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

• 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.

• 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

An example:Big Monkey and Little Monkey

Big monkey w

w w

c

cc

0,0

Little monkey

9,1 6-2,4 7-2,3What 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)

• 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?

An example:Big Monkey and Little Monkey

Big monkey w

w w

c

cc

0,0

Little monkey

9,1 6-2,4 7-2,3

Now Little Monkey has to choose before he sees Big Monkey moveTwo Nash equilibria (c,w), (w,c)Also a third Nash equilibrium: Big Monkey chooses between c & wwith probability 0.5 (mixed strategy)

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

An example:Big Monkey and Little Monkey

c

c v

v

5,3 4,4

0,09,1

Little Monkey

Big Monkey

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.

Big monkeyw

w w

c

cc

0,0

Little monkey

9,1 6-2,4 7-2,3

Little Monkey will

Never choose this path.Or this one

w c

9,1 4,4

We can see that Big Monkey will always choosew.So the tree reduces to:

9,1

Eliminating Dominated Strategies

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

a

a b

b 5,3

4,4

0,0

9,1

Row

Column

Eliminating Dominated Strategies

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

a

a b

b 5,3

4,4

0,0

9,1

For any column action, row will prefer a.

Eliminating Dominated Strategies

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

a

a b

b 5,3

4,4

0,0

9,1

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

cooperate defect

defect 0,-10

-10,0

-8,-8

-1,-1

Row

Column

cooperate

Prisoner’s Dilemma

• Each player can cooperate or defect

cooperate defect

defect 0,-10

-10,0

-8,-8

-1,-1

Row

Column

cooperate

Defecting is a dominant strategy for row

Prisoner’s Dilemma

• Each player can cooperate or defect

cooperate defect

defect 0,-10

-10,0

-8,-8

-1,-1

Row

Column

cooperate

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:– IT budgeting

– Bandwidth and resource usage, spam

– Shared communication channels

– Environmental laws, overfishing, whaling, 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

– Mechanism design

Coming next time

• How to select an optimal strategy

• How to deal with incomplete information

• How to handle multi-stage games