Mechanism design for Multi-Agent Systems

Mechanism Design for Multiagent Systems Vincent Conitzer Assistant Professor of Computer Science and Economics Duke University [email_address]

Introduction Often, decisions must be taken based on the preferences of multiple, self-interested agents Allocations of resources/tasks Joint plans … Would like to make decisions that are “good” with respect to the agents’ preferences But, agents may lie about their preferences if this is to their benefit Mechanism design = creating rules for choosing the outcome that get good results nevertheless

Part I: “Classical” mechanism design Preference aggregation settings Mechanisms Solution concepts Revelation principle Vickrey-Clarke-Groves mechanisms Impossibility results

Preference aggregation settings Multiple agents … humans, computer programs, institutions, … … must decide on one of multiple outcomes … joint plan, allocation of tasks, allocation of resources, president, … … based on agents’ preferences over the outcomes Each agent knows only its own preferences “ Preferences” can be an ordering ≥ i over the outcomes, or a real-valued utility function u i Often preferences are drawn from a commonly known distribution

Elections Outcome space = { , , } > > > >

Resource allocation Outcome space = { , , } v( ) = $55 v( ) = $0 v( ) = $0 v( ) = $0 v( ) = $32 v( ) = $0

So, what is a mechanism? A mechanism prescribes: actions that the agents can take (based on their preferences) a mapping that takes all agents’ actions as input, and outputs the chosen outcome the “rules of the game” can also output a probability distribution over outcomes Direct revelation mechanisms are mechanisms in which action set = set of possible preferences

Example: plurality voting .5 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5 Every agent votes for one alternative Alternative with most votes wins random tiebreaking

Some other well-known voting mechanisms In all of these rules, each voter ranks all m candidates (direct revelation mechanisms) Other scoring mechanisms Borda : candidate gets m-1 points for being ranked first, m-2 for being ranked second, … Veto : candidate gets 0 points for being ranked last, 1 otherwise Pairwise election between two candidates: see which candidate is ranked above the other more often Copeland : candidate with most pairwise victories wins Maximin : compare candidates by their worst pairwise elections Slater : choose overall ranking disagreeing with as few pairwise elections as possible Other Single Transferable Vote (STV) : candidate with fewest votes drops out, those votes transfer to next remaining candidate in ranking, repeat Kemeny : choose overall ranking that minimizes the number of disagreements with some vote on some pair of candidates

The “matching pennies” mechanism Winner of “matching pennies” gets to choose outcome

Mechanisms with payments In some settings (e.g. auctions), it is possible to make payments to/collect payments from the agents Quasilinear utility functions: u i (o, π i ) = v i (o) + π i We can use this to modify agents’ incentives

A few different 1-item auction mechanisms English auction: Each bid must be higher than previous bid Last bidder wins, pays last bid Japanese auction: Price rises, bidders drop out when price is too high Last bidder wins at price of last dropout Dutch auction: Price drops until someone takes the item at that price Sealed-bid auctions (direct revelation mechanisms): Each bidder submits a bid in an envelope Auctioneer opens the envelopes, highest bid wins First-price sealed-bid auction: winner pays own bid Second-price sealed bid (or Vickrey ) auction: winner pays second highest bid

What can we expect to happen? In direct revelation mechanisms, will (selfish) agents tell the truth about their preferences? Voter may not want to “waste” vote on poorly performing candidate (e.g. Nader) In first-price sealed-bid auction, winner would like to bid only ε above the second highest bid In other mechanisms, things get even more complicated…

A little bit of game theory Θ i = set of all of agent i ’s possible preferences (“types”) Notation: u i ( θ i , o ) is i ’s utility for o when i has type θ i A strategy s i is a mapping from types to actions s i : Θ i -> A i For direct revelation mechanism, s i : Θ i -> Θ i More generally, can map to distributions, s i : Θ i -> Δ ( A i ) A strategy s i is a dominant strategy if for every type θ i , no matter what the other agents do , s i ( θ i ) maximizes i ’s utility A direct revelation mechanism is strategy-proof (or dominant-strategies incentive compatible ) if telling the truth ( s i ( θ i ) = θ i ) is a dominant strategy for all players (Another, weaker concept: Bayes-Nash equilibrium )

The Vickrey auction is strategy-proof! 0 b = highest bid among other bidders What should a bidder with value v bid? Option 1: Win the item at price b , get utility v - b Option 2: Lose the item, get utility 0 Would like to win if and only if v - b > 0 – but bidding truthfully accomplishes this!

