Non-Negative Function

f(x) is a non-negative function defined on [a,b]

⇔

f(x) ≥0 for all x in [a,b]

Cramer-Rao Lower Bound

いかなる最尤推定量もCramer-Rao Lower Boundで与えられる推定誤差の分散より小さくなることはない。

w2.gakkai-web.net/gakkai/ieice/vol4no1pdf/vol4no1_32.pdf

Based on - synonyms

according to

as shown by

as evidenced by

Bertrand Model

Model
Two firms with the same marginal cost (MC=c), producing a homogeneous product and competing in prices.

homogeneous product → consumers purchase from cheapest firms

Bertrand's theorem
Let (p1*,p2*) be a Nash Equilibrium. Then, p1*=p2*=c.

If each firm has a different marginal cost c1, c2 (c1<c2), the Nash equilibria are (p1, p2) such that c1≤p1=p2≤c2.

http://en.wikipedia.org/wiki/Bertrand_competition

RHS/ LHS

RHS: right-hand side

LHS: left-hand side

Dominance and Nash Equilibria

Definition

Strategy s is strictly dominant if strategy s strictly dominates every other possible strategy.

Strategy s is weakly dominant if strategy s dominates all other strategies, but some are only weakly dominated.

Strategy s is strictly dominated if some other strategy exists that strictly dominates s.

Strategy s is weakly dominated if some other strategy exists that weakly dominates s.


Important!

1. A strictly dominant strategy must be played in Nash Equilibria.

2. Strictly dominated strategies cannot be played in Nash Equilibria.

3. Weakly dominated strategies may be played in Nash Equilibria.

4. If, after completing iterated elimination of strongly dominated strategies, there is only one strategy for each player remaining, that strategy set is the unique Nash equilibrium.

5. If, after completing iterated elimination of weakly dominated strategies, there is only one strategy for each player remaining, that strategy set is also a Nash equilibrium. (The Nash equilibrium found by eliminating weakly dominated strategies may not be the only Nash equilibrium.)

 http://en.wikipedia.org/wiki/Strategic_dominance

Integration by Parts

http://en.wikipedia.org/wiki/Integration_by_parts

Homothetic/ Quasilinear Preference

Homothetic Preference

A monotone preference relation such that if x ∼ y, then αx ∼ αy for any α ≥ 0.

Quasilinear Preference (with respect to commodity 1)

A preference relation such that

1. if x∼y, then (x+αe1)∼(y+αe1) for e1 = (1,0,...,0) and any α ∈ R.

2. Good1 is desirable; that is,x+αe1 ≻x for all x and α>0.

Note that commodity 1 can be negative (commodity 1 ∈ R).

Mas-Colell, Whiston and Green "Microeconomic Theory" (p.45)

Upper/ Lower Hemi-Continuous

1. f: X → Y is upper hemi-continuous if

  1) it has a closed graph and

  2) the image of is compact.


2. f: X → Y is lower hemi-continuous if

  1) x_n ∈ X converges to x* ∈ X and

  2) y* ∈ f (x*),

then there exists a sequence {y_n}_n and N such that y_n ∈ f(x_n) for all n ≧ N, and y_n converges to y*.


3. f: X → Y is continuous if it is both upper and lower hemi- continuous.

www.econ.yale.edu/~go49/pdfs/ECON 201/NoteUHC.pdf

Open/ Closed Map

A function f : X → Y is open if for any open set U in X, the image f(U) is open in Y.

A function f : X → Y is closed if for any closed set U in X, the image f(U) is open in Y.

http://en.wikipedia.org/wiki/Open_and_closed_maps

Function/ Map/ Image/ Range/ Domain/ Codomain

f: X → Y

f: Function (Map)

X: Domain

Y: Codomain

Image: In mathematics, an image (range) is the subset of a function's codomain which is the output of the function on a subset of its domain.

http://en.wikipedia.org/wiki/Map_%28mathematics%29

http://en.wikipedia.org/wiki/Image_%28mathematics%29

http://en.wikipedia.org/wiki/Codomain

Graph of a Function

In mathematics, the graph of a function f is the collection of all ordered pairs (x, f(x)).

f: X → Y

G(f)={(x,y)∈ X×Y|y∈ f(x)}

http://en.wikipedia.org/wiki/Graph_of_a_function