Robust Implementation: The Role of Large Type Spaces

Transcription

1 Robust Implementaton: The Role of Large Type Spaces Drk Bergemann y Stephen Morrs z Frst Verson: March 2003 Ths Verson: Aprl 2004 Abstract We analyze the problem of fully mplementng a socal choce functon when the planner does not know the agents belefs about other agents types. We dentfy an ex post monotoncty condton that s necessary and - n economc envronments - su cent for full mplementaton n ex post equlbrum; we also dentfy an ex post monotoncty no veto condton that s su cent. These results are the ex post equlbrum analogues of Jackson s (99) results about Bayesan mplementaton. We show by example that ex post monotoncty mples nether Maskn monotoncty (necessary and almost su cent for complete nformaton mplementaton) nor - for some type spaces - nterm monotoncty (.e., the Bayesan monotoncty condton that s necessary and almost su cent for Bayesan mplementaton). We dentfy a robust monotoncty condton that s equvalent to nterm monotoncty on all type spaces; robust monotoncty mples both Maskn monotoncty and ex post monotoncty. Robust monotoncty s necessary for nterm mplementaton on all type spaces and s suf- cent for nterm mplementon on all common support type spaces when there are at least three agents and an economc condton s sats ed. Wthout a common support restrcton, we show that nterm mplementaton on all type spaces s equvalent to mplementaton under an ex post verson of domnance solvablty. Keywords: Mechansm Desgn, Implementaton, Common Knowledge, Unversal Type Space, Interm Equlbrum, Ex-Post Equlbrum, Domnant Strateges. Jel Classfcaton: C79, D82 Ths research s supported by NSF Grant #SES The rst author gratefully acknowledges support through a DFG Mercator Research Professorshp at the Center of Economc Studes at the Unversty of Munch. We bene ted from dscusson wth Amanda Fredenberg, Matt Jackson and Mke Rordan. We would lke to thank semnar audences at Caltech, Columba Unversty, Cornell Unverty, New York Unversty and the Unversty of Mchgan for helpful comments. Parts of ths paper were reported n early drafts of our work on Robust Mechansm Desgn (Bergemann and Morrs (200)). y Department of Economcs, Yale Unversty, 28 Hllhouse Avenue, New Haven, CT 065, drk.bergemann@yale.edu. z Department of Economcs, Yale Unversty, 30 Hllhouse Avenue, New Haven, CT 065, stephen.morrs@yale.edu.

2 Robust Implementaton May 4, Contents Introducton 3 2 Example A 4 2. Ex Post Implementaton Interm Implementaton The Implementaton Problem 7 3. Ex Post Equlbrum Interm Equlbrum Maskn Monotoncty and Ex Post Monotoncty 9 4. Maskn Monotoncty Ex Post Monotoncty Ex Post Implementaton 6 Interm Monotoncty and Robust Monotoncty 5 6. Interm Monotoncty Robust Monotoncty Comparng Monotoncty Propertes: Results Comparng Monotoncty Notons: Examples Example B Example C Implementaton on All Type Spaces Iteratve Implementaton Example D Example A Revsted Characterzaton Prvate Values and Domnant Strateges 32 9 Dscusson In nte Acton Games Fnte Type Spaces The Pure Strategy Restrcton Concluson

3 Robust Implementaton May 4, Introducton Ths paper looks at the problem of fully mplementng a socal choce functon when agents have nterdependent values. Thus each agent has a payo type. The agents have preferences over outcomes that depend on the pro le of payo types. The planner does not know the agents types but must choose a mechansm such that n every equlbrum of the mechansm, agents play of the game results n the outcome spec ed by the socal choce functon at every payo type pro le. Ths problem has been analyzed under the assumpton of complete nformaton,.e., there s common knowledge among the agents of ther payo types (e.g., Maskn (999)). It has also been analyzed under the assumpton of ncomplete nformaton, on the assumpton that there s a xed type space and there s common knowledge among the agents of the pror (or the prors) accordng to whch agents form ther belefs (e.g., Jackson (99)). We want to analyze the problem of full mplementaton under the assumpton that the planner knows nothng about what agents know or beleve about other agents payo types, or ther hgher order belefs. We beleve that by xng a small type space and assumng common knowledge among the agents of the type space and agents belefs on the type space, researchers have been makng very strong mplct assumptons. We would lke to relax those assumptons. There has recently been much nterest n the lterature on usng the concept of ex post equlbrum snce t seems unrealstc to allow the mechansm to depend on the planner s knowledge of the type space (e.g., Dasgupta and Maskn (2000)). We provde a complete analyss of full mplementaton n ex post equlbrum. We ntroduce an ex post monotoncty condton that - along wth ex post ncentve compatblty - s necessary for ex post mplementaton. We show that a slght strengthenng of ex post monotoncty - the ex post monotoncty no veto condton - s su cent for mplementaton wth at least three agents. The latter condton reduces to ex post monotoncty n economc envronments. These results are the ex post analogues of the Bayesan mplementaton results of Jackson (99), and we employ smlar arguments to establsh our results. However, for full mplementaton usng a strong soluton concept does not necessarly mply stronger results: the fact that non truth-tellng behavor may fal the strngent requrement of beng an ex post equlbrum may make mplementaton easer. We show n an economc example that ex post monotoncty may hold even when both Maskn monotoncty (the necessary condton for complete nformaton mplementaton) and nterm monotoncty on a xed type space (the necessary condton for nterm mplementaton) fal. Thus ex post mplementaton s possble even when complete nformaton mplementaton and nterm ncomplete nformaton mplementaton are mpossble. We therefore nd a condton - robust monotoncty - that s equvalent to requrng nterm monotoncty on every type space. Suppose that we x a "decepton" spec yng, for each payo type of each agent, a set of types that he mght msreport hmself to be. We requre that for some agent and a type msreport of agent under the decepton, for every msreport 0 that that the other agents mght make under the decepton, there exsts an outcome y whch s strctly preferred by agent to the outcome he would receve under the socal choce functon for every possble payo type pro le that mght msreport 0 ; where ths outcome y sats es the extra restrcton that no payo type of agent prefers outcome y to the socal choce functon f the other agents were really types 0. Ths condton - whle a lttle convoluted - s a somewhat easer to nterpret than the nterm (Bayesan) monotoncty condtons. It s very strong and mples both Maskn monotoncty and ex post monotoncty condtons (but s strctly weaker than domnant strateges). Robust monotoncty s necessary for nterm mplementaton on all type spaces and s su cent for nterm mplementaton on all common support type spaces when there are at least three agents and an economc condton s sats ed. We show that nterm mplementaton on all type spaces s possble f and only f t s possble to mplement the socal choce functon usng an ex post teratve deleton procedure: we x a mechansm and teratvely delete messages for each payo type that

