Agent-Based Systems Coalition formation The core and the Shapley - PowerPoint PPT Presentation

Agent-Based Systems Agent-Based Systems Where are we? Agent-Based Systems • Coalition formation • The core and the Shapley value • Different representations Michael Rovatsos • Simple games mrovatso@inf.ed.ac.uk • Qualitative coalitional games Today . . . Lecture 11 – Resource Allocation • Resource Allocation 1 / 18 2 / 18 Agent-Based Systems Agent-Based Systems Auctions The English Auction (EA) • Auctions = method for allocating scarce resources in a society given preferences of agents • Each bidder raises freely his bid (in public), auction ends if no • Most common types of auctions: bidder is willing to raise his bid anymore - English (first-price open-cry ascending), Dutch (reverse), first-price • Bidding process public in correlated auctions, it can be sealed bid, Vickrey auction (second-price sealed bid) worthwhile to counterspeculate • Additional variations depending on following characteristics: • In correlated value auctions, often auctioneer increases price at - private-value, public-value, correlated value auctions constant/appropriate rate, also use of reservation prices - risk-neutral, risk-seeking, risk-averse bidders/auctioneer • Dominant strategy in private-value EA: bid a small amount above • Some interesting issues/problems: highest current bid until one’s own valuation is reached - Lying (lying bidders, lying auctioneer) - Bidder collusion - Incentive for counterspeculation 3 / 18 4 / 18

Agent-Based Systems Agent-Based Systems The English Auction (EA) Dutch/First-Price Sealed Bid Auctions • Dutch (descending) auction: seller continuously lowers prices until • Advantages: one of the bidders accepts the price - Truthful bidding is individually rational & stable - Auctioneer cannot lie (whole process is public) • First-price sealed bid: bidders submit bids so that only auctioneer • Disadvantages: can see them, highest bid wins (only one round of bidding) - Can take long to terminate in correlated/common value auctions • DA/FPSB strategically equivalent (no information given away - Information is given away by bidding in public during auction, highest bid wins) - Use of shills (in correlated-value EA) and “minimum price bids” • Advantages: possible, to drive prices - Efficient in terms of real time (especially Dutch) - Bidder collusion self-enforcing (once agreement has been - No information is given away during auction reached, it is safe to participate in a coalition) and identification of - Bidder collusion not self-enforcing, and bidders have to identify each partners easily possible other 5 / 18 6 / 18 Agent-Based Systems Agent-Based Systems Dutch/First-Price Sealed Bid Auctions – Problems The Vickrey Auction (VA) • Second-price sealed bid: Highest bidder wins, but pays price of • No dominant strategy, individually optimal strategy depends on second-highest bid assumptions about others’ valuations • Advantages: • One would normally bid less than own valuation but just enough to - Truthful bidding is dominant strategy win Incentive to counter-speculate - No incentive for counter-speculation • Without incentive to bid truthfully, computational resources might - Computational efficiency be wasted on speculation • Disadvantages: • Another problem: lying auctioneer - Bidder collusion self-enforcing - Lying auctioneer • Would be nice to combine efficiency of Dutch/FPSB with incentive compatibility of English auction Vickrey auction can be seen as • Unfortunately, VA is not very popular in real life attempt to achieve this • But very successful in computational auction systems 7 / 18 8 / 18

Agent-Based Systems Agent-Based Systems Further issues in auctions Further issues in auctions (II) • Some properties of protocols change • Pareto efficiency : all protocols allocate auction item to the bidder - if there is uncertainty about own valuations who values it most (in isolated private value/common value - if one can pay to obtain information about others’ valuations auctions) - if we are looking at sequential (multiple) auctions - But this result requires risk-neutrality if there is some uncertainty • Undesirable private information revelation about own valuations - Example: truthful bidding in EA/VA may lead sub-contractors to • Revenue equivalence in terms of expected revenue among all re-negotiate rates after finding out that price was lower than they protocols if valuations independent, bidders risk-neutral and thought auction is private value • In terms of communication, auctions are not a very expressive • Winner’s curse in correlated/common value auctions method of negotiation - If I win, I always know I won’t get to re-sell at the same price, - Solely concerned with determining a selling price for some item because others value the goods less! - Will look at bargaining and argumentation in next two lectures 9 / 18 10 / 18 Agent-Based Systems Agent-Based Systems Combinatorial Auctions Combinatorial Auctions (II) • Generalised model of resource allocation, auctioning bundles of goods Z = { z 1 , . . . , z n } instead of single items • Winner determination: computing the optimal allocation • A valuation function v i : 2 Z → R indicates how much Z ⊆ Z is Z ∗ 1 , . . . Z ∗ n given valuations submitted by bidders worth to agent i • Prone to strategic manipulation as agents may not reveal their true • Sensible properties of valuation functions: valuations (e.g. may overstate the value of possible bundles) - Normalisation: v ( ∅ ) = 0 - Free disposal: Z 1 ⊆ Z 2 implies v ( Z 1 ) ≤ v ( Z 2 ) • Representational complexity: exponential in the number of goods (imagine listing all possible valuations of all bundles) • The outcome is an allocation Z 1 , Z 2 , . . . , Z n of goods being auctioned among the agents • Computational complexity: winner determination is NP-hard even under restrictive assumptions • Maximising social welfare: - Z ∗ 1 , . . . Z ∗ n = arg max ( Z 1 ,..., Z n ) ∈ alloc ( Z , Ag ) sw ( Z 1 , . . . , Z n , v 1 , . . . , v n ) where sw ( Z 1 , . . . , Z n , v 1 , . . . , v n ) = � n i = 1 v i ( Z i ) 11 / 18 12 / 18

