CS 15-892 Foundations of Electronic Marketplaces

Summary & future research directions

Tuomas SandholmComputer Science Department

Carnegie Mellon University

Systems with self-interested agents (computational or human)

• Mechanism (e.g., rules of an auction) specifies legal actions for each agent & how the outcome is determined as a function of the agents’ strategies

• Strategy (e.g., bidding strategy) = Agent’s mapping from known history to action• Rational self-interested agent chooses its strategy to maximize its own expected

utility given the mechanism => strategic analysis required for robustness => noncooperative game theory

• But … computational complexity– In executing the mechanism

• E.g. combinatorial auctions NP-complete & inapproximable to clear– In determining the optimal strategy

• E.g. NP-complete valuation calculations• E.g. uncomputable best-response strategies in repeated games

– In executing the optimal strategy• E.g. chess: how much space needed to represent an optimal strategy?

• Has significant impact on prescriptions– Has received little attention in game theory

Is there a desirable way to aggregate agents’ truthful preferences?

• Set of outcomes O

• Each agent i has a most-to-least-preferred ordering Ri of O

• R = (R1, R2, ... , R|A| )

• Social choice functional G (R, O ) = R

• To avoid unilluminating technicalities in proof, assume Ri and R are strict total orders

• Some possible (weak) desiderata of G– 1. Pareto principle: If every agent prefers x to y , then x is preferred to y in R– 2. Independence of irrelevant alternatives: If x is preferred to y in G (R, O ), and if R’ is another preference

profile s.t. each agent’s preference between x and y is the same as in R, then x is preferred to y in G (R’, O )

– 3. Nondictatorship: No agent is decisive for every pair of outcomes in O• Arrow’s impossibility theorem: If |O | ≥ 3, then no G satisfies desiderata 1-3• The impossibility holds even if only the highest ranked outcome is sought:• Thrm. Let |O | ≥ 3. If a social choice function f: R -> outcomes is monotonic and Paretian, then f is

dictatorial– f is monotonic whenever x = f(R) and x maintains its position in R’, then f(R’) = x– x maintains its position whenever x >i y => x > i’ y

Goal of mechanism design• Implementing a social choice function f(R) using a game

– Actually, say we want to implement f(u1, …, u|A|)

• Center = “auctioneer” does not know the agents’ preferences• Agents may lie

– unlike in the theory of social choice which we discussed in class before• Goal is to design the rules of the game (aka mechanism) so that in

equilibrium (s1, …, s|A|), the outcome of the game is f(u1, …, u|A|)

• Mechanism designer specifies the strategy sets Si and how outcome is determined as a function of (s1, …, s|A|) (S1, …, S|A|)

• Variants– Strongest: There exists exactly one equilibrium. Its outcome is f(u1, …, u|A|)– Medium: In every equilibrium the outcome is f(u1, …, u|A|)– Weakest: In at least one equilibrium the outcome is f(u1, …, u|A|)

Revelation principle• Any outcome that can be supported in Nash (dominant

strategy) equilibrium via a complex “indirect” mechanism can be supported in Nash (dominant strategy) equilibrium via a “direct” mechanism where agents reveal their types truthfully in a single step

Agent 1’s preferences

Agent |A|’s preferences

...

Strategy formulator

Strategy formulator

Strategy

Strategy Original“complex”“indirect”mechanism Outcome

Constructed “direct revelation” mechanism

Uses of the revelation principle• Literal: “Only direct mechanisms needed”

– Problems: • Strategy formulator might be complex

– Complex to determine and/or execute best-response strategy– Computational burden is pushed on the center (assumed away)– Thus the revelation principle might not hold in practice if these computational problems are hard– This problem traditionally ignored in game theory

• Even if the indirect mechanism has a unique equilibrium, the direct mechanism can have additional bad equilibria

