So, one nice thing about first-price auction sealed with options,

is you can have people bid asynchronously.

So a lot of procurement auctions might be done this way.

So, for instance, the government might say maybe no,

you've been on a certain contract.

Put your bid in an envelope and send it to us.

Actually in procurement auctions if you're trying to sell something,

the government usually is the lowest bid that wins.

But those can be held to asynchronously.

Everybody can just mail their things in then you open up this the things and

whichever is the winning bid is unpicked out of the envelopes.

Dutch auctions in contrast you get to have the people together and

sitting there watching the clock go down.

But the nice thing about those auctions is you don't need as much communication in

the sense that, all you need to do is have this thing drop and

then somebody just say yes.

So only one bit needs to be transmitted from the bidders of the auctioneer so

it's very efficient in terms of the information and communication.

You have the clock dropping down and someone saying give it to me.

So, you know, there's different auctions in term of actual implementation, okay.

How should people bid in these auctions?

Well, as we mentioned just a few minutes ago bid less than your valuation.

Because in basically deciding how low to bid is going to be the tricky part because

you've got a trade off.

The lower that you make your bid, the lower the amount you pay, but

also the lower the probability that you win.

So you're trading off probability of winning against amount that you're

having to pay.

And so now, you don't have a dominant strategy.

How low you want to go actually depends on what other people are doing.

So, if I think other people are going to bid very very low,

then I'm going to be willing to lower my bid a lot.

If I think other people are going to be bidding fairly aggressively,

I might have to keep my bid higher.

So I can't make my decision on how I should bid without knowing exactly what

the bid of others are.

And so, now we don't have a dominate strategy,

I don't have one thing which is best to do regardless of what other people are doing.

I have to think about what they're doing in calculating my bid, okay?

So let's have a quick look at an equilibrium of one of these and

see how we can at least verify something in the equilibrium and

then we'll talk a little bit about how you might find the equilibrium in these

auctions which is not going to be always extremely easy.

So let's think of a first price option, so

let's start with a very simple case to analyze first.

Two bidders, both risk neutral, and

they get an independent draw from a uniform distribution at 0,1, okay.

So each of these two bidder, so we've got two bidders,

each person uniforms year one valuations and those are drawn independently.

And the claim here is that, if we look at,

we want to get a Bayes-Nash equilibrium of this game, what are we going have?

The bids are just going to be each person drops their value by half, so

if my values three-quarters, I'll say three-eighths.

If my value is a half, I'll say a quarter.

So I just take whatever my value is and I just shade it by a half and that's my bid.

And so let's go through and just verify that that's the Bayes-Nash equilibrium so

we want to check whether this is equilibrium, so let's,

given the symmetry here, we can check for one of the bidders.

So let's suppose that the other bidder is actually, whatever their value is,

they bid a half of it.

And now, we think about bidder one, and let's let bidder one choose

a strategy as to one of how high they're going to bid as a function of their value.

So, if I'm bidding s1, so let's suppose I put a bid of s1 in and

the other bidder is bidding half of their value.

So when am I going to win?

I'm going to win when half of v2 is less and s1 or v2 is less than 2s1.

Okay, and what's my payoff then?

My payoff is V1 minus S1.

But I'm going to lose whenever that the other individual bid

when v2 over 2 is bigger s1 or v2 is bigger than 2s1.

And then I get a utility of 0, okay.

So in terms of figuring out what my value is, there's cases where I win the auction.

So I can integrate over those, where v2 is up to 2 over s1, and

then I'm getting this, and otherwise,

if the value of the other person is bigger than 2s1, I'm going to get 0.

So this is my expected utility, the simple integral,

integrate that thing, what do you get?

You get 2v1 times s1- 2s1 squared, okay.

Very simple expression so

we have an expected utility as a function of what my s1 is.

Conditional on the other person following this prescribed strategy.

So let's differentiate that with respect to s1 set at equal to 0.

