ISOTONE EQUILIBRIUM IN GAMES OF INCOMPLETE INFORMATION. By David McAdams 1

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1 ISOTONE EQUILIBRIUM IN GAMES OF INCOMPLETE INFORMATION By Davd McAdams 1 An sotone pure strategy equlbrum exsts n any game of ncomplete nformaton n whch each player s acton set s a fnte sublattce of multdmensonal Eucldean space, types are multdmensonal and atomless, and each player s nterm expected payoff functon satsfes two non-prmtve condtons whenever others adopt sotone pure strateges: () sngle-crossng n own acton and type and () quassupermodularty n own acton. Condtons (,) are satsfed n supermodular and log-supermodular games gven afflated types, and n games wth ndependent types n whch each player s ex post payoff satsfes supermodularty n own acton and non-decreasng dfferences n own acton and type. Ths result s appled to provde the frst proof of pure strategy equlbrum exstence n the unform prce aucton when bdders have mult-unt demand, non-prvate values, and ndependent types. Keywords: Games of ncomplete nformaton, strategc complementarty, pure strategy equlbrum, sotone strateges, multunt auctons, unform prce aucton 1. ntroducton monotone methods have proven to be powerful n the study of games wth strategc complementarty. For example, Mlgrom and Roberts (1990) and Vves (1990) show that supermodular games possess several useful propertes, ncludng exstence of pure strategy equlbrum, monotone comparatve statcs on equlbrum sets, and concdence of the predctons of varous soluton concepts such as Nash equlbrum, correlated equlbrum, and ratonalzablty. Mlgrom and Shannon (1994) generalze these results to games wth strategc complementarty ncludng, as Athey (1998) shows, logsupermodular games wth afflated types. Ths paper adds to ths lterature by provdng suffcent condtons for exstence of monotone pure strategy equlbrum n games of ncomplete nfor- 1

2 maton n whch players have multdmensonal actons and multdmensonal types. A player s pure strategy s monotone (techncally sotone ) when hs acton s non-decreasng along every dmenson of hs acton space as hs type ncreases along any dmenson of hs type space. The suffcent condtons for these exstence results are satsfed n the two most wdely studed sorts of games wth strategc complementarty, supermodular games and logsupermodular games, gven afflated types. Isotoncty s mportant snce t often provdes testable emprcal mplcatons. For nstance, n the Cournot wth advertsng example dscussed n Secton 2, lower producton and advertsng costs are each assocated wth (weakly) hgher sales and advertsng levels. Ths paper departs from the usual strategc complements framework, however, and consders a broad class of games n whch only some of the requrements of strategc complementarty are satsfed. For nstance, Mlgrom and Shannon (1994) requre that each player s expected payoff functon must satsfy sngle-crossng n own acton and others actons (nformally, complementarty across actons ) and quassupermodularty wthn own acton (nformally, complementarty wthn own acton ). Ths paper extends a new approach poneered by Athey (2001) to develop monotone methods that apply to games of ncomplete nformaton whch may fal to exhbt complementarty across actons but n whch ncremental expected payoffs to hgher actons satsfy sngle-crossng n own type (nformally, monotone ncremental returns n own type ) when others adopt monotone strateges. Mlgrom and Shannon (1994) do not requre monotone ncremental returns n own type to prove exstence of a pure strategy equlbrum but, naturally, they can not guarantee exstence of an sotone equlbrum. In a settng wth fntely many one dmensonal actons and atomless one dmensonal types, Athey (2001) shows that a non-decreasng pure strategy equlbrum exsts when each player s nterm expected payoff satsfes monotone ncremental returns n own type gven any non-decreasng strateges by others. Ths paper generalzes her result n a settng wth multdmensonal actons and multdmensonal types, showng that an sotone pure strategy equlbrum exsts when each player s nterm expected payoff satsfes complementarty n own acton and monotone ncremental returns n own type gven any sotone strateges by others. Ths result extends to games wth a contnuum acton space whenever each player s ex post payoff s also contnuous n hs and others actons, just as Athey (2001) s results extend to ths case. 2

3 The rest of the paper s organzed as follows. Secton 2 llustrates the man exstence result by applyng t to games wth a contnuum acton space and dfferentable payoffs. Secton 3 lays out the basc model of ncomplete nformaton games wth fnte acton spaces and atomless types whle Secton 4 states the man theorem and gves three sets of suffcent prmtve condtons. Secton 5 apples the man theorem to provde the frst general pure strategy equlbrum exstence result for the unform prce aucton when bdders have non-prvate values and ndependent types. Secton 6 explores the heart of the contrbuton whle the Appendx provdes proofs. 2. llustraton gven dfferentable payoffs Consder an ncomplete nformaton game n whch n players each receve a sgnal t = (t 1,..., t h ) [0, 1] h and choose an acton a = (a 1,..., a k ) [0, 1] k. Defne each player s nterm expected payoff functon π nt gven others pure strateges a ( ) as π nt (a, t ; a ( )) (a, a (t ), t, t )f(t t )dt π post [0,1] h(n 1) where π post s hs ex post payoff and f( t ) s the condtonal p.d.f. of others types gven that player s type s t. Suppose also that π post (a, t), f(t t ) are smooth functons of a, t and of t, respectvely so that π nt s dfferentable n a, t. (Bold notaton s used to refer to vectors of all players actons and types.) A specalzed verson of Corollary 1 of the man theorem stated on page 18 apples to ths class of games. Corollary: Suppose that, for each bdder = 1,..., n and all actons a, types t, and sotone strategy profles a ( ) of others, (1) 2 π nt a j 1 a j 2 (, a j1,j2, t ; a ( )) 0 (1 j 1 < j 2 k) 2 π nt (2) a j tm Then an sotone pure strategy equlbrum exsts. 2 (, a j, t m ; a ( )) 0 (1 j k, 1 m h) For llustraton purposes t s smplest to consder examples n whch player types are ndependent, snce then the cross-partal nequaltes (1, 2) on expected payoffs are mpled drectly by the correspondng cross-partal nequaltes on ex post payoffs. 3

