Combinatorial auctions Maria Serna Fall 2016 AGT-MIRI - - PowerPoint PPT Presentation

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Combinatorial auctions Maria Serna Fall 2016 AGT-MIRI - - PowerPoint PPT Presentation

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Selling sets of items Bidding Languages The winner determination problem Combinatorial auctions Maria Serna Fall 2016 AGT-MIRI Combinatorial auctions Selling sets of items Bidding Languages The winner determination problem 1 Selling sets

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Selling sets of items Bidding Languages The winner determination problem

Combinatorial auctions

Maria Serna Fall 2016

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem

1 Selling sets of items 2 Bidding Languages 3 The winner determination problem

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem

References

  • P. Cramton et al. (eds.),

Combinatorial Auctions MIT Press, 2006. Chapters 9 - 8

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem

Sealed-bid for bundles of items

Objectives:

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem

Sealed-bid for bundles of items

Objectives: Combinatorial auctions are mechanisms for deciding on the allocation of sets of items. Sealed-bid format.

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem

Sealed-bid for bundles of items

Objectives: Combinatorial auctions are mechanisms for deciding on the allocation of sets of items. Sealed-bid format. Bidding languages. How to express your bid on bundles? Combinations of atomic bids.

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem

Sealed-bid for bundles of items

Objectives: Combinatorial auctions are mechanisms for deciding on the allocation of sets of items. Sealed-bid format. Bidding languages. How to express your bid on bundles? Combinations of atomic bids. The winner determination problem which bidder should obtain which items? Mostly as an optimal resource allocation problem.

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem

Complements and substitutes

The value an agent assigns to a bundle of goods may relate to the value it assigns to the individual goods in a variety of ways . . . .

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem

Complements and substitutes

The value an agent assigns to a bundle of goods may relate to the value it assigns to the individual goods in a variety of ways . . . . Bundles can be:

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem

Complements and substitutes

The value an agent assigns to a bundle of goods may relate to the value it assigns to the individual goods in a variety of ways . . . . Bundles can be: Complements: The value assigned to a set is greater than the sum of the values assigns to its elements. For example a pair of shoes.

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem

Complements and substitutes

The value an agent assigns to a bundle of goods may relate to the value it assigns to the individual goods in a variety of ways . . . . Bundles can be: Complements: The value assigned to a set is greater than the sum of the values assigns to its elements. For example a pair of shoes. Substitutes: The value assigned to a set is lower than the sum

  • f the values assigned to its elements.

For example a ticket to the theater and another to a football match for the same night.

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem

Complements and substitutes

The value an agent assigns to a bundle of goods may relate to the value it assigns to the individual goods in a variety of ways . . . . Bundles can be: Complements: The value assigned to a set is greater than the sum of the values assigns to its elements. For example a pair of shoes. Substitutes: The value assigned to a set is lower than the sum

  • f the values assigned to its elements.

For example a ticket to the theater and another to a football match for the same night. Prices and valuations cannot be decided per item as the bidding strategy for one item may depend on the expectation of getting

  • ther items.

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem

Sealed-bid auction protocol

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem

Sealed-bid auction protocol

We need to decide on all items.

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem

Sealed-bid auction protocol

We need to decide on all items. Setting: one seller (auctioneer) and several potential buyers (bidders) also there are many goods to be sold.

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem

Sealed-bid auction protocol

We need to decide on all items. Setting: one seller (auctioneer) and several potential buyers (bidders) also there are many goods to be sold. Bidding: the bidders submit their valuations to the auctioneer. How?

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem

Sealed-bid auction protocol

We need to decide on all items. Setting: one seller (auctioneer) and several potential buyers (bidders) also there are many goods to be sold. Bidding: the bidders submit their valuations to the auctioneer. How? Clearing: the auctioneer announces the winning bids.

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem

Sealed-bid auction protocol

We need to decide on all items. Setting: one seller (auctioneer) and several potential buyers (bidders) also there are many goods to be sold. Bidding: the bidders submit their valuations to the auctioneer. How? Clearing: the auctioneer announces the winning bids. The winning bids determine which bidder obtains which good, and how much each bidder has to pay.

