Out of all commonly described types of auction, the Vickrey auction seems to be the one that sounds the weirdest, most complex and least useful. It is defined as type of auction using offers sealed in envelopes. As one would expect, the highest bid wins the auction, but, weirdly enough, the winner does not have to pay the price he offered, instead he/she pays only the price that has been offered in the second highest bid.
While this is obviously not the most effective type of auction from the view of the seller, around 30 billion dollars were reassigned using this type of auction (and derived types) in the year 2010 and this sum is probably rising every year. The reason for this is the fact the Google, Yahoo and other mayor players at the field of internet advertising are using this type of auction (or more specifically generalized Vickrey Auction). Thus, every time you browse internet, dozens of Vickrey actions happen to determine which specific advert you will see. Using modified Vickreys auction, Google claims to aim for win-win-win situation by reaching ideal ratio of the advertiser’s benefit, Google’s profit and thanks to some modifications to the auction also user’s experience, as this auction might not be the most profitable for the seller, but it is socially-optimal. 
Vickrey auction is often also called Second Price auction because of the above-mentioned fact that not the highest, but second highest price is to be paid by the winner of the auction. The concept of this auction was first thoroughly described by the Canadian Economy Nobel Prize holder William Vickrey in 1961. It was however used much earlier, for example, it is used by stamp collectors since 1893.
Aside from its use in specific real-life scenarios, it is very interesting for theoretical research and demonstrating several matters commonly mentioned in Game Theory. 
Vickrey auction some interesting perks that make this type of auction very popular in academic and other theoretic circles. First of all, this type of auction is known to be truth revealing, or more specifically, the equilibrium strategy for this auction is truth revealing. This is because the equilibrium strategy for Vickerey auction is to offer the true value of the auction’s object, thus reveal the ‘truth’. This is also the dominant strategy (weakly). The true value of the object also includes secondary considerations of the value, such as the possible loss of profit in case a bidder competitor wins the object.
The first point leads to second – if all bidders (players) follow the equilibrium (dominant) strategy, it will lead to the maximal possible economic efficiency for all of the bidders taking part in the auction. There is no possible loss of value for those who did not win the auction, because there is no chance that a bidder will wrongly estimate the completion and overpay, unlike in case of First Price (English) sealed offer auction. This is however theory, while in the macro scope these traits of Vickrey auction are being observed, it can not be said the real value is going to unfold during just one instance of Vickrey auction. 
The interesting truth-revealing trait of Vickrey auction was discovered by famous German poet Johann Wolfgang Goethe. Following situation also provides an example of Vickrey auction instance:
""I am inclined to offer Mr. Vieweg from Berlin an epic poem, Hermann and Dorothea, which will have approximately 2000 hexameters…. Concerning the royalty we will proceed as follows: I will hand over to Mr. Counsel Bottiger a sealed note which contains my demand, and I wait for what Mr. Vieweg will suggest to offer for my work. If his offer is lower than my demand, then I take my note back, unopened, and the negotiation is broken. If, however, his offer is higher, then I will not ask for more than what is written in the note to be opened by Mr. Bottiger.""
(the only difference is the Goethe is not willing to pay less than he demands) 
Proof that Vickrey action dominant strategy is to offer real value of the auction’s object
As mentioned, the truth revealing property of Vickrey action is what makes it really interesting and sometimes useful. Let’s prove this statement:
When dealing in Vickrey auction, it is a dominant strategy to offer the real value of the object.
Proof. John wants to buy a car in an auction. The value of the car to John 100$. John is considering to offer more than the car’s value, let’s say 105$. The other bidders highest bid is to John practically a random number. The auction can end in three ways for John – a) Someone else offers more, b) John offers the most, someone else offered more than is the perceived car value (eg. 102$), c) John offers the highest bid, the second highest bid is less that the car’s value to John (eg. 98$).
In first case a), John is not getting the car. In case b), John gets the car, but he is overpaying the perceived value of the car – even though he is only paying the second highest bid. That would not happen if he only offered the car’s perceived value (Case c)). Thus, it is more reasonable to offer only the perceived value, as offering more can only lead to overpaying. Offering less than the perceived value is a similar case. This also show the main benefit of Vickrey auction – the bidder is never overpaying if playing the dominant strategy. This makes it useful in cases where the auctioneer profit is not the most important aspect of the auction, as other aspect has to be acknowledged, such as stability of a network of happiness of players. 
Uses of Vickrey auction
Even though Vickrey auction is not as know or popular as other, more straightforward types of auctions, it still sees numerous uses in real-life, especially in environments where auction is processed automatically, extremely quickly and where socially-optimal outcome is preferable to maximization of the auction owner profit.
As already mentioned, Vickrey auction in its modified version (Vickrey-Clark-Groves auction, to be more thoroughly described later) the main mechanism that Google uses to sell advertising in macro-scope. 
Word of Warcraft Auctions
Word of Warcraft is a well-known online Massively multiplayer online role-playing game (MMORPG) published in 2004 but still somehow popular. Players are often part of guilds and need specific items to achieve their in-game goals. As items are distributed somewhat randomly among players, players need to exchange them for different items, or, more commonly, for in game currency. Such exchange can be done though auction set in a player’s guild. Different types of auctions can be used, but Vickrey Auction is especially popular.
In such in-guild auction, an item is offered to the other guild members. After that, a standard Vickrey auction takes place: All interested players offer a price, after which the second-best price is paid to the highest bidder. Compared to the other commonly used system – fixed prices – this allows better effectivity of the guild and higher joy from the game. 