Collusion in the Vickrey auction 0 b = highest bid among other bidders Example: two colluding bidders price colluder 1 would pay when colluders bid truthfully v 2 = second colluder’s true valuation v 1 = first colluder’s true valuation price colluder 1 would pay if colluder 2 does not bid gains to be distributed among colluders

The revelation principle For any (complex, strange) mechanism that produces certain outcomes under strategic behavior… … there exists an incentive compatible direct revelation mechanism that produces the same outcomes! “ strategic behavior” = some solution concept (e.g. dominant strategies) mechanism outcome actions P 1 P 2 P 3 types new mechanism

The Clarke mechanism [Clarke 71] Generalization of the Vickrey auction to arbitrary preference aggregation settings Agents reveal types directly θ i ’ is the type that i reports , θ i is the actual type Clarke mechanism chooses some outcome o that maximizes Σ i u i ( θ i ’, o) To determine the payment that agent j must make: Choose o’ that maximizes Σ i≠j u i ( θ i ’, o’) Make j pay Σ i≠j (u i ( θ i ’, o’) - u i ( θ i ’, o)) Clarke mechanism is: individually rational : no agent pays more than the outcome is worth to that agent (weak) budget balanced : agents pay a nonnegative amount

Why is the Clarke mechanism strategy-proof? Total utility for agent j is u j ( θ j , o) - Σ i≠j (u i ( θ i ’, o’) - u i ( θ i ’, o)) = u j ( θ j , o) + Σ i≠j u i ( θ i ’, o) - Σ i≠j u i ( θ i ’, o’) But agent j cannot affect the choice of o’ Hence, j can focus on maximizing u j ( θ j , o) + Σ i≠j u i ( θ i ’, o) But mechanism chooses o to maximize Σ i u i ( θ i ’, o) Hence, if θ j ’ = θ j , j ’s utility will be maximized! Extension of idea: add any term to player j ’s payment that does not depend on j ’s reported type This is the family of Groves mechanisms [Groves 73]

The Clarke mechanism is not perfect Requires payments + quasilinear utility functions In general money needs to flow away from the system Vulnerable to collusion, false-name manipulation Maximizes sum of agents’ utilities, but sometimes we are not interested in this E.g. want to maximize revenue

Impossibility results without payments Can we do without payments (voting mechanisms)? Gibbard-Satterthwaite [Gibbard 73, Satterthwaite 75] impossibility result: with three or more alternatives and unrestricted preferences , no voting mechanism exists that is deterministic strategy-proof onto (every alternative can win) non-dictatorial (more than one voter can affect the outcome) Generalization [Gibbard 77] : a randomized voting rule is strategy-proof if and only if it is a randomization over unilateral and duple rules unilateral = at most one voter affects the outcome duple = at most two alternatives have a possibility of winning

Single-peaked preferences [Black 48] Suppose alternatives are ordered on a line a 1 a 2 a 3 a 4 a 5 Every voter prefers alternatives that are closer to her most preferred alternative Let every voter report only her most preferred alternative (“peak”) v 1 v 2 v 3 v 4 v 5 Choose the median voter’s peak as the winner Strategy-proof!

Impossibility result with payments Simple setting: v( ) = x v( ) = y We would like a mechanism that: is efficient (trade iff y > x) is budget-balanced (seller receives what buyer pays) is strategy-proof (or even weaker form of incentive compatible) is individually rational (even just in expectation) This is impossible! [Myerson & Satterthwaite 83]

Part II: Enter the computer scientist Computational hardness of executing classical mechanisms New kinds of manipulation Computationally efficient approximation mechanisms Automatically designing mechanisms using optimization software Designing mechanisms for computationally bounded agents Communication constraints

How do we compute the outcomes of mechanisms? Some voting mechanisms are NP-hard to execute (including Kemeny and Slater) [Bartholdi et al. 89, Dwork et al. 01, Ailon et al. 05, Alon 05] In practice can still solve instances with fairly large numbers of alternatives [Davenport & Kalagnanam AAAI04, Conitzer et al. AAAI06, Conitzer AAAI06] What about Clarke mechanism? Depends on setting

Inefficiency of sequential auctions Suppose your valuation function is v( ) = $200, v( ) = $100, v( ) = $500 (complementarity) Now suppose that there are two (say, Vickrey) auctions, the first one for and the second one for What should you bid in the first auction (for )? If you bid $200, you may lose to a bidder who bids $250, only to find out that you could have won for $200 If you bid anything higher, you may pay more than $200, only to find out that sells for $1000 Sequential (and parallel ) auctions are inefficient

Combinatorial auctions v( ) = $500 v( ) = $700 v( ) = $300 Simultaneously for sale: , , bid 1 bid 2 bid 3 used in truckload transportation, industrial procurement, radio spectrum allocation, …

The winner determination problem ( WDP ) Choose a subset A (the accepted bids) of the bids B, to maximize Σ b in A v b , under the constraint that every item occurs at most once in A This is assuming free disposal , i.e. not everything needs to be allocated

WDP example Items A, B, C, D, E Bids: ({A, C, D}, 7) ({B, E}, 7) ({C}, 3) ({A, B, C, E}, 9) ({D}, 4) ({A, B, C}, 5) ({B, D}, 5)

An integer program formulation x b equals 1 if bid b is accepted, 0 if it is not maximize Σ b v b x b subject to for each item j, Σ b : j in b x b ≤ 1 If each x b can take any value in [0, 1], we say that bids can be partially accepted In this case, this is a linear program that can be solved in polynomial time This requires that each item can be divided into fractions if a bidder gets a fraction f of each of the items in his bundle, then this is worth the same fraction f of his value v b for the bundle

Weighted independent set Choose subset of the vertices with maximum total weight, Constraint: no two vertices can have an edge between them NP-hard (generalizes regular independent set) 2 2 3 4 3 2 4

The winner determination problem as a weighted independent set problem Each bid is a vertex Draw an edge between two vertices if they share an item Optimal allocation = maximum weight independent set Can model any weighted independent set instance as a CA winner determination problem (1 item per edge (or clique)) Weighted independent set is NP-hard, even to solve approximately [Håstad 96] - hence, so is WDP [Sandholm 02] noted that this inapproximability applies to the WDP

Polynomial-time solvable special cases Every bid is on a bundle of size at most two items [Rothkopf et al. 98] ~maximum weighted matching With 3 items per bid, NP-hard again (3-COVER) Items are organized on a tree & each bid is on a connected set of items [Sandholm & Suri 03] More generally, graph of bounded treewidth [Conitzer et al. AAAI04] Even further generalization given by [Gottlob & Greco EC07] item A item B item C item D item E item F item G item H

Clarke mechanism in CA (aka. Generalized Vickrey Auction, GVA) v( ) = $500 v( ) = $700 v( ) = $300 $500 $300

Clarke mechanism in CA… v( ) = $700 v( ) = $300 $700 pays $700 - $300 = $400

Collusion under GVA v( ) = $1000 v( ) = $700 v( ) = $1000 $0 $0 E.g. [Ausubel and Milgrom 06] ; general characterization in [Conitzer & Sandholm AAMAS06]

False-name bidding [Yokoo et al. AIJ2001, GEB2003] v( ) = $800 v( ) = $700 v( ) = $300 v( ) = $200 loses wins, pays $0 wins, pays $200 wins, pays $0 A mechanism is false-name-proof if bidders never have an incentive to use multiple identifiers No mechanism that allocates items efficiently is false-name-proof [Yokoo et al. GEB2003]

Characterization of false-name-proof voting rules Theorem [Conitzer 07] Any (neutral, anonymous, IR) false-name-proof voting rule f can be described by a single number k f in [0,1] With probability k f , the rule chooses an alternative uniformly at random With probability 1- k f , the rule draws two alternatives uniformly at random; If all votes rank the same alternative higher among the two, that alternative is chosen Otherwise, a coin is flipped to decide between the two alternatives

Alternative approaches to false-name-proofness Assume there is a cost to using a false name [Wagman & Conitzer AAMAS08] Verify some of the agents’ identities after the fact [Conitzer TARK07]

Strategy-proof mechanisms that solve the WDP approximately Running Clarke mechanism using approximation algorithms for WDP is generally not strategy-proof Assume bidders are single-minded (only want a single bundle) A greedy strategy-proof mechanism [Lehmann, O’Callaghan, Shoham JACM 03] : 1. Sort bids by (value/bundle size) {a}, 11 {b, c}, 20 {a, d}, 18 {a, c}, 16 {c}, 7 {d}, 6 2. Accept greedily starting from top 3. Winning bid pays bundle size times (value/bundle size) of first bid forced out by the winning bid 1*(18/2) = 9 2*(7/1) = 14 Worst-case approximation ratio = (#items) Can get a better approximation ratio, √(#items) , by sorting by value/ √(bundle size) 0

Clarke mechanism with same approximation algorithm does not work {a}, 11 {b, c}, 20 {a, d}, 18 {a, c}, 16 {c}, 7 {d}, 6 {b, c}, 20 {a, d}, 18 {a, c}, 16 {c}, 7 {d}, 6 Total value to bidders other than the {a} bidder: 26 Total value: 38 { a} bidder should pay 38 - 26 = 12, more than her valuation!