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem

Sealed-bid auction protocol

We need to decide on all items. Setting: one seller (auctioneer) and several potential buyers (bidders) also there are many goods to be sold. Bidding: the bidders submit their valuations to the auctioneer. How? Clearing: the auctioneer announces the winning bids. The winning bids determine which bidder obtains which good, and how much each bidder has to pay. No good may be allocated to more than one bidder.

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem OR Language XOR Language

1 Selling sets of items 2 Bidding Languages 3 The winner determination problem

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem OR Language XOR Language

Bidding Languages

As there are 2n − 1 non-empty bundles for n goods, submitting complete valuations may not be feasible.

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem OR Language XOR Language

Bidding Languages

As there are 2n − 1 non-empty bundles for n goods, submitting complete valuations may not be feasible. A compact preference representation language is used: OR/XOR, weighted formulas, k-additive form, . . . .

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem OR Language XOR Language

Bidding Languages

As there are 2n − 1 non-empty bundles for n goods, submitting complete valuations may not be feasible. A compact preference representation language is used: OR/XOR, weighted formulas, k-additive form, . . . . Based on atomic bids (Bi, pi), specifying the price pi the bidder is prepared to pay for a particular bundle Bi.

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem OR Language XOR Language

Bidding Languages

As there are 2n − 1 non-empty bundles for n goods, submitting complete valuations may not be feasible. A compact preference representation language is used: OR/XOR, weighted formulas, k-additive form, . . . . Based on atomic bids (Bi, pi), specifying the price pi the bidder is prepared to pay for a particular bundle Bi. The bidding language determines a sort of conflict graph: atomic bids are vertices and edges connect bids that cannot be accepted together.

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem OR Language XOR Language

Bidding Languages

Standard bidding languages belong to the OR/XOR family. Basic OR and XOR languages. Combinations of OR and XOR OR∗ with dummy items

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem OR Language XOR Language

Assumptions

Let G be a finite set of goods.

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem OR Language XOR Language

Assumptions

Let G be a finite set of goods. We use bidding languages to encode valuations v : 2G → R.

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem OR Language XOR Language

Assumptions

Let G be a finite set of goods. We use bidding languages to encode valuations v : 2G → R. We assume that valuations are both normalized and monotonic:

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem OR Language XOR Language

Assumptions

Let G be a finite set of goods. We use bidding languages to encode valuations v : 2G → R. We assume that valuations are both normalized and monotonic:

v is normalized iff v({}) = 0. v is monotonic iff v(X) ≤ v(Y ) whenever X ⊆ Y .

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem OR Language XOR Language

Assumptions

Let G be a finite set of goods. We use bidding languages to encode valuations v : 2G → R. We assume that valuations are both normalized and monotonic:

v is normalized iff v({}) = 0. v is monotonic iff v(X) ≤ v(Y ) whenever X ⊆ Y .

Valuations are non-negative and there are no substitutes.

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem OR Language XOR Language

Atomic bids

An atomic bid is a pair (S, p) where S ⊂ G is a bundle of goods and p ∈ R+ is a price.

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem OR Language XOR Language

Atomic bids

An atomic bid is a pair (S, p) where S ⊂ G is a bundle of goods and p ∈ R+ is a price. Expressing how much the agent is prepared to pay in return for receiving S.

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem OR Language XOR Language

Atomic bids

An atomic bid is a pair (S, p) where S ⊂ G is a bundle of goods and p ∈ R+ is a price. Expressing how much the agent is prepared to pay in return for receiving S. The atomic bid (S, p), due to monotonicity, defines the valuation v: v(X) =

  • p

if S ⊆ X

  • therwise

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem OR Language XOR Language

Atomic bids

An atomic bid is a pair (S, p) where S ⊂ G is a bundle of goods and p ∈ R+ is a price. Expressing how much the agent is prepared to pay in return for receiving S. The atomic bid (S, p), due to monotonicity, defines the valuation v: v(X) =

  • p

if S ⊆ X

  • therwise

Atomic bids alone cannot express very interesting bundle’s valuations.

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem OR Language XOR Language

OR: Valid combinations

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem OR Language XOR Language

OR: Valid combinations

How can bidders combine several atomic bids?

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem OR Language XOR Language

OR: Valid combinations

How can bidders combine several atomic bids? The auctioneer may accept any combination of non-conflicting atomic bids (bundles don’t overlap) and charge the sum of the associated prices.

