Intro to Game Theory
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Transcript of Intro to Game Theory
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