4 Robust Implementaton May 4, are strctly domnated by another message for each payo type pro le and message pro le that has survved the procedure. Ths requrement s stronger than robust monotoncty. Ths last result about teratve deleton llustrates a general pont well-known from the lterature on epstemc foundatons of game theory (e.g., Brandenburger and Dekel (987), Battgall and Snscalch (2003)): equlbrum soluton concepts only have bte f we make strong assumptons about type spaces,.e., we assume small type spaces where the common pror assumpton holds. Our unform mplementaton result says that equlbrum has no bte (relatve to terated deleton of strctly domnated strateges) f we allow for su cently rch type spaces. The results n ths paper concern full mplementaton. An earler companon paper of ours (Bergemann and Morrs (2003)) addresses the analogous questons of robustness to rch type spaces, but lookng at the queston of partal mplementaton,.e., does there exst a mechansm such that some equlbrum mplements the socal choce functon. We showed that ex post (partal) mplementaton of the socal choce functon s a necessary and su cent condton for partal mplementaton on all type spaces. Ths paper establshes that an analogous result does not hold for full mplementaton. In that paper, we also looked at the partal mplementaton of socal choce correspondences, but showed that partal mplementaton on all type spaces was sometmes easer than ex post partal mplementaton. We leave for future work the queston of full mplementaton of socal choce correspondences on large type spaces. In the specal case of prvate values, ex post ncentve compatblty s equvalent to domnant strateges ncentve compatblty and thus partal mplementaton on all type spaces mples domnant strategy mplementaton. But strctly domnant strategy mplementaton s a su cent condton for full mplementaton. Thus n the prvate values case, movng to the stronger soluton concept of ex post equlbrum / domnant strateges s always (up to the domnant / strctly domnant strateges dstncton) a more strngent requrement. Ths paper shows that ths well known observaton does not translate to an nterdependent values settng. The paper s organzed as follows. Secton 2 descrbes a smple example that llustrates some of the key ponts n the paper. Secton 3 descrbes the formal envronment and soluton concepts. Secton 4 ntroduces our noton of ex post monotoncty and compares t to Maskn monotoncty. Secton 5 reports our analyss of the ex post mplementaton problem. Secton 6 ntroduces nterm monotoncty and robust monotoncty, and characterzes how the monotoncty condtons relate to each other usng propostons and examples. Secton 7 presents our results on nterm mplementaton on all type spaces and reports results on unform mplementablty. Secton 9 concludes. 2 Example A Consder the followng nterdependent values socal choce settng. There are two agents and 2. Each agent has two possble payo types, = ; 0 and 2 = 2 ; 0 2. There are four possble socal outcomes, A = fa; b; c; dg. The payo s of the two agents are gven by: a ; 3 0; 0 0 0; 0 ; b ; 0 3; 3 0 ; 0; 0 c ; 0 ; 0 3; 3 0; 0 d ; 0; 0 0 0; 0 3; 3 Notce that the agents have dentcal nterests and, for each payo type pro le, have a unque preferred outcome. The socal choce functon f wll select that outcome: f a b 0 c d

5 Robust Implementaton May 4, We are nterested n a settng where all ths nformaton s common knowledge among the agents and the planner, but the planner knows nothng about the agents belefs and hgher order belefs about each others types. What can the planner do? 2. Ex Post Implementaton One approach to ths problem s to focus attenton on ex post mplementaton. That s, suppose the planner seeks a mechansm whose ex post equlbra mplement f. Snce ex post equlbra are ndependent of agents belefs about other agents types, ths s one way of dealng wth the lack of common knowledge. We rst analyze ths approach. Observe that the socal choce functon s ex post ncentve compatble. Thus f the planner smply nvtes the agents to announce ther payo types, each agent wll have an ncentve to tell the truth as long as he expect others to do so, whatever hs belefs about the other agents types. Thus truth tellng s an ex post equlbrum of the payo type drect mechansm. However, ths game also has another ex post equlbrum where each type of each agent always msreports hs type. But t s easy to construct a smple augmented mechansm where all (pure strategy) ex post equlbra yeld desrable outcomes. Consder the mechansm where agent 2 smply announces hs payo type; and agent announces hs payo type and also announces ether "truth" or "le" (wth the nterpretaton that the latter announcement s agent s announcement about whether he beleves agent 2 has told the truth). Ths mechansm can be represented by the followng table: (, truth) a b ( 0, truth) c d (, le) b a ( 0, le) d c What are the (pure strategy) ex post equlbra of ths game? In any ex post equlbrum, type 2 of agent 2 must announce 2 or 0 2. If type 2 of agent 2 announces 2, then type of agent must announce (, truth) and type 0 of agent must announce ( 0, truth); so type 0 2 of agent 2 must announce 0 2. On the other hand, f type 2 of agent 2 announces 0 2, then type of agent must announce (, le) and type 0 of agent must announce ( 0, le); so type 0 2 of agent 2 must announce 2. Thus there are two possble ex post equlbra and both mplement the socal choce functon. Thus for ths example, we have shown the possblty of ex post mplementaton. Theorem n Secton 5 dent es an ex post monotoncty condton that s necessary for ex post mplementaton; we also show that ths condton s su cent f there are at least three agents n an economc envronment and that a slghtly stronger ex post monotoncty no veto condton s su cent n non-economc envronments. Mechansms of ths form - where the augmented mechansm contans a copy of the drect mechansm - are common n the mplementaton lterature; Mookerjee and Rechelsten (990) refer to them as "augmented drect mechansms." ()

