Chapter 14

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Chapter 14 Game Theory

• 14.1 Nash Equilibrium
• 14.2 Repeated Prisoners Dilemma
• 14.3 Sequential-Move Games and Strategic
  Moves

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Game Theory and Life

• You are on a first date with the love of your
  dreams. You can propose 2 activities
• Safe activity (Coffee)
• Exciting Activity (Waterpark)
• Your date could either want a safe activity or an
  exciting activity. There are different results
  if your ideas match up or clash

3
First Date Game
What is the outcome of this game? Payoff format
is (Left, Top)
Chapter Fourteen
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Game Theory Components
Players agents participating in the game (You
and Your Date Strategies Actions that each
player may take under any possible circumstance
(Coffee, Waterpark) Outcomes The various
possible results of the game (four, each
represented by one cell of the payoff
matrix) Payoffs The benefit that each player
gets from each possible outcome of the game (the
profits entered in each cell of the payoff matrix)
Chapter Fourteen
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Best Responses

• In all game theory games, players choose
  strategies without knowing with certainty what
  the opposing player will do.

Players construct BEST RESPONSES -optimal
actions given all possible actions of other
players
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First Date Game Best Responses
If you know your date will pick coffee, you
should pick coffee, since 10 gt -5 If you know
your date will pick waterpark, you should pick
waterpark, since 20 gt 0
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First Date Game Best Responses
If your date knows you will pick coffee, they
should pick coffee, since 10 gt -5 If your date
knows you will pick waterpark, they should pick
waterpark, since 20 gt 0
Note that this game is SYMMETRICAL
Chapter Fourteen
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Nash Equilibrium
Definition A Nash Equilibrium occurs when each
player chooses a strategy that gives him/her the
highest payoff, given the strategy chosen by the
other player(s) in the game. ("rational
self-interest") Nash Equilibria occur when best
responses line up The Date Game Nash
equilibria Each proposes coffee or each proposes
waterpark.
Chapter Fourteen
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Game Theory

• A special kind of Best Response

DOMINANT STRATEGY

• Strategy that is best no matter what the other
  player does.

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Advertising
As STRATEGY
Dont advertise
Advertise
Bs STRATEGY
As profit 50 000
As profit 75 000
Dont advertise
Bs profit 50 000
Bs loss 25 000
As loss 25 000
As profit 10 000
Advertise
Bs profit 10 000
Bs profit 75 000
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Dominant Strategy
As dominant strategy is advertise
Bs dominant strategy is advertise
Dont advertise
Advertise
As profit 50 000
As profit 75 000
Dont advertise
Bs profit 50 000
Bs loss 25 000
As loss 25 000
As profit 10 000
Advertise
Bs profit 10 000
Bs profit 75 000
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Prisoners Dilemma

• This is an example of a prisoners dilemma type
  of game.
• There is dominant strategy.
• The dominant strategy does not result in the best
  outcome for either player.
• It is hard to cooperate even when it would be
  beneficial for both players to do so
• Cooperation between players is difficult to
  maintain because cooperation is individually
  irrational.
• eg., The dominant strategy advertise

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Classic Prisoners Dilemma
Rockys strategies
Confess
Deny
Dominant strategy confess, even though they
would both be better off if they both kept their
mouths shut.
Deny
Gingers strategies
Confess
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Dominant Strategy Equilibrium
Definition A Dominant Strategy Equilibrium
occurs when each player uses a dominant strategy.
Toyota
Honda
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Dominated Strategy
Definition A player has a dominated strategy
when the player has another strategy that gives
it a higher payoff no matter what the other
player does.
Toyota
Honda
Chapter Fourteen
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Dominant or Dominated Strategy
Why look for dominant or dominated strategies? A
dominant strategy equilibrium is particularly
compelling as a "likely" outcome Similarly,
because dominated strategies are unlikely to be
played, these strategies can be eliminated from
consideration in more complex games. This can
make solving the game easier.
Chapter Fourteen
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Dominated Strategy
Toyota
Honda
"Build Large" is dominated for each player By
eliminating the dominated strategies, we can
reduce the game matrix.
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Finding Nash Equilibrium Cases

1. Nash Equilibrium where Dominant Strategies
   overlap
2. Nash Equilibrium with one Dominant Strategy
3. Nash Equilibrium by eliminating Dominated
   Strategy
4. Nash Equilibrium through Best Responses

Chapter Fourteen
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Nash Equilibrium Dominant Overlap
Professor
Student
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Nash Equilibrium One Dominant
Professor
Student
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Nash Equilibrium Eliminate Dominated
Professor
Student
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Nash Equilibrium Best Responses
Professor
Student
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Nash Equilibrium

• However it is found, a Nash Equilibrium ALWAYS
  occurs where Best Responses line up
• If Multiple Nash Equilibria exist, we cant
  conclude WHICH outcome will occur, only the
  possible outcomes that can occur
• Also, it is often APPEARS that no Nash Equilibria
  exist

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No Nash Equilibrium
Fred
Barney
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Mixed Strategies
Pure Strategy A specific choice of a strategy
from the players possible strategies in a game.
(ie Rock) Mixed Strategy A choice among two
or more pure strategies according to
pre-specified probabilities. (ie Rock, Paper or
Scissors each 1/3rd of the time) If Pure
Strategies cant produce a Nash Equilibrium,
Mixed Strategies can If both players randomize
each choice 1/3rd of the time, nether have an
incentive to deviate.