- A Coconut yields 10 Calories
- Big Monkey expends 2 Calories climbing the tree.
- Little Monkey expends 0 Calories climbing the tree.

An example:Big Monkey and Little Monkey

- If BM climbs the tree
- BM gets 6 C, LM gets 4 C
- LM eats some before BM gets down
- If LM climbs the tree
- BM gets 9 C, LM gets 1 C
- BM eats almost all before LM gets down
- If both climb the tree
- BM gets 7 C, LM gets 3 C
- BM hogs coconut
- How should the monkeys each act so as to maximize their own calorie gain?

An example:Big Monkey and Little Monkey

- Assume BM decides first
- Two choices: wait or climb
- LM has four choices:
- Always wait, always climb, same as BM, opposite of BM.
- These choices are called actions
- A sequence of actions is called a strategy

An example:Big Monkey and Little Monkey

c

w

Big monkey

c

w

c

Little monkey

w

0,0

9,1

6-2,4

7-2,3

- 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)

Simultaneous games

- Def. simultaneous game: it is a game where both players move at the same time without possibility to communicate their choices

An example:Big Monkey and Little Monkey

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

Little Monkey

c

v

Big Monkey

5,3

4,4

c

v

9,1

0,0

Choosing Strategies

- How can a monkey maximize its payoff, given that it knows the other monkeys will play a Nash strategy?

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

Tragedy of the Commons

- Game theory can be used to explain overuse of shared resources.
- Extend the Prisoner’s Dilemma to more than two players.
- A cow costs a dollars and can be grazed on common land.
- The value of milk produced (f(c) ) depends on the number of cows on the common land.
- Per cow: f(c) / c

Tragedy of the Commons

- To maximize total wealth of the entire village: max f(c) – ac.
- Maximized when marginal product = a
- Adding another cow is exactly equal to the cost of the cow.
- What if each villager gets to decide whether to add a cow?
- Each villager will add a cow as long as the cost of adding that cow to that villager is outweighed by the gain in milk.

Tragedy of the Commons

- When a villager adds a cow:
- Output goes from f(c) /c to f(c+1) / (c+1)
- Cost is a
- Notice: change in output to each farmer is less than global change in output.
- Each villager will add cows until output- cost = 0.
- Problem: each villager is making a local decision (will I gain by adding cows), but creating a net global effect (everyone suffers)

Tragedy of the Commons

- Problem: cost of maintenance is externalized
- Farmers don’t adequately pay for their impact.
- Resources are overused due to inaccurate estimates of cost.
- Relevant to:
- Bandwidth and resource usage,
- Spam
- Overfishing, pollution, etc.

Avoiding Tragedy of the Commons

- Private ownership
- Prevents TOC, but may have other negative effects.
- Social rules/norms, external control
- Nice if they can be enforced.
- Taxation
- Try to internalize costs; accounting system needed.
- Solutions require changing the rules of the game
- Change individual payoffs

Duopolists’ Dilemma

- The Duopolists’ Dilemma
- Each firm has two strategies. It can produce airplanes at the rate of:
- 3 a week
- 4 a week

GAME THEORY

- Because each firm has two strategies, there are four possible combinations of actions:
- Both firms produce 3 a week (monopoly outcome).
- Both firms produce 4 a week.
- Airbus produces 3 a week and Boeing produces 4 a week.
- Boeing produces 3 a week and Airbus produces 4 a week.

Equilibrium of the Duopolists’ Dilemma

- Both firms produce 4 a week.

- Like the prisoners, the duopolists do not cooperate and get a worse outcome than the one that cooperation would deliver.

Conclusion from Duopolist Game

- Collusion is Profitable but Difficult to Achieve
- The duopolists’ dilemma explains why it is difficult for firms to collude and achieve the maximum monopoly profit.
- Even if collusion were legal, it would be individually rational for each firm to cheat on a collusive agreement and increase output.
- In an international oil cartel, OPEC, countries frequently break the cartel agreement and overproduce.

Other Oligopoly Games

- Other Oligopoly Games
- Advertising campaigns by Coke and Pepsi, and research and development (R&D) competition between Procter & Gamble and Kimberly-Clark are like the prisoners’ dilemma game.
- Over the past almost 40 years since the introduction of the disposable diaper, Procter & Gamble and Kimberly-Clark have battled for market share by developing ever better versions of this apparently simple product.

P&G and Kimberly-Clark have two strategies: spend on R&D or do no R&D.

- Table shows the payoff matrix as the economic profits for each firm in each possible outcome.

The Nash equilibrium for this game is for both firms to undertake R&D.

- But they could earn a larger joint profit if they could collude and not do R&D.

Repeated Games

- Repeated Games
- Most real-world games get played repeatedly.
- Repeated games have a larger number of strategies because a player can be punished for not cooperating.
- This suggests that real-world duopolists might find a way of learning to cooperate so they can enjoy monopoly profit.
- The larger the number of players, the harder it is to maintain the monopoly outcome.

Is Oligopoly Efficient?

- Is Oligopoly Efficient?
- In oligopoly, price usually exceeds marginal cost.
- So the quantity produced is less than the efficient quantity.
- Oligopoly suffers from the same source and type of inefficiency as monopoly.
- Because oligopoly is inefficient, antitrust laws and regulations are used to try to reduce market power and move the outcome closer to that of competition and efficiency.

Examples

- Battle of the sexes (game of coordination)
- In the game above there are two Nash equilibria:

NE1={(Go to football match), (Go to football match)}

NE2={(Go to opera), (Go to opera)}

Girlfriend

Boyfriend

Examples

- Penalty kicks
- In the game above there are no Nash equilibria (in pure strategy!)

Goal keeper

Striker

Price cutting game

- Dominant strategy: Low

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