6 Robust Implementaton May 4, Interm Implementaton We can also analyze whether nterm mplementaton s possble on d erent type spaces. Suppose that agents had the followng type space: t t 0 t 00 t 000 t 2 t 0 2 t 00 2 t ( ") 8 ( ") 8 " 8 " 8 ( ") 8 ( ") 8 " 8 " 0 8 " 8 " 8 ( ") 8 ( ") 8 " 8 " 8 ( ") 8 ( ") where " < 2. The four types of agent are represented as rows, the four types of agent 2 are represented as columns and the numbers represent the pror on type pro les. The payo type of a gven type s recorded at the end of hs row/column. If ths s the true type space and agents are nvted to play the augmented mechansm (), then there s clearly a strct pure strategy nterm equlbrum where agents follow strateges: and 8 >< s () = >: 8 >< s 2 () = >: ( ; truth) f t ( 0 ; truth) f t 0 ( ; le) f t 00 ( 0 ; le) f t f t f t f t f t To see why ths s an equlbrum, note that f " = 0, then we have dsjont type spaces consstng of types (t ; t 0 ; t 2 ; t 0 2); and types (t 00 ; t 000 ; t 00 2; t ), respectvely and the above type space reduces to: t 8 t 0 8 t 2 t 0 2 t 00 2 t t t In ths new type space, the types n the rst dsjont type space (t ; t 0 ; t 2 ; t 0 2) play accordng to one ex post equlbrum of the augmented mechansm (), whereas the types n the second dsjont type space (t 00 ; t 000 ; t 00 2; t ) play accordng to the other ex post equlbrum. Gven the strct ncentves, allowng " to be postve but small does not stop these strateges beng an equlbrum. But now, wth probablty ", there s mscoordnaton. Ths example llustrates one mportant message of ths paper: there s a sgn cant gap between ex post mplementaton and nterm mplementaton. It s sometmes easer to ex post mplement than to nterm mplement. In ths example, there s no mechansm that nterm mplements f on every type space. Here s an nformal argument by contradcton (the formal argument appears n Secton 7). We rst clam that a mechansm nterm mplements f on every type space f and only f t teratvely mplements f n the followng sense. Iteratvely delete for each payo type all messages that were not best responses to some belef over payo type - message pars of the opponent that have not yet been deleted. There s teratve mplementaton of f f, for any payo type pro le,

7 Robust Implementaton May 4, every survvng message pro le s consstent wth f. To prove the harder "only f" part of the clam, construct a type space where each player has a type correspondng to every payo type - message par that survve the terated elmnaton. For each payo type-message par survvng the terated deleton, there s a belef over the survvng payo type - message pars of the opponent such that that message s a best response for that payo type. Thus we can construct belefs on the type space such that t s an equlbrum for each type to send the message wth whch t s labelled. Now we argue that teratve mplementaton s not possble for our example. Frst, note that for each type, there s at least one message (call t m ( )) wth the property that g (m ()) = f () for each. Also observe that message m ( ) s never deleted for type. There must be a rst round - call t round n - when message m ( ) s deleted for type 0, for some. Thus n the prevous round, m j ( j) had not been deleted for type 0 j. Now f type 0 conjectures that hs opponent s type 0 j sendng message m j ( j), then hs payo to sendng message m ( ) s. Snce ths s not a best response, there must exst another message bm such that g bm ; m j ( j) = f 0 ; 0 j. But now ths message bm can never be deleted for type 0, a contradcton. 3 The Implementaton Problem We x a nte set of agents, ; 2; :::; I. Agent s payo type s 2, where s a nte set. We wrte 2 = ::: I. There s a set of outcomes Y. Each agent has utlty functon : Y! R. Thus we are n the world of nterdependent types, where an agent s utlty depends on other agents payo types. A socal choce functon s a mappng f :! Y. If the true payo type pro le s, the planner would lke the outcome to be f (). In ths paper, we restrct our analyss to the mplementaton of a socal choce functon rather than a socal choce correspondence or set. We are nterested n analyzng behavor n a varety of type spaces, ncludng rcher sets of types than payo types. For ths purpose, we shall refer to agent s type as t 2 T, where T s a nte set. 2 A type of agent must nclude a descrpton of hs payo type. Thus there s a functon b : T! wth b (t ) beng agent s payo type when hs type s t. A type of agent must also nclude a descrpton of hs belefs about the types of the other agents; thus there s a functon b : T! (T ) wth b (t ) beng agent s belef type when hs type s t. Thus b (t ) [t ] s the probablty that type t of agent assgns to other agents havng types t. A type space s a collecton: T = T ; b I ; b. = The type space s a common support type space f there exsts T T such that b (t ) [t ] > 0, (t ; t ) 2 T. The type space f a common pror type space f there exsts p 2 (T ) such that b (t ) [t ] = p (t ; t ) P. p t ; t 0 The type space s a payo type space f, for each, T = and b s the dentty map. A planner must choose a game form or mechansm for the agents to play n order the determne the socal outcome. Let M be the countably n nte set of messages avalable to agent. 3 Let 2 The nte set restrcton clar es the relaton to the exstng lterature. In Secton 9, we dscuss what happens f we allow for uncountable type spaces. 3 Ths assumpton clar es the relaton wth the exstng lterature. We dscuss n Secton 9 what happens f we restrct attenton to nte messages or allow larger sets of messages. t 0

8 Robust Implementaton May 4, g (m) be the outcome f acton pro le m s chosen. Thus mechansms do not nvolve randomzaton contngent on the message pro le. But randomzaton can be bult nto the outcome space Y. Thus a mechansm s a collecton M = (M ; :::; M I ; g ()) ; where g : M! Y. Now holdng xed the payo envronment, we can combne a type space T wth a mechansm M to get an ncomplete nformaton game (T ; M). We are nterested n a settng where the planner does not know the payo types of the agents and knows nothng about agents belefs and hgher order belefs about other agents types. Two approaches to ths problem are to look at ex post equlbra of the game wth payo types; or we can look at nterm (Bayesan Nash) equlbra on a varety of rcher type spaces. We consder each n turn. 3. Ex Post Equlbrum Consder the "payo types game" where each agent s possble types are. Thus we have an ncomplete nformaton game where agent s payo f message pro le m s sent and payo type pro le s realzed s (g (m) ; ). A pure strategy n ths game s a functon s :! M. De nton (Ex post equlbrum) A pure strategy pro le s = (s ; :::; s I ) s an ex post equlbrum of the payo types game f for all, and m. 4 (g (s ()) ; ) (g ((m ; s ( ))) ; ) De nton 2 (Ex post mplementaton) Socal choce functon f s ex post mplementable f there exsts a mechansm M such that every (pure strategy) ex post equlbrum s of the game M sats es g (s ()) = f (). We restrct attenton to pure strategy equlbra. Ths helps make comparsons wth the exstng lterature (where the assumpton s standard). However, when we conduct analyss allowng rch type spaces, the restrcton wll not bte. Ths ssue s dscussed n detal n Secton Interm Equlbrum Next we consder an ncomplete nformaton game wth an arbtrary type space T and a mechansm M. The payo of agent f message pro le m s chosen and type pro le t s realzed s then gven by g (m) ; b (t) : A pure strategy for agent n the ncomplete nformaton game (T ; M) s gven by s : T! M. Pure strategy (nterm, or Bayesan Nash) equlbra are de ned n the usual way. 4 Ex post ncentve compatblty was dscussed as "unform ncentve compatblty" by Holmstrom and Myerson (983). Ex post equlbrum s ncreasngly studed n game theory (see Kala (2002)) and s often used n mechansm desgn as a more robust soluton concept (Cremer and McLean (985), Dasgupta and Maskn (2000), Perry and Reny (2002), Bergemann and Valmak (2002)).

