(High school) algebra is a set of definitions for working with equations - these equations apply to anything that can be counted or
measured
problem: you have some decision makers you want them to behave in a certain way -- but that depends on information you can imagine, but you don't have -- and they do! - eg, stockholders (owners) vs. managers (agency) - stockholders in principle have all rights to the company --
telling it what to do, and receiving any profits it makes
A mechanism has several parts: - uncertainty that managers know (or can learn), but stockholders
won't (until too late)
there are many definitions of "simple" for networks
for the purpose of economics, the simplest definition is "the network is a star network" - if the center is just a connection point, then it's like the
"Internet cloud": all users are equally distant from each other (one hop to the center, and one more to the other user)
in a star network, the "value" to user A is the set of other users she can reach via the network (note: this is not a number!) - it might matter who is connected -- that's why phone
companies offer "family plans"
in the 1980s and 1990s, economists often assumed it doesn't matter who is connected, only how many are connected (this is a number!)
if everybody is the same, then the value to society of N people in a network is W = N u(N), where u() is usually an increasing function in N
if u() is linear, u(N) = aN, and W = aN^2 - eg, profit to marketing with customers joining randomly,
each having the same probability of buying
thus networks have strong increasing returns to scale (if adding members is constant marginal cost)
returns to scale can be a barrier to entry (why aren't we all learning Chinese, when so many more people speak Chinese natively than English? because english is good enoughl; there's no need to learn chinese (most internationalized Chinese know English!)
Write an email, send to ises-hw@turnbull.sk.tsukuba.ac.jp. Subject is "ISES Network Homework".
3. What is the cost of connecting to the network? (In English...)
Not all networks are star networks
School networks: Oxford, Harvard, U. Tokyo have more "central" positions than graduates of other schools
Marriage networks of feudal lords Why did the Medici become so important in Italy? A network-based analysis - More connections by marriage than other families - The Medici formed a unique "bridge" between two parts of the
network - In a star network, the Center is the bridge between any two
"leaves"
- Being on the "shortest path" can give power
There are also global parameters of a network, that sometimes have surprising properties
- Diameter of the whole network
- Diameter of a circle: goes through the center, connecting two points on the circle - but networks may not have a well-defined center
- Diameter of a circle is the longest line segment that intersects the circle
- Define the diameter of a network as the longest "shortest path" connecting any two members of the network
- Useful fact: a random link will reduce the diameter of a "sparse" network dramatically.