Incremental Mechanism Design

Transcription

1 Incremental Mechansm Desgn Vncent Contzer Duke Unversty Department of Computer Scence Abstract Mechansm desgn has tradtonally focused almost exclusvely on the desgn of truthful mechansms. There are several drawbacks to ths: 1. n certan settngs (e.g. votng settngs), no desrable strategyproof mechansms exst; 2. truthful mechansms are unable to take advantage of the fact that computatonally bounded agents may not be able to fnd the best manpulaton, and 3. when desgnng mechansms automatcally, ths approach leads to constraned optmzaton problems for whch current technques do not scale to very large nstances. In ths paper, we suggest an entrely dfferent approach: we start wth a naïve (manpulable) mechansm, and ncrementally make t more strategyproof over a sequence of teratons. We gve examples of mechansms that (varants of) our approach generate, ncludng the VCG mechansm n general settngs wth payments, and the pluralty-wth-runoff votng rule. We also provde several basc algorthms for automatcally executng our approach n general settngs. Fnally, we dscuss how computatonally hard t s for agents to fnd any remanng benefcal manpulaton. 1 Introducton In many multagent settngs, we must choose an outcome based on the preferences of multple self-nterested agents, who wll not necessarly report ther preferences truthfully f t s not n ther best nterest to do so. Typcal settngs n whch ths occurs nclude auctons, reverse auctons, exchanges, votng settngs, publc good settngs, resource/task allocaton settngs, rankng pages on the web [1], etc. Research n mechansm desgn studes how to choose outcomes n such a way that good outcomes are obtaned even when agents respond to ncentves to msreport ther preferences (or manpulate). For the most part, researchers have focused smply on creatng truthful (or strategy-proof) mechansms, n whch no agent ever has an ncentve to msreport. Ths approach s typcally justfed by appealng to a result known as the revelaton prncple, whch states that for any mechansm that does well n the face of strategc msreportng by agents, there s a truthful mechansm that wll perform just as well. Ths materal s based upon work supported by the Natonal Scence Foundaton under ITR grants IIS and IIS , a Sloan Fellowshp, and an IBM Ph.D. Fellowshp. Tuomas Sandholm Carnege Mellon Unversty Computer Scence Department sandholm@cs.cmu.edu The tradtonal approach to mechansm desgn has been to try to desgn good mechansms that are as general as possble. Probably the best-known general mechansm s the Vckrey- Clarke-Groves (VCG) mechansm [16; 4; 10], whch chooses the allocaton that maxmzes the sum of the agents utltes (the socal welfare), and makes every agent pay the externalty that he 1 mposes on the other agents. Ths s suffcent to ensure that no ndvdual agent has an ncentve to manpulate, but t also has varous drawbacks: for example, the surplus payments can, n general, not be redstrbuted, and the desgner may have a dfferent objectve than socal welfare, e.g. she may wsh to maxmze revenue. Other general mechansms have ther own drawbacks, and there are varous mpossblty results such as the Gbbard-Satterthwate theorem [8; 15] that show that certan objectves cannot be acheved by truthful mechansms. The lack of a general mechansm that s always satsfactory led to the creaton of the feld of automated mechansm desgn [5]. Rather than try to desgn a mechansm that works for a range of settngs, the dea s to have a computer automatcally compute the optmal mechansm for the specfc settng at hand, by solvng an optmzaton problem. A drawback of that approach s that current technques do not scale to very large nstances. Ths s n part due to the fact that, to ensure strategy-proofness, one must smultaneously decde on the outcome that the mechansm chooses for every possble nput of revealed preferences, and the strategy-proofness constrants nterrelate these decsons. Another observaton that has been made s that n complex settngs, t s unreasonable to beleve that every agent s endowed wth the computatonal abltes to compute an optmal manpulaton. Ths nvaldates the above-mentoned revelaton prncple, n that restrctng attenton to truthful mechansms may n fact come at a cost n the qualty of the outcomes that the mechansm produces. Addng to ths the observaton that n some domans, all strategy-proof mechansms are unsatsfactory (by the Gbbard-Satterthwate theorem), t becomes mportant to be able to desgn mechansms that are not strategy-proof. Recent research has already proposed some manpulable mechansms. There has been work that proposes relaxng the constrant to approxmate truthfulness (n varous senses). Approxmately truthful mechansms can be easer to execute [12; 2], or can crcumvent mpossblty results that apply to truthful mechansms [14; 9]. Other work has studed manpulable mechansms n whch 1 We wll use she for the center/desgner, and he for an agent.

