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.

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Eco and Soc Networks Notiz am Chapter Notes [2], erstellt von cheekymonky52 am 12/04/2013.