9 Robust Implementaton May 4, De nton 3 (Interm equlbrum) A pure strategy pro le s = (s ; :::; s I ) s an nterm equlbrum of the game (T ; M) f g (s (t)) ; b (t) b (t ) [t ] g ((m ; s ( ))) ; b (t) b (t ) [t ] t 2T for all, t and m. t 2T De nton 4 (Interm Implementaton) Socal choce functon f s nterm mplementable on type space T f there exsts a mechansm M such that every (pure strategy) equlbrum s of the game (T ; M) sats es g (s (t)) = f b (t) for all t. 4 Maskn Monotoncty and Ex Post Monotoncty The exstng lterature on complete nformaton and Bayesan mplementaton dent es "monotoncty" condtons that are necessary and "almost su cent" for mplementon. It s useful to ntroduce our noton of ex post monotoncty by comparng t wth Maskn monotoncty. 4. Maskn Monotoncty Maskn (999) ntroduced a celebrated monotoncty noton for the complete nformaton envronment whch consttutes a necessary and almost su cent condton for complete nformaton mplementaton. De nton 5 (Maskn monotoncty) Socal choce functon f s (Maskn) monotone, f f 0 ; 0 y; 0 ) f 0 ; (y; ) for all and y, then f 0 = f (). In words, monotoncty requres that f alternatve x s f optmal wth respect to some pro le of preferences and the pro le s then altered so that, n each ndvdual s orderng a does not fall below any alternatve that t was not below before, then x remans f optmal wth respect to the new pro le. (Maskn (999)). Maskn monotoncty s necessary for complete nformaton mplementaton and, when there are at least three agents and no veto power holds, also su cent. To motvate the monotoncty notons of ths paper, t s useful to re-wrte ths statement. Frst, we can gve the equvalent contrapostve statement: f f () 6= f 0, then there exsts and y such that f 0 ; 0 y; 0 and (y; ) > f 0 ;. Also, t s useful to thnk of the agents n a complete nformaton settng engagng n a "decepton" where they msreport the true type pro le n a coordnated way. Wrte :!

10 Robust Implementaton May 4, for the common decepton strategy. Now Maskn monotoncty requres that for every decepton wth f 6= f, then there exsts,, and y such that whle (y; ) > (f ( ()) ; ) ; (2) (f ( ()) ; ()) (y; ()) : (3) Ths alternatve statement suggests a rather ntutve descrpton why monotoncty s a necessary condton for mplementaton. Suppose that f s complete nformaton mplementable. Then f the agents were to deceve the desgner by msreportng () rather than reportng truthfully and f the decepton () would lead to a d erent allocaton,.e. f ( ()) 6= f (), then the desgner should be able to fend o the decepton. Ths requres that there s some agent and pro le such that the desgner can o er agent a reward y for denouncng the decepton () by the agents f the true type pro le s. Yet, at the same tme, the desgner has be aware that the reward could be used n the wrong crcumstances, namely when the true payo type pro le s () and t s ndeed reported to be (). The rst strct nequalty (2) then guarantees the exstence of a whstle-blower, whereas the second weak nequalty (3) guarantees ncentve compatble behavor by the whstle-blower. Both these features wll re-appear n all the monotoncty condtons studed n ths paper. 4.2 Ex Post Monotoncty Wth ncomplete nformaton, a decepton -.e., a non truth-tellng strategy n the drect mechansm a decepton, s a collecton = ( ; :::; I ), each :! and () = ( ( ) ; :::; I ( I )) : In a drect revelaton game would ndcate s reported type as a functon of hs true type. For a drect revelaton mechansm, f agents report the decepton rather than truthfully, then the resultng socal outcome s gven by f ( ()) rather than f (). We wrte f () f ( ()). For any pro le of payo types of agents other than, we wrte Y ( ) for the set of allocatons that make agent worse o than under the socal choce functon at all of hs payo types. So Y ( ) y : f 0 ; ; 0 ; u y; 0 ; ; :. (4) De nton 6 (Ex-post monotoncty) Socal choce functon f sats es ex post monotoncty (EM) f for every decepton wth f 6= f, there exsts ; and y 2 Y ( ( )) such that (y; ) > (f ( ()) ; ) : (5) At rst glace, ex post monotoncty looks lke a stronger requrement than Maskn monotoncty, snce the whstle-blowng constrant for Maskn monotoncty (2) stays the same, whle the sngle ncentve compatblty constrant (3) s replaced by the requrement that y 2 Y ( ( )), whch mples a famly of constrants, f 0 ; ( ) ; 0 ; ( ) y; 0 ; ( ) : (6) But because of the coordnaton bult nto the complete nformaton deceptons, t becomes harder to nd a reward y for Maskn monotoncty than for ex post monotoncty. In Secton 6, we wll descrbe an example that s Maskn monotonc and not ex post monotonc; and another example that s ex post monotonc but not Maskn monotonc.

