Questions in Section General

Let `N(t)` denote a city population of individuals at time `t` (in years). If this population is subject to immigration, and if the initial population `N(0) = 1000000`, then `N(t)` can be given by the following equation: ` N(t)=1000000e^(lambdat)+v/lambda(e^(lambdat)-1)`.

Let `f(x)=cos(x)-x`.

  1. Write out the iteration formula of Newton's Method for this choice of `f(x)`.
  2. With initial approximation `p_0=pi/4`, compute the first five terms of the sequence `{p_n}_(n=1)^oo`. Use six digits of accuracy in your computations.
  3. What is `f(p_5)`?

The Borda Count is susceptible to strategic misreporting of preferences. Here are some examples to practice how this works.

  1. Suppose you are one of three people voting on a set of four alternatives named `A`, `B`, `C`, and `D`. The Borda Count will be used as the voting system. The other two voters have the rankings

Consider a betting market with two horses `A` and `B` and two bettors 1 and 2. Let’s suppose that each bettor has wealth `w`. Bettor 1 believes there is a probability of `1/2` that horse `A` will win, and a probability of `1/2` that horse `B` will win. Bettor 2 believes there is a probability of `1/4` that horse `A` will win, and a probability of `3/4` that horse `B` will win. Both bettors have logarithmic utility for wealth, and they each choose bets to maximize expected utility of wealth given their beliefs.

Imagine that you're advising a group of agricultural officials who are investigating measures to control the outbreak of an epidemic in its early stages within a livestock population. On short notice, they are able to try controlling the extent to which the animals come in contact with each other, and they are also able to introduce higher levels of sanitization to reduce the probability that one animal passes the disease to another.

Suppose a search engine has two ad slots that it can sell. Slot a has a clickthrough rate of 10 and slot b has a clickthrough rate of 5. There are three advertisers who are interested in these slots. Advertiser x values clicks at 3 per click, advertiser y values clicks at 2 per click, and advertiser z values clicks at 1 per click. Compute the socially optimal allocation and the VCG prices for it.

Let's consider the model of information cascades. Assume that the probability that the state is Good (G) is `p = 1/2`. The probability of a High signal given a Good state is `q = 2/3`. The probability of a Low signal given a Bad state is `q = 2/3`. Remember that each person observes a signal and the choices (but not the signals) of all those who chose before him. Each person chooses between Accept (A) and Reject (R). Suppose that you are the tenth person to make a choice and you have observed that everyone before you chose R, that is, we are in an R-cascade.

In the basic six degrees of separation question, one asks whether most pairs of people in the world are connected by a path of at most six edges in the social network, where an edge joins any two people who know each other on a first-name basis. Now lets consider a variation on this question. For each person in the world, we ask them to rank the 30 people they know best, in descending order of how well they know them. (Lets suppose for purposes of this question that each person is able to think of 30 people to list.) We then construct two different social networks:

Suppose that some researchers studying educational institutions decide to collect data to address the following two questions:

  1. As a function of `k`, what fraction of UC Denver classes have `k` students enrolled?
  2. As a function of `k`, what fraction of 3rd-grade elementary school classrooms in Colorado have `k` pupils?

Which one of these would you expect to more closely to follow a power-law distribution as a function of `k`?

Consider a product that has network effects. Consumers are named using real numbers between 0 and 1; the reservation price for consumer `x` when a `z` fraction of the population uses the product is given by the formula `r(x)f(z)`, where `r(x)=1-x` and `f(z)=z`.

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