• As an analysis tool– Best direct mechanism gives tight upper bound on how well any indirect

mechanism can do• Space of direct mechanisms is smaller than that of indirect ones• One can analyze all direct mechanisms & pick best one• Thus one can know when one has designed an optimal indirect mechanism (when it is as

good as the best direct one)

Solution concepts from noncooperative game theory

• Tools for building robust, nonmanipulable systems with self-interested agents and different agent designers

• Different solution concepts– For existence, use strongest equilibrium concept– For uniqueness, use weakest equilibrium concept

Nash eq Dominant strategy eq

Strength

Strength against collusion

Subgame perfect eq Perfect Bayesian eqBayes-Nash eq Sequential eq

Strong Nash eq

Coalition-Proof Nash eq

Implementation in dominant strategies

Strongest form of mechanism design

Implementation in dominant strategies

• Goal is to design the rules of the game (aka mechanism) so that in dominant strategy equilibrium (s1, …, s|A|), the outcome of the game is f(u1, …, u|A|)

• Nice in that agents cannot benefit from counterspeculating each other– Others’ preferences– Others’ rationality– Other’s endowments– Other’s capabilities …

Gibbard-Satterthwaite impossibility

• Thrm. If |O | ≥ 3 (and each outcome would be the social choice under f for some input profile (u1, …, u|A|) ) and f is implementable in dominant strategies, then f is dictatorial

Ways around the Gibbard-Satterthwaite impossibility

• Use a weaker equilibrium notion– In practice, agent might not know others’ revelations

• Design mechanisms where computing a beneficial manipulation is hard – E.g. Manipulation by insincerely ranking the outcomes is NP-complete in

“second order Copeland” voting mechanism [Bartholdi, Tovey, Trick 1989]• Copeland score: Number of competitors an outcome beats in pairwise competitions • 2nd order Copeland: Copeland, and break ties based on the sum of the Copeland

scores of the competitors that the outcome beat

• Randomization

• Agents’ preferences have special structure

Need almost this much randomness

General preferences

Quasilinear preferences

Quasilinear preferences: Groves mechanism• Outcome (x1, x2, ..., xk, m1, m2, ..., m|A| )• Quasilinear preferences: ui(x, m) = mi + vi(x1, x2, ..., xk)• Utilitarian setting: Social welfare maximizing choice

– Outcome s(v1, v2, ..., v|A|) = maxx i vi(x1, x2, ..., xk)

• Thrm. Assume every agent’s utility function is quasilinear. A utilitarian social choice function f: v -> (s(v), m(v)) can be implemented in dominant strategies if mi(v)= ji vj(s(v)) + hi(v-i) for arbitrary function h

Uniqueness of Groves mechanism• Thrm. Assume every agent’s utility function is

quasilinear. A utilitarian social choice function f: v -> (s(v), m(v)) can be implemented in dominant strategies for all v: A x O -> R only if mi(v)= ji vj(s(v)) + hi(v-i) for some function h

Clarke tax “pivotal” mechanism

• Special case of Groves mechanism: hi(v-i) = - ji vj(s(v-i))

• So, agent’s payment mi = ji vj(s(v)) - ji vj(s(v-i)) 0 is a tax• Intuition: Agent internalizes the negative externality he

imposes on others by affecting the outcome– Agent pays nothing if he does not change (“pivot”) the outcome

Clarke tax mechanism… • Pros

– Social welfare maximizing outcome– Truth-telling is a dominant strategy– Feasible in that it does not need a benefactor (i mi 0)

• Cons– Budget balance not maintained (in pool example, generally i mi < 0)

• Have to burn the excess money that is collected• Thrm. [Green & Laffont 1979]. Let the agents have quasilinear preferences u i(x, m) =

mi + vi(x) where vi(x) are arbitrary functions. No social choice function that is (ex post) welfare maximizing (taking into account money burning as a loss) is implementable in dominant strategies