11 Robust Implementaton May 4, Ex Post Implementaton We present necessary and su cent condtons for a socal choce functon f to be ex-post mplementable n the payo type space. Our results extend the work of Maskn (999) for complete nformaton mplementaton and Jackson (99) on Bayesan mplementaton (.e., nterm mplementaton on a xed type space) to the noton of ex post equlbrum. If we were just nterested n partally mplementng f -.e., constructng a mechansm wth an ex post equlbrum achevng f - then by the revelaton prncple we could restrct attenton to drect mechansms and a necessary and su cent condton s the followng ex post ncentve compatblty condton. De nton 7 (Ex Post Incentve Compatblty) Socal choce functon f s ex post ncentve compatble (EPIC) f (f () ; ) f 0 ; ; for all, and 0. Ex post ncentve and monotoncty condtons are necessary condtons for ex post mplementaton. Theorem (Necessty) If f s ex post mplementable, then t sats es (EP IC) and (EM). Proof. Let (M; g) mplement f wth equlbrum strateges s :! M. Consder any ; 0 2. Snce s s an equlbrum, (g (s ()) ; ) g s 0 ; s ( ) ; for all 2. Notng that g s 0 ; s ( ) = f 0 ; establshes (EP IC). Suppose that for some decepton, f 6= f. It must be that s s not an equlbrum at some 2. Therefore there exsts and m 2 M such that we have (g (m ; s ( ( ))) ; ) > (g (s ( ())) ; ) Let y, g (m ; s ( ( ))). Then, from above, But snce s s an equlbrum t follows that (y; ) > (f ( ()) ; ) : f 0 ; ( ) ; 0 ; ( ) = g s 0 ; ( ) ; 0 ; ( ) Ths establshes that y 2 Y ( ). g (m ; s ( ( ))) ; 0 ; ( ) = y; 0 ; ( ) ; We proceed by showng that n a wde class of envronments, to be referred to as economc envronments, ex post ncentve and monotoncty condton are also su cent condtons for ex post mplementaton.

12 Robust Implementaton May 4, De nton 8 (Economc envronment) An envronment s economc at state 2 f, for every allocaton a 2 Y, there exst 6= j and allocatons x and y respectvely such that and (x; ) > (a; ) u j (y; ) > u j (a; ). An envronoment s economc f t s economc at every state. We shall prove the su cency of the ex post monotoncty condton by usng the followng augmented mechansm. It s smlar to mechansms used to establsh su cency n the complete nformaton mplementaton lterature (e.g., Maskn (999)). Each agent sends a message of the form m = ( ; z ; y ), where 2, z s a non-negatve nteger and y 2 Y. The mechansm s descrbed by three rules.. If z = 0 for all, then g (m) = f (). 2. If z j = and z = 0 for all 6= j, then outcome y j s chosen f y j 2 Y j ( j); otherwse outcome f () s chosen. 3. In all other cases, y e j(z) s chosen, where ej (z) s the agent wth the hghest value of z (and, n the event of a te, the lowest label). A strategy pro le n ths game s a collecton s = (s ; :::; s I ), wth s :! M and we wrte s () = s () ; s 2 () ; s 3 () 2 Z + Y ; and s k () = s k () I. We shall refer to ths mechansm as the augmented mechansm. = Theorem 2 (Economc Envronment) If I 3 and f sats es ex post ncentve compatblty and ex post monotoncty and the envronment s economc, then f s ex post mplementable. Proof. The proposton s proved n three steps, usng the above mechansm. Step. There s an ex post equlbrum s wth g (s ()) = f () for all. Any strategy pro le s of the followng form s an ex post equlbrum: s ( ) = ( ; 0; ). Suppose agent thnks that hs opponents are types and devates to a message of the form s ( ) = 0 ; z ; y ; f ether z = 0 or z > 0 but y =2 Y ( ), then the payo gan s f 0 ; ; f ( ; ) (f ( ; ) ; f ( ; )), whch s non-postve by (EPIC); f z = and y 2 Y ( ), then the payo gan s (y ; ( ; )) (f ( ; ) ; f ( ; )), whch s non-postve by the de nton of Y ( ).

13 Robust Implementaton May 4, Step 2. In any ex post equlbrum, s 2 ( ) = 0 for all and. Suppose that rule 2 or rule 3 apples to the message pro le sent at payo type pro le, so that there exsts such that s 2 ( ) =. Gven the strateges of the other agents, any agent j 6= of type j who thought hs opponents were types j could send any message of the form (; z j ; y j ) and obtan utlty u j (y j ; ). Thus we must have u j (g (s ()) ; ) u j (a; ) for all a and all j 6=. Ths contradcts the economc envronment assumpton. Step 3. In any ex post equlbrum wth s 2 ( ) = 0 for all and, f s = f. Suppose that f s 6= f. By (EM), there exsts ; and y 2 Y s ( ) such that (y; ) > f s () ;. Now suppose that type of agent beleves that hs opponents are of type and sends message m = (; ; y), whle other agents send ther equlbrum messages, then from the de nton of g () : so that and ths completes the proof of su cency. g (m ; s ( )) = y; (g (m ; s ( )) ; ) = (y; ) > f s () ; = (g (s ()) ; ), The economc envronment condton was used to show that n the augmented mechansm n equlbrum, the nteger reports z all have to say z = 0, or else any agent j could pro tably change hs report z and obtan a more desrable allocaton to f (), where the economc envronment guaranteed the exstence of agent j wth a preferred allocaton. We now proceed to establsh su cent condtons for ex post mplementaton outsde of economc envronments. We begn by establshng an mplcaton of non-economc envronments. Lemma The envronment s non-economc at f and only f there exsts j and b 2 Y such that (b; ) (a; ) for all a 2 A and 6= j. Proof. The envronment s non-economc (by de nton) f and only f there exsts an allocaton b, such that f u j (y; ) > u j (b; ) for some j, y 2 Y, then there does not exst 6= j and a 2 Y such that (a; ) > (b; ). Thus (b; ) (a; ) for all a 2 Y and 6= j. The ex post analogue of Jackson s "no veto hypothess" s smply the requrement that the state be non-economc. De nton 9 (No Veto Power) Socal choce functon f sats es no veto power at f (b; ) (a; ) for all a 2 Y and all 6= j mples that f () = b. De nton 0 (Ex Post Monotoncty No Veto (EMNV)) A socal choce functon f sats es ex post monotoncty no veto f the followng s true. Fx any decepton and sets (wrte = I = ). Suppose that the envronment n non-economc at each =2. Suppose also that ether f ( ()) 6= f () for some 2 or the no veto power property fals for some =2. Then there exsts, 2 and y 2 Y ( ( )) such that (y; ) > (f ( ()) ; ).