2 fndng a benefcal manpulaton s computatonally dffcult n varous senses [3; 13; 6; 7]. In ths paper, we ntroduce a new approach. We start wth a naïvely desgned mechansm that s not strategy-proof (for example, the mechansm that would be optmal n the absence of strategc behavor), and we attempt to make t more strategy-proof. Specfcally, the approach systematcally dentfes stuatons n whch an agent has an ncentve to manpulate, and corrects the mechansm to take away ths ncentve. Ths s done teratvely, and the mechansm may or may not become (completely) strategy-proof eventually. The fnal mechansm may depend on the order n whch possble manpulatons are consdered. One can conceve of ths approach as beng a computatonally more effcent approach to automated mechansm desgn, nsofar as the updates to the mechansm to make t more strategy-proof can be executed automatcally (by a computer). Indeed, we wll provde algorthms for dong so. It s also possble to thnk about the results of ths approach theoretcally, and use them as a gude n tradtonal mechansm desgn. We wll pursue ths as well, gvng varous examples. Fnally, we wll argue that f the mechansm that the approach produces remans manpulable, then any remanng manpulatons wll be computatonally hard to fnd. Ths approach bears some smlarty to how mechansms are desgned n the real world. Real-world mechansms are often ntally naïve, leadng to undesrable strategc behavor; once ths s recognzed, the mechansm s amended to dsncent the undesrable behavor. For example, some naïvely desgned mechansms gve bdders ncentves to postpone submttng ther bds untl just before the event closes (.e., snpng); often ths s (partally) fxed by addng an actvty rule, whch prevents bdders that do not bd actvely early from wnnng later. As another example, n the 2003 Tradng Agent Competton Supply Chan Management (TAC/SCM) game, the rules of the game led the agents to procure most of ther components on day 0. Ths was deemed undesrable, and the desgners tred to modfy the rules for the 2004 competton to dsncent ths behavor [11]. 2 As we wll see, there are many varants of the approach, each wth ts own merts. We wll not decde whch varant s the best n ths paper; rather, we wll show for a few dfferent varants that they can result n desrable mechansms. 2 Mechansm desgn background In a mechansm desgn settng, we are gven: A set of agents N ( N = n); A set of outcomes O (here, f payments are used n the settng, an outcome ncludes nformaton on payments to be made by/to the agents); For each agent N, a set of types Θ (and we denote by Θ = Θ 1... Θ n the set of all type vectors,.e. the set of all possble nputs to the mechansm); 2 Interestngly, these ad-hoc modfcatons faled to prevent the behavor, and even an extreme modfcaton durng the 2004 competton faled. Later research suggests that n fact all reasonable settngs for a key parameter would have faled [17]. For each N, a utlty functon u : Θ O R; 3 An objectve functon g : Θ O R. For example, n a sngle-tem aucton, N s the set of bdders; O = S Π, where S s the set of all possble allocatons of the tem (one for each bdder, plus potentally one allocaton where no bdder wns), and Π s the set of all possble vectors π 1,..., π n of payments to be made by the agents (e.g., Π = R n ); assumng no allocatve externaltes (that s, t does not matter to a bdder whch other bdder wns the tem f the bdder does not wn hmself), Θ s the set of possble valuatons that the bdder may have for the tem (for example, Θ = R 0 ); the utlty functon u s gven by: u (θ, (s, π 1,..., π n )) = θ π f s s the outcome n whch wns the tem, and u (θ, (s, π 1,..., π n )) = π otherwse. (In stuatons n whch a type conssts of a sngle value, we wll typcally use v rather than θ for the type.) 4 A (determnstc) mechansm conssts of a functon M : Θ O, specfyng an outcome for every vector of (reported) types. 