Nash equilibrium

I came across the concept of Nash equilibrium for games. As far as now I associated the name Nash to the movie "A beautiful mind", which I liked very much.
My professor of game theory at the university was so weird that I never really understood what he said. He looked like Bakunin and always had a stinky pipe in his hands.
But going back to the equilibrium, a game is said to be in such state when even if a player knew the strategies of all other players, he would not find any benefit in changing his strategy.
Described like this it sounds strange, but imagine I have two roads to drive to reach my office. If I know that everybody who wants to go there is taking one path and the other is empty (no traffic), I will choose the path without traffic. This game is not in Nash equilibrium, because I benefit by knowing other players' strategies. But if I know that the other drivers will equally distribute on both paths, than it does not really matter if I choose one or the other. This is an example of game in Nash equilibrium.

Probability distributions

I have to admit that my mathematics knowledge is really rusted... too bad!
But at least nowadays I do not have to search for information in my old books anymore: we have Google and Wikipedia! Hurraa!

The last thing which gave me troubles were probability distributions. I could still figure out what a discrete probability distribution could be (e.g. the probability of getting a '5' rolling a dice in the discrete interval 1 to 6 is 1/6), but I had no clue of what a continuous distribution is :(
Wikipedia helped again.
Probability of a continuous distribution is for example the probability that a tree leaf is 3,5cm long on a given continuous interval like between 3 and 4 cm.
With discrete distributions, the probability of an impossible event is 0 ( e.g. getting the result 3,5 on a dice), but the weird things with continuous distributions is that the probability can be 0 even if an event is not impossible!
Going back to the 3,5cm long leaf: how many chances do you have of finding such a long leaf? well... it is difficult to find out, because we can't count and add the probabilities.
Formally the value 3,5 has an infinitesimally small probability, which is statistically equivalent to 0. WOW!

eBay auctions on weekends...

I read an article "PENNIES FROM eBay: THE DETERMINANTS OF PRICE IN ONLINE AUCTIONS" by David Lucking-Reiley, Doug Bryan, and Daniel Reeves.
The article is quite old: it was written in 1999, when eBay was only 4 years old and available only in USA (I think, but I may be wrong).
What caught my eye was this chart!
It is a histogram of auction closings (doesn't matter wich kind) by day-of-the-week.
The authirs comment that "As one might expect for a consumer-oriented site like eBay, volume is heaviest on weekends."
This data contraddict my observations of what is going on today. I do not have as reliable data as they had, but the auctions I have seen are definetly less succesful on weekends!
Bidders tend to bid last minute (either they don't know the concept of proxy bidding or they refuse to use it) and on weekends they are not as much online as on weekdays from their offices!
I will try to investigate this points to get reliable data to check if I am right ;)

Payoff

I came across the concept of payoff... I never tought about it!
For example in a English (ascending-bid) auction the bidders who lose the auction have a payoff of 0. This makes sense, since they don't earn and don't pay anything.
What I never thought about is that the bidder who wins the auction has a payoff which is not equal to the amount he pays, but to the difference between how much he values the item and how much he pays for it!
Let's make an example: I always buy a beauty cream on eBay, which costs 19,99€ in the normal shops. If I can get that cream on eBay for 15€ (+1,85€ shipment), my payoff will be 3,34€.
The 3,34€ are the difference between my estimated value of the item and the money I pay for it on eBay. The question is: does it make sense to but the cream on eBay, take the risk of being cheated by the unknown seller or by the postal service which may lose the packet to save only 3,34€?
Well, for me yes, consider it is more than 15% discount on the shop price!
Perhaps the real question is if I really need the cream... but women can do almost anything for their beauty ;)

Trust

[eBay founder] Pierre M. Omidyar… hit on the idea of building a flea market in cyberspace – where people could buy and sell anything to anybody. There was one snag, though. How could he persuade complete strangers to trust one another enough to hand over merchandise or cash without ever having met? Omidyar’s solution was to devise a system where buyers and sellers can rate their experiences with different traders… That provided the assurance people needed to feel comfortable trading with one another – and it helped Omidyar’s eBay become the largest person-to-person auction site on the Web.

Source: Green and Browder [1998].

Study material

I am currently reading this: Why study auctions.

Most common auction types

Perhaps it's worth to mention that there are different kind of auctions and the most famous are:

1. the ascending-bid auction (also called English aution, open or oral) is perhaps the most famous, in which the bidders offers higher and higher prices until nobody offers more
2. the descending-bid auction (also called Dutch auction, since its most famous application is the sale of flowers in Nederlands) in which the price sinks until a bidder offers to pay the current price
3. the first-price sealed auction in which the bids are conveied in private form and the auction is won by the bidder who offers and pays the highest price
4. the second-price sealed auction (or Vickrey auction) in which the bids are conveied in private form and the auction is won by the bidder who offers the highest price, but pays just the price of the second highest bid plus a small increment.

There is an important theorem, the revenue equivalence theorem, that proofs that under some circumstances these 4 auctions have the same results. I am trying to understand it, but I am still not done (I need to refresh some mathematics :o !!!) I will let you know...