Agent-Based Systems Agent-Based Systems Bidding Languages XOR bids • We specify a number of bids, but we will par for at most one • As before, we want to have succinct representation schemes for • β = ( Z 1 , p 1 ) XOR · · · XOR ( Z k , p k ) valuation functions if Z ′ does not satisfy any of  0 • Atomic Bid: β = ( Z , p ) , where Z ⊆ Z and p ∈ R + is the price   v β ( Z ′ ) = ( Z 1 , p 1 ) , . . . , ( Z k , p k ) • A bundle of goods Z ′ satisfies ( Z , p ) if Z ⊆ Z ′  max { p i | Z i ⊆ Z ′ } otherwise  - Bundle { a , b , c } satisfies the atomic bid ( { a , b } , 4 ) • Example: β = ( { a , b } , 3 ) XOR ( { c , d } , 5 ) - Bundle { b , d } does not satisfy the atomic bid ( { a , b } , 4 ) - v β ( { a } ) = 0 • An atomic bid β = ( Z , p ) defines a valuation function v β - v β ( { a , b } ) = 3 if Z ′ satisfies ( Z , p ) � p - v β ( { c , d } ) = 5 v β ( Z ′ ) = - v β ( { a , b , c , d } ) = 5 0 otherwise • XOR bids are fully expressive, number of bids may be exponential • Not sufficient to express any valuation function in |Z| , v β ( Z ) can be computed in polynomial time 13 / 18 14 / 18 Agent-Based Systems Agent-Based Systems OR bids The VCG Mechanism (I) • Terminology: - ‘Indifferent’ valuation function: v 0 ( Z ) = 0 for all Z ⊆ Z • Combine more than one atomic statement disjunctively - sw − i ( Z 1 , . . . , Z n ) = � j ∈ Ag : j � = i v j ( Z j ) , social welfare of all agents but i • β = ( Z 1 , p 1 ) OR · · · OR ( Z k , p k ) • The Vickrey-Clarke-Groves mechanism (VCG Mechanism): • The valuation for Z ′ ⊆ Z is determined w.r.t. atomic bids W s.t.: 1 Every agent declares a valuation function ˆ v i (may not be true) - every bid in W is satisfied by Z ′ 2 Mechanism choses the allocation that maximises the social welfare: - each pair of bids in W has mutually disjoint sets of goods - there is no other subset of bids W ′ from W satisfying the first two Z ∗ 1 , . . . , Z ∗ n = arg ( Z 1 ,..., Z n ) ∈ alloc ( Z , Ag ) sw ( Z 1 , . . . , Z n , ˆ v 1 , . . . , ˆ v i , . . . , ˆ v n ) max conditions that � ( Z i , p i ) ∈ W ′ p i > � ( Z j p j ) ∈ W p j 3 Every agent pays to the mechanism an amount p i • Example: β = ( { a , b } , 3 ) OR ( { c , d } , 5 ) (‘compensation’ for the utility other agents lose by i participating) - v β ( { a } ) = 0 , v β ( { a , b } ) = 3, v β ( { c , d } ) = 5, v β ( { a , b , c , d } ) = 8 p i = sw − i ( Z ′ 1 , . . . , Z ′ v 1 , . . . , v 0 , . . . , ˆ • Not fully expressive, consider: n , ˆ v n ) − sw − i ( Z ∗ 1 , . . . , Z ∗ - v ( { a } ) = 1, v ( { b } ) = 1, v ( { a , b } ) = 1 n , ˆ v 1 , . . . , ˆ v i , . . . , ˆ v n ) , where • Can be exponentially more succinct than XOR bids Z ′ 1 , . . . , Z ′ v 1 , . . . , v 0 , . . . , ˆ ( Z 1 ,..., Z n ) ∈ alloc ( Z , Ag ) sw ( Z 1 , . . . , Z n , ˆ n = arg max v n ) 15 / 18 16 / 18

Agent-Based Systems Agent-Based Systems The VCG Mechanism (II) Summary • The VCG mechanism is incentive compatible: - telling the truth is the dominant strategy • Different auction types and properties • Generalisation of the Vickrey auction: for a single good VCG • Combinatorial Auctions reduces to the Vickrey mechanism • Bidding Languages - p i would be the amount of the second highest valuation • The VCG mechanism • Shows that social welfare maximisation can be implemented • Next time: Bargaining in dominant strategies in combinatorial auctions • Computing VCG payments is NP-hard 17 / 18 18 / 18