5 Gven a mechansm M, a benefcal manpulaton 6 conssts of an agent N, a type vector θ 1,..., θ n Θ, and an alternatve type report ˆθ for agent such that u (θ, M( θ 1,..., θ n )) < u (θ, M( θ 1,..., θ 1, ˆθ, θ +1,..., θ n )). In ths case we say that manpulates from θ 1,..., θ n nto θ 1,..., θ 1, ˆθ, θ +1,..., θ n. A mechansm s strategyproof or (domnant-strateges) ncentve compatble f there are no benefcal manpulatons. (We wll not consder Bayes-Nash equlbrum ncentve compatblty here.) In settngs wth payments, we enforce an ex-post ndvdual ratonalty constrant: we cannot make an agent worse off than he would have been f he had not partcpated. That s, we cannot charge an agent more than he reported the outcome (dsregardng payments) was worth to hm. 3 Our approach and technques In ths secton, we explan the approach and technques that we consder n ths paper. We recall that our goal s not to (mmedately) desgn a strategy-proof mechansm; rather, we start wth some manpulable mechansm, and attempt to ncrementally make t more strategy-proof. Thus, the basc template of our approach s as follows: 1. Start wth some (manpulable) mechansm M; 2. Fnd some set F of manpulatons (where a manpulaton s gven by an agent N, a type vector θ 1,..., θ n, and an alternatve type report ˆθ for agent ); 3. If possble, change the mechansm M to prevent (many of) these manpulatons from beng benefcal; 4. Repeat from step 2 untl termnaton. Ths s merely a template; at each one of the steps, somethng remans to be flled n. Whch ntal mechansm do we 3 The utlty functon s parameterzed by type; whle the u are common knowledge, the types encode (prvate) preferences. 4 In general, we may have addtonal nformaton, such as a pror over the types, but we wll not use ths nformaton n ths paper. 5 In general, a mechansm may be randomzed, specfyng dstrbutons over outcomes, but we wll not consder ths n ths paper. 6 Benefcal here means benefcal to the manpulatng agent.

3 choose n step 1? Whch set of manpulatons do we consder n step 2? How do we fx the mechansm n step 3 to prevent these manpulatons? And how do we decde to termnate n step 4? In ths paper, we wll not resolve what s the best way to fll n these blanks (t seems unlkely that there s a sngle, unversal best way), but rather we wll provde a few nstantatons of the technque, llustrate them wth examples, and show some nterestng propertes. One natural way of nstantatng step 1 s to choose a naïvely optmal mechansm, that s, a mechansm that would gve the hghest objectve value for each type vector f every agent would always reveal hs type truthfully. For nstance, f we wsh to maxmze socal welfare, we smply always choose an outcome that maxmzes socal welfare for the reported types; f we wsh to maxmze revenue, we choose an outcome that maxmzes socal welfare for the reported types, and make each agent pay hs entre valuaton. In step 2, there are many possble optons: we can choose the set of all manpulatons; the set of all manpulatons for a sngle agent; the set of all manpulatons from or to a partcular type or type vector; or just a sngle manpulaton. The structure of the specfc settng under consderaton may also make certan manpulatons more natural than others; we can dscover whch manpulatons are more natural by ntuton, by hrng agents to act n test runs of the mechansm, by runnng algorthms that fnd manpulatons, etc. Whch set of manpulatons we choose wll affect the dffculty of step 3. Step 3 s the most complex step. Let us frst consder the case where we are only tryng to prevent a sngle manpulaton, from θ = θ 1,..., θ n to θ = θ 1,..., θ 1, ˆθ, θ +1,..., θ n. We can make ths manpulaton undesrable n one of three ways: (a) make the outcome that M selects for θ more desrable for agent (when he has type θ ), (b) make the outcome that M selects for θ less desrable for agent (when he has type θ ), or (c) a combnaton of the two. We wll focus on (a) n ths paper. There may be multple ways to make the outcome that M selects for θ suffcently desrable to prevent the manpulaton; a natural way to select from among these outcomes s to choose the one that maxmzes the desgner s orgnal