Game Theory

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Game Theory

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Game to play
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GAME THEORY

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

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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).

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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.

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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.

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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.

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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

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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.

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Prisoners dilemma payoff matrix for Art and Bob.
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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

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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.

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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.

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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.

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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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Simultaneous games

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

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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
  (simultaneous game)

Little Monkey
c
v
Big Monkey
5,3
4,4
c
v
0,0
9,1
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Choosing Strategies

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

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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
• Bandwidth and resource usage,
• Spam
• Overfishing, 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

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Duopolists Dilemma

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

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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.

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Duopolists Dilemma
The payoff matrix as the economic profits for
each firm in each possible outcome.
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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.
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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.

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Other Oligopoly Games

• Other Oligopoly Games
• Advertising campaigns by Coke and Pepsi, and
  research and development (RD) 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.

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• PG and Kimberly-Clark have two strategies spend
  on RD or do no RD.
• Table shows the payoff matrix as the economic
  profits for each firm in each possible outcome.

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The Nash equilibrium for this game is for both
firms to undertake RD.
But they could earn a larger joint profit if
they could collude and not do RD.
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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.

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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.

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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
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Examples

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

Goal keeper
Striker
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Matching Pennies
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Price cutting game

• Dominant strategy Low

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Avoidance
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Two-persons Game
In the game above there are 3 Nash equilibria
NE1A,A NE2B,B
NE3C,B