4 Example (Cournot wth 2 advertsng channels, n frms): Consder an undfferentated product Cournot competton game n whch n rsk neutral frms each choose a quantty q and levels of two sorts of advertsng e 1, e 2 to expand the sze of the total market. In the pharmaceutcal context, for example, drug companes advertse to patents through meda advertsng and to doctors through detalng (such as offce vsts from company reps). Frms also receve multdmensonal ndependent prvate nformaton t, where hgher own type mples (weakly) lower own advertsng and producton costs. In partcular, suppose that () D (p; e) = D(p) + γ(e) s total demand, () φ (e, t) s frm s advertsng cost functon, and () c (q ; t) s frm s producton cost functon, 3 where q, e, t refer to vectors of all frms quanttes, advertsng levels, and types. Frm s ex post payoff s π post (q, e, t) = π (q, e, t) q p (q, e) c (q ; t) φ (e, t) where p (q, e) s the market clearng prce. If there were just one advertsng channel, an sotone pure strategy equlbrum would always exst n ths example snce 2 π q e 0, 2 π q t 0, 2 π e t 0. Gven two advertsng channels, smlarly, an sotone equlbrum exsts as long as 2 π 0. Note though that exstence even of a unque sotone equlbrum does not provde the bass for monotone comparatve statcs. For e 1 e2 example, suppose that a change n the tax code lowers all frms producton costs. In the new sotone equlbrum, some frms may produce and/or advertse less than they dd n the orgnal equlbrum. 3. model: ncomplete nformaton games Ths Secton lays out the model of ncomplete nformaton games wth atomless types and fnte acton spaces Actons and Lattces Defnton (, ): Let (L, ) be a partally ordered set and let S L. The least upper bound of S, S, s the unque element of L (f t exsts) satsfyng S c a c for all a S and all c L. The greatest lower bound of S, S, s the unque element of L satsfyng S c a c for 4

5 all a S and all c L. When S = {a, b}, I use the standard notaton a b and a b. Defnton (Lattce, Sublattce, Complete): A lattce (L,,, ) s a partally ordered set (L, ) such that a b, a b L for all a, b L. L 1 L s a sublattce of L f and only f a b, a b L 1 for all a, b L 1. L 1 s complete f and only f S, S L 1 for every subset S L 1. Every fnte sublattce s complete (Brkhoff (1967)). Assumpton 1: Each player = 1,..., n has a common acton set L R k that s a fnte sublattce of k dmensonal Eucldean space wth respect to the product order on R k. 4 ) ( ) a m, a m L for m = 1,..., k. A A typcal acton s a ( a 1,..., a k typcal acton profle s a (a 1,..., a n ) (a, a ) Π n =1L. Smlar subscrpt, superscrpt, and bold notaton wll be used consstently throughout the paper to refer to types and strateges as well as actons. For each m = 1,..., k, defne L m { a m R : ( a m, a m ) L for some a m R k 1}. By defnton, L Π k m=1l m, though I do not assume that L = Π k m=1l m. Wthout loss, let L m = {0, 1,..., L m 1} Types and Strateges Assumpton 2: Player s type t s drawn from common support T = [0, 1] h. f : R nh R + +, the jont densty on type profles (or states) t = (t 1,..., t n ), s bounded above by K and bounded below by K > 0. 5 The type space s endowed wth the product order and the usual Eucldean topology and measure. Defnton (Pure strategy, Isotone pure strategy): A pure strategy (PS) a ( ) : T L s a measurable functon mappng each type nto an acton a (t ). In an sotone pure strategy (IPS), t > t mples a (t ) a (t ). S denotes the space of all of player s PS, S = Π j S j the space of others PS profles, and S = Π n =1S the space of full PS profles. Smlarly, I, I, and I are the spaces of own IPS, others IPS profles, and full IPS profles. 5

6 3.3. Payoffs Gven a profle of actons a and types t, player s ex post payoff (or utlty) s Π post (a, t). Assumpton 3: Π post s bounded and measurable. Interm expected payoff Π nt (, ; ) : L T S R, smlarly, depends on hs own acton, own type, and others strateges: Π nt (a, t ; a ( )) = E [ ] t t Π post (a, a (t ), t) t. For the most part, I restrct attenton to settngs n whch others follow IPS, a ( ) I. Defnton (Quassupermodular n x (or QSP M(x))): Let (L,,, ) be a lattce. g : L R s quassupermodular n x f and only f g(x ) (>)g(x x) g(x x) (>) g(x) for all x, x L. (Weak nequalty mples weak nequalty and strct nequalty mples strct.) Assumpton 4: Π nt (a, t ; a ( )) satsfes QSP M(a ) for all t T and all a ( ) I. Defnton (Sngle-crossng n (x, t) (or SC(x, t)) and n t (or SC(t))): Let (L,,, ) be a lattce and (T, ) a partally ordered set. g : L T R satsfes sngle-crossng n (x, t) f and only f g(x, t) (>)g(x, t) g(x, t ) (>) g(x, t ) for all x > x L and all t > t T. Smlarly, g : T R satsfes sngle-crossng n t f and only f for all t > t. g(t) (>)0 g(t ) (>) 0 Assumpton 5: Π nt satsfyng SC(a, t ) for all a ( ) I. (Ths s equvalent to Π nt (a, t ; a ( )) Π nt (a, t ; a ( )), the ncremental expected payoff to a versus a, satsfyng SC(t ) for all a > a and a ( ) I.) 6

7 3.4. Best Response and Equlbrum Let BR (t, a ( )) arg max a L Π nt (a, t ; a ( )) denote player s best response acton set when others follow pure strateges a ( ). When t can not cause confuson, I smplfy ths notaton to BR (t ). Defnton (Isotone pure strategy equlbrum): a ( ) S s a pure strategy equlbrum (PSE) f and only f a (t ) BR ( t, a ( ) ) for all, t. Any PSE a ( ) I s called an sotone pure strategy equlbrum (IPSE). 4. exstence of sotone equlbrum Theorem 1: Under assumptons 1-5, an IPSE exsts n games of ncomplete nformaton. The proof s n the Appendx. A straghtforward extenson n whch actons sets L = [0, 1] k and ex post payoffs are contnuous n actons a s also provded n the Appendx Suffcent prmtve condtons I gather here three sets of prmtve condtons that others work proves are suffcent for nterm expected payoff to satsfy quassupermodularty n own acton and sngle-crossng n own acton and type (Assumptons 4,5). I refer the reader to ths other work for the formal defntons of such standard terms as afflated, supermodular, log-supermodular, and non-decreasng dfferences. 1. Types are afflated and Π post (a, t) s supermodular n (a, t j ) for all j. In ths case, Athey (2002) proves that Π nt (a, t ; a ( )) s supermodular n (a, t ) when a ( ) I. 2. Types are afflated and Π post (a, t) s log-supermodular n (a, t). In ths case, Athey (1998) proves that Π nt (a, t ; a ( )) s log-supermodular n (a, t ) when a ( ) I. 3. Types are ndependent and Π post (a, t) s supermodular n a wth nondecreasng dfferences n (a, t ). Then expected payoff Π nt (a, t ; a ( )) s supermodular n a and has non-decreasng dfferences n (a, t ) when a ( ) S. (Others may follow any strateges.) See Topks (1979). 7