objectve. Note that these modfcatons may ntroduce other benefcal manpulatons. When we are tryng to prevent a set of manpulatons, we are confronted wth an addtonal ssue: after we have prevented one manpulaton n the set, we may rentroduce the ncentve for ths manpulaton when we try to prevent another manpulaton. Resolvng ths would requre solvng a potentally large constraned optmzaton problem, consttutng an approach smlar to standard automated mechansm desgn rentroducng some of the scalablty problems that we wsh to avod. Therefore, when addressng the manpulatons from one type vector, we wll smply act as f we wll not change the outcomes for any other type vector. Formally, for ths partcular nstantaton of our approach, f M s the mechansm at the begnnng of the teraton and M s the mechansm at the end of the teraton (after the update), and F s the set of manpulatons under consderaton, we have M (θ) arg max o O(M,θ,F ) g(θ, o) (here, θ = θ 1,..., θ n ), where O(M, θ, F ) O s the set of all outcomes o such that for any benefcal manpulaton (, ˆθ ) (wth (, θ, ˆθ ) F ), u (θ, o) u (θ, M( θ 1,..., θ 1, ˆθ, θ +1,..., θ n )). It may happen that O(M, θ, F ) = (no outcome wll prevent all manpulatons). In ths case, there are varous ways n whch we can proceed. One s not to update the outcome at all,.e. set M (θ) = M(θ). Another s to mnmze the number of agents that wll have an ncentve to manpulate from θ after the change, that s, to choose M (θ) arg mn o O { N : ( (, θ, ˆθ ) F : u (θ, o) < u (θ, M( θ 1,..., θ 1, ˆθ, θ +1,..., θ n )))} (and tes can be broken to maxmze the objectve g). Many other varants are possble. For example, nstead of choosng from the set of all possble outcomes O when we update the outcome of the mechansm for some type vector θ, we can lmt ourselves to the set of all outcomes that would result from some benefcal manpulaton n F from θ that s, the set {o O : (( (, ˆθ ) : (, θ, ˆθ ) F ) : o = M( θ 1,..., θ 1, ˆθ, θ +1,..., θ n ))} n addton to the current outcome M(θ). The motvaton s that rather than consder all possble outcomes every tme, we may wsh to smplfy our job by consderng only the ones that cause the falure of strategy-proofness n the frst place. We next present examples of some of the above-mentoned varants. 4 Instantatng the methodology In ths secton, we llustrate the potental benefts of the approach by exhbtng mechansms that t can produce n varous standard mechansm desgn settngs. We wll demonstrate a settng n whch the approach ends up producng a strategy-proof mechansm, as well as a settng n whch the produced mechansm s stll vulnerable to manpulaton (but n some sense more strategy-proof than naïve mechansms). (A thrd settng that we studed decdng on whether to produce a publc good s omtted due to space constrant.) We emphasze that our goal n ths secton s not to come up wth spectacularly novel mechansms, but rather to show that the approach advocated n ths paper produces sensble results. Therefore, for now, we wll consder the approach successful f t produces a well-known mechansm. In future research, we hope to use the technque to help us desgn novel mechansms as well. 7 We do emphasze, however, that although the mechansms that the approach eventually produces were already known to us, the approach smply follows local updatng rules wthout any knowledge of what the fnal mechansm should be. In other words, the algorthm s not even gven a hnt of what the fnal mechansm should look lke. 4.1 Settngs wth payments In ths subsecton, we show the followng result: n general preference aggregaton settngs n whch the agents can make payments (e.g. combnatoral auctons), (one varant of) our technque yelds the VCG mechansm after a sngle teraton. We recall that the VCG mechansm chooses an outcome that 7 Certanly, f we apply the approach to a prevously unstuded mechansm desgn doman, t wll produce a novel mechansm. However, t would be dffcult to evaluate the qualty of such a mechansm, snce there would be nothng to compare the mechansm to.

