Game Theory

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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

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An exampleBig 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.

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An exampleBig 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?

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An exampleBig 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

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An exampleBig Monkey and Little Monkey
c
w
Big monkey
c
w
c
w
Little monkey
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 well never get to
  the right side
• of the tree)

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An exampleBig 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.

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An exampleBig Monkey and Little Monkey

• What if the monkeys have to decide simultaneously?

c
w
Big monkey
c
w
c
w
Little monkey
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)
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An exampleBig 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
0,0
9,1
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Choosing Strategies

• In the simultaneous game, its 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.

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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
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Eliminating Dominated Strategies

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

Column
a
b
4,4
9,1
a
Row
b
0,0
5,3
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Eliminating Dominated Strategies

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

a
b
4,4
9,1
a
b
0,0
5,3
For any column action, row will prefer a.
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Eliminating Dominated Strategies

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

a
b
4,4
9,1
a
b
0,0
5,3
Given that row will pick a, column will pick
b. (a,b) is the unique Nash equilibrium.
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Prisoners Dilemma

• Each player can cooperate or defect

Column
cooperate
defect
-10,0
-1,-1
cooperate
Row
defect
-8,-8
0,-10
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Prisoners Dilemma

• Each player can cooperate or defect

Column
cooperate
defect
-10,0
-1,-1
cooperate
Row
defect
-8,-8
0,-10
Defecting is a dominant strategy for row
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Prisoners Dilemma

• Each player can cooperate or defect

Column
cooperate
defect
-10,0
-1,-1
cooperate
Row
defect
-8,-8
0,-10
Defecting is also a dominant strategy for column
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Prisoners 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

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Prisoners 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

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Tragedy of the Commons

• Game theory can be used to explain overuse of
  shared resources.
• Extend the Prisoners 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

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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.

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Tragedy of the Commons

• When a villager adds a cow
• Output goes from f(c) /c to f(c1) / (c1)
• 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)

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Tragedy of the Commons

• Problem cost of maintenance is externalized
• Farmers dont 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.

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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

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Coming next time

• How to select an optimal strategy
• How to deal with incomplete information
• How to handle multi-stage games