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem OR Language XOR Language

OR: Valid combinations

How can bidders combine several atomic bids? The auctioneer may accept any combination of non-conflicting atomic bids (bundles don’t overlap) and charge the sum of the associated prices. Such bids can be expressed using only OR operations.

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem OR Language XOR Language

OR: Valid combinations

How can bidders combine several atomic bids? The auctioneer may accept any combination of non-conflicting atomic bids (bundles don’t overlap) and charge the sum of the associated prices. Such bids can be expressed using only OR operations. We have to define the valuation associated to any potential combination.

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem OR Language XOR Language

The OR Language

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem OR Language XOR Language

The OR Language

An OR-combination of two bids defining valuations v1 and v2 defines the following valuation: (v1 or v2)(X) = max

X1⊆X(v1(X1) + v2(X \ X1)).

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem OR Language XOR Language

The OR Language

An OR-combination of two bids defining valuations v1 and v2 defines the following valuation: (v1 or v2)(X) = max

X1⊆X(v1(X1) + v2(X \ X1)).

If there are k atomic bids defining valuations v1, . . . , vk, then the overall bid defines the valuation v1 or (v2 or · · · (vk−1 or vk))).

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem OR Language XOR Language

The OR Language

An OR-combination of two bids defining valuations v1 and v2 defines the following valuation: (v1 or v2)(X) = max

X1⊆X(v1(X1) + v2(X \ X1)).

If there are k atomic bids defining valuations v1, . . . , vk, then the overall bid defines the valuation v1 or (v2 or · · · (vk−1 or vk))). This is the standard bidding language. If an author doesn’t say what language they are using, it’s probably this one.

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem OR Language XOR Language

OR: Expressive Power

The OR language is not fully expressive.

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem OR Language XOR Language

OR: Expressive Power

The OR language is not fully expressive. Theorem The OR language can represent all supermodular valuations, and

  • nly those.

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem OR Language XOR Language

OR: Expressive Power

The OR language is not fully expressive. Theorem The OR language can represent all supermodular valuations, and

  • nly those.

A valuation v is supermodular iff we have v(X ∪ Y ) ≥ v(X) + v(Y ) − v(X ∩ Y ), for all X, Y ⊆ G.

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem OR Language XOR Language

Basic valuation problems

The most basic problem:

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem OR Language XOR Language

Basic valuation problems

The most basic problem: Bundle Evaluation (BunEval) Given a valuation v in language L, bundle S, and K ∈ Q. Is it the case that v(S) ≥ K?

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem OR Language XOR Language

Basic valuation problems

Theorem BunEval is NP-complete for the OR language

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem OR Language XOR Language

Basic valuation problems

Theorem BunEval is NP-complete for the OR language NP-hardness: by reduction from Set Packing.

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem OR Language XOR Language

The XOR Language

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem OR Language XOR Language

The XOR Language

Another interpretation of a set of atomic bids by the same bidder would be that the auctioneer can accept at most one of these bids.

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem OR Language XOR Language

The XOR Language

Another interpretation of a set of atomic bids by the same bidder would be that the auctioneer can accept at most one of these bids. An XOR-combination of valuations v1 and v2 is: (v1 xor v2)(X) = max{v1(X), v2(X)}.

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem OR Language XOR Language

The XOR Language

Another interpretation of a set of atomic bids by the same bidder would be that the auctioneer can accept at most one of these bids. An XOR-combination of valuations v1 and v2 is: (v1 xor v2)(X) = max{v1(X), v2(X)}. If there are k atomic bids defining valuations v1, . . . , vk, then the overall bid defines the valuation v1 xor (v2 xor · · · (vk−1 xor vk))).

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem OR Language XOR Language

XOR: Expressive Power

The XOR language is fully expressive.

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem OR Language XOR Language

XOR: Expressive Power

The XOR language is fully expressive. Theorem The XOR language can represent all monotonic valuations, and

  • nly those.

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem OR Language XOR Language

XOR: succictness

We have assumed that each bidder submits a set of atomic bids and that the operator to be applied (OR or XOR) is implicit.

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem OR Language XOR Language

XOR: succictness

We have assumed that each bidder submits a set of atomic bids and that the operator to be applied (OR or XOR) is implicit. The size of a bid is the number of atomic bids in it.

