Chapter Notes [2]

Ch 19 - Cascading Behaviour in Networks Individuals make decisions based on the choices of their neighbours. Each individual has a threshold level and exactly this fraction of more than this fraction of his neighbours adopt A then he too will switch to behaviour A. There is assumed to be a set of initial adopters and each node is assumed to have the same threshold level q = b/a+b. There are two equilibriums; when all nodes adopt A and when all nodes adopt B. Whether a complete cascade occurs depends on network structure, threshold level and initial adopters. Tightly-knit communities can stop a complete cascade but can be overcome by reducing threshold level by increasing payoffs of product A or by convincing a small number of nodes in the tightly-knit community to adopt A. Blocking cluster - a set of nodes which have a density p if every node in the subset S has at least a fraction p of its edges go to nodes S i.e. number of edges in S/total number of edges. If you have a blocking cluster with a density greater than 1-q then a complete cascade will not form and vice versa. If each node has a different threshold level, the role of threshold level plays an important role in choice of initial adopters i.e. choose those nodes that are connected to nodes with low threshold levels as they are seen to be easily influenced.

Ch 20 - Small World Phenomenon Watts-Strogatz Model - Assuming that all your friends are connected to new friends then the phenomenon that everyone is connected by a short path is quite understandable. However, assuming that triadic closure exists within a network this can be seen to limit the number of people you can reach by following short paths. Model shows how homophily creates a highly clustered network through triadic closure and how weak ties can create short paths. Two dimensional grid and each node can have two types of links; short links to neighbours within a distance of r and long range links to other random nodes. Clustering exponent can be used to determine whether two nodes v,w are connected by edge. If exponent is small then it is more likely to be a long range link i.e. more random but if exponent is small then it is more likely to be a short range link. The most optimal choice for exponent is 2 for decentralized search (how to find the short paths). Model can be adapted to not be placed in a grid i.e. using rank between two nodes which is the number of nodes closer to v then w, rank(w)-1 or social distance between two nodes which is the smallest size foci that both nodes share, s(v,w)-1.

Ch 21 - Epidemics 1. Branching Process - Disease spreads in waves. So the first person to enter into the network with the disease can pass on the disease to each person he meets with some probability, let’s say he meets k people. These k people then have a probability of infected a set of different people they meet, lets say they each meet k people so the maximum number of people infected become k2. The disease can die out after a finite number of steps if none of the k new people infected pass on the disease or it can continue infinitely. Basic Reproductive Number is the number of new cases of the disease caused by a single individual. If R < 1 then disease will die out with probability of 1 i.e. it is certain but if R > 1, the disease will permit with some probability greater than 0 i.e. it is not certain unless probability of infecting a new person is 1. 2. SIR Epidemic - Stages on the disease is susceptible, infected and removed. Progress of the epidemic is controlled by the probability of infection ‘p’ and the length of time an individual is infected ‘t’. 3. SIS Epidemic - There is no removed state meaning that there is not a bounded set of nodes. The epidemic can run for a very long time or if at any point all nodes are in the susceptible state then the disease will have died out. 4. SIRS Epidemic - Individuals can be temporarily immune but can become susceptible again after some time length. Transient Contacts - contact between two nodes can be between some time interval. Concurrency - a node is involved in two or more active partnerships that overlap in time. So disease can spread from both directions.

