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## PowerPoint Slideshow about 'Internet Economics כלכלת האינטרנט' - kristy

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### Internet Economicsכלכלת האינטרנט

Class 4 – Optimal Auctions

Last week (1/4)

- How to sell a single item to n bidders?
- Seller doesn’t know how much bidders are willing to pay
- vi is the value of bidder i for the item.
- Getting this information via an auction.
- Game with incomplete information.

Last week (2/4)

- The English Auction:
- Price starts at 0
- Price increases until only one
- bidder is left.

- Vickrey (2nd price) auction:
- Bidders send bids.
- Highest bid wins, pays 2nd highest bid.

- Private value model: each person has a privately known value for the item.
- We saw: the two auctions are equivalent in the private value model.
- Auctions are efficient:dominant strategy for each player: truthfulness.

Last week (3/4)

- The Dutch Auction:
- Price starts at max-price.
- Price drops until a bidder agrees to buy.

- 1st-price auction:
- Bidders send bids.
- Highest bid wins, pays his bid.

- Dutch auctions and 1st price auctions are strategically equivalent. (asynchronous vs simple & fast)
- No dominant strategies. (tradeoff: chance of winning, payment upon winning.)
- Analysis in a Bayesian model:
- Values are randomly drawn from a probability distribution.
- Strategy: a function. “What is my bid given my value?”

Last week (4/4)

- We considered the simplest Bayesian model:
- 2 bidders.
- Values drawn uniformly from [0,1].

Then:

In a 1st-price auction, it is a (Bayesian) Nash

equilibrium when all bidders bid

- An auction is efficient, if in (Bayesian) Nash equilibrium the bidder with the highest value always wins.
- 1st price is efficient!

Remark: Efficiency

- We saw that both 2nd –price and 1st –price auctions are efficient.
- What is efficiency (social welfare)?The total utility of the participants in the game (including the seller).

For each bidder: vi – pi

For the seller: (assuming it has 0 value for the item)

- Summing:

Optimal auctions

- Usually the term optimal auctions stands for revenue maximization.
- What is maximal revenue?
- We can always charge the winner his value.
- Maximal revenue: optimal expected revenue in equilibrium.
- Assuming a probability distribution on the values.
- Over all the possible mechanisms.
- Under individual-rationality constraints (later).

Example: Spectrum auctions

- One of the main triggers to auction theory.
- FCC in the US sells spectrum, mainly for cellular networks.
- Improved auctions since the 90’s increased efficiency + revenue considerably.
- Complicated (“combinatorial”) auction, in many countries.
- (more details further in the course)

New Zealand Spectrum Auctions

- A Vickrey (2nd price) auction was run in New Zealand to sale a bunch of auctions. (In 1990)
- Winning bid: $100000

Second highest: $6 (!!!!)

Essentially zero revenue.

- NZ Returned to 1st price method the year after.
- After that, went to a more complicated auction (in few weeks).
- Was it avoidable?

1st or 2nd price?

- Assume 2 bidders, uniform distribution on [0,1].
- Facts: (1) E[ max(v1,v2) ] = 2/3 (2) E[ min(v1,v2) ] = 1/3 (in general, k’th highest value of n is (n+1-k)/n+1)

- Revenue in 2nd price:
- Bidders bid truthfully.
- Revenue is 2nd highest bid.
- Expected revenue = 1/3

- Revenue in 1st price:
- bidders bid vi/2.
- Revenue is the highest bid.
- Expected revenue = E[ max(v1/2,v2/2) ]
- = ½ E[ max(v1,v2)]
- = ½ × 2/3 = 1/3

Revenue equivalence theorem

- No coincidence!
- Somewhat unintuitively, revenue depends only on the way the winner is chosen, not on payments.
- Auction for a single good.
- Values are independently drawn from distribution F (increasing)

Theorem (“revenue equivalence”):

All auctions where:

- the good is allocated to the bidder with the highest value
- Bidders can guarantee a utility of 0 by bidding 0.

yield the same revenue!

(more general: two auction with the same allocation rule yield the same revenue)

Remark: Individual rationality

- The following mechanism gains lots of revenue:
- Charge all players $10000000
- Bidder will clearly not participate.
- We thus have individual-rationality (or participation) constraints on mechanisms:bidders gain positive utility in equilibrium .
- This is the reason for condition 2 in the theorem.

