Set, Path, and Phat Dependence (speaking math to metaphor)

Set, Path, and Phat Dependence(speaking math to metaphor) Scott E Page University of Michigan Santa Fe Institute

Unofficial Coauthors Aaron Bramson, Jenna Bednar, John Jackson, Ken Kollman, Burt Monroe, Nolan McCarty, Carl Simon, Ken Zwick, Anonymous Undergrad

Big Picture • ``History Matters’’ • Unpacking Path Dependence • Markov Processes, Chaos, Independence • Path Dependence vs Phat Dependence • Outcomes vs Equilibria • Increasing Returns

New Idea Some situations may be characterized by increasing returns and or positive feedbacks and these may create path dependence. Some scholars even equate path dependence with increasing returns.

Path Dependence Examples • Technology: QWERTY (David) • Policy Feedbacks: Health care (Hacker) • Common Law (Hathaway) • Institutional Choices (North) • Trust & Social Capital (Putnam)

Closer to Home • Ann Arbor: Largest public university • Jackson: Largest four walled prison Cities went on different paths.

Outline • Dynamical Systems • Recursive Processes • Lots of Urn Models • An Externality Model • Big Think

Dynamical System • Set of States S • Outcomes X • Outcome function G(S) -> X • State Transition Function F(S,X) - > S • Deterministic or random

Example: Bad Parenting Set of States S: {Happy, Irate} Outcomes X: {Candy, Art Project} Outcome function G(S) -> X G(Happy) = Art, G(Irate) = Candy State Transition Function F(S,X) -> S F(Happy, Art) = 1/2 Irate, 1/2 Happy F(Irate, Candy) = Irate F(Happy, Candy) = Irate F(Irate, Art) = Happy

Basic Markov Theory A1: Finite set of states A2: Transition probabilities between states fixed A3: Can get from any state to any other through a sequence of transitions

Basic Markov Theory A1: Finite set of states A2: Transition probabilities between states fixed A3: Can get from any state to any other through a sequence of transitions Result: System attains a unique stable equilibrium.

Example A1: Finite set of states {engaged, bored} A2: Transition probabilities between states fixed If engaged become bored with probability 1/4 If bored become engaged with probability 1/2 A3: Can get from any state to any other through a sequence of transitions? Yes

Math t+1 engaged bored engaged 3/4 1/4 t bored 1/2 1/2

Math t+1 engaged bored engaged 3/4 1/4 t bored 1/2 1/2 Equilibrium: engaged = 2/3

Implications If history matters (in the long run) then one of the three assumptions - finite states - fixed transition probabilities - can get from anywhere to anywhere else must be violated.

Recursive Function Outcome at time t, x(t) Outcome function F: X -> X Example: F(X) = x+2 1,3,5,7,9,11,…..

Chaos 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 arbitrary amounts. Example: Tent Map

Chaos 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 arbitrary amounts. Example: Tent Map This is not path dependence!

The Amazing Urn Models

Basic Urn Model Urn contains balls of various colors. The outcome equals the color of the ball selected.

Urn 1: Bernoulli • Assumptions • U = {B blue ,R red} • Select ball and return • Result • P(red) = R/(B+R) • Outcomes independent

Urn 2: Polya • Assumptions • U = {1 Blue, 1 Red} • Select and return • Add new ball that is the same color as the ball selected

Polya in Action

Polya Results • Any probability of red balls is an equilibrium and equally likely • Any history of B blue and R red balls is equally likely

Polya Results • Any probability of red balls is an equilibrium and equally likely • Any history of B blue and R red balls is equally likely P(RBBB) = (1/2)(1/3)(2/4)(3/5) = 1/20 P(BBBR) = (1/2)(2/3)(3/4)(1/5) = 1/20

Polya Oversell The Polya Process is the canonical example of path dependence. It is also used to show why increasing returns (drawing red balls increases the likelihood of red balls) creates path dependence.

States, Paths, Phats Process is State Dependent if outcome probabilities depend upon some finite set of states Process is Path Dependent if outcome probabilities depend upon the sequence of outcomes Process is Phat Dependent if outcome probabilities depend upon the set of outcomes {p,a,t,h} but not their order.

Examples How much weight I gained over lunch is phat dependent on the food I ate. The order is irrelevant. Where I went for lunch was probably path dependent.

Polya is Phat The Polya process has an infinite number of possible combinations of red and blue balls so it cannot be state dependent. However, outcome probabilities do not depend upon the order in which outcomes occurred. They only depend on the set of outcomes that occurred.

Phat Combinatorics Thirty rounds of the Poly process creates over one billion possible paths but only thirty one possible sets of outcomes. Paths = 2*2*2*2*2…*2 Sets = {0,1,2,…30}

Question: Is Phat dependence a sufficient condition for multiple equilibria.

Urn 3: Balancing • Assumptions • U = {1 Blue, 1 Red} • Select and return • Add new ball of opposite color as ball selected • Rotation Schemes

Balancing in Action

Equilibria vs Outcomes Outcome probabilities in the balancing process are phat dependent but the equilibrium distribution is unique. The balancing process converges to equal probabilities of choosing red and blue balls.

Note Well The fact that an outcome depends upon the past set (or order) of outcomes does not imply equilibrium dependence. History can matter at each moment but not matter in the long run. Example: Manifest Destiny, Railroads

Increasing Returns? Polya process may not be path dependent in the strongest sense, but it does exhibit increasing returns. Are increasing returns sufficient or necessary for path dependence?

Urn 4: One Sided Polya • Assumptions • U = {1 Blue, 1 Red} • Select and return • Add another Blue if a blue is selected

Urn 4: One Sided Polya • Assumptions • U = {1 Blue, 1 Red} • Select and return • Add another Blue if a Blue is selected P(Blue) ->1

IR Not Sufficient Blue clearly satisfies increasing returns. If a blue is selected it increases the probability that a blue is selected in the future. That is also true of selecting a red ball. Picking a red ball increases (relatively) the probability of selecting a red ball in the future.

Cute or Relevant? An example like this might be just a cute mathematical trick or it could point to a relevant insight. Here it does the latter.

Cute or Relevant? An example like this might be just a cute mathematical trick or it could point to a relevant insight. Here it does the latter. At the turn of the past century, gas, electric, and steam cars all had increasing returns, yet it is well accepted that gas would win because it had much larger increasing returns.

Urn 5. Double Polya • Assumptions • U = {1 Blue,1 Red, 1 Blue Sq, 1 Red Sq.} • Select and return • Add new ball of same color but other shape

Double Polya in Action

What We Know So Far • Increasing returns neither necessary (Double Polya) or sufficient (One Sided Polya) for path dependence. • (negative externalities are) • Polya Process not even path dependent

Set, Path, and Phat Dependence (speaking math to metaphor)