8 In Mlgrom and Roberts (1990) and Vves (1990), a supermodular game s one n whch Π post (a, t) s supermodular n a, wth no condtons placed on the dstrbuton of types. Thus, the prmtve condtons of cases 1,2 are only satsfed n a subclass of supermodular (and log-supermodular) games. Ths stands to reason, of course, snce I prove that an sotone PSE exsts whereas Mlgrom and Roberts (1990) and Vves (1990) only prove exstence of PSE. Case 3 makes the very strong requrement of ndependence but allows that players payoffs may fal to exhbt complementarty across actons. If ndependence s replaced by afflaton n case 3, an sotone equlbrum may not exst. For example, McAdams (2002) provdes a unform prce aucton example wth afflated prvate values n whch all case 3 requrements on ex post payoffs are satsfed but n whch some bdders reduce ther bds on all unts as ther values ncrease n all equlbra. 5. example: unform prce aucton Provng exstence of PSE n the unform prce aucton wth mult-unt demand s partcularly challengng snce payoffs fal to satsfy strategc complementarty (Mlgrom and Shannon (1994) does not apply) and fal to satsfy dagonal quasconcavty (Reny (1999) 6 does not apply). Indeed, the only general PSE exstence theorems that I am aware of that apply to the unform prce aucton requre prvate values. Jackson and Swnkels (2001) prove exstence of PSE wth postve probablty of trade n two sded 7 or one sded unform prce auctons gven prvate values and a very general correlaton structure. Bresky (2000) proves exstence of IPSE gven ndependent prvate values. Lke Bresky (2000), my applcaton of Theorem 1 requres ndependent types but proves IPSE exstence n a settng that allows for a much more general structure of values. model: n bdders and S dentcal objects (or unts) for sale. Informaton and Payoffs: Bdder receves value V (q, t) from the allocaton q = (q 1,..., q n ) n the state t = (t 1,..., t n ), where t are..d. wth common support [0, 1] h. V s pecewse contnuous n t and V (q, t) V (q, t) s non-decreasng n t whenever q > q and q j q j for all j. (No other assumptons on values.) Bdders seek to maxmze expected surplus, the dfference between ther value and payment. 8

9 Bds: A permssble bd s a vector b = (b (1),..., b (S)) such that b (q 1 ) b (q 2 ) when q 1 < q 2 and b (1),..., b (S) {p mn, p mn + 1,..., p max }. Allocaton: Let b k (b( )) (shorthand b k ) be the kth hghest unt-bd across all bd schedules. Defne q max{q : b (q) > b S } and q max{q : b (q) b S }. q (q ) s the least (greatest) quantty that bdder can receve n any market clearng allocaton. n =1 q S n =1 q and quantty s ratoned n the followng manner: 8 Each bdder s assgned at least q and randomly ordered nto a rankng ρ to raton the remanng quantty r S n =1 q. If r = 0, stop. Else the frst bdder n order, 1 = ρ(1), receves q 1 = q 1 + mn{q 1 q 1, r}. Decrement r by q 1 q 1 and repeat ths process wth bdder 2 = ρ(2) and so on untl all quantty has been assgned. Payment: A varety of unform prce payment rules have been consdered n the lterature. I study here the two most common: n the Sth (or S + 1st) prce auctons, all bdders pay the lowest wnnng bd b S (or hghest losng bd b S+1 ) on all unts that they wn,.e. total payment Z = q b S (or = q b S+1 ). Several features of the model are worthy of note: 1. The formulaton of bdder values ncludes as a specal case the benchmark nterdependent values n whch bdder s value for q unts takes the form V (q, t) and all ncremental values V (q, t) V (q, t) (for q > q ) are typcally assumed to be non-decreasng n t and strctly ncreasng n t. 2. Some sorts of externaltes are permtted. Values take the form V (q, t), wth the only monotoncty restrcton beng that ncremental values V (q, t) V (q, t) are non-decreasng n own type whenever q > q and q j q j for all j. In other words, bdders may care about what other bdders wn wth the caveat that own margnal values are nondecreasng n own type. (Such a monotoncty assumpton s present, for nstance, n Jehel, Moldavanu, and Stacchett (1996).) 3. Bdders receve multdmensonal prvate nformaton and values need not be strctly ncreasng n own type. For example, part of a bdder s nformaton may be relevant to own values gven certan allocatons but not gven other allocatons. 4. Values need not be monotone at all n others prvate nformaton. 9

10 5. Margnal values may be ncreasng n own quantty, allowng for ncreasng returns to scale n consumpton. On the other hand, I contnue to make the standard requrement that bds be non-ncreasng n quantty to guarantee exstence of a market clearng allocaton. 9 Theorem 2: An IPSE exsts n ths model of the unform prce aucton. Proof sketch: The set of all non-ncreasng bd schedules forms a lattce wth respect to the product order. Thus, t suffces to check that Assumptons 4,5 are satsfed. In fact, I prove n the Appendx that two stronger condtons hold: () expected payoff Π nt (, t ; b ( ; )) s modular n own bd (see below) for all types t and all profles of others strateges (sotone or not) and () Π nt (, ; b ( ; )) has non-decreasng dfferences n own bd and type for all profles of others strateges. 10 Non-decreasng dfferences (NDD): The ntuton behnd NDD of expected payoffs s clear. Frst, note that ex post payment has zero dfferences n own bd and type snce payment does not depend on type. Next, ex post values have NDD snce submttng a hgher bd (holdng others bds fxed) always leads one to wn weakly greater quantty and others to wn weakly less quantty. And by assumpton ex post ncremental value from such a change n the allocaton s non-decreasng n own type. Fnally, NDD s preserved under ntegraton 11 so expected payoffs have NDD no matter what strateges others follow. (Independence s crucal n ths step.) Defnton (Modular (or Addtvely Separable)): Let (L,,, ) be a lattce. g : L R s modular (or addtvely separable) n x f and only f for all x, x L. g(x x) + g(x x) = g(x ) + g(x) Modularty: Consder two bd schedules, b 1 ( ) and b 2 ( ). Ther least upper bound b 1 2 ( ) and greatest lower bound b 1 2 ( ) are ther upper and lower envelopes, respectvely. Gven any state t, profle of others bds b ( ; t ) and ratonng orderng ρ, the aucton mechansm maps each of these four bds nto four allocatons (shorthand q 1 q (b 1 ( ), b ( ; t ), ρ) and so on) and nto four unform prces (shorthand p 1 p (b 1 ( ), b ( ; t )) and so on). The key step of the modularty proof s to show that {(q 1, p 1 ), (q 2, p 2 )} = {(q 1 2, p 1 2 ), (q 1 2, p 1 2 )}. 10

