Introduction

A central topic of economics is price finding in markets.
How does a buyer and a seller agree on the price of a good?

Supply / demand for BIG markets

Supply-demand for BIG markets

Supply-demand model works well for big markets.
We have lots of buyers, sellers, lots of info.
Predictable market price, balancing supply and demand.
Only sensible price at which a buyer and seller can trade.

What about difficult markets?

Today we focus on markets with few buyers or sellers, few trades, or non-standardized goods.

Gov't sells radio frequencies to broadcasters.
A painting by a famous artist trades hands.
An oil well goes on sale with unknown amount of oil.

A simple model

Selena wants to sell one item of a good.
Two buyers, Alice and Bob.
Each buyer has a secret valuation $v_A$ and $v_B$ :
how much they are willing to pay for the item.

How does Selena optimally sell the item?

Let's assume...

All of Selena's income is profit.
Alice and Bob each have random valuations between 0 and 1.
The valuations are secret but their distribution is well-known.

We saw the notion of an abstraction already in game theory class: we lose a bit in realism, but gain in structure which allows us to make precise statements, and develop a richer theory. The intuition and conclusions still apply in real life.
In particular, we are assuming that each of the three characters has a certain amount of information about the valuations of the other players, namely the distribution they are sampled from, and they can use this info to strategize their actions.
Assuming that each valuation $v_i$ follows a uniform distribution between 0 and 1 simplifies Selena's life, because she knows that for any value $x\in [0,1]$ , the probability that a bidder's valuation is below $x$ is exactly $x$ , i.e., $P[v_i\leq x]=x$ .
However, assuming a uniform distribution is just for pedagogical reasons: most results in auction theory can be extended to more general distributions.

Price posting

Simplest solution: Selena posts a price $p$ , sells item to first taker.
What is her optimal price $p$ ? One maximizing her expected revenue.
Expected revenue: price $\times$ prob. of making a sale.
For price $p$ , prob. of making a sale is $(1-p^2)$ .
Hence her expected revenue is $p \cdot ( 1-p^2)$ .

If both buyers accept the posted price, we assume the firm selects a random buyer to trade with.
The proof of this and all other math formulas can be found in the slides, for anyone interested.
Assuming that 100% of the sale price is profit for the seller, this profit is
- $p$ in case of selling, $0$ otherwise, where
- $(1-p^2)$ is the probability of selling the item.
- To see this probability, consider conversely that the probability that none of the buyers accepts the price must be $p^2$ , because each valuation must be below $p$ , and these are two individual events each happening with probability $p$ (recall: valuations are uniformly distributed between 0 and 1).
$Pr{v_i \leq x}$ is the probability that the randomly drawn value of valuation $v_i$ is below $x$ .
Independent values will be important: when is this (not) a sensible assumption? (Example: oil drilling rights auctions, ...)
Uniform distribution is just for pedagogical reasons; standard model allows for more general distributions.

Price posting

At the optimum point, expected revenue is $\approx 0.38$ .
We will see that an auction can do better.

Auctions

Lesson summary

Auctions can be superior to posting a price.
We discuss four important auction formats.
How to make buyers reveal their secret valuations.
Bid shilling.
The revenue-equivalence theorem.
Guides on what auction format to use in practice.

Auctions

Alice and Bob (bidders) submit bids based on own valuations.
Selena (auctioneer) selects winner to allocate the item to.
The winner's payment is a function of the bids.

An auction is a competitive game for buyers, where the seller makes the rules.

Auction formats

There are two broad classes:
1. Static Auctions: bidders submit their bids at the same time.
2. Dynamic Auctions: bidders submit bids over time.

The difference is whether or not bidders can react to the bids of others and adjust their own.
Static auctions are also called sealed-bid auctions.

Auction formats

Four auction formats we discuss today:

Static auctions
- with a first-price payment rule
- with a second-price payment rule
Dynamic Auctions
- with ascending price (a.k.a. English auctions)
- with descending price (a.k.a. Dutch auction)

Example, Selena auctions off a painting. Suppose in a static auction, Alice bids 1000 and Bob bids 2000. In a first-price format, Bob wins and pays 2000. In a second-price format, Bob still wins but only pays 1000.
The English format became popular in England. Selena continuously raises the price as long as at least 2 bidders accept it. As soon as only one bidder remains, they win and pay the current price.
The Dutch format became popular in the Netherlands, particularly in the tulip market. Selena starts at a high price and continuously lowers it as long as there are no takers. As soon as a bidder accepts the price, they win and pay the current price.
We are gonna analyze each of these four formats, and take the opportunity to discuss relevant notions in auction theory along the way.
For me, the second-price auction is the least intuitive format. But it turns out to be one of the nicest and easiest to analyze, so we will start there.

