Selling sets of items OR Language Bidding Languages XOR Language The winner determination problem Assumptions Let G be a finite set of goods. We use bidding languages to encode valuations v : 2 G → R . We assume that valuations are both normalized and monotonic: AGT-MIRI Combinatorial auctions
Selling sets of items OR Language Bidding Languages XOR Language The winner determination problem Assumptions Let G be a finite set of goods. We use bidding languages to encode valuations v : 2 G → 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
Selling sets of items OR Language Bidding Languages XOR Language The winner determination problem Assumptions Let G be a finite set of goods. We use bidding languages to encode valuations v : 2 G → 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
Selling sets of items OR Language Bidding Languages XOR Language The winner determination problem 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
Selling sets of items OR Language Bidding Languages XOR Language The winner determination problem 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
Selling sets of items OR Language Bidding Languages XOR Language The winner determination problem 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 : � p if S ⊆ X v ( X ) = 0 otherwise AGT-MIRI Combinatorial auctions
Selling sets of items OR Language Bidding Languages XOR Language The winner determination problem 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 : � p if S ⊆ X v ( X ) = 0 otherwise Atomic bids alone cannot express very interesting bundle’s valuations. AGT-MIRI Combinatorial auctions
Selling sets of items OR Language Bidding Languages XOR Language The winner determination problem OR: Valid combinations AGT-MIRI Combinatorial auctions
Selling sets of items OR Language Bidding Languages XOR Language The winner determination problem OR: Valid combinations How can bidders combine several atomic bids? AGT-MIRI Combinatorial auctions
Selling sets of items OR Language Bidding Languages XOR Language The winner determination problem 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
Selling sets of items OR Language Bidding Languages XOR Language The winner determination problem 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
Selling sets of items OR Language Bidding Languages XOR Language The winner determination problem 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
Selling sets of items OR Language Bidding Languages XOR Language The winner determination problem The OR Language AGT-MIRI Combinatorial auctions
Selling sets of items OR Language Bidding Languages XOR Language The winner determination problem The OR Language An OR-combination of two bids defining valuations v 1 and v 2 defines the following valuation: ( v 1 or v 2 )( X ) = max X 1 ⊆ X ( v 1 ( X 1 ) + v 2 ( X \ X 1 )) . AGT-MIRI Combinatorial auctions
Selling sets of items OR Language Bidding Languages XOR Language The winner determination problem The OR Language An OR-combination of two bids defining valuations v 1 and v 2 defines the following valuation: ( v 1 or v 2 )( X ) = max X 1 ⊆ X ( v 1 ( X 1 ) + v 2 ( X \ X 1 )) . If there are k atomic bids defining valuations v 1 , . . . , v k , then the overall bid defines the valuation v 1 or ( v 2 or · · · ( v k − 1 or v k ))) . AGT-MIRI Combinatorial auctions
Selling sets of items OR Language Bidding Languages XOR Language The winner determination problem The OR Language An OR-combination of two bids defining valuations v 1 and v 2 defines the following valuation: ( v 1 or v 2 )( X ) = max X 1 ⊆ X ( v 1 ( X 1 ) + v 2 ( X \ X 1 )) . If there are k atomic bids defining valuations v 1 , . . . , v k , then the overall bid defines the valuation v 1 or ( v 2 or · · · ( v k − 1 or v k ))) . 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
Selling sets of items OR Language Bidding Languages XOR Language The winner determination problem OR: Expressive Power The OR language is not fully expressive. AGT-MIRI Combinatorial auctions
Selling sets of items OR Language Bidding Languages XOR Language The winner determination problem OR: Expressive Power The OR language is not fully expressive. Theorem The OR language can represent all supermodular valuations, and only those. AGT-MIRI Combinatorial auctions
Selling sets of items OR Language Bidding Languages XOR Language The winner determination problem OR: Expressive Power The OR language is not fully expressive. Theorem The OR language can represent all supermodular valuations, and only 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
Selling sets of items OR Language Bidding Languages XOR Language The winner determination problem Basic valuation problems The most basic problem: AGT-MIRI Combinatorial auctions
Selling sets of items OR Language Bidding Languages XOR Language The winner determination problem 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
Selling sets of items OR Language Bidding Languages XOR Language The winner determination problem Basic valuation problems Theorem BunEval is NP-complete for the OR language AGT-MIRI Combinatorial auctions
Selling sets of items OR Language Bidding Languages XOR Language The winner determination problem Basic valuation problems Theorem BunEval is NP-complete for the OR language NP-hardness: by reduction from Set Packing. AGT-MIRI Combinatorial auctions
Selling sets of items OR Language Bidding Languages XOR Language The winner determination problem The XOR Language AGT-MIRI Combinatorial auctions
Selling sets of items OR Language Bidding Languages XOR Language The winner determination problem 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
Selling sets of items OR Language Bidding Languages XOR Language The winner determination problem 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 v 1 and v 2 is: ( v 1 xor v 2 )( X ) = max { v 1 ( X ) , v 2 ( X ) } . AGT-MIRI Combinatorial auctions
