A Monte Carlo Study of A Generalized Urn Problem1 1' Arthur, B' et' al', A Generalized Urn Problem a

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A Monte Carlo Study ofA Generalized Urn
Problem11. Arthur, B. et. al., A Generalized Urn
Problem and its Application, Cybernetics and
Systems Analysis, 1983

• J. Kader Hyer
• STAT 789

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• This study looks at the paper A Generalized
  Urn Problem and its Application (Arthur, B. et.
  al., 1983) and a conducts a Monte Carlo
  simulation in order to test the theory outlined
  by the authors. I first review the paper and
  restate the theories outlined. Then I conduct a
  Monte Carlo simulation of various scenarios to
  check the authors conclusions.
•  

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• What happen to the steam engine?
• Does anybody remember betamax?
• How did Blu-Ray beat out HD-DVD?
• Why do we use a QWERTY board?

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• These questions all have something similar in
  common. Market share in these products niches
  did not accommodate dual technologies, but rather
  the market shifted to a single technology.
  Economic theory tells us that the market is full
  of rational agents that choose the best product,
  thus making the market efficient. Often
  competing technologies cannot be distinguished
  solely on the basis of efficiency and sometimes
  the most efficient choice isnt allows what the
  market adopts.

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• This phenomenon is examined in the form of an
  urn problem. Given technologies A and B, we can
  think of these a white and black balls,
  respectively, in an urn of infinite capacity.
  Allowing an agent to make a choice only by
  observing what others are doing. What happens to
  the market share for a given product niche as the
  number of agents making this type of decision
  grows?

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• The authors consider three rules Rn 1, 2, 3,
• R1 One removes an odd numbered sample of r balls
  from the urn. Given the dominate color of the
  sample (white or black), one adds a ball of that
  color to the sample and returns the balls to the
  urn.
• R2 One randomly selects a single ball at a time
  from the urn until m balls of a certain color is
  obtained. Then one adds a ball of the same color
  as the m balls sampled and returns them all to
  the urn.
• R3 One removes an odd numbered sample of r
  balls from the urn. Given the dominate color of
  the sample, one adds a ball of the opposite color
  to the sample and returns the balls to the urn.
  

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• The Authors conjecture that for R1 and R2,
• P y(t) 0 U y(t) 1 1
• And for R3,
• Py(t) ? 1,
• for a unique ? ½.

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