All-pay auction (1/3)

- Rules:
- Sealed bid
- Highest bid wins
- Everyone pay their bid
- Equilibrium with the uniform distribution: b(v)=
- Does it achieve more or less revenue?
- Note: Bidders shade their bids as the competition increases.

All-pay auction (2/3)

- expected payment per each player: herbid.
- Each bidder bids
- Expected payment for each bidder:
- Revenue: from n bidders
- Revenue equivalence!

All-pay auction (3/3)

- Examples:
- crowdsourcing over the internet:
- First person to complete a task for me gets a reward.
- A group of people invest time in the task. (=payment)
- Only the winner gets the reward.
- Advertising auction:
- Collect suggestion for campaigns, choose a winner.
- All advertiser incur cost of preparing the campaign.
- Only one wins.
- Lobbying
- War of attrition
- Animals invest (b1,b2) in fighting.

What did we see so far

- 2nd-price, 1st-price, all pay: all obtain the same seller revenue.
- Revenue equivalence theorem:Auctions with the same allocation decisions earn the same expected seller revenue in equilibrium.
- Constraint: individual rationality (participation constraint)
- Many assumptions:
- statistical independence,
- risk neutrality,
- no externalities,
- private values,
- …

Next: Can we get better revenue?

- Can we achieve better revenue than the 2nd-price/1st price?
- If so, we must sacrifice efficiency.
- All efficient auction have the same revenue….
- How?
- Think about the New-Zealand case.

Vickrey with Reserve Price

- Seller publishes a minimum (“reserve”) price R.
- Each bidder writes his bid in a sealed envelope.
- The seller:
- Collects bids
- Open envelopes.
- Winner: Bidder with the highest bid, if bid is above R. Otherwise, no one winsPayment: winner pays max{2nd highest bid, R}

Yes. For bidders, exactly like an extra bidder bidding R.

Still Truthful?

Can we get better revenue?

1

- Let’s have another look at 2nd price auctions:

2 wins

v2

1 wins

x

1 wins and pays x

(his lowest winning bid)

0

x

0

v1

1

Can we get better revenue?

1

- I will show that some reserve price improve revenue.

Revenue increased

2 wins

v2

1 wins

R

Revenue increased

0

0

v1

1

Revenue loss here (efficiency loss too)

R

Can we get better revenue?

1

We will be here with probability R(1-R)

v2

- Gain is at least 2R(1-R) R/2 = R2-R3
- Loss is at most R2 R = R3

2 wins

We will be here with probability R2

1 wins

Average loss is R/2

Loss is always at mostR

0

v1

0

1

- When R2-2R3>0, reserve price of R is beneficial.(for example, R=1/4)

Reservation price

- Can increase revenue!
- 2 bidders, uniform distribution: optimal reserve price = ½
- Revenue: 5/12=0.412 > 1/3
- n bidders, uniform distribution: optimal reserve price = ½

Theorem: (Myerson ‘81)Vickrey auction with a reserve price maximizes revenue.

- For a general family of distributions (uniform, exponential, normal, and many others).
- Reserve price is independent of n.

(Nobel prize, 2007)

Reservation price

Let’s see another example:How do you sell one item to one bidder?

- Assume his value is drawn uniformly from [0,1].
- Optimal way: reserve price.
- Take-it-or-leave-it-offer.
- Let’s find the optimal reserve price:E[revenue] = ( 1-F(R) ) × R = (1-R) ×R

R=1/2

- Surprising? No. We said that the optimal reserve price does not depend on n.

Probability that the buyer will accept the price

The payment for the seller

Back to New Zealand

- Recall: Vickrey auction.Highest bid: $100000. Revenue: $6.
- Two things to learn:
- Seller can never get the whole pie.
- “information rent” for the buyers.
- Reserve price can help.
- But what if R=$50000 and highest bid was $45000?
- Of the unattractive properties of Vickrey Auctions:
- Low revenue despite high bids.
- 1st-price may earn same revenue, but no explanation needed…

Summary: Efficiency vs. revenue

Positive or negative correlation ?

- Always: Revenue ≤ efficiency
- Due to Individual rationality.
- More efficiency makes the pie larger!
- However, for optimal revenue one needs to sacrifice some efficiency.
- Consider two competing sellers: one optimizing revenue the other optimizing efficiency.
- Who will have a higher market share?
- In the longer terms, two objectives are combined.

Next week

- Designing dominant-strategy mechanisms for more general environments.
- the magic of the VCG mechanism.

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