From auctions to graph coloring Nicolas Bousquet Journ ees du - - PowerPoint PPT Presentation

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From auctions to graph coloring Nicolas Bousquet Journ ees du - - PowerPoint PPT Presentation

May 09, 2023 379 likes •820 views

From auctions to graph coloring Nicolas Bousquet Journ ees du G-SCOP 2017 1/17 2011-2014 : Th` ese en th eorie et algorithmique des graphes ` a Montpellier. 2014-2015 : Post-doctorat en th eorie des jeux economiques ` a Montr

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From auctions to graph coloring

Nicolas Bousquet Journ´ ees du G-SCOP 2017

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2011-2014 : Th` ese en th´ eorie et algorithmique des graphes ` a Montpellier. 2014-2015 : Post-doctorat en th´ eorie des jeux ´ economiques ` a Montr´ eal. 2015-2016 : ATER en combinatoire et th´ eorie des graphes ` a Lyon.

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What is an auction ?

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Auctions today

Ad auctions. ≈ 150 billions a year. Spectrum auctions. ≈ 40/50 billions a year.

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Auctions today

Ad auctions. ≈ 150 billions a year. Spectrum auctions. ≈ 40/50 billions a year. Ad auctions : when you access a website, an immediate auction is

  • rganized to sell ad slots on the webpage.

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Auctions today

Ad auctions. ≈ 150 billions a year. Spectrum auctions. ≈ 40/50 billions a year. Ad auctions : when you access a website, an immediate auction is

  • rganized to sell ad slots on the webpage.

Spectrum auctions : a seller (the state) sells frequencies to telecommunication companies trying to maximizing the revenue (of the state) and, if possible global welfare.

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Let’s design an auction !

Your valuation : 1000$. Opponent’s valuation : Between 950$ and 1000$. Item + discount Item + no discount No item

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Let’s design an auction !

Your valuation : 1000$. Opponent’s valuation : Between 950$ and 1000$. Item + discount Item + no discount No item The bidder with the higher price has the item and he pays the price he announces for the item. First price auction

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Efficency ?

You bid 1000$ and your opponent bids 950$. instead of

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Efficency ?

You bid 1000$ and your opponent bids 950$. instead of You bid 970$ and your opponent bids 980$. instead of

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Efficency ?

You bid 1000$ and your opponent bids 950$. instead of You bid 970$ and your opponent bids 980$. instead of Even worse from the seller ! He does not maximize his profit !

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Truthfulness and efficiency

An auction is truthful if no bidder has any incentive to lie.

(His welfare can only decrease if he is lying on his valuation)

A truthful auction is “better” (for both sellers and buyers). Informal claim

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Truthfulness and efficiency

An auction is truthful if no bidder has any incentive to lie.

(His welfare can only decrease if he is lying on his valuation)

A truthful auction is “better” (for both sellers and buyers). Informal claim The ultimate goal : Design the best possible truthful auction... ... that can be explained to human beings... ... and whose “proof” is simple otherwise they won’t trust you.

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A truthful auction

Your valuation : 1000$. Opponent’s valuation : Between 950$ and 1000$. Item + discount Item + no discount No item

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A truthful auction

Your valuation : 1000$. Opponent’s valuation : Between 950$ and 1000$. Item + discount Item + no discount No item The bidder with the higher price has the item and he pays the price of the second highest bid for the item. Second price auction

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A truthful auction

Your valuation : 1000$. Opponent’s valuation : Between 950$ and 1000$. Item + discount Item + no discount No item The bidder with the higher price has the item and he pays the price of the second highest bid for the item. Second price auction ⇒ This auction is truthful !

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Proof on an example

You bid 1000$ and your opponent bids 950$. since you pay 950$.

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Proof on an example

You bid 1000$ and your opponent bids 950$. since you pay 950$. The seller maximizes his profit (under reasonable conditions).

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Spectrum auctions

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Spectrum auctions

Specificities :

  • The bidders discover their own valuations’ functions.

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Spectrum auctions

Specificities :

  • The bidders discover their own valuations’ functions.
  • Valuation functions admit complementarities.