• If there is some party that has no private information to reveal and no preferences over x, welfare maximization and budget balance can be obtained by having that party’s payment be m0 = - i=1.. mi

– E.g. auctioneer could be agent 0– Might still not work if participation is voluntary

– Vulnerable to collusion• Even by coalitions of just 2 agents

Implementation in Bayes-Nash equilibrium

Implementation in Bayes-Nash equilibrium

• Goal is to design the rules of the game (aka mechanism) so that in Bayes-Nash equilibrium (s1, …, s|A|), the outcome of the game is f(u1, …, u|A|)

• Weaker requirement than dominant strategy implementation– An agent’s best response strategy may depend on others’ strategies

• Agents may benefit from counterspeculating each others’– preferences– rationality– endowments– capabilities …

– Can accomplish more than under dominant strategy implementation• E.g., budget balance & Pareto efficiency (social welfare maximization)

under quasilinear preferences …

Expected externality mechanism [d’Aspremont & Gerard-Varet 79; Arrow 79]

• Like Groves mechanism, but sidepayment is computed based on agent’s revelation vi , averaging over possible true types of the others v-i

• Outcome (x1, x2, ..., xk, m1, m2, ..., m|A| )• Quasilinear preferences: ui(x, m) = mi + vi(x1, x2, ..., xk)• Utilitarian setting: Social welfare maximizing choice

– Outcome s(v1, v2, ..., v|A| ) = maxx i vi(x1, x2, ..., xk)

• Others’ expected welfare when agent i announces vi is (vi) = v-i p(v-i) ji vj(s(vi , v-i))– Measures change in expected externality as agent i changes her revelation

• Thrm. Assume quasilinear preferences and statistically independent valuation functions vi. A utilitarian social choice function f: v -> (s(v), m(v)) can be implemented in Bayes-Nash equilibrium if mi(vi)= (vi) + hi(v-i) for arbitrary function h

• Unlike in dominant strategy implementation, budget balance achievable – Intuitively, have each agent contribute an equal share of others’ payments– Formally, set hi(v-i) = - [1 / (|A|-1)] ji (vj)

• Does not satisfy participation constraints (aka individual rationality constraints) in general

– Agent might get higher expected utility by not participating

Myerson-Satterthwaite impossibility• Avrim is selling a car to Tuomas, both are risk neutral,

quasilinear• Each party knows his own valuation, but not the other’s

valuation• The probability distributions are common knowledge• Want a mechanism that is

– Ex post budget balanced– Ex post Pareto efficient: Car changes hands iff vbuyer > vseller

– (Interim) individually rational: Both Avrim and Tuomas get higher expected utility by participating than not

• Thrm. Such a mechanism does not exist (even if randomized mechanisms are allowed)– This impossibility is at the heart of more general exchange

settings (NYSE, NASDAQ, combinatorial exchanges, …) !

Auctioning one item

Auction settings• Private value : value of the good depends only on

the agent’s own preferences– E.g. cake which is not resold or showed off

• Common value : agent’s value of an item determined entirely by others’ values– E.g. treasury bills

• Correlated value : agent’s value of an item depends partly on its own preferences & partly on others’ values for it– E.g. auctioning a transportation task when

bidders can handle it or reauction it to others

Common auction mechanisms

• “First-price mechanisms”: First-price sealed-bid, Dutch– Strategic underbidding in (Nash) equilibrium

• “Second-price mechanisms”: English, Vickrey, Japanese (= open-exit)– Truth-telling as a dominant strategy in private-value

auctions• If bidders know their own valuations

Results for private value auctions

• Dutch strategically equivalent to first-price sealed-bid• Risk neutral agents => Vickrey strategically equivalent

to English• All four protocols allocate item efficiently

– (assuming no reservation price for the auctioneer)• English & Vickrey have dominant strategies => no

effort wasted in counterspeculation• Which of the four auction mechanisms gives highest

expected revenue to the seller?– Assuming valuations are drawn independently & agents

are risk-neutral• The four mechanisms have equal expected revenue!

