model_thinking.2

Model Thinking
==============

https://www.coursera.org/modelthinking/class/


Section 2 - Segregation and Peer Effects
========================================


Sorting and Peer Effects Introduction
-------------------------------------

People that live in the same area tend to look, act, think alike 
	sorting - homophily - we choose to hang out with people who are like us
	peer effects - we start to act like the people we hang out with

	both of these create groups that look alike

	study and understand these effects using multiple models - unexpected results
		schelling's tipping model
		granovetter
		standing ovation

		identification problem - sorting or peer effects

we will use agent-based models in this section
	we study agents (individuals), 
	who follow rules (have behaviors), 
	which result in outcomes

	contrast with equation-based models

	game theory studies individuals behaving optimally
	we will study agents behaving normally


Schelling's Segregation Model
-----------------------------

(spatial segregation model)

studied racial and income segregation in NYC

choosing where to live - should i stay or should i go?
	actor - person living in a neighborhood (the middle of a tic-tac-toe grid)
	behavior - (threshold based rule) 
		move if < 33% of neighbors are like me
	outcome - ended up with significantly higher segregation than expected

	key insight - micromotives !== macrobehavior
		what you see at macro level may not represent what's going on at micro level 

		33% threshhold yields 72% segregation
		52%        -->        94%
		80%        -->        no equilibrium, continuous churn

	tipping phenomenon - 
		exodus tip - one person moving out causes another person to move out
		genesis tip - one person coming in causes another person to move out


Measuring Segregation
---------------------

TID - total index of dissimilarity
	how segregated is a city - rich people (blue) vs poor people (yellow)

	| b/B - y/Y |
		determines how distorted the distribution is on a given block
		b = # blue on block, B = blue total
		y = # yellow on block, Y = yellow total

		measure each block individually
		sum the block measurements 

		rationalize to number between 0-1 
			divide by number of populations in test - in this case 2 (blue, yellow)

	test cases for model
		case where all blocks are 50/50 - TID = 0
		case where half of blocks are all blue, other half all yellow - TID = 2/2 = 1


Consider a city consisting of four city blocks of equal population. One block consists of all rich people. One block consists of all poor people, and two blocks consist of half rich and half poor people. What is the index of dissimilarity?

B = 20                - total population of rich people
Y = 20                - total population of poor people

10/20 - 0/20  = 10/20 - all rich people
0/20  - 10/20 = 10/20 - all poor people
5/20  - 5/20  = 0     - 50/50 mix
5/20  - 5/20  = 0     - 50/50 mix

20/20 = 1             - sum of all blocks
1/2 = 0.5             - divide by # of populations in test

There are equal numbers of rich and poor people. For the block of all rich people, the contribution to the index of dissimilarity equals |1-0.5| = 0.5. For the block of all poor people, the contribution to the index equals |0-0.5| = 0.5. For the other two blocks the contribution equals |0.5-0.5| = 0. Therefore, the index of dissimilarity equals (0.5+0.5)/2 = 0.5. Don't forget to divide by two.


Peer Effects
------------

Contagion effect/phenomenon
	sometimes the tail wags the dog
		the people at the ends of the distribution (extremists) really drive what happens
		makes these things incredibly difficult to predict

Granovettor's model
	N individuals
	each has a threshold (e.g. for joining in on a collective activity)
		Tj for person J
		Join if Tj others join

	growth determined by successive threshold triggering

	collective action more likely if
		lower thresholds
		more variation in thresholds

		need to know 
			avg level of discontent
				sum(T0..Tj)/N
					determine individual thresholds
					sum the thresholds
					divide by N
			distribution of discontent scores

		are there a quantity of people willing to act, and are they connected?

	collective action will not happen if nobody has a threshold of zero

use to model virality, growth, adoption...

Suppose that there are ten people who have the following thresholds for joining a volunteer project: Two will volunteer even if no one else does. Six require five others. And two will volunteer so long as anyone else does. How many people will volunteer total?

0, 0, 1, 1 will act
5, 5, 5, 5, 5, 5 will not act

Correct answer is 4. The two who will volunteer regardless of others, volunteer Since these two volunteer, the two that will volunteer as long as at least one other person has volunteered will also volunteer. However, the other 6 will not volunteer because they will only volunteer if 5 others have volunteered and only 4 have. So 4 of the 10 will volunteer


Standing Ovation Model
----------------------

builds on granovettor
allows us to look at threshold models and peer effects in more subtle ways

two decision points (whether or not you're going to stand up)
	initially, at end of performance (quick judgement)
	after seeing larger audiences decision

	thus, this has 
		peer effects
		information effects
			ovation gives you info about the quality of the show

threshold based on quality (granovettor threshold was based on quantity)

rules
	initial rule
		stand if S > T

		T - threshold to stand
		S = Q + E - signal (quality plus error/diverity)
			Q - quality of performance 
				Q = T means that 50% of the people will stand
			E - either error OR diversity
				error - variation in what we perceive, noise
				diversity - everybody brings different background/experiences

	subsequent rule
		stand if > X% stand (peer effect)

claims
	1 - the higher the Q, the more people stand
	2 - the lower the T, the more people stand
	3 - the lower the X, the more people stand
	4 - if Q < T, then the more variation in E, the more people stand

example
	N = 1000
	T = 60
	Q = 50

	E = 0 --> S = 50
		nobody stands
	E = [-15, 15] --> S = 35-65
		only those whose E > 10 will stand
	E = [-50, 50]
		those whose E > 10 will stand
		those whose X <  will stand

what causes a big E - 
	audience - diverse, or unsophisticated
	performance - multidimensional, or complexh

variance in X
	big X - people really secure in who they are
	small X - people ready to jump on any bandwagon


If X%, the percentage of others you need to stand, is less than 50%, does increasing the noise increase the chance of a standing ovation if Q is bigger than the threshold? (Hint: Draw the picture)

X < 50 
Q > T
E

No. Q>t means that more than half of the audience perceive the quality of the show to be above their threshold (more than 50% will stand up). if X% is less than 50%, there will always be a standing ovation if Q>t regardless of the variance.


advanced ovations

	factors not modeled in the first example
		theatre
			who you can see/who can see you
				those in the front ("celebrities") can't see many other people
					they influence many others, but are not influenced by others
				those in the back ("acdemics") can see almost everybody else
					they don't influence others, but are influenced by others
		grouping
			people in groups more liekly to stand if one member stands
				groups or pairs will exhibit localized peer effects
				leading to larger and more probable ovations

how do we increase probability of ovation?
	higher quality (claim 1 - bigger Q)
	lower threshold (claim 2 - lower T)
	larger peer effects (claim 3 - lower X)
	more diverse (claim 4 - bigger E)
	use celebrities
	big groups

fertility
	collective action/participation problems
		who are celebrities in that context?
		what groups would have an impact?
	academic performance
	urban renewal
	fitness/health


The Identification Problem
--------------------------

how do you tell whether something happened because of Schelling or standing ovation?
	sorting (homophily) or peer effect?

	some things are clearly sorting
		neighborhood segregation
		middle school kids - four groups white/black boys/girls

	some things are clearly peer effect
		soda/pop/coke

	why we can't tell from looking at the picture
		two types of people - A's and B's
		two example poulations - AABBAA, BBABBA
			sorting - minorities would migrate
			peer effect - minorities would assimilate 

			both effects yield AAAAAA and BBBBBB

	in sorting, we can actually see (find evidence) people move

	have to have dynamic, micro level data
		you can't tell if you just have the snapshot