11 prce p 2 p 1 S j b j(, t j ) b 2 ( ) b 1 ( ) q 1 q 2 quantty Fgure 1: Modularty of ex post payoff gven sgnals t Fgure 1 llustrates why ths s true n a specal case n whch () bds are made over a contnuum of quanttes and prces and () bd schedules are contnuous and strctly downward slopng. Smplfcaton () elmnates all subtletes that arse from the detals of the prcng and/or ratonng rules, but stll the Fgure conveys the basc dea. Two bd schedules for bdder, b 1 ( ) and b 2 ( ), are labelled whle ther upper envelope b 1 2 ( ) s traced wth open crcles. Prce and bdder s quantty are determned by where s bd schedule crosses the resdual supply schedule. In ths case, prce equals p 1 and bdder gets quantty q 1 whether he submts b 1 ( ) or b 1 2. Furthermore, t s easy to observe that all other bdders receve the same quantty after ether bd as well. Smlarly, prce and the allocaton are dentcal whether bdder submts b 2 ( ) or b 1 2 ( ). In other words, the par of outcomes (allocaton and prce) of the aucton after submttng bds b 1 ( ), b 2 ( ) s dentcal to those after b 1 2 ( ), b 1 2 ( ). Consequently, bdder s ex post surplus also matches up n ths sense. Thus the expectaton of any functon of ex post surplus, taken wth respect to any dstrbuton over types, wll tself be modular. In partcular, as long as bdder utlty takes the form u (V, Z ), then expected utlty wll be modular n own bd regardless of the type dstrbuton. 11

12 6. heart of the contrbuton Theorem 1, the paper s man result, s essentally a corollary of the powerful Monotoncty Theorem of Mlgrom and Shannon (1994) (hereafter MS). Indeed, n my vew, the man contrbuton of ths paper s to uncover the structure possessed by arg max x g(x, t) when x and t are multdmensonal and g satsfes the condtons of the Monotoncty Theorem. Ths structure n turn happens to be exactly what s requred to extend Athey (2001) s ngenous approach to provng exstence of monotone pure strategy equlbrum to a settng wth multdmensonal actons and multdmensonal types. It seems worthwhle, then, to dscuss these pror contrbutons and thereby ndcate what I feel s at the heart of my contrbuton Monotoncty Theorem Frst, I state a weakened verson of the Monotoncty Theorem, for whch two more defntons are needed. 12 Defnton (Strong set order): Let (L,,, ) be a lattce. The strong set order L s a partal orderng on P(L), the space of subsets of L. For A, A L, A L A f and only f a A, a A mples that a a A, a a A. Defnton (Increasng n the strong set order): Let (L,,, ) be a lattce and (T, ) a partally ordered set. A correspondence g : T P(L) s ncreasng n the strong set order f and only f g(t ) L g(t) whenever t > t. Theorem (Mlgrom and Shannon (1994)): Let g : L T R, where (L,,, ) s a complete lattce and (T, ) a partally ordered set. Then arg max x L g(x, t) s a complete sublattce for all t and ncreasng n the strong set order f g satsfes QSP M(x) and SC(x, t). Gven the Monotoncty Theorem, t s not surprsng that Assumptons 4 and 5 of my model (QSPM(own acton) and SC(own acton, own type) of expected payoffs whenever others follow sotone strateges) are assocated wth a result provng exstence of IPSE. These condtons guarantee that each player always has an sotone pure best response strategy whenever others follow sotone pure strateges: For a gven profle of others sotone pure strateges a ( ), Assumptons 4,5 and the Monotoncty Theorem mply that player s set of best response actons, BR (t, a ( )), s a complete sublattce 12

13 for all types t and that BR (, a ( )) s ncreasng n the strong set order. Snce acton sets are fnte, also, these sets are non-empty. Consequently, an sotone selecton exsts from BR (, a ( )) Athey s Vector Representaton Of course, exstence of an sotone best response s far from guaranteeng sotone equlbrum. At the heart of the exstence result s an extenson of Athey (2001) s remarkable proof that there s a sense n whch each bdder s set of sotone pure best response strateges s convex. Ths convexty then s used to apply Glcksberg (1952) s Fxed Pont Theorem to a best response correspondence whose doman and range are restrcted to the set of IPS profles. To be more precse, gven one dmensonal fnte acton sets (say {0, 1, 2,..., z}) and one dmensonal atomless types (say drawn from [0, 1]), Athey observes that any non-decreasng strategy can be dentfed, up to the actons played by a zero measure set of types, wth an z dmensonal non-decreasng vector of types (perhaps wth repetton) at whch the player ncreases hs acton. For nstance, when z = 3, the set of all strateges a : [0, 1] {0, 1, 2, 3} such that a (t ) = 0 for t < 1/2, a (t ) = 2 for t (1/2, 3/4), and a (t ) = 3 for t > 3/4 gets mapped to the vector (1/2, 1/2, 3/4). Say that two sotone strateges a ( ), a ( ) are equvalent f and only f Pr t (a (t ) = a (t )) = 1. It s easy to see that each such equvalence class of sotone strateges maps to a dfferent vector, and that the range of ths bjecton s a compact, convex subset of R z. Furthermore, ths mappng s a homeomorphsm wth respect to the usual Eucldean topology on R z and the topology on strateges correspondng to the metrc a ( ) a ( ) = Pr t (a (t ) a (t )). An mportant property of ths topology s that each bdder s expected payoff s contnuous n others strateges 13 whenever payoffs are bounded. (See Athey (2001).) 6.3. Convexty of Isotone Best Response Strateges Why s the mage of player s set of sotone pure best response strateges under the Athey map a convex subset of R z? Take as known that hs expected payoffs (gven others strateges a ( )) satsfy the requrements of the Monotoncty Theorem. Then the fact that hs best response acton set s ncreasng n the strong set order mples that a BR(t ) for all t < t < t whenever a BR(t ) BR(t ). Now, convexty of the mage of 13