Revenue equivalence theorem

• Even more generally: Thrm.– Assume risk-neutral bidders, valuations drawn independently from potentially different

distributions with no gaps– Consider two Bayes-Nash equilibria of any two auction mechanisms– Assume allocation probabilities yi(v1, … v|A|) are same in both equilibria

• Here v1, … v|A| are true types, not revelations

• E.g., if the equilibrium is efficient, then yi = 1 for bidder with highest vi

– Assume that if any agent i draws his lowest possible valuation vi, his expected payoff is same in both equilibria

• E.g., may want a bidder to lose & pay nothing if bidders’ valuations are drawn from same distribution, and the bidder draws the lowest possible valuation

– Then, the two equilibria give the same expected payoffs to the bidders (& thus to the seller)

Revenue equivalence ceases to hold if agents are not risk-neutral

• Risk averse bidders: – Dutch, first-price sealed-bid ≥ Vickrey, English

• Risk averse auctioneer: – Dutch, first-price sealed-bid ≤ Vickrey, English

Results for non-private value auctions

• Dutch strategically equivalent to first-price sealed-bid• Vickrey not strategically equivalent to English• All four protocols allocate item efficiently• Winner’s curse

– Common value auctions:

– Agent should lie (bid low) even in Vickrey & English Revelation to proxy bidders?

• Thrm (revenue non-equivalence ). With more than 2 bidders, the expected revenues are not the same: English ≥ Vickrey ≥ Dutch = first-price sealed bid

˜ v 1 E[v | ˆ v 1,b(ˆ v 2 ) b( ˆ v 1 ),...,b( ˆ v N ) b(ˆ v 1)]

Results for non-private value auctions...

• Common knowledge that auctioneer has private info – Q: What info should the auctioneer release ?

• A: auctioneer is best off releasing all of it– “No news is worst news”– Mitigates the winner’s curse

• Asymmetric info among bidders– E.g. first-price sealed-bid common value auction with

bidders A, B, C, D• A & B have same good info. C has this & extra signal.

D has poor but independent info• A & B should not bid; D should sometimes

• => “Bid less if more bidders or your info is worse”• Most important in sealed-bid auctions & Dutch

Vulnerability to bidder collusion

• v1 = 20, vi = 18 for others• Collusive agreement for English: e.g. 1 bids 6, others

bid 5. Self-enforcing• Collusive agreement for Vickrey: e.g. 1 bids 20, others

bid 5. Self-enforcing• In first-price sealed-bid or Dutch, if 1 bids below 18,

others are motivated to break the collusion agreement• Need to identify coalition parties

Vulnerability to shills

• Only a problem in non-private-value settings• English & all-pay auction protocols are

vulnerable– Classic analyses ignore the possibility of shills

• Vickrey, first-price sealed-bid, and Dutch are not vulnerable

Vulnerability to a lying auctioneer

• Truthful auctioneer classically assumed• In Vickrey auction, auctioneer can overstate 2nd

highest bid to the winning bidder in order to increase revenue

– Bid verification mechanisms, e.g. cryptographic signatures

– 3rd party auctionbots (reveal highest bid to seller after closing)

• In English, first-price sealed-bid, Dutch, and all-pay, auctioneer cannot lie because bids are public

Auctioneer’s other possibilities• Bidding

– Seller may bid more than his reservation price because truth-telling is not dominant for the seller even in the English or Vickrey protocol (because his bid may be 2nd highest & determine the price) => seller may inefficiently get the item

• In an expected revenue maximizing auction, seller sets a reservation price strategically like this [Myerson 81]

– Auctions are not Pareto efficient (not surprising in light of Myerson-Satterthwaite theorem)

• Setting a minimum price• Refusing to sell after the auction has ended

Multi-unit auctions & exchanges

(multiple indistinguishable units of one item for sale)