Designing mechanisms automatically Mechanisms such as Clarke are very general… … but will instantiate to something specific for specific settings This is what we care about Different approach: solve mechanism design problem automatically for setting at hand, as an optimization problem [Conitzer & Sandholm UAI02]

Small example: divorce arbitration Outcomes: Each agent is of high type with probability 0.2 and of low type with probability 0.8 Preferences of high type: u(get the painting) = 100 u(other gets the painting) = 0 u(museum) = 40 u(get the pieces) = -9 u(other gets the pieces) = -10 Preferences of low type: u(get the painting) = 2 u(other gets the painting) = 0 u(museum) = 1.5 u(get the pieces) = -9 u(other gets the pieces) = -10

Optimal dominant-strategies incentive compatible randomized mechanism for maximizing expected sum of utilities low high .96 .04 .96 .04 .47 .4 .13 high low

How do we set up the optimization? Use linear programming Variables: p(o | θ 1 , …, θ n ) = probability that outcome o is chosen given types θ 1 , …, θ n (maybe) π i ( θ 1 , …, θ n ) = i’s payment given types θ 1 , …, θ n Strategy-proofness constraints: for all i, θ 1 , … θ n , θ i ’ : Σ o p(o | θ 1 , …, θ n )u i ( θ i , o) + π i ( θ 1 , …, θ n ) ≥ Σ o p(o | θ 1 , …, θ i ’, …, θ n )u i ( θ i , o) + π i ( θ 1 , …, θ i ’, …, θ n ) Individual-rationality constraints: for all i, θ 1 , … θ n : Σ o p(o | θ 1 , …, θ n )u i ( θ i , o) + π i ( θ 1 , …, θ n ) ≥ 0 Objective (e.g. sum of utilities) Σ θ 1 , …, θ n p( θ 1 , …, θ n ) Σ i ( Σ o p(o | θ 1 , …, θ n )u i ( θ i , o) + π i ( θ 1 , …, θ n )) Also works for other incentive compatibility/individual rationality notions, other objectives, etc. For deterministic mechanisms, use mixed integer programming (probabilities in {0, 1} ) Typically designing the optimal deterministic mechanism is NP-hard

Computational limitations on the agents Will agents always be able to figure out what action is best for them? Revelation principle assumes this Effectively, does the manipulation for them! Theorem [Conitzer & Sandholm 04] . There are settings where: Executing the optimal (utility-maximizing) incentive compatible mechanism is NP-complete There exists a non-incentive compatible mechanism, where The center only carries out polynomial computation Finding a beneficial insincere revelation is NP-complete for the agents If the agents manage to find the beneficial insincere revelation, the new mechanism is just as good as the optimal truthful one Otherwise, the new mechanism is strictly better

Hardness of manipulation of voting mechanisms Computing the strategically optimal vote (“manipulating”) given others’ votes is NP-hard for certain voting mechanisms (including STV) [Bartholdi et al. 89, Bartholdi & Orlin 91] Well-known voting mechanisms can be modified to make manipulation NP-hard, #P-hard, or even PSPACE-hard [Conitzer & Sandholm IJCAI03, Elkind & Lipmaa ISAAC05] Ideally, we would like manipulation to be usually hard , not worst-case hard Several impossibility results [Procaccia & Rosenschein AAMAS06, Conitzer & Sandholm AAAI06, Friedgut et al. 07]

Preference elicitation Sometimes, having each agent communicate all preferences at once is impractical E.g. in a combinatorial auction, a bidder can have a different valuation for every bundle ( 2 #items -1 values) Preference elicitation: sequentially ask agents simple queries about their preferences, until we know enough to determine the outcome

Preference elicitation (CA) center/auctioneer/organizer/… “ v({A})?” “ 30” “ 40” “ What would you buy if the price for A is 30, the price for B is 20, the price for C is 20?” “ nothing” “ v({A,B,C}) < 70?” “ v({B, C})?” “ yes” gets {A}, pays 30 gets {B,C}, pays 40 [Parkes, Ausubel & Milgrom, Wurman & Wellman, Blumrosen & Nisan, Conen & Sandholm, Hudson & Sandholm, Nisan & Segal, Lahaie & Parkes, Santi et al, …]

Preference elicitation (voting) > center/auctioneer/organizer/… ?” “ “ yes” > ?” “ “ no” “ most preferred?” “ ” > ?” “ “ yes” wins [Conitzer & Sandholm AAAI02, EC05, Konczak & Lang 05, Conitzer AAMAS07, Pini et al. IJCAI07, Walsh AAAI07]

Benefits of preference elicitation Less communication needed Agents do not always need to determine all of their preferences Only where their preferences matter

Other topics Online mechanism design : agents arrive and depart over time [Lavi & Nisan 00, Friedman & Parkes 03, Parkes & Singh 03, Hajiaghayi et al. 04, 05, Parkes & Duong 07] Distributed implementation of mechanisms [Parkes & Shneidman 04, Petcu et al. 06]

Some future directions General principles for how to get incentive compatibility without solving to optimality Are there other ways of addressing false-name manipulation? Can we scale automated mechanism design to larger instances? One approach: use domain structure (e.g. auctions [Likhodedov & Sandholm, Guo & Conitzer] ) Is there a systematic way of exploiting agents’ computational boundedness? One approach: have an explicit model of computational costs [Larson & Sandholm] Thank you for your attention!

Mechanism design for Multi-Agent Systems

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