14 Robust Implementaton May 4, EPMV s almost equvalent to requrng ex post monotoncty and no veto power everywhere. More precsely, we have:. If ex post monotoncty holds and no veto power holds at every type pro le, then EMNV holds. 2. If EPMV holds, then () ex post monotoncty holds and (2) f the envronment s non-economc whenever =, then no veto power holds whenever =. To see (), set = for all ; to see (2), set to be the truth-tellng decepton and, for some, = n f g and j = j for all j 6=. Thus n an economc envronment, EMNV s equvalent to ex post monotoncty. Theorem 3 (Su cency) For I 3, f sats es (EP IC) and (EMNV ), then t s ex post mplementable. Proof. We use the same mechansm as before. The argument that there exsts an ex post equlbrum s wth g (s ()) = f () for all s the same as before. Now we establsh three clams that hold for all equlbra. Let = f : s ( ) = (; 0; )g Clam. In any ex post equlbrum, for each =2, (a) there exsts such that u j (g (s ()) ; ) u j (a; ) for all a and j 6= ; and thus (b) the envronment s non-economc at. Frst, observe that for each =2, there exsts such that s 2 ( ) > 0. Gven the strateges of the other agents, any agent j 6= who thought hs opponents were types j could send any message of the form (; z j ; y j ) and obtan utlty u j (y j ; ). Thus we must have u j (g (s ()) ; ) u j (a; ) for all a and j 6= ; thus the envronment s non-economc for all =2. Clam 2. In any ex post equlbrum, for all 2, f s () ; (y; ) for all y 2 Y s ( ). Suppose that y 2 Y s ( ) and that type of agent beleves that hs opponents are of type and sends message m = (; z ; y), whle other agents send ther equlbrum messages. Now g (m ; s ( )) = y; so ex post equlbrum requres that (g (s ()) ; ) = f s () ; (g (m ; s ( )) ; ) = (y; ). Clam 3. If EPMV s sats ed, then Clam and 2 mply that g (s ()) = f () for all. Fx any equlbrum. Clam (b) establshes that the envronment s non-economc at all 2. Suppose g (s ()) 6= f () for some 2. Now EPMV mples that there exsts, 2 and y 2 Y s ( ) such that (y; ) > f s () ;, contradctng Clam 2. Suppose g (s ()) 6= f () for some =2. By clam (a), there exsts such that u j (g (s ()) ; ) u j (a; ) for all a and j 6=. Ths establshes that no veto power fals at. So agan EPMV mples that there exsts, 2 and y 2 Y ( ( )) such that (y; ) > (f ( ()) ; ), contradctng Clam 2. The structure of the proof s smlar to Jackson (99). The mechansm used to prove su cency s smpler as we requre the strateges to be n an ex-post rather than an nterm equlbrum. The entre argument s more compact due to the smplfyng assumpton of a socal choce functon rather than socal choce set.

15 Robust Implementaton May 4, Interm Monotoncty and Robust Monotoncty 6. Interm Monotoncty A decepton for a type space T s a collecton = ( ; :::; I ), wth : T! T. Wrte (t) = ( (t )) I = ; let f b : T! A and f b : T! A be de ned by f b (t) = f b (t) and f b (t) = f b ( (t)) for all t. De nton (Interm Monotoncty) Socal choce functon f sats es nterm monotoncty on type space T f, for every decepton wth f b 6= f b, there exsts, t and y : T! Y such that t 2T y ( (t)) ; b (t) b (t ) [t ] > t 2T f b ( (t)) ; b (t) b (t ) [t ] ; (7) and t 2T t 2T f b (t 0 ; t ) ; b (t 0 ; t ) b (t 0 ) [t ] (8) y ( (t ) ; t ) ; b (t 0 ; t ) b (t 0 ) [t ] ; 8t 0 2 T : Condtons lke ths are known as Bayesan monotoncty n the lterature. We use the term nterm monotoncty both because we are nterested n the case when there s no common pror and to hghlght the comparson wth ex post monotoncty. Postlewate and Schmedler (986) showed that such an nterm monotoncty condton s necessary and su cent for full mplementaton n an exchange economy wth nonexclusve nformaton and at least three agents. Palfrey and Srvastava (989) provde separate necessary and su cent condtons for nterm mplementaton when there s exclusve nformaton. Jackson (99) showed that nterm monotoncty s necessary and su cent for nterm mplementaton n economc envronments and that a slghtly strengthened property (Bayesan monotoncty no veto) s su cent. 6.2 Robust Monotoncty We wll be nterested n another new monotoncty noton that s equvalent to nterm monotoncty on all type spaces. In de nng robust monotoncty, we therefore formalze a decepton as a pontto-set mappng. A decepton s a collecton = ( ; :::; I ) wth :! 2 and 2 ( ). The nterpretaton s that ( ) s the collecton of correct or ncorrect reports that payo type mght send. A decepton s acceptable f 0 2 () ) f 0 = f (). A decepton s unacceptable f t s not acceptable. We wrte 0 : 0 2 ( ) and 0 j 0 j. j6= Thus 0 s the collecton of who mght report themselves to be 0 under decepton.

16 Robust Implementaton May 4, De nton 2 (Robust Monotoncty) Socal choce functon f sats es robust monotoncty f for every unacceptable decepton, there exst,, 0 2 ( ) such that, for all 0 2 and 2 0, there exsts y 2 Y 0 such that ( ) (y; ( ; )) > ( ) f 0 ; 0 ; ( ; ). (9) 2 2 Note that the allocaton y s allowed to depend on the msreport 0 and the dstrbuton. The noton of robust monotoncty shares many features wth the ex post monotoncty condton. Lke ex post monotoncty, robust monotoncty refers only to payo types and does not refer to prors or posterors over payo types nor does t refer to any general type spaces. Robust monotoncty also requres that the ex post ncentve compatblty requrement y 2 Y 0 be sats ed. But the whstle-blower nequalty (9) s a stronger verson of the ex post requrement. 6.3 Comparng Monotoncty Propertes: Results In ths Subsecton, we establsh the relaton between varous monotoncty notons. We rst show that robust monotoncty s equvalent to nterm monotoncty on all type spaces. Theorem 4 Socal choce functon f sats es robust monotoncty f and only f t sats es nterm monotoncty on every type space. Proof. ((). We rst prove that nterm monotoncty on every type space mples robust monotoncty. It s convenent to work wth the followng contrapostve statement of robust monotoncty. Thus for all,, 0 2 ( ), there exsts a payo pro le 0 2, to be denoted by: ; 0, j ; 0 2 j6= and a condtonal probablty dstrbuton j ; 0 2 ; 0 such that f e ; ; 0 ; e ; ; 0 y; e ; ; 0 ; (0) for all e mples f 2 : ( ; 0 )2 ( )g f 2 : ( ; 0 )2 ( )g j ; 0 u f 0 ; ; 0 ; ( ; ) () j ; 0 u (y; ( ; )). Now we construct a type space based on the decepton such that f the socal choce functon sats es nterm monotoncty on ths type space, then must be acceptable. Frst, agent has a set of "decepton" types T whch are somorphc to = ; 0 : 2 and 0 2 ( ) and for smplcty we dentfy every type t 2 T smply by such a par of payo types ; 0, or T,. The type ; 0 has payo type and assgns probablty j ; 0 to the event that each agent j s type j ; j ; 0.