Auctions / reverse auctions / exchanges with multiple

indistinguishable units for sale

• Assume the supply/demand curves are reasonable • Non-discriminatory pricing is O(N log N) to clear

with piecewise linear supply/demand curves• Discriminatory pricing is NP-complete to clear

with piecewise linear supply/demand curves• Discriminatory pricing is O(N log N) to clear with

linear supply/demand curves

Multi-item auctions & exchanges

(multiple distinguishable items for sale)

Protocol design for multi-item auctions• Sequential auctions

– How should rational agents bid (in equilibrium)?• Full vs. partial vs. no lookahead• Need normative deliberation control methods

– Inefficiencies can result from future uncertainties• Parallel auctions

– Inefficiencies can still result from future uncertainties– Postponing & minimum participation requirements

• Unclear what equilibrium strategies would be• Methods to tackle the inefficiencies

– Backtracking via reauctioning (e.g. FCC [McAfee&McMillan96])– Backtracking via leveled commitment contracts

[Sandholm&Lesser95,96][Sandholm96][Andersson&Sandholm98a,b]

• Breach before allocation• Breach after allocation

Protocol design for multi-item auctions...

• Combinatorial auctions [Rassenti,Smith&Bulfin82]...– Bidder’s perspective

• Reduces the need for lookahead• Potentially 2#items valuation calculations

– Automated optimal bundling of items– Auctioneer’s perspective:

• Label bids as winning or losing so as to maximize sum of bid prices (= revenue social welfare)

– Each item can be allocated to at most one bid• Exhaustive enumeration is 2#bids

Combinatorial markets can be complex to clear

• Optimal clearing NP-complete– E.g. auctions & reverse auctions

• Approximation is NP-complete– E.g. auctions to #bids1- or #items0.5-

– E.g. reverse auctions to 1 + ln(#items that any one bid contains)– E.g. multi-unit reverse auctions to 1 + ln(#units that any one bid contains)

• Finding a feasible solution is NP-complete– E.g. reverse auctions with XOR-constraints (auctions with XORs are trivial)– E.g. auctions, reverse auctions & exchanges without free disposal

• However, can be solved fast in practice (at least for auctions & reverse auctions) using modern search algorithms

• Cases with extreme special structure can be solved in provably polynomial time

Generalizations of combinatorial auctions

• Free disposal• Substitutability• Multiple units of each item• Combinatorial exchanges (= many-to-many auctions)• Reservation prices

– On items– On combinations– With substitutability

• Combinatorial reverse auctions• Combinations of these generalizations

Bidding languages for combinatorial markets

• Bundle bids• ORs, XORs, OR-of-XORs [Sandholm 99]• XOR-of-ORs, OR* [Nisan 00]• Logical connectives on subformulae with

prices [Boutilier & Hoos 01]• Side constraints [Sandholm et al 01]• Price quantity discounts / rebates [Sandholm

et al 01]• …

Side constraints [Sandholm & Suri IJCAI-01 workshop on Distributed Constraint Reasoning]

• Side constraints increase expressiveness & make markets practical• Noncombinatorial multi-item auctions are solvable in polynomial time

– Thrm. Budget constraints: NP-complete• Max number of items per bidder: polynomial time [Tennenholtz 00]

– Thrm. Max winners: NP-complete even if bids can be accepted partially– Thrm. XORs: NP-complete & inapproximable even if bids can be accepted partially– These results hold whether or not seller has to sell all items

• Combinatorial auctions are polynomial time if bids can be accepted partially– Any side constraints from above make the problem NP-complete

• Also counting constraints– Other constraints allow polynomial time clearing

• Cost constraints: mutual business, trading volume, minorities, …• Unit constraints

• Some side constraints can make NP-hard combinatorial auction clearing easy !• Results apply to exchanges & reverse auctions also

Future research

• Expected revenue-maximizing multi-item auctions– Dominant strategy equilibrium– Bayes-Nash equilibrium– May not be GVA, and may not be efficient

• Reserve price setting agent for GVA so as to maximize expected revenue (within GVA)

• Optimal auction design without prior knowledge of the valuation distribution– Competitive analysis as in online algorithms– Multi-unit case is partially solved by [Hartline et al 01]

Bidding Agents with Complex Valuation Problems in Auctions

Kate Larson and Tuomas Sandholm

Bidders may need to compute their valuations for (bundles of) goods

• In many B2B applications, e.g.