4 maxmzes socal welfare (not countng payments), and mposes the followng tax on an agent: consder the total utlty (not countng payments) of the other agents gven the chosen outcome, and subtract ths from the total utlty (not countng payments) that the other agents would have obtaned f the gven agent s preferences had been gnored n choosng the outcome. Specfcally, we wll consder the followng varant of our technque (perhaps the most basc one): Our objectve g s to try maxmze some (say, lnear) combnaton of allocatve socal welfare (.e. socal welfare not takng payments nto account) and revenue. (It does not matter what the combnaton s.) The set F of manpulatons that we consder s that of all possble msreports (by any sngle agent). We try to prevent manpulatons accordng to (a) above (for a type vector from whch there s a benefcal manpulaton, make ts outcome desrable enough to the manpulatng agents to prevent the manpulaton). Among outcomes that acheve ths, we choose one maxmzng the objectve g. We wll use the term allocaton to refer to the part of the outcome that does not concern payments, even though the result s not restrcted to allocaton settngs such as auctons. Also, we wll refer to the utlty that agent wth type θ gets from allocaton s (not ncludng payments) as u (θ, s). The followng smple observaton shows that the naïvely optmal mechansm s the frst-prce mechansm, whch chooses an allocaton that maxmzes socal welfare, and makes every agent pay hs valuaton for the allocaton. Observaton 1 The frst-prce mechansm naïvely maxmzes both revenue and allocatve socal welfare. Proof: That the mechansm (naïvely) maxmzes allocatve socal welfare s clear. Moreover, due to the ndvdual ratonalty constrant, we can never extract more than the allocatve socal welfare; and the frst-prce mechansm (naïvely) extracts all the allocatve socal welfare, for an outcome that (naïvely) maxmzes allocatve socal welfare. Before showng the man result of ths subsecton, we frst characterze optmal manpulatons under the frst-prce mechansm. Lemma 1 The followng s an optmal manpulaton ˆθ from θ Θ for agent under the frst-prce mechansm: for the allocaton s that would be chosen under the frst-prce mechansm for θ, report a value equal to s VCG payment under the true valuatons (u( ˆθ (s )) = V CG (θ, θ )); for any other allocaton s s, report a valuaton of 0. 8 The utlty of ths manpulaton s u(θ, s ) V CG (θ, θ ). (Ths assumes tes wll be broken n favor of allocaton s.) 8 There may be constrants on the reported utlty functon that prevent ths for example, n a (combnatoral) aucton, perhaps only monotone valuatons are allowed (wnnng more tems never hurts an agent). If so, the agent should report valuatons for these outcomes that are as small as possble, whch wll stll lead to s beng chosen. Wthout the te-breakng assumpton, the lemma does not hold: for example, n a sngle-tem frst-prce aucton, bddng exactly the second prce for the tem s not an optmal manpulaton for the bdder wth the hghest valuaton f the te s broken n favor of the other bdder. However, ncreasng the bd by any amount wll guarantee that the tem s won (and n general, ncreasng the value for s by any amount wll guarantee that outcome). Proof: Frst, we show that ths manpulaton wll stll result n s beng chosen. Suppose that allocaton s s s chosen nstead. Gven the te-breakng assumpton, t follows that u j (θ j, s) > u ( ˆθ, s ) + u j (θ j, s ), or equvalently, V CG (θ, θ ) < u j (θ j, s) u j (θ j, s ). However, by defnton, V CG (θ, θ ) = max s u j (θ j, s ) u j (θ j, s ) u j (θ j, s) u j (θ j, s ), so we have the desred contradcton. It follows that agent s utlty under the manpulaton s u (θ, s ) V CG (θ, θ ). Next, we show that agent cannot obtan a hgher utlty wth any other manpulaton. Suppose that manpulaton ˆθ results n allocaton s beng chosen. Because utltes cannot be negatve under truthful reportng, t follows that u ( ˆθ, s)+ u j (θ j, s) max s u j (θ j, s ). Usng the fact that V CG (θ, θ ) = max s u j (θ j, s ) u j (θ j, s ), we can rewrte the prevous nequalty as u ( ˆθ, s) + u j (θ j, s) V CG (θ, θ ) + u j (θ j, s ), or equvalently u ( ˆθ, s) V CG (θ, θ )+ u j (θ j, s ) u j (θ j, s). Because u j (θ j, s ) u j (θ j, s), we can rewrte the prevous nequalty as u ( ˆθ, s) V CG (θ, θ ) u (θ, s ) + j j u (θ, s) + u j (θ j, s ) u j (θ j, s) V CG (θ, θ ) j u (θ, s ) + u (θ, s), or equvalently, u (θ, s) u ( ˆθ, s) u (θ, s ) V CG (θ, θ ), as was to be shown. Theorem 1 Under the varant of our approach descrbed above, the mechansm resultng after a sngle teraton s the VCG mechansm. Proof: By Observaton 1, the naïvely optmal mechansm s the frst-prce mechansm. When updatng the outcome for θ, by Lemma 1, each agent must receve a utlty of at least u (θ, s ) V CG (θ, θ ), where s s the allocaton that maxmzes allocatve socal welfare for type vector θ. One way of achevng ths s to choose allocaton s, and to charge agent exactly V CG (θ, θ ) that s, smply run the VCG mechansm. Clearly ths maxmzes allocatve socal welfare. But, under the constrants on the agents utltes, t also maxmzes revenue, for the followng reason. For any allocaton s, the most revenue that we can hope to extract s the allocatve socal welfare of s, that s, u (θ, s), mnus the sum of the utltes that we must guarantee the agents, that

5 s, u (θ, s ) V CG (θ, θ ). Because s = s maxmzes u (θ, s), ths means that the most revenue we can hope to extract s V CG (θ, θ ), and the VCG mecha- nsm acheves ths. 4.2 Ordnal preferences In ths subsecton, we address votng (socal choce) settngs. In such a settng, there s a set of outcomes (also known as canddates or alternatves) and a set of agents (also known as voters), and every agent s type s a complete rankng over the canddates. (We do not need to specfy numercal utltes here.) The mechansm (or votng rule) takes as nput the agents type reports (or votes), consstng of complete rankngs of the canddates, and chooses an outcome. The most commonly used votng rule s the pluralty rule, n whch we only consder every voter s hghest-ranked canddate, and the wnner s smply the canddate wth the hghest number of votes rankng t frst (ts pluralty score). The pluralty rule s very manpulable: a voter votng for a canddate that s not wnnng may prefer to attempt to get the canddate that currently has the second-hghest pluralty score to wn, by votng for that canddate nstead. In the real world, one common way of fxng ths s to add a runoff round, resultng n the pluralty-wth-runoff rule. Under ths rule, we take the two canddates wth the hghest pluralty scores, and declare as the wnner the one that s ranked hgher by more voters. By the Gbbard-Satterthwate theorem, ths s stll not a strategy-proof mechansm (t s nether dctatoral nor does t preclude any canddate from wnnng) for example, a voter may change hs vote to change whch canddates are n the runoff. Stll, the pluralty wth runoff rule s, n an ntutve sense, less manpulable than the pluralty rule (and certanly more desrable than a strategy-proof rule, snce a strategy-proof rule would ether be dctatoral or preclude some canddate from wnnng). In ths subsecton, we wll show that the followng varant of our approach wll produce the pluralty-wth-runoff rule when startng wth the pluralty rule as the ntal mechansm. The set F conssts of all manpulatons n whch a voter changes whch canddate he ranks frst. We try to prevent manpulatons as follows: for a type (vote) vector from whch there s a benefcal manpulaton, consder all the outcomes that may result from such a manpulaton (n addton to the current outcome), and choose as the new outcome the one that mnmzes the number of agents that stll have an ncentve to manpulate from ths vote vector. We wll change the outcome for each vote vector at most once (but we wll have multple teratons, for vote vectors whose outcome dd not change n earler teratons). We are now ready to present the result. (The remanng proofs are omtted due to space constrant.) Theorem 2 For a gven type vector θ, suppose that canddate b s ranked frst the most often, and a s ranked frst the second most often (s(b) > s(a) >..., where s(o) s the number of tmes o s ranked frst). Moreover, suppose that the number of votes that prefers a to b s greater than or equal to the number of votes that prefers b to a. Then, startng wth the pluralty rule, after exactly s(b) s(a) teratons of the approach descrbed above, the outcome for θ changes for the frst tme, to a (the outcome of the pluralty wth runoff rule). 