17 Robust Implementaton May 4, Second, agent has a set of "pseudo-complete nformaton types" T 2, whch are somorphc to, and for smplcty, agan let T 2 =. The type correspondng to has payo type and he s convnced that each other agent j s type. More formally, we have T = T [ T 2. If t 2 T and t = ; 0, then b (t) = and f t 2 T 2 and b (t ) [t ] = and t =, then j ; 0, f tj = j ; j ; 0 for each j 6= 0, otherwse; b (t ) = ; (2), f tj = ( b (t ) [t ] = j ; j ) for each j 6= 0, otherwse. Now we prove the proposton, by showng that nterm monotoncty on ths type space mples the decepton we started wth must be acceptable. Consder the decepton on the constructed type space where each type ; 0 reports hmself to be type 0 ; 0, and all other types report ther types truthfully. Thus: 0 (t ) = ; 0, f t = ; 0. t, otherwse Notce that type t = ( ; ) reports hs type truthfully under ths decepton for all. Now we apply the nterm monotoncty condton as presented n De nton to ths decepton. For any type t 2 T 2, the decepton changes nether hs acton nor hs belefs about hs opponents reportng behavor. Thus he cannot be the crtcal type t n the de nton who "reports the decepton". More formally, for any type t = 2 T 2, the nterm monotoncty condtons reduce to, after usng (2) and (3): (y () ; ) > (f () ; ) and for all t 0 = 0 2 T 2, we would have f 0 ; 0 y ; 0 ; 0, whch clearly leads to a contradcton for t 0 =. Thus there must exst, t 2 T such that (7) and (8) hold. Lettng bt = ; 0, (7) becomes: j ; 0 u y 0 ; 0 ; j ; 0 ; j ; 0 j6= f 2 : ( ; 0 )2 ( )g > f 2 : ( ; 0 )2 ( )g j ; 0 u f 0 ; ; 0 ; ( ; ). (3) and y : T! Y ; ( ; ) In the specal case of the pseudo complete nformaton types wth t 0 = e ; ; 0, the nterm ncentve compatblty condton (8) becomes f e ; ; 0 ; e ; ; 0 y 0 ; 0 ; j ; 0 ; j ; 0 ; e ; j6= ; 0 ; 8 e : (4) (5)

18 Robust Implementaton May 4, But now (0), () and (5) mples that (4) fals. Thus nterm monotoncty on ths type space requres that f b (t) = f b ( (t)) for all t. Ths requres s acceptable. Ths completes the proof of robust monotoncty. ()) Suppose f sats es robust monotoncty. Fx any type space T and any decepton wth f b (t) 6= f b ( (t)) for some t. De ne by: n ( ) = 0 : 9t such that b (t ) = and b o ( (t )) = 0. For every, ( ) s the collecton of payo types 0 whch wll be reported by some type t when he s usng the decepton and has a true payo type. Decepton s unacceptable, so by robust monotoncty, there exst,, 0 2 ( ) such that, for all 0 2 and for all wth 2 2 : 0 2 ( ) ; there exsts y 0 ; such that ( ) y 0 ; ; ( ; ) (6) f 2 : 0 2 ( )g > ( ) f 0 ; 0 ; ( ; ) f 2 : 0 2 ( )g and f e ; 0 ; e ; 0 y 0 ; ; e ; 0 ; (7) for all e. We emphasze that the dstrbuton only generates postve probabltes over 2 whch could lead to a decepton 0 for some types t 2 T. Thus n the followng we omt the set spec caton 2 : 0 2 ( ) n the summaton whenever we take expectatons wth respect to ( ) as pro les 00 wth 0 =2 00 receve probablty zero anyhow. Now choose any t such that b (t ) = and b ( (t )) = 0. Let, b (t ) [t ] (8) and 0 ft 2T : b ( (t ))= 0 g j 0 Pft 2T : b (t )= and b ( (t ))=, 0 g b (t ) [t ] P ft 2T : b ( (t ))= 0 g b. (9) (t ) [t ] For a gven type space T and type t, 0 s the probablty that agent attaches to a payo type report 0 gven the decepton. Consequently, j 0 s the condtonal probablty that the true payo type pro le s f the announced type pro le s 0. We construct a reward functon y (t) on the type space T by settng: y ( (t ) ; t ), y b (t ) ; b (t ). (20) Usng the probabltes dstrbutons de ned n (8) and (9), and the reward functon de ned n (20) we have the followng equaltes useful to establsh the nterm reward nequalty: y ( (t)) ; b (t) b (t ) [t ] (2) = t 2T 0 2 y 0 ; j 0 ; 2 j 0 0

19 Robust Implementaton May 4, and = t 2T 0 2 f b ( (t)) ; b (t) b (t ) [t ] (22) f 0 ; 2 j 0 0 : As the nequalty (6) holds for every 0, we can nfer from (6) that > y 0 ; 0 ; 2 f 0 ; 2 j 0 0 j 0 0 holds when we take the expectaton wth respect to 0. By appealng to the equaltes (2) and (22), we establsh that: y ( (t)) ; b (t) b (t ) [t ] (23) > t 2T t 2T f b ( (t)) ; b (t) b (t ) [t ]. Usng agan the probabltes dstrbutons de ned n (8) and (9), the reward functon de ned n (20), we have the followng equaltes useful to establsh the nterm ncentve nequaltes: f b (t 0 ; t ) ; b (t 0 ; t ) b (t 0 ) [t ] (24) = t 2T f b (t 0 ) ; ; b (t 0 ) ; j 0 0 and = t 2T 0 2 y ( (t ) ; t ) ; b (t 0 ; t ) b (t 0 ) [t ] (25) 2 y ( ; ( j )) ; b (t 0 ) ; j 0 0 ; 8t 0 : By appealng the ex post ncentve nequaltes of robust monotoncty, (7), we know that f b (t 0 ) ; ; b (t 0 ) ; y 0 ; ( j ) ; b (t 0 ) ; ; (26) for all t 0. The nequaltes (26) then reman vald when we take expectatons wth respect to the condtonal and margnal dstrbutons j 0 and 0 respectvely. By usng the equaltes (24) and (25) we can then establsh the nterm ncentve compatblty condtons: f b (t 0 ; t ) ; b (t 0 ; t ) b (t 0 ) [t ] (27) t 2T t 2T y ( (t ) ; t ) ; b (t 0 ; t ) b (t 0 ) [t ] ; 8t 0 :

