Preference elicitation and multistage/iterative mechanismsUnnecessary communicationSingle-stage mechanismsMultistage mechanismsA (strange) example multistage auctionConverting single-stage to multistageExample: an elicitation protocol for the Bucklin voting rule [Conitzer & Sandholm EC 05]Funky strategic phenomena in multistage mechanismsEx-post equilibriumHow do we know that we have found the best elicitation protocol for a mechanism?Combinatorial auction WDP requires exponential communication [Nisan & Segal JET 06]iBundle: an ascending CA [Parkes & Ungar AAAI 00]Restricted valuationsRestricted valuations…Preference elicitation and multistage/iterative mechanismsVincent Conitzer [email protected] communication•We have seen that mechanisms often force agents to communicate large amounts of information–E.g. in combinatorial auctions, should in principle communicate a value for every single bundle!• Much of this information will be irrelevant, e.g.:–Suppose each item has already received a bid >$1–Bidder 1 values the grand bundle of all items at v1(I) = $1–To find the optimal allocation, we need not know anything more about 1’s valuation function (assuming free disposal)–We may still need more detail on 1’s valuation function to compute Clarke payments…–… but not if each item has received two bids >$1•Can we spare bidder 1 the burden of communicating (and figuring out) her whole valuation function?Single-stage mechanisms•If all agents must report their valuations (types) at the same time (e.g. sealed-bid), then almost no communication can be saved–E.g. if we do not know that other bidders have already placed high bids on items, we may need to know more about bidder 1’s valuation function–Can only save communication of information that is irrelevant regardless of what other agents report•E.g. if a bidder’s valuation is below the reserve price, it does not matter exactly where below the reserve price it is•E.g. a voter’s second-highest candidate under plurality rule •Could still try to design the mechanism so that most information is (unconditionally) irrelevant–E.g. [Hyafil & Boutilier IJCAI 07]Multistage mechanisms•In a multistage (or iterative) mechanism, –bidders communicate something, –then find out something about what others communicated,–then communicate again, etc.•After enough information has been communicated, the mechanism declares an outcome•What multistage mechanisms have we seen already?•[Conitzer & Sandholm LOFT 04] gives an example where–the optimal single-stage mechanism requires exponential communication,–there is an equivalent two-stage mechanism that requires only linear communicationA (strange) example multistage auctionbidder 1: is your valuation greater than 4? bidder 2: is your valuation greater than 6? bidder 2: is your valuation greater than 2? yesyes yesyes yes yesbidder 1: is your v. greater than 8? bidder 1: is your v. greater than 8? bidder 1: is your v. greater than 3? nonono no nono1 wins, pays 61 wins, pays 61 wins, pays 42 wins, pays 41 wins, pays 22 wins, pays 11 wins, pays 0•Can choose to hide information from agents, but only insofar as it is not implied by queries we ask of themConverting single-stage to multistage•One possibility: start with a single-stage mechanism (mapping o from Θ1 x Θ2 x … x Θn to O)•Center asks the agents queries about their types–E.g. “Is your valuation greater than v?”–May or may not (explicitly) reveal results of queries to others•Until center knows enough about θ1, θ2, …, θn to determine o(θ1, θ2, …, θn)•The center’s strategy for asking queries is an elicitation protocol for computing o•E.g. Japanese auction is an elicitation protocol for the second-price auction–English is too, roughlyExample: an elicitation protocol for the Bucklin voting rule [Conitzer & Sandholm EC 05]•Bucklin: for candidate j, let k(j) be the smallest k such that more than half of the voters rank j in the top k; winner w is candidate that minimizes k(w)•Idea: binary search for k(w)•Maintain lower bound l, upper bound u on k(w)•Ask each voter which candidates are among their top (l+u)/2–Already know each voter’s top l and bottom m-u candidates at this point, so only need u-l bits of information per voter•If exactly one candidate is among top (l+u)/2 for > ½ of the voters, that must be the winner•If at least two candidates are, update u ← (l+u)/2•If no candidates are, update l ← (l+u)/2•Total communication is nm + nm/2 + nm/4 + … ≤ 2nm bits•Can show Ω(nm) lower bound as well•Single-stage protocol requires Ω(nm log m) communicationFunky strategic phenomena in multistage mechanisms•Suppose we sell two items A and B in parallel English auctions to bidders 1 and 2–Minimum bid increment of 1•No complementarity/substitutability•v1(A) = 30, v1(B) = 20, v2(A) = 20, v2(B) = 30, all of this is common knowledge•1’s strategy: “I will bid 1 on B and 0 on A, unless 2 starts bidding on B, in which case I will bid up to my true valuations for both.”•2’s strategy: “I will bid 1 on A and 0 on B, unless 1 starts bidding on A, in which case I will bid up to my true valuations for both.”•This is an equilibrium!–Inefficient allocation–Self-enforcing collusion–Bidding truthfully (up to true valuation) is not a dominant strategyEx-post equilibrium•In a Bayesian game, a profile of strategies is an ex-post equilibrium if for each agent, following the strategy is optimal for every vector of types (given the others’ strategies)–That is, even if you are told what everyone’s type was after the fact, you never regret what you did–Stronger than Bayes-Nash equilibrium–Weaker than dominant-strategies equilibrium•Although, single-stage mechanisms are ex-post incentive compatible if and only if they are dominant-strategies incentive compatible•If a single-stage mechanism is dominant-strategies incentive-compatible, then any elicitation protocol for it (any corresponding multistage mechanism) will be ex-post incentive compatible•E.g. if we elicit enough information to determine the Clarke payments, telling the truth will be an ex-post equilibrium (but not dominant strategies)How do we know that we have found the best elicitation protocol for a mechanism?•Communication complexity theory: agent i holds input xi, agents must communicate enough information to compute some f(x1, x2,