So when we do that what do we end up with?

We end up s1 = a half v1 as being the solution to setting that equal to 0.

So if you want to maximize this and you want to check the second order conditions,

you want to maximize this indeed here we ended up with a condition where my optimal

bid given that the other person is bidding half their value is to bid half my value

as well.

Okay, so and again, the calculation for

the other person is exactly symmetric to this.

So this is Bayesian mash equilibrium on this game.

So what we've got is people bidding down and they're trading off a calculation.

Implicitly, in terms of what we're doing here,

if we want to look at the calculation that we were doing, by increasing your or

by decreasing your bid, the gain is that you pay less conditional on winning but

also, win less of the times.

So the trade-off is coming at your winning less of the time but

then you're paying less when you do win.

And so that trade-off is exactly captured to this maximization problem and

you want to shade your bid by half.

Okay, this is obviously a very narrow result.

We did two bidders, uniform valuations.

So we need to solve for this thing because this is not incentive compatible.

It's a direct mechanism in terms of dominant strategies, right.

So it's not a dominant.

We have no dominant strategies here.

We're not getting dominant strategy incentive compatibility we need to solve

for the equilibrium.

And more generally you could solve for if we had n bidders instead of two,

what would the equilibrium look like then.

It's going to look like instead of shading your bid by a half,

the general formula is going to be n-1 over n right.

So when n is 2, this is a half.

When n is 3, then it's going to be two thirds.

When n is 4, you go up to three quarters and so forth and so,

you keep climbing in terms of how much your bidding and what's happening.

Well, it gets harder and harder to win with more and more people in the auction.

You going to have to bid closer and closer to your value to have a chance to win.

And then the trade off is just that, as you lower your bid much beyond

something that's very close to your bid you have no chance of winning.

And so you're going to end up having this trade-off be closer to the actual value.

You can go through and do the calculation.

It's going to be very similar to what we did,

just a different calculation in terms of integrals.

Now you're going to win when everybody's below you,

not just the other person below you.

So that's a slightly more complicated integration, but

basically exactly the same logic we just did for the two bidder case.

Okay, so broader problem.

What we did here is we only verified that this was an equilibrium.

So we guessed the equilibrium and then verified it.

How do you actually find an equilibrium in these cases?

Well if it's not a nice uniform distribution, and

guessing that it's going to be a linear function of your value, then you've

got to guess more complicated functions and there's going to be basically some

integration problems which will give you some ideas of how this works for

certain distributions, for arbitrary distributions the bidding functions can be

much more complicated functions and basically will only be described up to

some integration conditions, so there are papers that give some solutions to this.

There's a paper by Milgrom and Weber in 1982.

There's some other papers out there which will give some background on solving these

kinds of auctions.

The first price auction has actually been solved quite early on in literature.

More generally, solving any of these ones where we don't have

strategies is going to involve some guesses and in some cases,

there might be some integration conditions that will give us nice solutions,

but they often won't exist in closed form.

And especially, here also, the symmetry helps a lot.

So if you have asymmetric auctions things can be difficult.

And more generally,

even finding whether equilibria exists in these worlds is not an easy problem.

There's literature, in fact that I've worked on a bit,

about existence of equilibria in auctions.

So one thing about equilibria in auctions is they're discontinuous.

I change my bid a little, and

I might be going from winning the object to not winning the object and

so suddenly my pay off goes from being positive to being zero.

And that this continuity means that I might not always

have a nicely defined best responses that are going to

move in ways that were used implicit proofs of existence before.

Made nice use of the best response correspondence, with that, it's not

going to have the properties it used to have if this continues games and getting

equilibrium existence in these kinds of auction settings it's quite tricky and

there's somethings that are known about setting to do exist equilibria.

There are also some examples where they don't exist equilibria so

that's actually an interesting project on it's own.

So, different kinds of auctions, different kinds of equilibria,

they're related to each other and we'll see more about that in a bit.