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The claim here is going to be that bidding your true value
is a dominant strategy at a second-price auction,
and it's worth contrasting that with what happens in a first-price auction.
In a first-price auction, you would not want
to bid your true value because if you win the first-price auction at that bid
you pay your true value and you get zero surplus.
You always want to shade your bid a little bit below your true value,
but not in a second-price auction.
Second-price auction, the bid you should turn in
is simply exactly what you value the object at.
Question is, why.
And to see that, let's imagine alternatives.
First, consider bidding more than your value.
Now, this clearly seems like a bad idea, but let's see
exactly why it's a bad idea.
So why would it matter if you bid more than your value
rather than bidding your value?
When can that actually affect your payoff?
Well, it's only going to matter if your bid is the highest bid.
So it's got to be greater than the maximum of everyone else's
bid, and that highest of the other's bid is greater than your value
because that's the only time that you would win with a bid above your value
when you would not have won with a bid at your value.
But think about what winning means in that case.
If you win here, what you've done is you've
won by paying more than your value for the item.
So you've won the item, but you've actually
lost the auction because you paid too much.
So you never want to bid more than your value.
And that's no surprise.
What, perhaps, is a surprise is that on a second-price auction
you also never want to bid less than your value.
Let's think about that logic.
When would it matter if you bid less than your value rather
than your true value?
Well, it's only going to matter if you would have won the item by bidding
your true value and you don't win by bidding below your value.
So it only matters when the maximum of the other's bids
are somewhere between your value and your low value bid.
But think about what that means.
It means that you could have won had you bid your true value,
and you would have paid the maximum of the other's bids.
Remember, you don't pay your bid, you pay the maximum of the other's bids.
So in this case, you lost the auction when you should've won it.
You've given up some surplus that you didn't need to give up.
So what we've got now is you don't want to bid more than your value,
you don't want to bid less than your value, you just want to bid your value.
A very simple strategy here at a second-price auction.
And notice that in doing this, we never talked about how others are bidding.
It doesn't matter what others are doing, how
they're coming up with their bidding strategies.
You always want to bid your true value.
That's exactly saying that bidding your true value
is a dominant strategy in a second-price auction, and that was our claim.
That's what's going to happen at second-price auctions.
Everybody should bid their true value.
So who wins in the second-price auction?
Well, suppose we line these values up from highest to lowest,
and think about the bids.
Everyone's bid is just their value.
So clearly, the person with the highest value has the highest bid and they win.
Remember, they don't pay their bid, they pay the second highest bid,
which is the second highest value.
So at a second-price auction, the item is
sold to the person who values it the most
and they pay the second highest value.
Notice that that was the same outcome we had in the ascending bid auction.
The form of the auction was completely different,
but the outcome is exactly the same.
These two auctions are equivalent, second-price options
and ascending bid auctions.
We're going to see that both of these auctions are used.
You see ascending bid auctions all the time, on eBay for example.
Second-price auctions are actually also often used,
and they're used far more than you might imagine.
For example, Google uses a version of the second-price auction
in selling its keywords.
That's how they make all those billions of dollars.
And we'll come to exactly how they do that a bit later.