A modified version of Vickrey auction, Vickrey-Clark-Groove auction, is used in one of the methods to fulfil networking request in the field of Network Routing. A scheme to assign a route through nodes in the network is necessary, as the nodes are not always able to fulfil all request given their technical limitations. Such nodes are known as “Selfish” nodes and are programmed to fulfil something that could be described as their utilities. The are pairs in the network which need a path with a given bandwidth to be assigned to them, and nodes are able to offer this bandwidth. The pair need to offer a payment to the nodes for the needed bandwidth. Given the fact that such transactions are done with automatically set prices, the Vickrey auction is a suitable option, because it prevents the pairs from overpaying and allows the most efficient paths in the network to be found. 
Closed groups auctions
Vickrey auctions is sometimes used in groups where the seller prefers the group welfare and happiness to his own profit. As this requires an amount of solidarity, this is not so common. Of of such groups are philatelist groups (stamp collectors), where Vickrey auction is the preferred type of auction since the year 1893 when it was first used.
Derived types of auctions
From Vickrey auction, another several types of auction can be derived to simplified it or use it more real-life scenarios.
Uniform Price auction
Vickrey auction is not suitable for trading divisible goods, such as water, oil or so. A modified version of Vickrey auction exists to be used when auctioning such goods. This type of auction is called Uniform Price auction. In such auction, every contestant declares how many pieces of good he wants and how much he is willing to pay for these pieces. After that, the number of pieces that is the subject of the auction is divided between the highest bidders according to their declared demands. The winners however do not pay their bid, but (mostly) only the lowest winning bid. Other versions of this auction exist, such a version in which the winners pay the highest non-winning bid or the highest winning bid. The latter has been successfully used to sell some of marketable treasury securities by the USA National Treasury. 
The uniform price auction is not as interesting as usual Vickrey auction, as in neither of the mentioned cases, the dominant strategy is to offer the real valuation of the auction’s object – this happens only in case a single, non-divisible object is being sold. 
Already mentioned, Vickrey-Clark-Groves auction (abbreviated as VCG auction) is a derivative of Vickrey auction, or technically speaking, Vickrey auction is a derivative of Vickrey-Clark-Groves auction, even though Vickrey auction has been described and used first. Vickrey–Clarke–Groves auction allows the seller to sell multiple items at once, unlike Vickrey auction.
In Vickrey–Clarke–Groves auction instance, a finite number of identical items is being sold. All bidders make offer, in such manner so that others do not see how much they are offering. The bids could also be described as (N, P) pair of numbers, where N is the desired number of items and P is the price for all these products.
After all offers are set, all possible combinations of bids are calculated by the auction owner (in practice this is often a computer system that calculates everything in an instant). Out of these calculations, the one which would mean the highest profit for the auction owner is selected and items are distributed to the bidders who offered the best price per item until there are any items left. For example, three people (John, Stacy, Clarinda) want three oranges. John offers one dollar for one orange, Stacy offers four dollars for one orange and Clarinda offers four dollars for three oranges (but refuses to have only one or two oranges). Even though Clarinda offered better price per orange than John, Clarinda is not going to get any oranges, because the sum of Johns and Stacy’s offers is better than Clarinda’s offer.
The bidders however do not need to pay the price they offered in their bid. They pay a different number instead, a number that is called Harm. This Harm is calculated a difference between the sum of bids of the auction from the second-best combination of bids and what other bidders have bid in the current combination of bids.
Using this principle, Vickrey-Clark-Groves auction allows to use the most important propriety of Vickrey auction – the truth revealing – in more macroscopic sense. This allows to auction to be socially-optimal. This is desirable in automatically functioning net of macroscopic size, such as mentioned network routing or internet. 
Even though some partial examples have been provided above, here you can find another examples to sum up the topic.
A historical painting is being auctioned. There are four bidders, John, Abraham, Simon and Nathaniel. Each of them offer a price they are willing to pay. They submit the offers in closed envelopes so the other cannot see how much they are willing to pay. The auction master opens the envelopes and sees that John offered 400 dollars, Abraham offered 450 dollars, Simon offered 350 dollars and Nathaniel offered 500 dollars. The painting goes to Nathaniel, because he offered the best price. However, Nathaniel only needs to pay 450 dollars, because 450 is the second best price offered.
Uniform Price auction
There are 50 apples and three bidders for these apples, Jack, Isaac and Sean. Jack offers 5 dollars per apple and wants 30 apples; Isaac wants 20 apples and offers 8 dollars per apple. Sean wants 15 apples for 7 dollars per apple. There offers are sealed so it is not possible to see other bidders offer. To determine who will get the apples, we need to calculate to maximum possible profit in case full price was paid – which is 30 apples for Jack and 20 apples for Isaac. They are however not going to pay the price they offered, but only 5 dollars per apple – because it is the second best price offered. In another version of Uniform Price auction, they might also pay 7 dollars per apple because it the best price that did not win.
Let’s reuse the scenario of Uniform Price auction so the difference is clearly visible: There are 50 apples and three bidders for these apples, Jack, Isaac and Sean. Jack offers 5 dollars per apple and wants 30 apples; Isaac wants 20 apples and offers 8 dollars per apple. Sean wants 15 apples for 7 dollars per apple. There offers are sealed so it is not possible to see other bidders offer. To determine who will get the apples, we need to calculate to maximum possible profit in case full price was paid – which is 30 apples for Jack and 20 apples for Isaac. They are however not going to pay the price they offered – and the price they are going to pay is the difference between Vickrey-Clark-Groves auction and Uniform Price auction. The price each of them is going to pay is calculated as a difference between the offers of the other two. Jack offered 150 dollars totally, Isaac offered 160 dollars and Sean offered 105 dollars. Thus, Jack is going to pay 160 – 105 = 55 dollars, while Isaac is going to pay 150 – 105 = 45 dollars.
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