14 the sotone best responses s clear. For example, when z = 2, suppose that w 1 = (1/2, 3/4) and w 2 = (0, 1/4) both correspond to sotone best response strateges; a convex combnaton of these vectors s w 1 /2+w 2 /2 = (1/4, 1/2). Revealed preference mples drectly that all types n [0, 1/2] fnd 0 to be a best response, all n [0, 1/4) (1/2, 3/4) fnd 1 a best response, and all n (1/4, 1] fnd 2 a best response. To conclude that w 1 /2 + w 2 /2 corresponds to an sotone best response, however, we also need to know that types n (1/4, 1/2) fnd 1 to be a best response. But the Monotoncty Theorem mples ndrectly that all types n [0, 3/4) fnd 1 to be a best response. It s easy to see that the same logc apples to all convex combnatons of any two sotone pure best response strateges. (See Athey (2001).) Indeed, ths approach apples as well to settngs wth multdmensonal types and multdmensonal actons. Surprsngly, the extenson to multdmensonal types s relatvely straghtforward whle that to multdmensonal actons s much more subtle and dffcult. Multdmensonal types, one dmensonal actons: When a player follows an IPS, hs type space s dvded nto regons n whch each acton s played such that no type n the a -regon s less than any type n the a -regon whenever a > a. To represent player s strategy as a vector, I partton player s type space nto many one dmensonal subsets of the form C(t 1 ) = [0, 1] {t 1 }. One may characterze the strategy over C(t 1 ), up to what acton s chosen by fntely many types, by a fnte dmensonal vector exactly as n Athey. And the vector of such vectors characterzes the strategy over the whole type space, agan up to a zero measure set of types. After care s taken n properly defnng the relevant topologes, I show n the Appendx that the nduced bjecton remans a homeomorphsm between equvalence classes of strateges and a compact, convex subset of a convex lnear topologcal space. A convex combnaton of elements n the mage of the Athey map corresponds to takng the convex combnaton, lne by lne for each C(t 1 ), of the boundares between the type regons who play each acton. For example, Fgures 2 and 3 llustrate a convex combnaton of two strateges when T = [0, 1] 2 and L = {0, 1, 2}. The number 0,1,2 n each regon of the type space s the acton played by types n that regon. Suppose that IPS correspondng to the two vectors llustrated n Fgure 2 are both best responses. Revealed preference drectly mples that all types play a best response acton n any strategy correspondng to the convex combnaton vector llustrated n Fgure 3 except for the upper left part of the type regon playng acton 1. On the other hand, 14

15 note that for every type t n the nteror of ths 1-regon there s a par of types {t, t } contaned wthn the unon of the 1-regons correspondng to the two orgnal strateges such that t < t < t. Thus, agan, the fact that BR (t, a ( )) s ncreasng n the strong set order mples that every such type t must fnd 1 to be a best response snce both t, t do. Note that ths verfcaton of convexty, lke Athey s, does not at all leverage the fact that BR (t, a ( )) s a lattce. t 1 2 t t t 1 Fgure 2: Two sotone strateges t 1 Fgure 3: Convex combnaton t 1 Multdmensonal actons, one dmensonal types: Whle the extenson of Athey (2001) s exstence result to multdmensonal types s relatvely straghtforward once vewed n the approprate lght, the generalzaton to multdmensonal actons s remarkably subtle and complex. Defnng a homeomorphsm from the space of strateges to a convex, compact subset of a vector space s the relatvely easy part. Consder the set of projectons of an acton onto each dmenson of the acton space. (Under projecton φ m : L L m, φ(a ) a m ts mth coordnate.) Each strategy a ( ) s characterzed by ts projectons {a m ( ) : m = 1,..., k}, where a m (t ) φ m (a (t )) for all types. Furthermore, a pure strategy a ( ) s sotone f and only f each a m ( ) s nondecreasng. We may therefore represent any sotone strategy as a vector of k vectors, each of whch characterzes an sotone functon mappng types nto a one dmensonal acton space, as n the prevous case. The subtle and dffcult part s provng that the mage of the sotone pure best response strateges s convex. An example hghlghts some of the ssues nvolved. Example: L = {0, 1, 2} {0, 1} {0, 1, 2}, T = [0, 1]. Let a ( ) be a 15

16 gven profle of others IPS. Consder two best response IPS of player : a (t ) = (0, 0, 1) for all t [0, 1/2) = (1, 1, 2) for all t [1/2, 1] a (t ) = (2, 0, 0) for all t [0, 1/2) = (2, 1, 0) for all t [1/2, 3/4) = (2, 1, 1) for all t [3/4, 1]. a ( ) ((1/2, 1), 1/2, (0, 1/2)) = w 1 and a ( ) ((0, 0), 1/2, (3/4, 1)) = w 2 under the Athey map. w 1 /2 + w 2 /2 = ((1/4, 1/2), 1/2, (3/8, 3/4)) s a convex combnaton of these vectors and maps back to an equvalence class of strateges havng representatve member a (t ; w 1 /2 + w 2 /2) = (0, 0, 0) for all t [0, 1/4) = (1, 0, 0) for all t [1/4, 3/8) = (1, 0, 1) for all t [3/8, 1/2) = (2, 1, 1) for all t [1/2, 3/4) = (2, 1, 2) for all t [3/4, 1]. Several new actons are played n the strategy a (t ; w 1 /2 + w 2 /2) and no type t plays an acton that he played under ether orgnal strategy. Thus, revealed preference tells us nothng about whether the new strategy s a best response. Even the fact that the set of types who fnd each acton to be a best response s convex does not help us at all. Rather, to conclude that each type plays a best response acton, one must repeatedly apply both the fact that the best response acton set s a lattce and that t s ncreasng n the strong set order. For example, consder a type t (3/8, 1/2). By the lattce property, type t fnds (2, 0, 1) = (0, 0, 1) (2, 0, 0) to be a best response whereas types t (1/2, 3/4) (greater than t ) fnd (1, 1, 0) = (1, 1, 2) (2, 1, 0) to be a best response. Now we can use ncreasngness n the strong set order to conclude that (2, 0, 1) (1, 1, 0) = (1, 0, 0) BR (t ). Fnally, agan usng the lattce property, (1, 0, 0) (0, 0, 1) = (1, 0, 1) BR (t ) and we are done. The proof for the general case s very smlar. For each type t, I prove by nducton that the requred acton a = (a 1,..., a k ) BR (t ): BR (t ) contans an element whose frst coordnate equals a 1. Then, gven that BR (t ) contans an element whose frst j coordnates equal (a 1,..., a j ), BR (t ) contans an element whose frst j + 1 coordnates equal (a 1,..., a j+1 ). 16