Second-price auction

Why make the winner pay the second highest bid?
Least intuitive format, but strategically simplest for bidders.
Simply bid your valuation truthfully!
A bidder wants to maximize their expected profit:
- own valuation - price paid, in case they win
- zero, otherwise

Second-price auction

Truthful equilibrium: a dominant strategy to bid truthfully.
Dominant strategy: outcome is at least as good as the outcome of any other strategy, no matter what the other bidder bids.

We can immediately say that you should never overbid (i.e., bid above your valuation), because in that case your profit can only be zero (if you lose) or negative (if you win). So the key question is whether you should ever underbid.
In the case that Bob's bid is higher than your valuation, bidding truthfully or underbidding both lead to losing and having a revenue of zero.
But if Bob's bid is lower than your valuation, any bid between these two values leads to you winning and having the same profit (namely the difference of these values), hence there is no gain in underbidding. However if you underbid too much, you run the risk of losing and having zero profit.
Hence in both cases we see that underbidding does not increase your profit, and can only decrease your chance of winning. So it is better to bid exactly your valuation.
The proof can be easily generalized to any number of bidders (to obtain that in a second-price auction, it is a dominant strategy for each bidder to bid truthfully.

TIP: use arrow chars in unicode for style: https://www.htmlsymbols.xyz/arrow-symbols

Second-price auction

Expected revenue

Selena's expected revenue is expected value of 2nd highest valuation.
For two independent variables uniformly sampled from $[0,1]$ , the expected value of the minimum is $1/3\approx 0.33$ .
This is not quite as good as posting a price (which provided expected revenue $0.38$ ). Why not?

Reserve price

Because the format we considered is not optimal for the auctioneer!
The optimal auction involves a reserve price r>0:
- If no bid is above $r$ , nobody wins.
- If one bid is above $r$ , the payment is $r$ .
- If both bids are above $r$ , the payment is the second-highest bid.

Reserve price

Fact: Under any reserve price $r$ , it is still optimal to bid truthfully, and if Selena sets $r=0.5$ , her expected revenue is $\approx 0.42$ , so it is better than posted price (where expected revenue was $\approx 0.38$ ).

Reserve price

The proof idea is that if $r=0.5$ , half of the time one valuation is above it and the other is below it, and the reserve price increases the paying price. On the other hand, if both valuations are below $r$ then there is no sale, which decreases Selena's revenue, but this case only happens one quarter of the time. Finally, if both valuations are above $r$ , the presence of a reserve price does no affect the result. Overall, there is a positive net gain in Selena's expected revenue. Below we present the formal proof.
The probability that both valuations are below $r$ is $r^2$ , the prob. that $r$ is in between the valuations is $2r(1-r)$ , and the prob.that both valuations are above $r$ is $(1-r)^2$ . You can check that these probabilities sum up to one.
In the first case Selena's revenue is zero, and in the second case it is $r$ . Hence the expected revenue for the first two cases is $2r^2(1-r)$ .
To compute the expected revenue of the third case, we simply copy the integral formula for the expected value (from a couple of slides ago), but we restrict the integration limits to $[r, 1\]$ : $\begin{align} \int_r^1 x\cdot f(x) dx &= 2\int_r^1 (x-x^2)dx\ &=2\cdot \left[\frac{x^2}{2}-\frac{x^3} {3}\right\]_r^1 \&=\frac{1}{3} - r^2 + \frac{2}{3}r^3. \end{align}$
Putting all terms together, we obtain that the expected revenue is $R(r)=0+2r^2(1-r)+\left(\frac{1}{3} - r^2 + \frac{2}{3}r^3\right)=\frac{1}{3}+r^2-\frac{4}{3}r^3.$
To maximize the auctioneer's expected revenue function R(r) above:
- We evaluate the function at all the local extrema (minima and maxima).
- Since the function is continuous and differentiable, the local extrema are found at the borders $r=0$ and $r=1$ , and at the roots of the derivative $R'(r)=2r-4r^2=0$ . These roots are $r=0$ and $r=1/2$ .
- By inspecting all these points, we find that the global maximum is found at $r=1/2$ , with a value of $R(1/2)=5/12\approx 0.42$ .
Literature: Myerson, Roger B. "Optimal auction design." Mathematics of Operations Research 6, No. 1 (1981): 58-73.

