- Mathematics: the scientific use of symbols (abbreviations)
- What about "1 + 1 = 2"?
- Husband + wife = family (including children > 0!)
- If you watch closely, you see that a child is born -- for a period the family is only 2, and the appearance of 3 or more is a biological process, not "math"
- "1 + 1 = 2" is about a certain class of objects/situations, and we define others cases to be "not math"
- "1 + 1 = 2" is a fact (theorem) of set theory!

(High school) algebra is a set of definitions for working with equations - these equations apply to anything that can be counted or

measured

- but some sets are literally unmeasurable: they must be excluded from consideration, even implicitly if a function is written f(x) for x in [0,1], then {x | f(x) < y} must be measurable for all y, or f cannot be integrated

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

- Eg, micro uncertainty vs macro uncertainty: if the company makes a loss, is the manager bad, or is the economy bad?

A mechanism has several parts: - uncertainty that managers know (or can learn), but stockholders

won't (until too late)

- choices that managers make, leading to outcomes that stockholders value
- choices have costs for managers
- stockholders can convert outcomes into lower costs for the managers
- we define an outcome function: determines the outcome given the uncertainty and the manager's choice
- the manager's behavior is an optimal choice, given the information have, and the reward they expect to get
- the reward is a function of the outcome (but not the uncertain information, because the stockholder doesn't know it)
- the stocjkholders have a choice function, which determines the outcome they want to get for each condition of information (they can imagine this, but don't know the condition)

- We talked about representing functions as "arrows" connecting one "object" (usually a set) to another "object"
- In mathematics, this is called a "directed graph" (there may be a connection from A to B, but not from B to A: consider links on the web, or one-way streets)
- There are also undirected graphs (if you can go from A to B, you can always go back from B to A)
- Graphs are used to represent networks (eg, "jinmyaku", or the Tokyo rail network, or the telephone network)

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 phonecompanies 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".

- Give
*one*network you are connected to (not the phone company, not the Internet, and not Microsoft Office). - Does your u(N) have decreasing, increasing, or constant returns to N?

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.