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Path Dependence

Something is path dependent if its outcome probabilities depend on the sequence of past outcomes.

If there  is path dependence a Markov process is not running as the outcome probabilities change.

Urn Models

Polya Process - when a ball of one colour is drawn out the ball is returned along with a ball of the same colour. Any probability of balls of a given colour is an equilibrium and equally likely. Also, any history of balls of one colour (say red) and balls of another colour (say blue) is equally likely.

Balanced Process - when a ball of one colour is drawn it is returned along with a ball of the opposite colour. This converges to equal percentages of balls of each colour.

A path dependent outcome is where the colour of balls in a given period depends on the path.

A path dependent equilibrium is where the percentage of balls in the long run depends on the path.

The polya process has both path dependent equiliria and outcomes whereas the balancing process only has path dependent outcomes.

PHAT processes have outcome probabilities whose that depend on the outcomes but not the order of the outcomes. The polya process is one such process.

Number of possible paths >> number of sets of outcomes.

A Markov process is not path dependent because it has fixed transition probabilities.

Chaos depends on extreme sensitivity to initial conditions. It is not path dependent.

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Co-Operation and Culture

Pure Co-Ordination Game There are two players. If they both do the same thing (co-ordinate) they both receive an equal payoff. If they don’t they both get nothing.

In an N-person co-ordination game is a Lyapunov function where the increment is 2.

Inefficient Co-Ordination is where the payoff for one sort of co-operation is less than the payoff for another sort of co-operation.

Axelrod’s Culture Model 

1. Pick a person

2. Pick a nighbour

3. Pick traits

4. Probability of interaction between person and neighbour = % of similar traits.

This leads to different cultures with thick boundaries. The thick boundaries occur because if they were not thick people would interact.

Bednar et al Model 

People do not only co-ordinate but also become consistent across features and also innovate or try something different. Small innovations can lead to quite a bit of variability across a culture. 

The process of co-ordination, becoming consistent and innovating can become a Markov process, due to the “error” associated with innovation.

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Externalities

An externality  is an action that materially affects someone not involved in that action.

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Lyapanov Function

A Lyapanov function tracks the level L of a resource R at time t.

If: 1. There is a finite resource (R0 < infinity and Rt >= 0);

2. The level of resource decreases with time (Rt+1 =  L(Rt). if L(Rt) != Rt, then Rt+1 < Rt). and 

3. The level of resource drops by at least a certain amount (There exists an amount X such if L(Rt) != Rt , then L(Rt) < Rt - X), then L is a Lyapanov function and will converge to a point where L(Rt) = Rt in finite time.

In other words a function will reach equilibrium if there is an amount below or above which it cannot go and if it changes by at least a finite amount each time.

A pure exchange market is an example of a Lyapanov function. 

HOTPO (half or three plus one) is not a Lyapanov function but can sometimes go to equilibrium. 

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Economic Growth Models

These notes come from notes on growth models as part of Scott E Page’s course Model Thinking, provided through Coursera.

Simple Growth Model 

1. Output (O) is consumed (E) or invested (I) so O = E + I

2. If there are L workers and M machines the output function with respect to the number of machines is F(M) = L*sqrt(M).

3. Machines depreciate at the rate d so Mt+1 = Mt +It - dMt

Equilibrium is reached when I/O*sqrt(M) = dM

Solow Growth Model

Output can be changed by improvement in technology.

L = labour, K = capital and A(t) = technological sophistication parameter that varies over time. 

F(L,K,A(t) =  A(t)*L^beta*K^(1 - beta), beta is between 0 and 1.

Innovation makes capital and labour more productive.

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Diversity Prediction Theorem

crowd average = individual average + diversity

crowd’s error = average error - diversity

or (c - theta)^2 = (1/n*sum i = 1 to n of (si - theta)^2) - (1/n*sum i = 1 to n of (si - c)^2) 

where c = crowd average, theta = true value and si are individual predictions.

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Replicator Dynamics

Replicator dynamics is how the proportion of types of something changes. Useful in psychology and economics for studying learning, and in the study of evolution.

Let each ith type of 1,2,3..,N types have a payoff of pi[i] and occur in proportion Pr[i]. 

The replicator equation is Pr[i]t+1 = Pr[i]t*pi[i]/sum of j=1 to N of Pr[j]*pi[j].

Fisher’s Theorem: The higher the variance, the higher the rate of adaptation.

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The Logic of Network Formation

Three ways of formation:

- random - nodes and edges form randomly;

- small world - edges in clusters of near neighbours with few links to other clusters; 

- preferential - nodes preferentially connect to nodes with more connections. 

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Network Structure

Networks are also graphs so have nodes (points) and edges (lines) 

Degree of a node - how many edges connected to a node.

Degree of a network - average degree of all nodes = 2xedges/nodes

The average degree of neighbours on network will be at least as large as the average degree of the network.

Path length - the minimum number of edges to get from one specified node to another.

Average path length - total path length/number of edges

Clustering coefficient - the percentage of triples of nodes that have edges between all three nodes.

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The Rule of 150

This note is from Gladwell, M, The Tipping Point, 2000 (2014 reprint), Abacus, London.

Groups of humans do not perform very well as a group if the group is bigger than 150.

This was the conclusion of anthropologist Robin Dunbar, who studied the size of societies of various primates and related it the mean brain size of each species studied.