English auction

Recall the rules:

Selena continually raises the price.
At any price, you decide whether to stay or leave.
If you leave, you may not return.
If you are the last one in the auction you win
and pay the price at which the second-to-last
bidder left.

English auction

English auction is strategically equivalent to static second-price auction.
It is a dominant strategy to stay until the price reaches one's valuation.
The expected revenue for Selena is the also the same!
Consequently, these two formats are also revenue equivalent.

Shill bidding

Second-price and English auctions popular among theorists, not so popular among practitioners.
One reason is that they are prone to shill-bidding: bidder that acts on behalf of the auctioneer to drive up the price.

First-price auction

The winning bidder pays her bid.
Other rules same as in the second-price auction; i.e.,
- all bidders submit their bids simultaneously,
- the highest bid wins.

First-price auction

Bidding in the first-price auction is not truthful.

Bidding truthfully can never be optimal: if you win, you earn nothing.
Underbidding is strictly better, you win sometimes and when you do you have a positive utility.

First-price auction

Equilibrium strategy: It is a Nash equilibrium for each bidder to bid half their own valuation.

Nash equilibrium: A set of strategies, one per player, where no one has an incentive to change their strategy.

First-price auction

Intuition: suppose you are Alice

If you bid 0, winning prob. is zero.
If you bid your valuation, profit is zero.
Hence, there is a sweet spot between 0 and your valuation where your expected profit is maximal.
It turns out this is bidding half your valuation, at which point you and Bob each wins half of the time.

First-price auction

Expected revenue

Reasonable to assume each bidder bids half their valuation.
Hence, Selena's revenue is $\frac{1}{2}\max{v_A, v_B}$ .
The expected value of $\max{v_A, v_B}$ is $2/3$ .
Hence, her expected revenue is $1/3$ .

The same as in second-price auction!

I find this result fascinating. The optimal strategies are so different, but somehow the fact that you underbid is exactly compensated by the fact that you pay more when you win.
Is there a deeper connection going on here? Or is it just a coincidence that the expected revenues are all the same so far?
Formal proof of the expected revenue: recall that the valuations $v_A$ and $v_B$ are assumed to be independent variables uniformly drawn from $[0,1]$ .
The cumulative density function (CDF) of the higher valuation, $\max{v_A, v_B}$ , is $F(x)=x^2$ . This is because $F(x)$ is lower than $x$ only if both valuations are below $x$ , and these are two independent events each happening with probability $x$ .
Then, the probability density function (PDF) is $f(x)=F'(x)=2x$ .
And the formula for the expected value of $\max{v_A, v_B}$ is $R = \int\limits_0^1 x\cdot f(x)\mathrm d x = \int\limits_0^1 2x^2 \mathrm d v = \frac{2}{3}.$

Revenue Equivalence

Fact: When valuations are secret and independent, there is no reserve price, and item goes to highest bidder, then all auction mechanisms are revenue equivalent.

Dutch auctions

Selena continually lowers the price.
As soon as a bidder accepts the price,
they are declared winners and auction is over.
Winner pays the price they accepted.

Dutch Auction

Recall the rules:

The auctioneer continually lowers the price.
At any price, you can decide whether or not to accept the price.
If you are the first to accept the price, you win and pay the price you just accepted.

Dutch Auction

It turns out that the Dutch auction is strategically equivalent and revenue equivalent to the static first-price auction.
The price that you accept in the Dutch auction corresponds to the price that you'd bid in a static first-price auction.
The tradeoffs that the bidders face are very similar: take the current price or wait a bit at the risk of another bidder accepting first. It is an equilibrium to wait till the price is half your valuation.

Recap

Analyzed important auction formats:

Static first-price auction.
Static second-price auction.
English auction.
Dutch auction.

Learned under standard assumptions:

First-price and Dutch auctions are strategy equivalent.
Second-price and English auctions are strategy equivalent.
All four actions are revenue equivalent.
Having a reserve price increases the expected revenue, and it beats posting a price.

Independence of valuations

In our analysis, it was important to assume that bidders' valuations are independent from one another.

Can you think of examples where this assumption isn't sensible?

Independence of valuations

Answer:

Sensible: - a piece of art, where the bidders are final clients.

Not sensible: drilling rights to an oil well. Bidders will have similar estimates of amount of oil, hence valuations are highly correlated.

Common value auctions

Special scenario: there is a unique valuation of item, but each bidder only has a private estimate of it.

In these cases, it is observed that sealed-bid auctions tend to give higher revenue than dynamic auctions.

Why do you think this is the case?