17 7. concluson Ths paper shows how two non-prmtve condtons, quassupermodularty n own acton and sngle-crossng n own acton and type of nterm expected payoff whenever others follow sotone strateges, are suffcent for exstence of an sotone pure strategy equlbrum n a very general settng wth fntely many multdmensonal actons and a contnuum of multdmensonal types. Furthermore, these condtons are satsfed n a varety of mportant classes of games such as supermodular and log-supermodular games wth afflated types as well as n some games n whch strategc complementarty fals. For nstance, as an applcaton of the man theorem, I provde the frst proof of equlbrum exstence n pure strateges (ndeed, sotone pure strateges) when bdders have non-prvate values and ndependent types n the unform prce aucton. Sloan School of Management, MIT, 50 Memoral Drve, Cambrdge, MA 02142, U.S.A.; mcadams@mt.edu, mcadams. 17

18 APPENDIX Assumpton 1 : Player s acton set s [0, 1] k. Assumpton 6: Π post (a, t) s contnuous n a for all t. Corollary 1: Under assumptons 1,2-6, an IPSE exsts n games of ncomplete nformaton. Proof. The proof closely follows that of Theorem 2 n Athey (2001), and I refer to the reader to ths proof for most detals. The only potentally substantal dfference s that each player s acton s multdmensonal, so one must argue that any sequence of IPS profles a j ( ) n a sequence of games havng fner and fner acton spaces has a subsequence that converges to an IPS profle a ( ) n the lmtng game havng a contnuum acton space. 14 But t s straghtforward to apply Helly s Selecton Theorem to the sequences a m j ( ) separately, each of whch has a subsequence convergng to a m ( ). Proof of Theorem 1 Athey map: The Athey map A sends each IPS a ( ) I to a vector, where A :I A a ( ) ( A (a ( ); m, j, t 1 A (a ( ); m, j, t 1 ) ) m=1,...,k,j L m,t 1 [0,1] h 1 ) sup{t 1 [0, 1] : a m (t 1, t 1 ) < j}. (a m ( ) : [0, 1] h L m was defned on page 15.) To avod a notatonal mess, I wll often refer to a typcal element n the range of the Athey map, ( A (a ( ); m, j, t 1 ) ) m=1,...,k,j L m,t 1 [0,1] h 1, smply as A (a ( )) or even more smply as A f there can be no confuson wth the map tself. Bjecton between equvalence classes: Two strateges a ( ), a ( ) are equvalent f and only f Pr t (a (t ) = a (t )) = 1. Two vectors A (a ( )), A (a ( )) n the range of the Athey map are equvalent f and only f (3) E t 1 [ max A (a m,j L m ( ); m, j, t 1 ) A (a ( ); m, j, t 1 ) ] = 0. 18

19 The Athey map nduces a bjecton between equvalence classes of IPS n I and equvalence classes of vectors n A. (When I refer to the Athey map from here on I mean to refer to ths nduced bjecton although for smplcty I wll use notaton as f the doman s I and the range A ). To see why, note frst by the model s assumptons on the dstrbuton of types there exst 0 < K K so that f (t 1 t 1 ) [K, K] for all t 1, t 1. Thus, A (a ( ); m, j, t 1 ) A (a ( ); m, j, t 1 ) = Pr t 1 t 1 ( a m (t 1, t 1 ) a m (t 1, t 1 ) ) [ K, K] for all t 1. Consequently, (3) holds f and only f Pr t (a (t ) = a (t )) = 1. Homeomorphsm: Indeed, Pr t (a,n (t ) = a, (t )) n 1 f and only f [ E t 1 max A (a,n ( ); m, j, t 1 ) A (a, ( ); m, j, t 1 ) ] n 0. m,j L m Thus, the Athey map s a homeomorphsm wth respect to the topologes on equvalence classes n I havng metrc d (a ( ), a ( )) Pr t (a (t ) a (t )) and on equvalence classes n A A (I ) havng metrc d (A, A ) E t 1 [ max A (a,n ( ); m, j, t 1 m,j L m ) A (a, ( ); m, j, t 1 ) Closed Range: By homeomorphsm, t suffces to show that any lmt pont of I s an element of I. So suppose that {a,n ( )} s a sequence of IPS convergng to a, ( ). Clearly, a, ( ) s sotone when restrcted to the set of types at whch t prescrbes exactly the same acton as a,n ( ) (for all n > N and any N ). By convergence of {a,n ( )}, then, a, ( ) must be sotone when restrcted to some full measure set of types. And any such strategy can be modfed on a zero measure set so that t becomes an IPS,.e. the equvalence class contanng a, ( ) ncludes an IPS. Compact Range: By Tychonoff s Theorem, 15 closedness mples that the range s compact wth respect to the topology of pontwse convergence. But the topology that I am usng s coarser than ths one, so the range must be compact wth respect to my topology as well. 16 Convex Range: Lemma 1 below characterzes the range of the Athey map. Ths range s convex snce (4, 5) are preserved under convex combnaton. That s to say, f A (a ( )), A (a ( )) each satsfy (4, 5), then so does αa (a ( ))+ (1 α)a (a ( )) for all α (0, 1). 19 ].

20 Lemma 1: a ( ) s an sotone strategy f and only f (4) (5) A (a ( ); m, j, t 1 ) A (a ( ); m, j, t 1 ) for all m, j > j L m, t 1 A (a ( ); m, j, t 1 ) A (a ( ); m, j, t 1 ) for all m, j L m, t 1 > t 1. Proof. a ( ) sotone f and only f a m ( ) non-decreasng for m = 1,..., k. : Suppose a m ( ) non-decreasng. Then (4, 5) hold for all j > j L m and all t 1 To prove (4), suppose otherwse that A (a ( ); m, j, t 1 ( ) t 1, t 1 j > j > a m ) < t 1 < A (a ( ); m, j, t 1 ). ( ) t 1, t 1 by defnton of the Athey In ths case, a m map, a contradcton. To prove (5), suppose otherwse that A (a ( ); m, j, t 1 ) < t 1 < A (a ( ); m, j, t 1 ). ( ) ( ) In ths case, smlarly, a m t 1, t 1 a m > a m t 1, t 1. But a m ( ) a m t 1, t 1 snce a m ( ) s non-decreasng, a contradcton. ( t 1, t 1 ) : Suppose that vector A satsfes (4, 5). Then a ( ) I exsts so that A = A (a ( )). Consder the pure strategy a ( ) defned as a m (t) max { } j L,m : A (m, j, t 1 ) t 1 for each m = 1,..., k. It s easy to verfy that A = A (a ( )) when a ( ) s so defned. Each such a m ( ) s non-decreasng: ) Let t 1 t 1 ) and t 1 t 1 and suppose otherwse frst that a m ( t 1, t 1 = j > a m ) ( t 1, t 1. By constructon ) of a m ( ), a m ( t 1, t 1 < j so that A (m, j, t 1 ) > t 1 and a m ( t 1, t 1 = j so that A (m, j, t 1 ) t 1, contradctng ( ) (4). Smlarly, ) A (m, j, t 1 ) t 1 and A (m, j, t 1 ) > t 1 when a m t 1, t 1 = j > a m ( t 1, t 1, contradctng (5). Closed Graph: Let A : I A and A : I A denote the composte Athey map takng IPS profles nto vector profles. For each IPS profle a ( ), let BR (a ( )) denote the set of player s best response IPS: a ( ) BR (a ( )) a ( ) I and a (t ) BR (t, a ( )) for all t. Recall from Secton 6.1 that BR s non-empty-valued. Furthermore, note that player s nterm expected payoffs are contnuous n a ( ) wth respect to our topology. Ths guarantees that BR has a closed graph. 20