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem OR Language XOR Language

XOR: succictness

We have assumed that each bidder submits a set of atomic bids and that the operator to be applied (OR or XOR) is implicit. The size of a bid is the number of atomic bids in it. An additive valuation assigns a value to each item and the sum of their element values to each bundle.

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem OR Language XOR Language

XOR: succictness

We have assumed that each bidder submits a set of atomic bids and that the operator to be applied (OR or XOR) is implicit. The size of a bid is the number of atomic bids in it. An additive valuation assigns a value to each item and the sum of their element values to each bundle. Additive valuations require linear size in the OR language, but may require exponential size in the XOR language as we need to assign a value to any combination.

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem OR Language XOR Language

XOR: succictness

We have assumed that each bidder submits a set of atomic bids and that the operator to be applied (OR or XOR) is implicit. The size of a bid is the number of atomic bids in it. An additive valuation assigns a value to each item and the sum of their element values to each bundle. Additive valuations require linear size in the OR language, but may require exponential size in the XOR language as we need to assign a value to any combination. Hence, while the XOR language is more expressive than the OR language, it is not more succinct (indeed, it will be significantly less succinct for many natural valuations).

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem

1 Selling sets of items 2 Bidding Languages 3 The winner determination problem

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem

The winner determination problem

The winner determination problem (WDP) is the problem of finding a set of winning atomic bids

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem

The winner determination problem

The winner determination problem (WDP) is the problem of finding a set of winning atomic bids that is feasible (e.g., no item allocated twice) and that will maximize the sum of the prices offered.

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem

The winner determination problem

The winner determination problem (WDP) is the problem of finding a set of winning atomic bids that is feasible (e.g., no item allocated twice) and that will maximize the sum of the prices offered. The sum of prices can be given different interpretations:

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem

The winner determination problem

The winner determination problem (WDP) is the problem of finding a set of winning atomic bids that is feasible (e.g., no item allocated twice) and that will maximize the sum of the prices offered. The sum of prices can be given different interpretations: If the simple pricing rule is used where bidders pay what they

  • ffered, then it is the revenue of the auctioneer.

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem

The winner determination problem

The winner determination problem (WDP) is the problem of finding a set of winning atomic bids that is feasible (e.g., no item allocated twice) and that will maximize the sum of the prices offered. The sum of prices can be given different interpretations: If the simple pricing rule is used where bidders pay what they

  • ffered, then it is the revenue of the auctioneer.

If the prices offered are interpreted as individual utilities, then it is the utilitarian social welfare of the selected allocation.

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem

The winner determination problem

The winner determination problem (WDP) is the problem of finding a set of winning atomic bids that is feasible (e.g., no item allocated twice) and that will maximize the sum of the prices offered. The sum of prices can be given different interpretations: If the simple pricing rule is used where bidders pay what they

  • ffered, then it is the revenue of the auctioneer.

If the prices offered are interpreted as individual utilities, then it is the utilitarian social welfare of the selected allocation. Let’s look at the problem as a resource allocation problem.

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem

Multiagent resource allocation

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem

Multiagent resource allocation

Consider a set of agents and a set of goods. Each agent has their

  • wn preferences regarding the allocation of goods to be selected.

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem

Multiagent resource allocation

Consider a set of agents and a set of goods. Each agent has their

  • wn preferences regarding the allocation of goods to be selected.

What constitutes a good allocation and how do we find it?

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem

Multiagent resource allocation

Consider a set of agents and a set of goods. Each agent has their

  • wn preferences regarding the allocation of goods to be selected.

What constitutes a good allocation and how do we find it? What goods? One or several goods? Available in single or multiple units? Divisible or indivisible? Can goods be shared? Static or changing properties (e.g., consumable or perishable goods)?

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem

Multiagent resource allocation

Consider a set of agents and a set of goods. Each agent has their

  • wn preferences regarding the allocation of goods to be selected.

What constitutes a good allocation and how do we find it? What goods? One or several goods? Available in single or multiple units? Divisible or indivisible? Can goods be shared? Static or changing properties (e.g., consumable or perishable goods)? What preferences? Ordinal or cardinal preferences? Are monetary side payments possible, and how do they affect preferences?

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem

Setting

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem

Setting

Set of agents N = {1, . . . , n} and finite set of indivisible goods G. An allocation A is a partitioning of G amongst the agents in N. Each agent i ∈ N has got a valuation function vi : 2G → R.