– Vehicle routing problem in transportation exchanges

– Manufacturing scheduling problem in procurement

• Value of a bundle of items (tasks, resources, etc) = value of solution with those items - value of solution without them

Performance profile tree

• Normative– Allows conditioning on path of solution quality– Allows conditioning on path of other solution features– Allows conditioning on problem instance features (different

trees to be used for different classes)• Constructed from statistics on earlier runs

0

4

26

4

5

10

3

15

20

A

P(B|A) B

5

CP(C|A)Solution quality

Theorems on strategic computing

yesyesno

Generalized VickreyOn which <bidder, bundle> pair to allocate next computation step ?

Multiple items for

sale

noEnglish (1st price ascending) yes

yes

no

nonoVickrey (2nd price sealed bid)

yesyesyesDutch (1st price descending)

yesyesyesFirst price sealed-bidSingle item for

sale

Costly computing

Limited computing

Strategic computing ?Counter-speculation by rational

agents ?

Auctionmechanism

If performance profiles are deterministic, only weak strategic computing can occur New normative deliberation control method uncovered a new phenomenon

Future research• In many B2B settings, automated bidders can

compute valuations dynamically faster than humans

• Some future research directions– Using our deliberation control method in systems

• Manufacturing planning, networks, …– New (market) mechanisms

• Game-theoretically engineered to work well under (different) models of bounded rationality

• Our results show that even the most common mechanism design principles (e.g., revelation principle) cease to hold

• Our normative deliberation control method = basis for new design principles ?

Preference Elicitation in Combinatorial Markets

Wolfram Conen Tuomas Sandholm

Another complex problem in (single-shot) combinatorial auctions:

“The revelation problem”• Bidders may need to bid on all 2#items combinations

– Need to compute the valuation for each combination• Each valuation computation can be NP-complete• For example if a carrier company bids on trucking tasks:

TRACONET [Sandholm AAAI-93] – Need to communicate the bids– Need to reveal the bids

Approaches for tackling the revelation problem• Classic single-shot full revelation mechanims (Vickrey-Clarke-Groves)

– Exponentially many valuations revealed• (Ascending) mechanisms with price feedback (iBundle, [Parkes et al 1999] ,

akBa [Wurman et al. 2000] , etc.)– Can save revelation– Need exponential revelation in worst case [Nisan 2001]

• Our new approach: an elicitor “agent”– Knows things that individual bidders don’t (others’ bids so far)– Asks non-redundant questions from bidders to focus their revelation – Can save revelation– Need exponential revelation in worst case [Nisan 2001]– Can be combined with price feedback mechanisms

Elicitor algorithms

• Query policy dependent elicitor algorithms– Algorithm & query policy are intertwined– Based on search algorithms where each search step

involves asking a bidder a question• Policy independent elicitor algorithms

– General framework & specific algorithms– Can support any query policy

• Note: Query policies are online control policies, i.e. contingency plans

Future research

• Good query policies• Evaluating the elicitor

– Savings in revelation (how many queries needed ?)• In general case / in cases with special preference structure• Worst / average case• Competitive analysis as in online algorithms

• Generalizing the elicitor– To (combinatorial) exchanges– To (combinatorial) markets with side constraints– To (combinatorial) markets with multiattribute

features

Thank you for your attention!

• It’s been a fun course (at least for me :-) )• You have learned a LOT• …it’s time for final project presentations