# Proving the Mysterious Vickrey Auction

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Vickrey auction is known for its “too good to be true” property. This type of auctions is designed in such a way that bidders will bid at their true valuations. In other words, they will not lie! How does that even work? Without further a do, let us use a simple game theory to prove that.

## The simple game theoretic proof¹

Consider the a hypothetical bidder that we call Albert. Albert has a valuation of ** v** and a bid

**. Remember that in Vickrey auction, each bidder will have to submit a sealed bid of their valuations in a closed envelope (or anything that is sealed). Let the highest bid among other bidders be**

*b***. In this situation, Albert will win the item if his bid of**

*B***is higher than**

*b***and he will pay a value of B for the item.**

*B*One technique in mathematics that we use to prove let say x = y value is by checking the boundary conditions. As such, to prove that Albert will bid at b = v, we need to check the boundary conditions. There are two boundary conditions here. One where ** c** =

**>**

*b***and the other where**

*v***=**

*c***<**

*b***.**

*v*## c = b > v

In this simple set up, there are three possible scenarios:

- B > c > v
- c > B > v
- c > v > B

In scenario number 1, Albert will not win the bid. Since his bid of cis lower than the highest bid among other people, another person will win the bid. Changing ** c** to

**does not help either. He will always lose. As such, he might as well bid**

*v***from the very start. In scenario number 2, his bid of cis higher than**

*v***. As such, he will win the item and pays a total amount of money that is equal to**

*B***. However,**

*B***is higher than his true valuation of**

*B***. Therefore even though he wins the item, he will incur a loss of**

*v***-**

*B***! In this scenario, it is better to bid at v and lose on purpose! In scenario number 3, the same thing happens. Albert will win the item and pays**

*v***. His gain will be**

*B***-**

*v***because he only values the item as much as**

*B***. But in this case, he might as well bid**

*v***instead of c in the first place because the end gain is the same! Therefore when**

*v***=**

*c***>**

*b***, he has the incentive to bid**

*v***.**

*v*## c = b < v

In this simple set up, there are three possible scenarios:

- v > c > B
- v > B > c
- B > v > c

In scenario number 1, Albert will win the item and pays ** B** for it. In this case, his surplus or gain will be

**-**

*v***. As such, he might as well bid**

*B***instead of**

*v***in the first place because it will yield the same gain anyway. In scenario number 2, Albert will lose the item. However if he bids**

*c***instead of**

*v***, he will turn from a loser to a winner! He could have gotten the item. So, he should have bid with**

*c***in the first place. In scenario number 3, Albert will lose the item regardless. As such, he might as well go with**

*v***. Therefore when**

*v***=**

*c***<**

*b***, under all possible scenarios Albert has the incentive to bid v instead of some other value.**

*v***Q.E.D**

[1] The proof can be found on wikipedia page. I also used my university notes to simply it.