# Single Value Combinatorial Auctions and Implementation in Undominated Strategies Moshe Babaioff*, Ron Lavi + and Elan Pavlov* + Social and Information.

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Single Value Combinatorial Auctions and Implementation in Undominated Strategies Moshe Babaioff*, Ron Lavi + and Elan Pavlov* + Social and Information Sciences Laboratory, California Institute of Technology * School of Computer Science and Engineering, Hebrew University

Outline  Basic definitions, motivation.  Describe the first main result, on a simplified model.  Define Single-Value combinatorial auctions  Examples, discussion  Describe the second main result.

Combinatorial Auctions m indivisible non-identical items for sale. n bidders compete for subsets of these items. Each bidder i has a valuation for each set of items: v i (S) = value that i assigns to acquiring the set S –v i is non-decreasing (“free disposal”). –v i (  ) = 0. Each bidder is strategic, aims to maximize his own utility: the value of the subset received minus the price. Objective: Find a partition (S 1 …S n ) of {1..m} that maximizes the social welfare:  i v i (S i )

Design goals Requirement 1: Computational feasibility. Design mechanisms that can be computed in reasonable time. –Going over all possible allocations of items to players can take more than a life time (!), even for medium-sized instances. Requirement 2: Design mechanisms that lead to “good” outcomes in the presence of strategic behavior. –Direct revelation mechanisms: The VCG mechanism always has dominant strategies, but sometimes violates requirement 1. Also has other disadvantages, described e.g. by [Ausubel-Milgron]. –Indirect mechanisms, mainly ascending auctions: players compete by raising prices; winners pay their “last offer”. Wish to satisfy 1+2 simultaneously.

Some Special Cases One item: –Vickrey’s 2 nd price auction leads to optimal welfare (in dominant strategies). –This has an equivalent ascending (English) auction. Multiple items: –Complementarities: v(S  T) > v(S) + v(T). –Substitutability: slightly more subtle than “no complementarities”. For substitutability, VCG is tractable, and there exist ascending auctions that are equivalent to VCG: –[Ausubel] Identical items; downward-sloping valuations. –[Demange-Gale-Sotomayor] Unit-demand bidders. –[Gul-Stacchetti] Gross-substitute bidders (leading to Walrasian prices).

The Case of Complementarities (I) In general, no mechanism can reach the optimal outcome with polynomial communication [Nisan-Segal]. The computational problem is hard: –For special cases that enable short communication, the computational problem is NP-hard [Lehmann-O’Callaghan-Shoham]. –Hence we allow approximations: (always) reaching an outcome with welfare at least (1/c)·(“optimal welfare”) (c is a parameter). –VCG is not truthful with approximations [Nisan-Ronen].

The Case of Complementarities (II) No “fast” ascending auction with strategic properties: –The proxy auction [Ausubel-Milgrom] and the i-bundle auction [Parkes] maintain exponential number of prices, and continues for exponential number of rounds. –No ascending auction with strategic properties that reaches an approximately optimal outcome.

Outline  Basic definitions, motivation.  Describe the first main result, on a simplified model.  Define Single-Value combinatorial auctions  Examples, discussion  Describe the second main result.

Single Minded Bidders Single-Minded Bidders [Lehmann-O’Callaghan-Shoham] : Player i desires one subset of items, S i, for a value v i –No algorithm can obtain better than √m approximation in polynomial time. –There exists a truthful mechanism that obtains a √m approximation in polynomial time. “Known” Single Minded Bidders [Mu’alem-Nisan] : The desired subset is public information; the value is private information. –They give variety of truthful mechanisms, under the “known” assumption.

Algorithm to Mechanism

The Japanese Wrapper (JW) 1 1 1 Same value Double value or retire W v1Tv1T vnTvnT v2Tv2T C-approx ALG W L v1v1 vnvn v2v2 The output at each iteration is improving and Pareto efficient T is the number of rounds A winner pays her last value v i T The dominant strategy of i is to increase her value when losing, as long as the increased value ≤v i Assumption: v i (T)=0 or v i (T)≥1

Analysis – First Step Lemma 1: T ≤ 2log(v max )+1 Proof: –Look at i, the last agent to retire. –Since ALG is Pareto efficient, i intersects some winner k. –Therefore at each iteration either i or k lose. –Each agent can lose at most log(v max ) times. sisi sksk

