Saturday, July 27, 2013

Path Dependence

Path Dependence

Lectures cover:
  • What is Path Dependence?
  • Construct Urn Models to understand
    • different types of path dependence
    • what causes path dependence
    • difference between path dependent outcomes and path dependent equilibria
  • Examples of Urn Models
    • Polya Process, Balancing Process,
  • Path dependencies is logically different from increasing returns
  • Path dependencies are caused by externalities i.e. interdependencies between choices/decisions
    • externalities with negative effects are more likely to be path dependent
  • Compare path dependence and Markov Processes, Tipping Points and Chaos
  • Key lessons - my own reflection of what this lecture means
Real-world examples:
  • QWERTY typewriter keyboard
    • increasing returns / virtuous cycle
      • the more QWERTYs lead to more QWERTYs
  • Technology
    • AC vs DC
    • Gasoline vs Electric Cars
  • Common Law
    • Influence of Precedent
  • Institutional Choices
    • Defined benefits
  • Economic Success
    • Ann Arbor vs Jackson
  • Manifest Destiny of America
  • Railroads 
 

Path Dependence

  • Path Dependent
    • Outcome probabilities depend upon the sequence of path outcomes.
      • Doesn't necessarily determine, merely affects the probabilities
    • What happens now depends on what happened along the path to get here
  • Phat Dependent
    • Outcome probabilities depend upon past outcomes outcomes but not their order
      • Polya process is Phat

Urn Models

  • Basic Urn Model
    • Urn contains balls of various colors
    • The outcome equals the color of the ball selected
  • Bernoulli Model
    • Fixed number of balls in the urn
    • U = {B blue, R red}     ; U stands for Urn is a set of B blue and R red
    • Process: Select ball and return to the urn
    • P(red) = R/(B+R)
    • Outcomes independent
  • Polya Process
    • U = {1 Blue, 1 Red}
    • Process: 
      • Select and return
      • Add a new ball that is the same color as the ball selected
    • Probabilities will change over time
    • Result
      • Any probability of red balls is an equilibrium and equally likely
      • Any history of B blue and R red balls is equally likely
        • Seeing just the set in the outcome doesn't tell you anything about the order (a Phat process) because any order is equally likely
    • Example:
      • Fashion: People buy leopard prints because there are more leopard prints
      • Technology: People buy iPhones because they see iPhones
  • Balancing Process
    • U = {1 Blue, 1 Red}
    • Process:
      • Select and return
      • Add a new ball that is the opposite color as the ball selected
    • Result
      • The balancing process converges to equal percentages of the two colors of balls
    •  Examples
      • Need to keep constituencies happy
        • Selection of site for political conventions - northern or southern state
        • Selection of site by Olympic committee - Asia, Europe, North America or South America
  • Sway Process
    •  U = {1 Blue, 1 Red}
    • Process
      • Select and return 
      • In period t, add a ball of the same color as the selected ball and add 2^(t-s) - 2(t-s-1) of color chosen in each period s < t
    • As you go back in time,the older events take on exponentially more weight over time
    • Early movers have a bigger effect
  • Distinguish between:
    • Path Dependent Outcomes
      • color of ball in a given period depends on the path
    • Path Dependent Equilibrium
      • percentage of red balls in long run depends on the path
  • Outcomes and equilibrium
    • Polya process: 
      • Path-dependent outcomes
      • Path-dependent equilibria
    • Balancing process: 
      • Path dependent outcomes
      • Equilibria independent of path
  • Examples of path-independent equilibria
    • Manifest Destiny - America will stretch from sea to shining sea
    • Railroads - Once railroads were invented, they will build themselves
    • Mobile - Once invented, mobiles is the future

Path Dependence and Chaos

  • Why is the difference between Path Dependence and Phat Dependence important
  • Markov processes
    • finite states
    • fixed transition probabilities
    • can get to any other state
    • not simple cycle
    • markov converges to a unique stochastic equilibrium
  • Chaos
    • Extreme Sensitivity to Initial Conditions (ESTIC)
    • If initial points x and x' differ by even a tiny amount after many iterations of the outcome function, they differ by arbitraty amounts.
  • Tent Map (an example of chaotic)
    • x in (0,1)
    • F(x) = 2x   if X <0.5,
    •         = 2 - 2x if X > 0.5
  • Tent Map is not path dependent
    • Nothing happens along the way/path will change the end
    • is deterministic
  • Path dependence means what happens along the way has an impact on the outcome
Types of outcomes
  • Independent
    • Outcome doesn't depend on starting point or what happens along the way
  • Chaotic
    • Outcome depends on initial conditions
  • Path dependent
    • Outcome probabilities depend upon sequence of past outcomes
  • Phat dependent 
    • Outcome probabilities depend upon past outcomes but not their order
History is path dependent. The future is being written today.
  • History/future is not independent i.e. what's happening now is happening regardless of what happened in the past. Independence means no structure.
  • History/future is not chaotic i.e. initial conditions matter but it isn't the only thing that matter. Once we write the Constitution, the rest plays out deterministically
  • History/future is path dependent not phat dependent because early events have a larger importance.

Path Dependence and Increasing Returns

  • Increasing Returns
    • More produces more
    • Positive feedback / Virtuous cycles
    • The more I have of something, the more I want the same thing 
    • The the more other people do something, the more that other people will do it
  • Example:
    • The more people get QWERTY typewriters, the more people will get QWERTY typewriters
  • Is increasing returns equivalent to path dependent equilibrium? No
    • Increasing returns without path dependent equilibrium
      • Example:
        • Gas / Electric
        • Always goes to equilibrium at Gas even though there is increasing returns
    • Path dependent equilibrium without increasing returns
      • Example
      • Symbiots
  • Path Dependencies comes from a different process - Externalities
    • Externalities:
      • interdependence between choices can create path dependence
    • Decisions create externalities
    • Externalities that big projects create path dependence
      • Choosing Project A first results in choosing Project C next
      • Choosing Project B first results in choosing Project D next
    • Yi J's decision to migrate to Australia affects parents & Dai J

Path Dependence or Tipping Point

  • Path Dependent Equilibrium
    • percentage of red balls in the long run depends on the path
  • Tipping points
    • direct tips
  • Comparison
    • Tipping points
      • a single instance in time where the long term equilibrium
      • a singular event that tips the event abruptly
      • diversity index - count of equilibria
        • diversity index (uncertainly) is reduced abruptly
      • entropy - how much information in the system
    • Path Dependent Equilibrium
      • accumulative effect of moving along the path
      • diversity index reduces gradually
        • unlike tipping point where diversity index (uncertainly) is reduced abruptly

Key Lessons

  • Externalities are the reason choices we make in the past will affect choices we make in the future
  • Pay more attention to choices with externalities e.g.
    • Where you choose to live
    • What line of work you choose to do
    • What language you choose to learn
  • Where possible make decisions where externalities are all positive 
    • including future externalities
    • this will reduce path dependencies i.e. keep your options open
  • History matters when it changes transition probabilities
    • Make decisions that increase transition probabilities to states you desire - to goal states
  • Pay attention to externalities when making decision especially negative externalities
    • Keep your options open

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