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Path Dependence

Path Dependence. Scott E Page University of Michigan Santa Fe Institute. Outline. Why we care Causes Outcomes or Equilibria Empirical Tests Conclusions. Why We Care. Efficiency History matters but how What can we predict . Causes. Multiple Peak Payoff learning evolution

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Path Dependence

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  1. Path Dependence Scott E Page University of Michigan Santa Fe Institute

  2. Outline • Why we care • Causes • Outcomes or Equilibria • Empirical Tests • Conclusions

  3. Why We Care • Efficiency • History matters but how • What can we predict

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

  5. Multiple Peaks Pure coordination game has two equilibria. Either one can be selected.

  6. 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 U’s AND decreases in the number of previous S’s.

  7. 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

  8. Local Interactions Senator A: Health Senator B: Environment Senator C: Defense

  9. 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

  10. Local Interactions Senator A: Health Defense Senator B: Environ Health Senator C: Defense Environ

  11. Path Dependence • Equilibria • Outcomes

  12. 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

  13. 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)

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

  15. 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. • RRBand BRR are equivalent • Therefore, this is an example of sequential set dependence.

  16. 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

  17. 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

  18. 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

  19. 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

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

  21. 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

  22. 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)

  23. 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.

  24. 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.

  25. 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.

  26. Empirical Testing III Unless we understand what is path dependent and what is not path dependent, we are likely to overfit our models.

  27. 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

  28. 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

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