Erlang or Stackless Python
I recently wrote a simple, event-driven, simulation framework to allow me to quickly prototype and test distributed protocols. Unfortunately it did not scale in the number of messages simulated. So I am planning to re-implement it.
To keep with the spirit of rapid prototyping, I want to use a dynamically typed language. My final two candidates are Erlang and Stackless Python (or rather PyPy). There certainly is a lot of buzz around Erlang these days: My advisor is very enthusiastic about it, and after reading through the tutorial and dummy protocols, I can see why. On the other hand, Stackless Python has the familiar syntax and the huge library of modules.
I read through many posts comparing the two and I finally decided to stick with Python for now. I will code up the framework (hopefully this week) and report back on my findings. However, I am personally still interested in coding something in Erlang, so who knows 
doors – one of them contains a prize, say a car and the other two door contain things you don’t care about, say goats. You choose a door. Then the TV host, who knows where the prize is, opens one door you haven’t chosen and that he knows has a goat. Then he asks if you want to stick to the door you have chosen or if you want to change to the other door. What should you do?
probability and in the other door with probability 
(note that the presenter can always open some door with a goat, so conditioning on that event doesn’t give you any new information).
doors. You choose one and the TV host open 
of them and asks if you want to change or stick with your first choice. Wouldn’t you change?
bucks or 
bucks. Then someone asks you if you wanted to pay 
bucks and change to the other envelope. Should you change?
, so I should change.
and one with 
, then my expected prize if I don’t change is 
. If I change, my expected prize is 
, so I should not change.
probability?
and 
(with 
). You pick one envelope at random and then you are able to look at the content of the first envelope you open and then decide to switch or not. Is there a strategy that gives you expected earnings greater than 
?
be a random variable on 
such that for each 
we have 
, for example, the normal distribution or the logistic distribution.
now, if 
say the other number is lower and if 
say the other number is higher. You get it right with probability
is:![\displaystyle \begin{aligned} &P(\text{picked }A) \mathop{\mathbb E}[Y \vert \text{picked }A] + P(\text{picked }B) \mathop{\mathbb E}[Y \vert \text{picked }B] = \\ & = \frac{1}{2} [P(X<A) A + P(X>A) B] + \frac{1}{2} [P(X<B) B + P(X>B) A] \\ &= \frac{1}{2}[A [P(X<A) + P(X>B)] + B [P(X>A) + P(X<B)]] > \frac{A+B}{2} \\ \end{aligned}](http://s.wordpress.com/latex.php?latex=%5Cdisplaystyle%20%5Cbegin%7Baligned%7D%20%26P%28%5Ctext%7Bpicked%20%7DA%29%20%5Cmathop%7B%5Cmathbb%20E%7D%5BY%20%5Cvert%20%5Ctext%7Bpicked%20%7DA%5D%20%2B%20P%28%5Ctext%7Bpicked%20%7DB%29%20%5Cmathop%7B%5Cmathbb%20E%7D%5BY%20%5Cvert%20%5Ctext%7Bpicked%20%7DB%5D%20%3D%20%5C%5C%20%26%20%3D%20%5Cfrac%7B1%7D%7B2%7D%20%5BP%28X%3CA%29%20A%20%2B%20P%28X%3EA%29%20B%5D%20%2B%20%5Cfrac%7B1%7D%7B2%7D%20%5BP%28X%3CB%29%20B%20%2B%20P%28X%3EB%29%20A%5D%20%5C%5C%20%26%3D%20%5Cfrac%7B1%7D%7B2%7D%5BA%20%5BP%28X%3CA%29%20%2B%20P%28X%3EB%29%5D%20%2B%20B%20%5BP%28X%3EA%29%20%2B%20P%28X%3CB%29%5D%5D%20%3E%20%5Cfrac%7BA%2BB%7D%7B2%7D%20%5C%5C%20%5Cend%7Baligned%7D&bg=T&fg=000000&s=0 "\displaystyle \begin{aligned} &P(\text{picked }A) \mathop{\mathbb E}[Y \vert \text{picked }A] + P(\text{picked }B) \mathop{\mathbb E}[Y \vert \text{picked }B] = \\ & = \frac{1}{2} [P(X<A) A + P(X>A) B] + \frac{1}{2} [P(X<B) B + P(X>B) A] \\ &= \frac{1}{2}[A [P(X<A) + P(X>B)] + B [P(X>A) + P(X<B)]] > \frac{A+B}{2} \\ \end{aligned}")
. Each player of the market controls one variable, say 
and is interested in maximizing one objective function 
. So, player 
is trying to set 
.
of commodities, 
of consumers and 
of producers. Now, we describe how to characterize each of them:
of commodities combinations he is interested (typically we take 
) and an utility function 
expressing his interest for this bundle of commodities. Consumer 
in a further restricted 
.
it has the capacity to produce.
, so for 
, 
is the amount of commodity 
that consumer 
, which is a vector indicating how much of each commodity originally exists in the market.
, 
, consumer 
shares of company 
.
. Consider the following scenario after prices 
are established to commodities:
, company 
, so each company will try to maximize its profit producing 
.
.
.
always exist that generates an equilibrium?
?
is an equilibrium, are there feasible 
such that 
and for at least one consumer 
? (This is called Pareto improvement)
and endowments 
to calculate the price vector 
. 
if there is 
such that: 
s.t. 
and 
, there is 
, 
such that 
. If 
, for example a linear function 
with some 
. In that case, 
and some player has money surplus and therefore he could increase his utility.
and 
. Now, under non-satiated utilities, by the last argument, we have that 
. Actually, if 
for each 
, then it becomes: 
, which means that the market clears:
is Pareto, i.e. there is no other feasible allocation 
such that for all 
with the inequality strict for at least one component. 
, because if 
then we could improve the utility of 
still within the budget, contradicting the optimality of 
for that budget. And clearly 
.
, what is a contradiction, because since 
and therefore 
. 
for 
. This gives us strong monotonicity and local nonsatiated preferences.
defined above: 
where 
is the bundle of best possible utility. Now, since we are using linear utilities we can’t guarantee there will be only one such bundle, so instead of considering a function, consider 
, i.e., 
is the set of all allocations that maximize 
. Since 
by 
and start buying in the cost-benefit order (the ones that provide more utility per buck spent). Most of the time there will be just one solution, but in points where 
, then 
is said to be upper hemicontinuous at the point 
if for any open neighbourhood 
of 
there exists a neighbourhood 
of a such that 
is a subset of 
in 
or: 
. To prove that we use 
. For any 
we know 
by the definition of 
is bounded from below, i.e., 
for some 
. Simply take 
with 
, then 
. That is clear from the fractional knapsack algorithm when one desirable item gets price zero.
is an upper hemicontinuous correspondence such that 
is a convex non-empty set for all 
has a fixed point, i.e., 
s.t. 
. 
-homogeneous, consider the simplex 
, its relative interior 
and the boundary 
. Now we define the following price correcting correspondence 
.
is set, it generates demand 
. For that demand, the price that would maximize profit would be 
, i.e. 
for all 
. It is natural to re-adjust the prices to 
:
– let’s give the intuition why this is true as we approach the boundary: if 
, then 
, therefore the demans explodes: 
such that 
. By the definition of 
. Now, I claim 
, still 
by Walras’ Law. So, if 
with 
and therefore 
for all 
, and 
. For this reason