Common value auctions

The auction may be used as a means of gathering information from other participants to triangulate a price

Answer: In a dynamic auction, a bidder can use the bids of others as additional signals of the correct valuation. If bids so far seem high, my initial estimate must be low, and vice versa, so I can adjust my personal estimate. Hence estimates converge.

In a static auction, there is no convergence of estimates, so it is more likely that some bidders keep unusually high estimates. As a result, there is a higher chance that the winner ends up paying more than the correct valuation. This is known as the winner's curse.

Equivalence of revenues

It is observed in practice that first-price auctions lead to higher revenue than second-price auctions.

This violates the equivalence of revenues, so an assumption in our analysis fails consistently.

What do you think it is?

Equivalence of revenues

Answer: Risk aversion. People prefer lower uncertainty games, even if this means lower expected profits.

Would you rather win a million dollars with a 50% chance, or 300 thousand with a 90% chance?

In Nash equilibrium analysis for first-price auctions, we claimed that if Bob bids half his valuation, then Alice should bid half hers, so each wins 50% of time. But we implicitly assumed that Alice is risk neutral. Yet she might prefer to bid more and win more often.

Front Running

Computer systems may have front runners: special nodes can see an incoming bid, react by creating their own bid, and make it look like their bid was created first.

If you run an auction on a system with front runners, which of the four auctions would you use? Why?

Front Running

Answer: Meet front runner Fred.

In a Dutch auction, if Fred is a bidder he waits for first bidder to signal accepting the price, and Fred makes the signal first. He's guaranteed to win with least possible price.

In second-price auction, if Fred is auctioneer he can shill bid successfully: when a highest bid arrives, he creates bid slightly under it and makes it appear as if it was created first.

Front Running

Answer: Meet front runner Fred.

In a first-price auction, if Fred is bidder and if he can "open the seals" he can win by slightly outbidding highest bid. (Poor data privacy, a common issue in computer systems)

Hence, it might be best to run an English auction.

Sniping

In a dynamic auction with a fixed bidding time window, sniping is placing a highest bid as late as possible, so other bidders can't react and overbid you. The practice leads to low revenue. Particularly bad when done by a front runner (microseconds).

How to protect an auction from snipers?

Sniping

Answer: candle auctions.
Dynamic first-price auction with random ending time.

Sniping

Answer: candle auctions.
Dynamic first-price auction with random ending time.

Similar to first-price auction
except that ending time is unpredictable.
At ending time, current highest bidder wins, pays own bid.
Sniping protection: the longer you wait to bid, the higher the chances the auction ends.

NFT Auction

You have the chance to bid on one of 25 unique NFTs that are created by a professional artist.
Use your budget that you accumulated during the last Academy Games.
Everything that you will not use for bidding (or if your bid was lower than your budget), you will receive in cash at the end of the Academy.
100% of the revenue of the auctions goes to the artist.
You are randomly assigned to one of three auction formats

The Artist & NFTs!

Jeremy Gluck (Nonceptualism)

Canadian Artist living in Great Britain.
Diverse Practice: Post-digital fine art in NFT, digital art, film, photography, sound and video art, installation, and performance.
Exhibitions: Works showcased offline in cities like London, Sydney, Bath, Anglesey, and Swansea, and online in various publications.
Art is exhibited in Tides Fine Art Gallery, Mumbles, Wales.

Format 1: Activity Rule Auction

The initial bidding phase lasts 30 seconds.
Every valid bid resets the timer.
You need to bid at least 30 cents more than the previous highest bid.
Whoever has the highest bid at the end, wins. Winners pay their bids.

Format 2: Candle Auction

Auction Format from the 16th Century.
The auction lasts for exactly 4 minutes.
A “candle mechanism” randomly determines, after the auction, when the auction ended
Grace-period of 1 minute.
Candle Phase of 3 minutes.
Whoever had the highest bid when the auction actually ended, wins.

Format 3: Hard Close Auction

Similar to eBay auctions.
Auction lasts for 4 minutes.
Whoever has the highest bid at the end of the auction, wins.
Winners pay their bids.

Auction 2: NFT Auction

Link will be distributed!

Auction 2: Discussion

Auction 2: Results!

Bonus Game: Dollar Auction

Each group of five players is bidding on a one dollar note.
Minimum bid is 5 cents, bid increments of 5 cents.
Auction lasts 90 seconds.
Highest bidder gets the prize.
Both highest and second highest bidders pay respective bids.

Dollar Auction

Discussion

Would you consider the auction paradoxical?
Did you catch yourself acting irrational?
Did you observe any phases in the game?
Why do you think the game could be considered an analogy for war?

Price Finding Mechanisms