Market price represents the average belief and is seen as the probability of an event occurring. Horse Races is a market where there is exogenous desirability  meaning that the outcome of the race is not influenced by the bettors bets. How a bettor spreads his wealth between his bets for two horses A and B will depend on his beliefs on the likelihood of each horse winning i.e. P(A) = a and P(B) = 1- a. Also the odds placed on each horse by the track (market) i.e. OA and OB. To model a bettors attitude towards risk we must determine the bettors payoff which is defined as the utility function U(w). Utility functions should be increasing as wealth increase but at a decreasing rate i.e. as the individual becomes wealthier the payoff for the bet becomes smaller because they have more to lose! Therefore, we would expect a bettor to reject a fair gamble. Logarithmic utility i.e. U(w) = ln(w) where w represents the wealth the bettor will receive, ensures that the bettor receives that same payoff from doubling his wealth regardless of how much wealth he currently has i.e. ln(mw) - ln(w) = ln(m). For a single bettor, given that bettors are using the logarithmic utility function to evaluate their payoffs given the wealth they can gain from each bet, it can be shown that their choice of the fraction of their wealth to place on each horse is not dependant on the odds because of the property of the logarithmic utility function. Therefore, in order to maximise their payoff a bettor should bet their beliefs. For multiple bettors each bettor has their own belief on the likelihood of each horse winning, a set wealth which may differ and they all use the logarithmic utility function. Given that all bettors bet all of their wealth and the track wishes to pay out exactly how much was bet i.e. it does not wish to make any money so the amount that all of the bettors are paid given that horse A wins is the sum of anwnOA = w. It can be computed that the track should set the odds to the inverse of OA i.e. OA-1 and are known as the ‘state prices’. The state prices reflect how much a bettor must bet in order to receive one dollar after the race. Therefore, even if the bettor does not wish to bet all of his wealth he can bet the inverse odds in order to ensure he does not lose money in the case that the horse he does not think will win does actually win. The state prices are determined by the weighted average of the bettors beliefs for each horse. Each individual has as much power to influence the odds as the amount of they contribute to the total wealth i.e. a larger weighted belief which represents the bettors share of the wealth means more influence. Wisdom of the Crowds is the idea that aggregating individual beliefs can actually converge towards the true belief. For state prices which are influenced by the bettors, if each bettor has a belief that horse A will win which is independent and the sum of each bettors belief is equal to the true probability of horse A winning. Also, each bettor has the same share of the total wealth then as the crowd grows the state price for A should converge to the true probability of A winning. Furthermore, overtime as bettors watch the races their belief on the probability that horse A will win should converge to a as horse A should win a probability a of the total races.In stock markets, the amount that the an individual who invests will receive will depend the state that the company is in i.e. the future value of a stock depends on the state the company. If you know how much a stock is worth in each state and the state prices of a stock in each state you can work out the stock prices by summing up the worth*state price for each state. Rearranging this equation you can work out the state prices if given the stock prices and the worth of a stock in each state. Asymmetric Information - one side of the market knows more information than the other. Market for Lemons (seller and buyers) and Labour Market (employees are sellers of skills having different skill levels therefore producing a different amount of revenue for the company which determines their value and the max amount the company is willing to pay and employers are buyers of skills valuing each type of employee differently depending on how skilled they are) are two examples of markets that show endogenous desirability and represent a market where there is asymmetric information. Signalling is when the seller in a market provides some information about the quality of the item they are selling. Sellers with good signals cause buyers to raise their expected value of the item being sold to them e.g. in labour markets education can provide a good or bad signal. Ebay uses a reputation system to provide buyers with signals on the quality of the sellers. Ebay give each seller a reputation score by using evaluations from previous buyers.

Voting involves opinions or preferences not numerical values.Individual Preferences - preference relation is a preference between TWO alternatives. Each individual has a set of preference relations for all pairs of alternatives. From this you can develop a ranking for all the alternatives. CONDORCET WINNER is the alternative that beats are other alternatives in a pairwise majority vote. Voting Systems -       1. Majority Rule: For each pair of alternatives count how many prefer X > Y or Y > X and keep the one that has the larger number. From the final set of preference relations you can produce a group ranking. This method can be subjected to the CONDORCET PARADOX which is when even if individual preferences are transitive the group ranking may not be. This voting systems can be used as part of an elimination tournament when the first pair of alternatives are compared and the winner is compared to the next alternative etc until there is one winner left. These tournaments can be arranged in a number of ways so therefore can be manipulated using strategic agenda setting.     2. Positional Voting: Each alternative is given a 'weight' depending on its position in each individual ranking, the alternatives are then ordered depending on their total weight to define the group ranking. This method can be used with the Borda Count where there are k alternatives and the top ranked alternative is given a weight equal to k - 1 and this is decreased by one for each lower ranked alternative with the last ranked alternative receiving 0. However, this method can be manipulated as the final group ranking depends on how the alternatives are ranked further down the list not including the highest ranked alternative in an individuals ranking, known as strategic misrepresenting of preferences. Arrow's Impossibility Theorem states that there is no voting system that can be free of pathologies i.e. has the following two principles, pareto principle and independence of irrelevant alternatives, that does not use dictatorship. A special circumstance where majority voting method can be used as the undesirable properties of pathologies and codercet paradox do not occur, which seems to meet both of the principles and does not use dictatorship is SINGLE PEAKED PREFERENCES which is used when voting on alternatives that correspond to numerical quantities or a linear ordering. In this voting method, each individual has a top favourite known as the 'peak' and all other alternatives 'fall off' on BOTH sides of this peak. In this circumstance no individual would benefit from lying about their ranking due to the medium individual favourite and the group ranking produced is complete and transitive. 

Ch 19

Ch 20

Ch 21

Ch 22

Ch 23