20 Robust Implementaton May 4, But by (23) and (27), we have con rmed nterm monotoncty on ths type space. The proof may appear rather ntrcate n ts detals. We next gve a bref outlne of the basc steps to show that nterm mples robust monotoncty. We start wth an arbtrary decepton whch sats es the nequaltes (0) and () and, crucally, do not nsst on beng acceptable. For the gven decepton, we then create a type space, consstng of two components for every agent. The rst component for agent s created by the set of pars of payo types ; 0, where the rst entry s the true payo type and the second entry s a feasble decepton (under ), or 0 2 ( ). For ths reason, we refer to these types as decepton types. For every such par ; 0 there exsts one partcular payo pro le 0 whch s salent for agent of type ; 0, as the decepton sats es (0) and (). Under the decepton, ths payo pro le could have been reported by all true payo pro les whch are n the support of. Consequently, the belef component of type ; 0 s gven by smply adoptng ; 0. The second component are pseudo complete nformaton types, descrbed by t = 2, whch have a probablty one belef that the true payo pro le s gven by and that all other agents report the decepton type ( j ; j ), and hence the pseudo n the labellng. Gven ths type space T, we then consder a partcular decepton : T! T. The decepton s localzed around the decepton types and the pseudo complete nformaton types report thruthfully. The decepton conssts of agent always reportng hs decepton type rather than hs true type, or ; 0 = 0 ; 0. We then verfy whether f s nterm monotone under. The exstence of the pseudo complete nformaton types forces the nterm ncentve compatblty condtons to reduce to ex post ncentve compatblty condtons. Ths guarantees the hypothess n the robust monotoncty noton, namely nequalty (0), and thus leads to the concluson n form of the nequaltes (). But then we obtan a contradcton to the reward condton of nterm monotoncty, unless the hypothess for the nterm monotoncty condton, namely f 6= f, s not sats ed,.e. f = f holds, but of course ths mples that s acceptable. For the second part of the proof we use the full strength of robust monotoncty to establsh nterm monotoncty. We start out wth a decepton on an arbtrary type space T such that f 6= f. We then extract from gven type t and assocated belef type (t ) [t ] a condtonal dstrbuton over payo types (t ) [ ]. For ths condtonal dstrbuton, we can then construct a reward by the robust monotoncty hypothess, whch we then employ for construct a reward allocaton o er to nduce type t to denounce the decepton. Theorem 5 If f sats es nterm monotoncty on the complete nformaton type space, then t sats es Maskn monotoncty. Proof. The proof s by contrapostve. Suppose then that f s not Maskn monotone, and hence there exsts b : I! I such that for all ; ; wth f (b ()) 6= f (), and all h such that we have (h (b ()) ; ) > (f (b ()) ; ) ; (f (b ()) ; b ()) < (h (b ()) ; b ()) : Consder then the complete nformaton type space T =. For every, let = b. To obtan the contradcton, let us then suppose that there exsts and t such that (h ( (t)) ; t) b (t ) [t ] > (f ( (t)) ; t) b (t ) [t ] (28) t 2T t 2T whle t 2T (f (t 0 ; t ) ; (t 0 ; t )) b (t 0 ) [t ] t 2T (h ( (t ) ; t ) ; t) b (t 0 ) [t ] ; 8t 0 6= t. (29)

21 Robust Implementaton May 4, Wth the complete nformaton type space and the symmetrc decepton strategy, the nequaltes (28) and (29) reduce to (h (b ()) ; ) > (f (b ()) ; ) (30) and f 0 ; 0 h b () ; 0 ; :::; 0 ; 0 ; 8 0 6=, (3) but naturally there exsts 0 = b (), and for ths pro le, the above nequalty reads (f (b ()) ; b ()) (h (b ()) ; b ()) ; 0 = b (), whch leads to the desred contradcton wth Maskn monotoncty. Theorem 6 If f sats es robust monotoncty, then t sats es ex post monotoncty. Proof. Let be an ex post decepton wth f 6= f. Let be a robust decepton wth ( ) = f g [ f ( )g. By the de nton of robust monotoncty, there exsts,, 0 2 ( ) such that, for all 0 2 and 2 0, there exsts y 2 Y 0 such that ( ) (y; ( ; )) > ( ) f 0 ; 0 ; ( ; ). 2 2 By the constructon of, we must have 0 = ( ). Thus there exsts,, 0 = ( ), 0 2 wth 0 = ( ) and y 2 Y 0 such that (y; ( ; )) > f 0 ; 0 ; ( ; ). But ths s ex post monotoncty. Whle Maskn monotoncty s mpled by nterm monotoncty on complete nformaton pror type spaces, we do not have an argument mplyng ex post monotoncty or robust monotoncty usng type spaces that have a common pror, full support or common support. Because the strct nequaltes n the de nton of nterm monotoncty gve rse to a non-compact set, t s not clear that such an argument s possble. The followng example shows how t s possble to have nterm monotoncty sats ed for every type space wth a sequence of full support prors, but fal n the lmt. 6.4 Comparng Monotoncty Notons: Examples 6.4. Example B The example sats es Maskn monotoncty and nterm monotoncty for all common prors over the payo type space. Yet t fals to satsfy ex post monotoncty, and thus robust monotoncty. There are three agents, = ; 2; 3 and each agent has a bnary payo type space 2 = f0; g. The entre payo type space s gven by = 3 =. For smplcty of the example, the allocaton space s dentcal to the payo type space, or A = and the socal choce functon f :! A s gven by the dentty mappng f () = for all 2. The payo s of the agents satsfy symmetry across allocatons a and payo types: (a; ) = (; a) for all, a 2 A and 2 and nvarance wth respect to symmetrc permutatons, for all ; a 2 A; 2 and :!, we have: (a; ) = ( (a) ; ()). The payo matrces below represent the payo s of the agents for allocaton a = (0; 0; 0) at all possble type pro les 2. Agent s the row player, agent 2 the column player and agent 3 the matrx player: