Path Dependence

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Path Dependence

• Scott E Page
• University of Michigan
• Santa Fe Institute

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Outline

• Why we care
• Causes
• Outcomes or Equilibria
• Empirical Tests
• Conclusions

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Why We Care

• Efficiency
• History matters but how
• What can we predict

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Causes

• Multiple Peak Payoff
• learning
• evolution
• Increasing AND decreasing returns
• Local interactions

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Multiple Peaks
Pure coordination game has two equilibria.
Either one can be selected.
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Increasing Decreasing Returns

• Repeated discrete choices among Urban (U) or
  Suburban (S)
• Marginal benefit of U at time t increases in the
  number of previous Us AND decreases in the
  number of previous Ss.

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Why Not Just Increasing Returns

• If all externalities are positive, then you can
  design an algorithm so that there is no path
  dependency. You can always optimize.
• Appending Efficiency Page (1996) Journal of
  Public Economics

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Local Interactions
Senator A Health Senator B Environment Senator
C Defense
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Example
Interaction Value A B C D A 3 2 4
-8 B 0 1 2 C 1 3 D
-1 B 2, BC 5, BCD 6, ABCD 7
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Local Interactions
Senator A Health Defense Senator B
Environ Health Senator C Defense Environ
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Path Dependence

• Equilibria
• Outcomes

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Polya Urn Process

• Initial Urn 1 red and 1 blue ball
• Each period
• pick a ball from urn.
• replace ball
• add ball of that color

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Example
Initial (1,1) Pick Red (2,1) Pick Red
(3,1) Pick Blue (3,2) Pick Red (4,2) Pick
Red (5,2) Pick Red (6,2)
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Theorem

• Any proportion of red balls is an equilibrium of
  this process
• All proportions are equally likely as equilibria

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Set or Path Dependence

• In each period, the probability of each type of
  ball does not depend upon the order that the
  balls were chosen but on the set of balls chosen.
  
• RRB and BRR are equivalent
• Therefore, this is an example of sequential set
  dependence.

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All Paths Equally Likely

• Given a set all paths have same probability.
• Example set 6R and 2B
• RRRRRB
• (1/2)(2/3)(3/4)(4/5)(1/6) 1/30
• BRRRRR
• (1/2)(1/3)(2/4)(3/5)(4/6) 1/30

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Importance of Initial Path

• Effect on final equilibrium is larger for early
  draws, but..
• Knowing the second draw is a B is as informative
  as knowing the first is a B
• RR (1/2)(2/3) 2/6
• RB (1/2)(1/3) 1/6
• BR (1/2)(1/3) 1/6
• BB (1/2)(2/3) 2/6

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Initial Path Dependence

• Lock in.
• After period ten
• select a ball
• put in another ball of same color
• remove a random ball
• There exists a T such that after period T there
  is only one color ball in the urn

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Fully Path Dependent Process

• Initial Urn 1 red and 1 blue ball
• In period t
• pick a ball from urn.
• replace ball
• add 2t balls of that color
• Now each path gives a unique probability
  distribution over balls

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Path Dependent Outcomes
A process might exhibit path dependent outcomes
but not generate path dependent equilibria.
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Path Dependent Process

• Initial Urn 1 red and 1 blue ball
• In period t
• pick a ball from urn.
• replace ball
• add ball of the opposite color

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Example
Initial (1,1) Pick Red (1,2) Pick Red
(1,3) Pick Blue (2,3) Pick Red (2,4) Pick
Blue (3,4) Pick Blue (4,4)
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Theorem

• The proportion of red balls equals the proportion
  of blue balls in equilibrium of this process
• But the outcome in any period depends upon the
  previous outcomes.

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Empirical Testing I
Empirical testing is easiest when you have a
panel of data. Why? You can test for set
dependence and for path dependence of outcomes
in each period and as equilibria. Explicit way
of saying that history matters.
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Empirical Testing II
Outcomes have many dimensions. Some dimensions
may be path dependent and others may not be. We
need theory to tell us which. We can test the
theory by using panels.
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Empirical Testing III
Unless we understand what is path dependent and
what is not path dependent, we are likely to
overfit our models.
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Summary/Conclusions

• Four levels of models
• Level 1 aggregate rules
• Level 2 selection of types based on
  fitness/evolutionary game theory
• Level 3 intelligent adaptation agent based
  modeling
• Level 4 cognitive closure game theory
• Complex Systems
• heterogeneity
• learning
• networks
• externalities
• Specific Results
• Learning models/empirical considerations Tim
• Behavioral Voting BDT
• Networks Troy
• Interpretations/uncertainty
• Culture and Path Dependence

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Meta Conclusions

• Level 4 not the only way to do science
• Not Too Complicated
• chain saws and arrows
• Two Levels to the Diversity Results
• diversity trumps ability
• Toulmin and see above