How do European bond auctions work?

In this post I described how the US government performed bond auctions.

The US government uses uniform pricing for its bond auctions.  Most European governments use discriminatory pricing for their bond auctions.

This post will describe how discriminatory price auctions work.

The table below from here shows that most European countries use discriminatory pricing for their bond auctions.  Some countries, like Italy and Slovenia, use both types.

Country	Auction Type
Austria	Custom
Belgium	Discriminatory
Cyprus	Discriminatory
France	Discriminatory
Germany	Discriminatory
Greece	Discriminatory
Hungary	Discriminatory
Ireland	Best price using Bloomberg auction system
Italy	Discriminatory for short term notes, Uniform for long term notes
Lithuania	Discriminatory
Macedonia	Discriminatory
Malta	Discriminatory
Norway	Uniform
Poland	Discriminatory
Portugal	Discriminatory
Slovenia	Uniform for short term notes, Discriminatory for long term notes
Sweden	Discriminatory
Switzerland	Uniform
Turkey	Discriminatory
United Kingdom	Uniform for index-based auctions, Discriminatory for conventional auctions

In uniform price auctions, like those done by the US government, all the winners receive the same price; hence the name uniform.  In discriminatory price auctions this is not the case.  The winners will pay different prices based on their bids at the auction.

For example, let's assume the French government is going to auction €100 billion of one year bills.

The bids are placed by stating the total amount to buy as well as the maximum price for a €100 note.  Below are the hypothetical bids placed for this auction:

Bid 1: €20 billion at €99.
Bid 2: €20 billion at €95.
Bid 3: €50 billion at €92.
Bid 4: €50 billion at €90.
Bid 5: €70 billion at €85.

A total of €210 billion is received in bids for an auction of €100 billion which makes the bid to cover ratio (€210/€100) 2.1.  (Please see here for explanations of bid-to-cover, compete versus noncompete, and other technical aspects of bond auctions).

The bonds are awarded to bidders as such:
Bid 1: €20 billion at €99.
Bid 2: €20 billion at €95.
Bid 3: €50 billion at €92.
Bid 4: €10 billion at €90.
Bid 5: No bonds awarded.

Notice that unlike the uniform auction, bidders will be awarded bonds at different prices depending on their bids.Computing the yield for this auction is slightly more complicate than computing it for a uniform price auction.  The yield is computed using a weighted average.  The weighted average price for this auction would be €93.8 ( 93.8 = (20/100 * 99) + (20/100 * 95) + (50/100 * 92) + (10/100 * 90 ).  Thus the yield for this auction would be 6.6% (€6.2 / €93.8 * 100).

Sources here, here, and here.