Preference Elicitation Partial-revelation VCG mechanism for Combinatorial Auctions and Eliciting Non-price Preferences in Combinatorial Auctions

Overview Introduction to Preference Elicitation Partial-revelation VCG mechanism for combinatorial auctions Model, Efficient Allocations, Optimality, Cost, Vickrey Payments, and Mechanism Model, Efficient Allocations, Optimality, Cost, Vickrey Payments, and Mechanism Eliciting Bid Taker Non-price Preferences in Combinatorial Auctions Non-Price-Based Utility, Utility Function Uncertainty, Computation of Minimax Regrets, and Preference Elicitation Non-Price-Based Utility, Utility Function Uncertainty, Computation of Minimax Regrets, and Preference Elicitation

Introduction to Preference Elicitation Def. Preference elicitation is the method in which bidders inform the auctioneer of their valuations of the different bundles in the combinatorial auction. Due to communication constraints and the fact that given the number of bundles may be exponential, it is desirable to find ways to express the preferences in way which is more efficient.

Partial-revelation VCG mechanism for combinatorial Auctions This paper describes an algorithm and data structure which uses a direct method with partial revelation of principles. This is done by exploiting the fact that many bidders have a preference order which can form a rank lattice

Model A set of bidders, n, and items, m, where the seller is not involved Preference order is rational and transitive A rank function can be formed such that assigns a unique value to every possible bundle The inverse of a rank function returns the bundle corresponding to rank given

Model (Cont.) A combination of ranks is an element of the n- ary cross product of ranks ie c = (r1,…,rn) A feasible combination is one where the bundles corresponding to the ranks form a partition of a subset of the set of items A dominance relation is a binary relation where the rank every item in a combination is greater than another combination The rank lattice is lattice formed by dominance relations and the set of possible combinations and lub(C)=(n,…,n) and the glb(C)=(1,…,1)

Efficient Allocations They present two algorithms which use the following ideas A combination that is feasible and not dominated by any other feasible combination is Pareto-efficient A combination that is feasible and not dominated by any other feasible combination is Pareto-efficient The welfare maximizing is one of the Pareto-efficient outcomes found The welfare maximizing is one of the Pareto-efficient outcomes found The first, PAR, finds all Pareto-efficient outcomes The second, EBF, finds a welfare-maximizing outcome and is a refinement of the first

Optimality Def. An admissible algorithm finds an efficient, feasible combination for every problem instance ie set of items and utilities. Elementary operations are A value function A value function A feasibility function A feasibility function A successor function A successor function Inserting a combination into the initial set Inserting a combination into the initial set Def. An algorithm is admissibly equipped if only elementary operations are used to obtain lattice- related information

Optimality EBF algorithms require the least amount of feasibility checks for every problem instance EBF algorithms check precisely one feasible combination for feasibility The MPAR algorithm requires n m valuation checks The EBF requires for the worst case (2 mn – n m )/2 valuation checks. This is due to the structure of the lattice

Vickrey Payments The algorithm requires no new valuation information to determine the Vickrey payments t(i) = V(E -i ) - ∑ jєN,j≠i u j (X j ) The reason for this is due to the lattice structure The most desirable combinations are checked first So all subsets that would have formed an efficient allocation if agent i had not participated have been viewed

The Mechanism The mechanism uses Two questions Two questions Give me the bundle with the next higher rank # Give me your valuation for bundle x An extended EBF algorithm with a deterministic method for selecting the next combination An extended EBF algorithm with a deterministic method for selecting the next combination A data structure A data structure A policy for asking questions A policy for asking questions

The Mechanism The data structure is a graph with nodes representing the agent, rank, bundle and value info Eliciting information happens during the algorithm when the algorithm needs more information to determine the next value- maximizing combination

Eliciting Bid Taker Non-Price Preferences in (Combinatorial) Auctions This paper models the winner determination problem in reverse combinatorial auctions where the bidders have other non-price preferences They use the notion that the bidder can specify tradeoff weights which represent how much a given feature is worth

Linear Utility for Non-price Features In an auction, the price may not be the only feature which is important Other possible features may include: The number of winners The number of winners Quality of certain items Quality of certain items Percentage of volume given to a specific supplier Percentage of volume given to a specific supplier Geographic diversity of the winners Geographic diversity of the winners

Linear Utility for Non-price Features The features can be summarized in a feature set F = {f1,…,fk} and used to calculated linear utility u(x) = ∑ i=1 k w i f i (x) – c(x) where c(x) is the cost of allocation and w i is the tradeoff weight Tradeoff weights represent how much a given feature is worth

Utility Function Uncertainty It is difficult to determine the exact set of tradeoff weights which should be used for the set of features considered by the agent To determine these weights the concept of minimax regret is used

Utility Function Uncertainty The regret of an allocation is the maximum amount of utility lost by choosing an allocation given a set of weights The maximum regret of an allocation is the maximum utility lost by choosing an allocation for any feasible set of weights The pairwise max regret is the amount of utility lost by choosing one allocation over another for any feasible set of weights The minimax optimal allocation is the minimum allocation such that there exists another allocation which maximizes pairwise regret with respect to the optimal allocation

Computation of Minimax Regret There are two settings for calculating minimax regret 1. The set of all possible weights is defined by an upper and lower bound for each weight 2. Arbitrary linear constraints on the weight define the weight set

Computation of Minimax Regret They define an algorithm for calculating minimax regret First let the generated allocation be some arbitrary allocation x’ First let the generated allocation be some arbitrary allocation x’ Solve the linear integer program with x’ and obtain the solution x* with objective value d* Solve the linear integer program with x’ and obtain the solution x* with objective value d* Solve the linear integer program with x* obtaining a regret level and an adversarial allocation x’’. If r* > d* then add x’’ to the generated allocations and repeat step two; otherwise terminate with optimal solution x* Solve the linear integer program with x* obtaining a regret level and an adversarial allocation x’’. If r* > d* then add x’’ to the generated allocations and repeat step two; otherwise terminate with optimal solution x*

Arbitrary Weight Constraints This method uses equation 6 to calculate minimax regret when W is given by a set of arbitrary linear constraints 1. Start with a weight vector w1 2. Given the fixed weight vector, solve equation 6 obtaining x i ’ and max regret mi’ 3. Given the x i ’ in step 2, solve equation 6 obtaining a new weight vector w i+1 and max regret m i+1 4. Repeat if m i+1 is not equal to m i

Empirical Evaluation They measure the amount of constraints generated for the minimax regret problems with bounded weights They found that for most problem sizes the vast majority of instances take between 3 and 4 rounds of constraint generation to generate a minimax optimal allocation This program generates a feasible allocation so that if the number of constraints generated was too large the algorithm could be stopped at any time

Preference Elicitation They propose two simple techniques for preference elicitation Adjustment of weight bounds via a graphical interface Adjustment of weight bounds via a graphical interface Querying the buyer with comparison queries to induce linear constraints Querying the buyer with comparison queries to induce linear constraints

Conclusion Two widely different methods for eliciting preferences from bidders 1. Exploiting a ranking of bundles in order to find optimal allocations while not necessarily requiring bidders to reveal all of their valuations 2. Using the notion of minimax regret to tradeoff weights when dealing with features in utilities to determine an optimal allocation