9 5 Computng the mechansm s outcomes In ths secton, we dscuss how to automatcally compute the outcomes of the mechansms that are generated by ths approach n general. It wll be convenent to thnk about settngs n whch the set of possble type vectors s fnte (so that the mechansm can be represented as a fnte table), although these technques can be extended to (some) nfnte settngs as well. (At the very least, types can be grouped together nto a fnte number; for specfc settngs, somethng better can often be done.) One potental upsde relatve to standard automated mechansm desgn technques s that we do not need to compute the entre mechansm (the outcomes for all type vectors); rather, we only need to compute the outcome for the type vector that s actually reported. Let M 0 denote the (naïve) mechansm from whch we start, and let M t denote the mechansm after t teratons. Let F t denote the set of benefcal manpulatons that we are consderng (and are tryng to prevent) n the tth teraton. Thus, M t s a functon of F t and M t 1. What ths functon s depends on the specfc varant of the approach that we are usng. When we try to prevent manpulatons by makng the outcome for the type vector from whch the agent s manpulatng more desrable for that agent, we can be more specfc, and say that, for type vector θ, M t (θ) s a functon of the subset Ft θ F t that conssts of manpulatons that start from θ, and of the outcomes that M t 1 selects on the subset of type vectors that would result from a manpulaton n Ft θ. Thus, to compute the outcome that M t produces on θ, we only need to consder the outcomes that M t 1 chooses for type vectors that dffer from θ n at most one type (and possbly even fewer, f Ft θ does not consder all possble manpulatons). As such, we n need to consder M t 1 s outcomes on at most Θ type vectors to compute M t (θ) (for any gven θ), whch s much smaller than the set of all type vectors ( n Θ ). Of course, to compute M t 1 (θ ) for some type vector θ, we need to consder M t 2 s outcomes on up to n Θ type vectors, etc. Because of ths, a smple recursve approach for computng M t (θ) for some θ wll requre O(( n Θ ) t ) tme. Ths approach may, however, spend a sgnfcant amount of tme recomputng values M j (θ ) many tmes. Another approach s to use dynamc programmng, computng and storng mechansm M j 1 s outcomes on all type vectors before proceedng to compute outcomes for M j. Ths approach wll requre O(t ( n Θ ) ( n Θ )) tme (for every teraton, for every type vector, we must nvestgate all possble manpula- 9 Ths s assumng that tes n the pluralty rule are broken n favor of a; otherwse, one more teraton s needed. (Some assumpton on te-breakng must always be made for votng rules.)

6 tons). We note that when we use ths approach, we may as well compute the entre mechansm M t (we already have to compute the entre mechansm M t 1 ). If n s large and t s small, the recursve approach s more effcent; f n s small and t s large, the dynamc programmng approach s more effcent. We can gan the benefts of both by usng the recursve approach and storng the outcomes that we compute n the process, so that we need not recompute them. All of ths s for fully general (fnte) domans; t s lkely that these technques can be sped up consderably for specfc domans. Moreover, as we have already seen, some domans can smply be solved analytcally. 6 Computatonal hardness of manpulaton We have demonstrated that our approach can change naïve mechansms nto mechansms that are less (sometmes not at all) manpulable. In ths secton, we wll argue that n addton, f the mechansm remans manpulable, the remanng manpulatons are computatonally dffcult to fnd. Ths s especally valuable because, as we argued earler, f t s too hard to dscover benefcal manpulatons, the revelaton prncple ceases to hold, and a manpulable mechansm can sometmes actually outperform all truthful mechansms. We frst gve an nformal, but general, argument for the clam that any manpulatons that reman after a large number of teratons of our approach are hard to fnd. Suppose that the only knowledge that an agent has about the mechansm s the varant of our approach by whch the desgner obtans t (the ntal naïve mechansm, the manpulatons that the desgner consders, how she tres to elmnate these opportuntes for manpulatons, how many teratons she