1. 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

2. 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.

3. 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?

4. 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

5. An example: Big Monkey and Little Monkeyc
Big monkey w 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)

6. 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.

7. An example: Big Monkey and Little Monkey
What if the monkeys have to decide simultaneously? c
Big monkey w 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)

8. 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

9. 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.

10. Eliminating Dominated Strategies
The first step is to eliminate actions that are worse than another action, no matter what. w c
Big monkey w c 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

11. 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

12. 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.

13. 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.

14. Prisoner’s Dilemma Each player can cooperate or defect Column
-1,-1
-10,0
Row
defect
-8,-8
0,-10

15. Prisoner’s Dilemma Each player can cooperate or defect Column
-1,-1
-10,0
Row
defect
-8,-8
0,-10
Defecting is a dominant strategy for row

16. Prisoner’s Dilemma Each player can cooperate or defect Column
-1,-1
-10,0
Row
defect
-8,-8
0,-10
Defecting is also a dominant strategy for column

17. 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

18. Prisoner’s Dilemma Relevant to: How do players escape this dilemma?
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

19. 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

20. 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.

21. 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)

22. 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.

23. 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

24. Coming next time How to select an optimal strategy
How to deal with incomplete information
How to handle multi-stage games