21 Fxed Pont: Defne Λ A BR A 1 : A A whch maps the vector A (a ( )) correspondng to a profle of others IPS nto the set of vectors correspondng to player s IPS best responses. Fnally, defne Λ : A A by Λ (A(a( ))) = (Λ 1 (A 1 (a 1 ( ))),..., Λ n (A n (a n ( )))). The arguments presented so far mply that () A s a convex, compact subset of a convex topologcal lnear space (ndeed, of a vector space) and () Λ s non-empty-valued () wth a closed graph. All that remans to be able to nvoke Glcksberg Fxed Pont Theorem s that Λ (or each Λ ) s convexvalued. Ths s the most mportant techncal result n the paper, so I label t as a theorem. Theorem 3: Λ s convex-valued for all players. IPSE exsts: To complete the proof of Theorem 1 gven Theorem 3, I need to show that the equvalence class of strateges correspondng to any fxed pont of Λ contans an IPSE. Suppose that A(a ( )) s a fxed pont of Λ (for a ( ) I). Ths does not mply that a ( ) s an IPSE snce some zero measure set of types may be playng a non-best response. But a ( ) â( ) for some profle â( ) such that â ( ) BR (a ( )) for all and â( ) s an IPSE: Player always plays a best response under â ( ) and all others best responses do not change as modfes hs acton over a zero measure set of types. Proof of Theorem 2 Proof. By the dscusson n the text, t suffces to show that each bdder s ex post valuaton has NDD n own bd and own type and both hs ex post valuaton and ex post payment are modular n own bd. In the followng, I consder bdder 1 only and fx the profle of others bds b 1 ( ), the ratonng rankng ρ, and the state t. The analyss focuses on propertes of the realzed allocaton and payment when bdder 1 submts one of two bds b 1 ( ) or b 2 ( ) or ther jon b 1 2 ( ) b 1 ( ) b 2 ( ) or meet b 1 2 ( ) b 1 ( ) b 2 ( ). Shorthand notaton: q 1 j q j (b 1 ( ), b 1 ( ); ρ) and so on for the other bds b 2 ( ), b 1 2 ( ), and b 1 2 ( ). (Note that whle bdder 1 s bd vares, others bds are held fxed.) Smlarly, defne p 1 b S (b 1 ( ), b 1 ( )), p 1 b S+1 (b 1 ( ), b 1 ( )) and so on, where b S (b( )) and b S+1 (b( )) are the Sth and S +1st hghest unt bds gven the profle of schedules b( ). Lastly, defne bdder j s range of 21

22 demand at each prce p by mn D j (p) = max{q : b j (q) > p} and max D j (p) = max{q : b j (q) p}. For bdder 1, I wll use shorthand D 1 1(p) to refer to hs range of demand gven bd b 1 ( ) and so on for the other bds b 2 ( ), b 1 2 ( ), and b 1 2 ( ). Characterzng the allocaton: Defne bdder 1 s ratonng functon to be R ρ 1(p) S ρ(j)=ρ(1) 1 ρ(j)=1 max D j (p) ρ(j)=n ρ(j)=ρ(1)+1 mn D j (p). R ρ 1(p) s the amount that would be left for bdder 1 f all ahead of hm n the ratonng rankng ρ were gven ther maxmum demand at prce p and all behnd hm were gven ther mnmal demand at that prce. By desgn of the assumed ratonng rule, 17 or, equvalently, q 1 = mn D 1 (b S ) f R ρ 1(b S ) mn D 1 (b S ) = R ρ 1(b S ) f R ρ 1(b S ) [ mn D 1 (b S ), max D 1 (b S ) ] = max D 1 (b S ) f R ρ 1(b S ) max D 1 (b S ). q 1 = mn D 1 (b S+1 ) f R ρ 1(b S+1 ) mn D 1 (b S+1 ) = R ρ 1(b S+1 ) f R ρ 1(b S+1 ) [ mn D 1 (b S+1 ), max D 1 (b S+1 ) ] = max D 1 (b S+1 ) f R ρ 1(b S+1 ) max D 1 (b S+1 ). Both approaches, based on b S and on b S+1, lead to the same ratonng outcome because ratonng only occurs when b S = b S+1 : b S+1 (b( )) b S (b( )) mples that there s a unque market clearng allocaton, and both approaches lead to that allocaton. More explctly, (6) q 1(b( ); ρ) = max (7) = max { { }} mn D 1 (b S (b( ))), mn R1(b ρ S (b( ))), max D 1 (b S (b( ))) { { }} mn D 1 (b S+1 (b( ))), mn R1(b ρ S+1 b( )), max D 1 (b S+1 (b( ))). 22