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem

Setting

Set of agents N = {1, . . . , n} and finite set of indivisible goods G. An allocation A is a partitioning of G amongst the agents in N. Each agent i ∈ N has got a valuation function vi : 2G → R. Recall that the utilitarian social welfare of an allocation is defined as follows: swu(A) =

  • i∈N

vi(A).

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem

Setting

Set of agents N = {1, . . . , n} and finite set of indivisible goods G. An allocation A is a partitioning of G amongst the agents in N. Each agent i ∈ N has got a valuation function vi : 2G → R. Recall that the utilitarian social welfare of an allocation is defined as follows: swu(A) =

  • i∈N

vi(A). The input to an allocation problem is a triple (N, G, V), where V is a set of valuation functions (+ possibly the initial allocation A0).

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem

Welfare Optimization

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem

Welfare Optimization

Welfare Optimisation (WO) Instance: (N, G, V) and K ∈ Q. Question: Is there an allocation A such that swu(A) > K?

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem

Welfare Optimization

Welfare Optimisation (WO) Instance: (N, G, V) and K ∈ Q. Question: Is there an allocation A such that swu(A) > K? Theorem The Welfare Optimisation problem is NP-complete.

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem

Welfare Optimization

Welfare Optimisation (WO) Instance: (N, G, V) and K ∈ Q. Question: Is there an allocation A such that swu(A) > K? Theorem The Welfare Optimisation problem is NP-complete. The problem, trivially, belongs to NP (assuming an explicit representation of valuations).

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem

WO: Hardness

AGT-MIRI Combinatorial auctions

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Selling sets of items Bidding Languages The winner determination problem

WO: Hardness

Set Packing Instance: Collection C of finite sets and K ∈ Q. Question: Is there a collection of disjoint sets C′ ⊆ C such that |C′| > K?

AGT-MIRI Combinatorial auctions

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SLIDE 84

Selling sets of items Bidding Languages The winner determination problem

WO: Hardness

Set Packing Instance: Collection C of finite sets and K ∈ Q. Question: Is there a collection of disjoint sets C′ ⊆ C such that |C′| > K? Given an instance C of Set Packing construct the following allocation instance;

AGT-MIRI Combinatorial auctions

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SLIDE 85

Selling sets of items Bidding Languages The winner determination problem

WO: Hardness

Set Packing Instance: Collection C of finite sets and K ∈ Q. Question: Is there a collection of disjoint sets C′ ⊆ C such that |C′| > K? Given an instance C of Set Packing construct the following allocation instance; Goods: each item in one of the sets in C is a good. Agents: one for each set in C and another agent (agent 0). Valuations: vC(S) = 1 if S = C and vC(S) = 0 otherwise; v0(S) = 0, for all bundles S. That is, every agent values “its” bundle at 1 and every other bundle at 0. Agent 0 values all bundles at 0.

AGT-MIRI Combinatorial auctions

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SLIDE 86

Selling sets of items Bidding Languages The winner determination problem

WO: Hardness

AGT-MIRI Combinatorial auctions

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SLIDE 87

Selling sets of items Bidding Languages The winner determination problem

WO: Hardness

Not every allocation immediately corresponds to a valid solution of Set Packing: the bundles owned by individual agents may not all be sets in C.

AGT-MIRI Combinatorial auctions

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SLIDE 88

Selling sets of items Bidding Languages The winner determination problem

WO: Hardness

Not every allocation immediately corresponds to a valid solution of Set Packing: the bundles owned by individual agents may not all be sets in C. But: for every given allocation there exists another allocation with equal social welfare that does directly correspond to a valid solution for Set Packing:

AGT-MIRI Combinatorial auctions

slide-89
SLIDE 89

Selling sets of items Bidding Languages The winner determination problem

WO: Hardness

Not every allocation immediately corresponds to a valid solution of Set Packing: the bundles owned by individual agents may not all be sets in C. But: for every given allocation there exists another allocation with equal social welfare that does directly correspond to a valid solution for Set Packing: Assign any goods owned by an agent with valuation 0 to agent 0.

AGT-MIRI Combinatorial auctions

slide-90
SLIDE 90

Selling sets of items Bidding Languages The winner determination problem

WO: Hardness

Not every allocation immediately corresponds to a valid solution of Set Packing: the bundles owned by individual agents may not all be sets in C. But: for every given allocation there exists another allocation with equal social welfare that does directly correspond to a valid solution for Set Packing: Assign any goods owned by an agent with valuation 0 to agent 0. This reallocation does not affect social welfare.