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Diffusion Models and Susceptible-Infected-Susceptible Models

Diffusion Models involve contact and spreading (eg the transmission of a disease or a rumour in a population). Suppose the population has N individuals and at time  t the number in the population infected is Wt. Let the transmission rate be beta and the contact rate be m. In a given time period the number of people affected will be:

N*m*beta*(Wt/N)*(N-Wt)/N

So Wt+1=Wt+ N*m*beta*(Wt/N)*(N-Wt)/N

or Wt+1=Wt(1+m*beta*(N-Wt)/N)

The greatest rate of increase is where Wt is approximately N-Wt

This is not a tipping point.

A Susceptible-Infected-Susceptible Model on the other hand can involve a tipping point. 

In this model a person may recover from an infection so again become susceptible. Therefore using the notation above

Wt+1=Wt+ N*m*beta*(Wt/N)*(N-Wt)/N - aWt

where a is the rate of recovery.

So, if m*beta(N-Wt)/N - a  <= 0 then the infection will not spread. The tipping point is where this is zero. Early in infection this is almost m*beta - a

ie. if m*beta/a > 1 the infection will spread. 

R0 = m*beta 

For vaccination

Let V = proportion vaccinated and r0 be the altered infection rate

r0 = R0*(1 - V). We want this to be <=1

ie. 1 - 1/R0 <= V.

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Percolation Models

These track the spread of something such as water in the soil, forest fires or bank failures. Percolation can be modelled as squares on a board. If two neighbouring squares are filled in, percolation will occur, otherwise it will stop. A tipping point occurs when the density is at about 59.3%.

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Linear Models

Models of the form Y=k+aX1+bX2+....

Find line of best fit by finding the coefficients that will give the least variance. 

As with categorical models Rsq = 1 - total variance of model/variance of all Y values.

Output of statistical analysis packages on regression model is typically:

                               Coef  SE p-value

Intercept

X1

X2

SE is the standard error of each coefficient

p-value is the probability that the coefficient has the wrong sign.

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Categorical Models and R-Squared

These notes are a summary of Scott E Page’s notes as part of the Model Thinking course offered through Coursera. The link to these notes is here:

https://d3c33hcgiwev3.cloudfront.net/_5252a1f8d86ec782d86e207b917b190e_modelthinking_03.06_Categorical_Models.pdf?Expires=1547510400&Signature=huNGd0K49ifUO-i0CIGujfEVU6bbReZsppcwKNp8eh2mv3~1vi2VM2wJ635RAEOpYUPEljFxHiZxgF4mqP5uH1vzMO8JzhNoPzzU7cwgyKT2aoaWXEZpkYdqpdVuvic67ID6nDrGugZQ4W9qeXyOQjdwSN6MrbQ4T6GJq0GT-Zs_&Key-Pair-Id=APKAJLTNE6QMUY6HBC5A

Categorical models are those which are based attributes of a sample where the sample is split into categories.

The goodness of such a model can be measured by R-squared (Rsq). It measures the proportion of the variance explained by the model.

Rsq = 1 - (total of variances of a variable in each category/variance of whole sample).

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Models Involving People

These notes are from Scott E Page’s lectures as part of the Model Thinking course offered through Coursera. The lectures that include this material are here:https://www.coursera.org/learn/model-thinking/home/week/3  

Rational Actor Models involve an individual optimising something (eg, profit)

An example is revenue = quantity x price. Let price = 50 - quantity (say).

Therefore, revenue = quantity(50 - quantity). 

To optimise revenue, set quantity = 25.

Another example is optimising consumption and donations from an income.

Suppose we want to optimise (consumption)^0.5x(donations)^0.5 = (consumptionX(income-consumption)) ^0.5. Split consumption and donations 50/50.

In a decision the objective depends only on the actions of an individual.

In a game the objective depends on the action of others.

Behavioural Models take account of human behaviour not always being rational

Prospect Theory states that people are risk-averse when making decisions about gains but risk-taking when making decisions about losses.

Hyperbolic Discounting means that people discount more in the near future than they do in the distant future.

Status Quo Bias is where people will not take an action because they do not want to change.

Base Rate Bias involves people guessing something similar to an answer to what they gave when asked about something else immediately beforehand.

Rule-Based Models  are those where people follow a rule (eg. Granovetter and standing ovation models)

There four main types of rule-based models - fixed decision, fixed game, adaptive decision, adaptive game.

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Decision Models

These notes are from Scott E Page’s lectures as part of the Model Thinking course offered through Coursera. The lectures that include this material are here:https://www.coursera.org/learn/model-thinking/home/week/2 

 Decision models can be used in a normative manner to aid making a decision. They can also be used to predict behaviour.

There are two types of decision models.

Multi-criterion models use criteria to make a choice. These include spatial models.

Probabilistic models involve assigning probabilities to choices. Markov models are probabilistic.

Decision trees are probabilistic decision models. Each branch should have a value and some will have probabilities as well. To make the decision:

1. Draw the tree;

2. Write down costs/payoffs and probabilities for branches; and

3. Solve backwards.

You can use decision trees to infer the probabilities of outcomes based on returns. Let the probability of an outcome be p, the cost be c and the return be r. For an action to be profitable p.r - c(1-p) > 0.  

Value of information = return with information - return without information.

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Aggregation

These notes are from Scott E Page’s lectures as part of the Model Thinking course offered through Coursera. The lectures that include this material are here:https://www.coursera.org/learn/model-thinking/home/week/2

More is Different Often things are not merely or do not behave simply like the sum of their individual parts. Attributed to Philip Anderson.

Preference Aggregation is not transitive.

Models can lead to four possible states if run for a long time:

- equilibrium;

- period;

- random; or

- complex.