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Spectrum auctions

Specificities :

  • The bidders discover their own valuations’ functions.
  • Valuation functions admit complementarities.

How best to allocate bandwidth dates back 100 years. Since the 1990s, auctions have become the standard way to allocate bandwidth. Two main auctions used worldwide :

  • SMRA (Simultaneous Multi-Round Auction).
  • CCA (Combinatorial Clock Auction).

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Clock Auctions

Clock auctions : the prices are initially set to zero At t = 0, the price of every item is 0.

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Clock Auctions

Clock auctions : the prices are initially set to zero and, periods after periods, prices are updated. At t = 0, the price of every item is 0. While all the bids are not “somehow” disjoint : Each bidder bids on her favorite set. If an item is in several bids, its price increases.

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Clock Auctions

Clock auctions : the prices are initially set to zero and, periods after periods, prices are updated. At t = 0, the price of every item is 0. While all the bids are not “somehow” disjoint : Each bidder bids on her favorite set. If an item is in several bids, its price increases. Return the “best possible” allocation.

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SMRA and CCA

Item vs package bidding :

  • Package bidding in the CCA : all or nothing bid at price p(S).

⇒ The bidder receives either all or none of the items.

  • Item bidding in the SMRA : a bid for S at price p(S) is the

union of single item bids for s at price p(s) for s ∈ S. ⇒ The bidder can be allocated a subset of her bid.

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SMRA and CCA

Item vs package bidding :

  • Package bidding in the CCA : all or nothing bid at price p(S).

⇒ The bidder receives either all or none of the items.

  • Item bidding in the SMRA : a bid for S at price p(S) is the

union of single item bids for s at price p(s) for s ∈ S. ⇒ The bidder can be allocated a subset of her bid. Advantage of package bidding : No exposure problem ⇒ the allocation is individually rational.

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SMRA and CCA

Item vs package bidding :

  • Package bidding in the CCA : all or nothing bid at price p(S).

⇒ The bidder receives either all or none of the items.

  • Item bidding in the SMRA : a bid for S at price p(S) is the

union of single item bids for s at price p(s) for s ∈ S. ⇒ The bidder can be allocated a subset of her bid. Advantage of package bidding : No exposure problem ⇒ the allocation is individually rational. Drawback : No market clearing ⇒ usually market clearing helps for finding guarantees.

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Our results

These auctions :

  • Work well in practice...

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Our results

These auctions :

  • Work well in practice...
  • ... But we do not theoretically understand why !

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Our results

These auctions :

  • Work well in practice...
  • ... But we do not theoretically understand why !

The CCA has a polylogarithmic guarantee (under technical as- sumptions). Almost tight. Theorem (B., Cai, Hunkenschr¨

  • der, Vetta)

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New auctions

  • Buy TV and audio useless frequencies.
  • Sell them back to telecommunication companies.

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New auctions

  • Buy TV and audio useless frequencies.
  • Sell them back to telecommunication companies.

First auction of that type (April 2017) :

  • 19.8 billions of revenue.
  • More than 200 companies bought or sold frequencies.
  • Second-highest grossing spectrum auction in FCC history.

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Constraints

“Sell them back to telecommunication companies”

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Constraints

“Sell them back to telecommunication companies”

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Constraints

“Sell them back to telecommunication companies”

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Constraints

“Sell them back to telecommunication companies”

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Constraints

“Sell them back to telecommunication companies” (source : FCC)

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A problem

Coloring a graph is NP-hard and hard to approximate. Problem

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A problem

Coloring a graph is NP-hard and hard to approximate. Problem What can we do ? Use the structure of the graph to derive efficient (approximation) algorithm to color graphs.

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Conclusion

Questions

  • Why is it working ?

Understand the shape of valuation functions.

  • Improve the “truthfulness process” of spectrum auctions.

Implementations are messy...

  • Improve coloring algorithms on geometric classes.

Hard problems open for decades.

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Conclusion

Questions

  • Why is it working ?

Understand the shape of valuation functions.

  • Improve the “truthfulness process” of spectrum auctions.

Implementations are messy...

  • Improve coloring algorithms on geometric classes.

Hard problems open for decades.

Thanks for your attention !

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