Analysis – Second Step Lemma 2: The JW is a (2 c T+1)-approximation. Proof: –W=winners. L=losers. OPT(W  L)≤OPT(W)+OPT(L) –R t = agents that retired at iteration t v t = the vector of values at time t. –OPT(R t ) < 2 OPT(R t | v t ) < 2 c ·V(W) R 1 R 2 R T L W

Summary of the Analysis Theorem: The JW mechanism obtains an O(c log(v max ))- approximation in dominant strategies for known single minded players. Remarks: The first ascending auction for this case, and the first general method to convert any given algorithm to an ascending auction. This is useful e.g. for: –Special cases that allow the √m barrier to be broken. –Converting heuristics that work well in practice to truthful mechanisms.

Outline  Basic definitions, motivation.  Describe the first main result, on a simplified model.  Define Single-Value combinatorial auctions  Examples, discussion  Describe the second main result.

What about Multi-Minded Players? The picture for multi-minded players is very partial: If we have B > 3 copies of each item, [Bartal-Gonen-Nisan] give a truthful -approximation mechanism. Very recently [Lavi-Swamy] give a randomized -approx. (for any B>1 ) which is truthful-in-expectation (in particular it assumes that players are risk-neutral). No tractable deterministic mechanism is known, and no tractable ascending auction is known.

Single-Value Players A Single-Value player has: S –a collection S i of several desired subsets of items, and, S –a value v i >0 for any s  S i. Two information models: S –“Known” domain: S i is publicly known. v i is private information. S –“Unknown” domain – Both v i, S i are private information.

Given a graph G=(V,E): Each agent i desires path from his source node s i to his target node t i for value v i. Edges (=goods) should be allocated to agents in order to maximize welfare. A natural Single Value multi minded CA. V 1 =10 V 2 =4 t1t1 s1s1 s2s2 t2t2 Example: Edge-Disjoint Paths

Single-Value Players A Single-Value player has: S –a collection S i of several desired subsets of items, and, S –a value v i >0 for any s  S i. Two information models: S –“Known” domain: S i is publicly known. v i is private information. S –“Unknown” domain – Both v i, S i are private information. Type may have exponential size -- direct revelation is intractable. [Nisan] shows that no algorithm can obtain better than Ω(√m) approximation with polynomial communication. Single Value Players Single Minded Multi Minded KnownMN, APTT AT UnknownLOS ?

Our Results Known domains: decoupling of algorithms and incentives. THM 1: There exists a method to convert any c-approx algorithm to an ascending auction that obtains O(log 2 (v max ) c)-approx in dominant strategies (assuming “appropriate oracle access”). Unknown domains: THM 2: There exists an ascending auction that obtains an O(log 2 (v max ) √m)-approximation in “undominated strategies”. We actually use “Computationally Feasible Undominated Strategies”, a slightly stronger notion that we define.

Outline  Basic definitions, motivation.  Describe the first main result, on a simplified model.  Define Single-Value combinatorial auctions  Examples, discussion  Describe the second main result:  A mechanism for unknown multi-minded players.  The switch to undominated strategies

Modifying the JW An iterative procedure, similar to before. Agents bid for a value and a bundle: –Start from value=1, bundle=G (all the items). –Value increases over time. –Bundle shrinks over time. Winner gets his final bundle, pays his final value. An agent focuses on one of his bundles.

Same value v i and set bundle=s’ i Multiply value and set bundle=s’ i, or retire ALG W,s’ L,s’ 1,G1,G 1,G1,G 1,G1,G v1,s1v1,s1 v2,s2v2,s2 vn,snvn,sn W v 1 T,s 1 T v 2 T,s 2 T v n T,s n T A winner gets s i T and pays her last value v i T The General JW

Same value v i and set bundle=s’ i Multiply value and set bundle=s’ i, or retire ALG W,s’ L,s’ 1,G1,G 1,G1,G 1,G1,G v1,s1v1,s1 v2,s2v2,s2 vn,snvn,sn W v 1 T,s 1 T v 2 T,s 2 T v n T,s n T s’ i is a subset of s i A winner gets s i T and pays her last value v i T The General JW

Same value v i and set bundle=s’ i Multiply value and set bundle=s’ i, or retire ALG W,s’ L,s’ 1,G1,G 1,G1,G 1,G1,G v1,s1v1,s1 v2,s2v2,s2 vn,snvn,sn W v1T,s1Tv1T,s1T v2T,s2Tv2T,s2T vnT,snTvnT,snT s’ i is a subset of s i A winner gets s i T and pays her last value v i T The General JW

The 1-CA algorithm for Single Minded Players [Halldorsson, Mu’alem-Nisan] Input: v,s Procedure Go over the agents in descending order of values. Let player i be the current agent. –If and |s i |≤√m add i to Greedy. Output: value-maximal THM [Halldorsson] : This is a 2√m-approx.

Example 3 3 5 4 10 Result: Black wins.

Example THM [Mu’alem-Nisan] : 1-CA can be implemented truthfully for “known” single minded players, but not for “unknown” minded players. 3 3 5 4 10 Green lies about bundle: 3 3 4 10 5 Result: Black wins. Result: Green+Red+Purple win.

Plugging the 1-CA in our Wrapper Input: v t,s t, W t-1 Procedure Go over the agents in descending order of values. Let player i be the current agent. –If i  W t-1 allow i to pick a bundle such that |s i t+1 |≤√m, in any other case s i t+1= s i t. –If |s i t+1 |≤√m, add i to any of the allocations for which he can be added. Output s t+1 and with maximal value. Allocation is also: Pareto efficient Improving over time Shrinking

“loser-if-silent” 3 3 5 4 10 But the actual scenario is… 5 Green thinks the scenario is… 10 DFN: Player i is a “loser-if-silent” if she will surely retire if she does not shrink her active set. If player i is a “loser-if-silent”, and her active set strictly contains a desired bundle, then she will shrink her active set.

What are the “reasonable” strategies here? On which bundle to bid? No dominant strategy! DFN: A strategy is “reasonable” if, –Never bids over her value. –Never retires before getting to v i /2. –Never bids on an undesired bundle. –If player i is a “loser-if-silent”, and her active set strictly contains a desired bundle, then she will shrink her active set. Theorem: For any combination of “reasonable” strategies, the mechanism achieves O(log 2 (v max ) √ m)-approximation.

Feasible implementation in undominated strategies

Dominant Strategies Implementation Definition: –A strategy s i dominates the strategy s’ i, if for any s -i, u i (s i,s -i ) ≥ u i (s’ i,s -i ), and this inequality is strict for at least one instance of s -i. –A strategy s i is dominant for agent i if it dominates any other strategy of agent i. Definition: A mechanism implements a c-approximation in dominant strategies if for the vector of dominant strategies s, it outputs a c-approximation outcome.

Undominated Strategies Implementation Definition: A mechanism implements a c-approximation in undominated strategies if for any combination of undominated strategies, it outputs a c-approximation outcome. Dominated strategies Undominated strategies

Feasible implementation in undominated strategies DFN: A mechanism has the “fast undominance recognition property” if there exists a polynomial-time procedure that, given any strategy s’ i, determines if s’ i is dominated, and, if so, finds an undominated strategy s i that dominates s’ i. Main Definition: A mechanism feasibly implements a c- approximation in undominated strategies if –For any undominated strategies vector s, it outputs a c- approximation outcome in polynomial time. –The mechanism has the fast undominance recognition property. The agent is never harmed by this transformation

Dominant vs. Undominated Dominant Strategies Mechanism Rational agents do not play dominated strategies: truthful. Straightforward to choose a strategy (bid truthfully). An agent does not assume rationality of others. No coordination between agents needed. Approximation achieved for truthful agents. An agent never regrets her action. Feasible US Implementation Rational agents do not play dominated strategies. Easy (polynomial) for an agent to choose a strategy. An agent does not assume rationality of others. No coordination between agents needed. Approximation achieved for any vector of undominated strategies. An agent might regret her action.

Unknown Multi Minded CA Results Main Theorem: The “1-CA Japanese Mechanism” feasibly implements an O(log 2 (v max ) √ m)- approximation in undominated strategies for unknown multi-minded CAs with single-value players. This also holds for players who are only “δ-close” to single value, with an additional approximation loss of δ.

Summary  We define and study single value Combinatorial Auctions in two information models (Known-Unknown).  We present a general way to create truthful mechanism from any algorithm for Known Single Value players.  We define the notion of a “feasible implementation in undominated strategies”.  We present such mechanisms for Unknown Single Value players.  The first strategic mechanisms with non trivial approximation for this case.

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