Selling sets of items OR Language Bidding Languages XOR Language The winner determination problem 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 v 1 and v 2 is: ( v 1 xor v 2 )( X ) = max { v 1 ( X ) , v 2 ( X ) } . If there are k atomic bids defining valuations v 1 , . . . , v k , then the overall bid defines the valuation v 1 xor ( v 2 xor · · · ( v k − 1 xor v k ))) . AGT-MIRI Combinatorial auctions
Selling sets of items OR Language Bidding Languages XOR Language The winner determination problem XOR: Expressive Power The XOR language is fully expressive. AGT-MIRI Combinatorial auctions
Selling sets of items OR Language Bidding Languages XOR Language The winner determination problem XOR: Expressive Power The XOR language is fully expressive. Theorem The XOR language can represent all monotonic valuations, and only those. AGT-MIRI Combinatorial auctions
Selling sets of items OR Language Bidding Languages XOR Language The winner determination problem 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
Selling sets of items OR Language Bidding Languages XOR Language The winner determination problem 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
Selling sets of items OR Language Bidding Languages XOR Language The winner determination problem 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
Selling sets of items OR Language Bidding Languages XOR Language The winner determination problem 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
Selling sets of items OR Language Bidding Languages XOR Language The winner determination problem 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
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
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
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
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
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 offered, then it is the revenue of the auctioneer. AGT-MIRI Combinatorial auctions
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 offered, 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
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 offered, 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
Selling sets of items Bidding Languages The winner determination problem Multiagent resource allocation AGT-MIRI Combinatorial auctions
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 own preferences regarding the allocation of goods to be selected. AGT-MIRI Combinatorial auctions
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 own preferences regarding the allocation of goods to be selected. What constitutes a good allocation and how do we find it? AGT-MIRI Combinatorial auctions
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 own 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
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 own 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
Selling sets of items Bidding Languages The winner determination problem Setting AGT-MIRI Combinatorial auctions
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 v i : 2 G → R . AGT-MIRI Combinatorial auctions
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 v i : 2 G → R . Recall that the utilitarian social welfare of an allocation is defined as follows: � sw u ( A ) = v i ( A ) . i ∈ N AGT-MIRI Combinatorial auctions
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 v i : 2 G → R . Recall that the utilitarian social welfare of an allocation is defined as follows: � sw u ( A ) = v i ( A ) . i ∈ N The input to an allocation problem is a triple ( N , G , V ), where V is a set of valuation functions (+ possibly the initial allocation A 0 ). AGT-MIRI Combinatorial auctions
Selling sets of items Bidding Languages The winner determination problem Welfare Optimization AGT-MIRI Combinatorial auctions
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 sw u ( A ) > K? AGT-MIRI Combinatorial auctions
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 sw u ( A ) > K? Theorem The Welfare Optimisation problem is NP-complete. AGT-MIRI Combinatorial auctions
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 sw u ( 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
Selling sets of items Bidding Languages The winner determination problem WO: Hardness AGT-MIRI Combinatorial auctions
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
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
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: v C ( S ) = 1 if S = C and v C ( S ) = 0 otherwise; v 0 ( 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
Selling sets of items Bidding Languages The winner determination problem WO: Hardness AGT-MIRI Combinatorial auctions
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
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
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
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
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
Selling sets of items Bidding Languages The winner determination problem Welfare Optimization AGT-MIRI Combinatorial auctions
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 sw u ( A ′ ) < sw u ( A ) ? AGT-MIRI Combinatorial auctions
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 sw u ( A ′ ) < sw u ( A ) ? Theorem The Welfare Improvement problem is NP-complete. AGT-MIRI Combinatorial auctions
Selling sets of items Bidding Languages The winner determination problem Pareto Optimality AGT-MIRI Combinatorial auctions
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
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
Selling sets of items Bidding Languages The winner determination problem Envy-Freeness AGT-MIRI Combinatorial auctions
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
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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