AGT-MIRI Combinatorial auctions

slide-91
SLIDE 91

Selling sets of items Bidding Languages The winner determination problem

WO: Hardness

Not every allocation immediately corresponds to a valid solution of Set Packing: the bundles owned by individual agents may not all be sets in C. But: for every given allocation there exists another allocation with equal social welfare that does directly correspond to a valid solution for Set Packing: Assign any goods owned by an agent with valuation 0 to agent 0. This reallocation does not affect social welfare. Note that social welfare is equal to |C′|.

AGT-MIRI Combinatorial auctions

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SLIDE 92

Selling sets of items Bidding Languages The winner determination problem

Welfare Optimization

AGT-MIRI Combinatorial auctions

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SLIDE 93

Selling sets of items Bidding Languages The winner determination problem

Welfare Optimization

Welfare Improvement (WI) Instance: (N, G, V) and allocation A. Question: Is there an allocation A′ such that swu(A′) < swu(A)?

AGT-MIRI Combinatorial auctions

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SLIDE 94

Selling sets of items Bidding Languages The winner determination problem

Welfare Optimization

Welfare Improvement (WI) Instance: (N, G, V) and allocation A. Question: Is there an allocation A′ such that swu(A′) < swu(A)? Theorem The Welfare Improvement problem is NP-complete.

AGT-MIRI Combinatorial auctions

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SLIDE 95

Selling sets of items Bidding Languages The winner determination problem

Pareto Optimality

AGT-MIRI Combinatorial auctions

slide-96
SLIDE 96

Selling sets of items Bidding Languages The winner determination problem

Pareto Optimality

Pareto Optimality (PO) Instance: (N, G, V) and and allocation A. Question: Is A Pareto optimal?

AGT-MIRI Combinatorial auctions

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SLIDE 97

Selling sets of items Bidding Languages The winner determination problem

Pareto Optimality

Pareto Optimality (PO) Instance: (N, G, V) and and allocation A. Question: Is A Pareto optimal? Theorem The Pareto Optimality problem is coNP-complete.

AGT-MIRI Combinatorial auctions

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SLIDE 98

Selling sets of items Bidding Languages The winner determination problem

Envy-Freeness

AGT-MIRI Combinatorial auctions

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SLIDE 99

Selling sets of items Bidding Languages The winner determination problem

Envy-Freeness

Envy-Freeness (EF) Instance: (N, G, V). Question: Is there a complete allocation A that is envy-free?

AGT-MIRI Combinatorial auctions

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SLIDE 100

Selling sets of items Bidding Languages The winner determination problem

Envy-Freeness

Envy-Freeness (EF) Instance: (N, G, V). Question: Is there a complete allocation A that is envy-free? Envy-freeness: No player will envy (any of) the other(s).

AGT-MIRI Combinatorial auctions

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SLIDE 101

Selling sets of items Bidding Languages The winner determination problem

Envy-Freeness

Envy-Freeness (EF) Instance: (N, G, V). Question: Is there a complete allocation A that is envy-free? Envy-freeness: No player will envy (any of) the other(s). Assuming all goods need to be allocated:

AGT-MIRI Combinatorial auctions

slide-102
SLIDE 102

Selling sets of items Bidding Languages The winner determination problem

Envy-Freeness

Envy-Freeness (EF) Instance: (N, G, V). Question: Is there a complete allocation A that is envy-free? Envy-freeness: No player will envy (any of) the other(s). Assuming all goods need to be allocated: Theorem The Envy-Freenessproblem is NP-complete.

AGT-MIRI Combinatorial auctions

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SLIDE 103

Selling sets of items Bidding Languages The winner determination problem

Envy-Freeness

Envy-Freeness (EF) Instance: (N, G, V). Question: Is there a complete allocation A that is envy-free? Envy-freeness: No player will envy (any of) the other(s). Assuming all goods need to be allocated: Theorem The Envy-Freenessproblem is NP-complete. Checking whether there is an allocation that is both Pareto

  • ptimal and envy-free is even harder: Σp

2-complete.

AGT-MIRI Combinatorial auctions