performs, etc.). Gven ths, the most natural algorthm for an agent to fnd a benefcal manpulaton s to smulate our approach for the relevant type vectors, perhaps usng the algorthms presented earler. However, ths approach to manpulaton s computatonally nfeasble f the agent does not have the computatonal capabltes to smulate as many teratons as the desgner wll actually perform. Unfortunately, ths nformal argument fals f the agent actually has greater computatonal abltes or better algorthms than the desgner. However, t turns out that f we allow for random updates to the mechansm, then we can prove hardness of manpulaton n a formal, complexty-theoretc sense. So far, we have only dscussed updatng the mechansm n a determnstc fashon. When the mechansm s updated determnstcally, any agent that s computatonally powerful enough to smulate ths updatng process can determne the outcome that the mechansm wll choose, for any vector of revealed types. Hence, that agent can evaluate whether he would beneft from msrepresentng hs preferences. However, ths s not the case f we add random choces to our approach (and the agents are not told about the random choces untl after they have reported ther types). In fact, we can prove the followng result. (As n most prevous work on hardness of manpulaton, ths s only a worst-case noton of hardness, whch may not prevent manpulaton n all cases.) Theorem 3 When the updates to the mechansm are chosen randomly, evaluatng whether there exsts a manpulaton that ncreases an agent s expected utlty s #P-hard. 7 Dscusson Whle we have gven a framework, and a portfolo of technques wthn that framework, for makng mechansms more strategy-proof, and llustrated ther usefulness wth examples, we have not yet ntegrated the technques nto a sngle, comprehensve approach. Ths suggests some mportant questons for future research. Is there a sngle, general method that obtans all of the benefts of the ndvdual technques that we have descrbed (possbly by makng use of these technques as subcomponents)? If not, can we provde some gudance as to whch technques are lkely to work best n a gven settng? Another drecton for future research s to consder other types of manpulaton, such as false-name bddng [18]. References [1] Alon Altman and Moshe Tennenholtz. Rankng systems: The PageRank axoms. ACM-EC, [2] Aaron Archer, Chrstos Papadmtrou, K Tawar, and Eva Tardos. An approxmate truthful mechansm for combnatoral auctons wth sngle parameter agents. SODA, [3] John Barthold, III and James Orln. Sngle transferable vote ressts strategc votng. Socal Choce and Welfare, 8(4): , [4] Ed H. Clarke. Multpart prcng of publc goods. Publc Choce, 11:17 33, [5] Vncent Contzer and Tuomas Sandholm. Complexty of mechansm desgn. UAI, pages , [6] Vncent Contzer and Tuomas Sandholm. Unversal votng protocol tweaks to make manpulaton hard. IJCAI, [7] Bo Faltngs and Quang Huy Nguyen. Mult-agent coordnaton usng local search. IJCAI, [8] Allan Gbbard. Manpulaton of votng schemes. Econometrca, 41: , [9] Andrew Goldberg and Jason Hartlne. Envy-free auctons for dgtal goods. ACM-EC, pages 29 35, [10] Theodore Groves. Incentves n teams. Econometrca, 41: , [11] Chrstopher Kekntveld, Yevgeny Vorobeychk, and Mchael Wellman. An analyss of the 2004 supply chan management tradng agent competton. IJCAI-05 Workshop on Tradng Agent Desgn and Analyss, [12] Anshul Kothar, Davd Parkes, and Subhash Sur. Approxmately-strategyproof and tractable mult-unt auctons. ACM-EC, pages , [13] Noam Nsan and Amr Ronen. Computatonally feasble VCG mechansms. ACM-EC, pages , [14] Davd Parkes, Jayant Kalagnanam, and Marta Eso. Achevng budget-balance wth Vckrey-based payment schemes n exchanges. IJCAI, pages , [15] Mark Satterthwate. Strategy-proofness and Arrow s condtons: exstence and correspondence theorems for votng procedures and socal welfare functons. Journal of Economc Theory, 10: , [16] Wllam Vckrey. Counterspeculaton, auctons, and compettve sealed tenders. Journal of Fnance, 16:8 37, [17] Yevgeny Vorobeychk, Chrstopher Kekntveld, and Mchael Wellman. Emprcal mechansm desgn: Methods, wth applcaton to a supply chan scenaro. ACM-EC, [18] Makoto Yokoo, Yuko Sakura, and Shgeo Matsubara. Robust combnatoral aucton protocol aganst false-name bds. Artfcal Intellgence, 130(2), 2004.