23 NDD n own bd and own type: By (6), observe that q1(b 1 ( ), b 1 ( ); ρ) s non-decreasng and qj (b( ); ρ) (j ) non-ncreasng n own bd b 1 ( ) for any gven b 1 ( ), ρ. Thus, bdder s ex post valuaton for the allocaton, V 1 (q (b( ); ρ), t), has non-decreasng dfferences n own bd and type. Modularty n own bd: Let b 1 ( ), b 2 ( ) be two permssble bds. By defnton of the, operatons, mn{p 1, p 2 } = p 1 2 and max{p 1, p 2 } = p 1 2. Smlarly, mn{p 1, p 2 } = p 1 2 and max{p 1, p 2 } = p 1 2. Wthout loss, suppose that p 1 = p 1 2 p 2 = p 1 2 and q1 1 q1. 2 (It s straghtforward to see that p 1 < p 2 mples q1 1 q1.) 2 By the dscusson n the text (page 10) note that, to prove modularty of ex post valuatons and ex post payments n the Sth prce aucton, t suffces to show that q1 1 = q1 1 2 and q1 2 = q For ths result, frst, note that q1 1 = q1 2 mples q1 1 = q1 2 = q1 1 2 = q1 1 2 : By (6), q1 1 = q1 2 f and only f ether (A), (B), or (C) s satsfed. (A) max{mn D 1 (p 1 ), mn D 2 (p 2 )} R ρ 1(b 1 ( )) (B) max D 1 (p 1 ) = max D 2 (p 2 ) < R ρ 1(b 1 ( )), (C) mn D 1 (p 1 ) = mn D 2 (p 2 )} > R ρ 1(b 1 ( )). mn{max D 1 (p 1 ), max D 2 (p 2 )}, Further, f condton (A) holds for bds b 1 ( ), b 2 ( ), then the analogous condton must hold for bds b 1 2 ( ), b 1 2 ( ), and smlarly for condtons (B,C). So, wthout loss, suppose that q1 1 < q1. 2 Second, mn D 1 (p 1 ) = mn D 1 2 (p 1 ): Else mn D 1 (p 1 ) > mn D 1 2 (p 1 ), so that there exsts some q < mn D 1 (p 1 ) q1 1 such that b 2 (q) < b 1 (q). But ths would mply q1 2 q1, 1 a contradcton. Thrd, ether both max D 1 (p 1 ), max D 1 2 (p 1 ) R1(p ρ 1 ) or both R1(p ρ 1 ): Else max D 1 (p 1 ) > R1(p ρ 1 ) > max D 1 2 (p 1 ), so that there exsts q < R1(p ρ 1 ) q1 1 such that b 2 (q) < p 1. But ths would mply that max D 2 (p 1 ) R1(p ρ 1 ) mplyng (snce p 2 p 1 ) that q1 2 q1, 1 a contradcton. All together, by (6), I have proven now that q1 1 = q The proof that q1 2 = q1 1 2 s entrely analogous, the key steps beng to show that max D 2 (p 2 ) = max D 1 2 (p 2 ) and that mn D 2 (p 2 ), mn D 1 2 (p 2 ) R1(p ρ 2 ) or both R1(p ρ 2 ). Ths completes the proof for the Sth prce aucton. The proof for the S + 1st prce aucton s entrely analogous, when p s replaced wth p. 23

24 Proof of Theorem 3 Prelmnares: The type space has partton T = {C(t 1 ) [0, 1] {t 1 C(t 1 )} t 1 [0,1] h 1, where }. Let a ( ; α) be any sotone strategy n the equvalence class n the pre-mage of αa (a ( )) + (1 α)a (a ( )) wth respect to the Athey map. All equvalent strateges specfy the same acton for all but the zero measure set of types D at whch player s acton ncreases along some dmenson n strategy a ( ; α): D {t : t 1 = A (a ( ; α); m, j, t 1 ) for some m, j L m }. I need to prove only that, for all types t D, a (t ; α) s a best response gven that both a ( ), a ( ) are sotone best response strateges. What I wll show s even stronger: a (t ; α) s a best response gven only that the actons played by types n C(t 1 ) n strateges a ( ), a ( ) are all best response actons. Wthout loss, then, I may focus entrely on the one dmensonal set of types C(t 1 ) and, ndeed, drop all reference to t 1. Thus, subsequently, I wll treat the notatonally smpler case n whch T = [0, 1]: I wll drop all superscrpts, and any reference to the full set of player s types refers nstead to the one dmensonal subset C(t 1 ). For a gven type ˆt / D, defne the shorthand a (ˆt ; α) a(α) = ( a 1 (α),..., a k (α) ) and a j 1,...,j 2 (α) (a j 1 (α), a j1+1 (α),..., a j 2 (α)). (Subscrpts denotng player dentty are dropped when referrng to actons for smplcty. Ths should not cause confuson snce I only refer to player throughout the entre proof.) Part 1: In ths part of the proof, I dentfy structure on bdder s best response actons BR ( ) that suffces for the convexty concluson. (Ths structure s lad out here as Workng Assumptons.) The second part then proves that ths structure s present as long as the condtons of the Monotoncty Theorem are satsfed. Workng Assumpton 1: Player s type ˆt has best response actons a, a such that a a(α) a. Workng Assumpton 2: For each dmenson j = 1,..., k of the acton space, there exst types t j, tj such that () t j ˆt t j, () type tj has a best response acton à such that à 1,...,j a 1,...,j (α), à j+1 = a j+1 (α), and à j+2,...,k a j+2,...,k (α), and () type t j has a best response acton ă such that ă 1,...,j a 1,...,j (α), ă j+1 = a j+1 (α), and ă j+2,...,k a j+2,...,k (α). 24

25 ă á ä = 1 j j + 1 k ã = 1 j + 1 k = 1 j k ȧ = 1 j k = 1 j + 1 k à = 1 j j + 1 k t j ˆt t j Fgure 4: Illustraton of nducton step Gven these two workng assumptons, an nducton argument proves that a(α) s a best response acton for player s type ˆt,.e. a(α) BR (ˆt ). Base step (j = 0): a, a BR (ˆt ), where a a (α) a. ) Inducton step: Suppose ȧ, á BR (ˆt, where ȧ m = a m (α) = á m for m = 1,.., j and ȧ m a) m (α) á m for m = j + 1,.., k. Then we may conclude ä, ã BR (ˆt, where ä m = a m (α) = ã m for m = 1,..., j + 1 and ä m a m (α) ã m for m = j + 2,.., k. Base step s satsfed by Workng Assumpton 1. By Workng Assumpton 2 and the fact that BR ( ) s ncreasng n the strong set order, ä à á BR (ˆt ) and ã ă ȧ BR (ˆt ). (It s easly checked that ä 1,...,j+1 = a 1,...,j+1 (α) and ä j+2,...,k a j+2,...,k (α) as well as that ã 1,...,j+1 = a 1,...,j+1 (α) and ã j+2,...,k a j+2,...,k (α).) Ths notaton heavy step s llustrated n Fgure 4. The block from 1 to j s labelled n the ă box to represent the fact that ă 1,...,j a 1,...,j (α), and so on. The four actons ã, á, ȧ, ä BR (ˆt ) whereas ă BR (t j ) and à BR (t j ). Ths completes the nducton step and hence the proof of Theorem 3 gven the two Workng Assumptons. Part 2: Now I prove that Workng Assumptons 1, 2 are satsfed gven that BR ( ) s non-empty- and lattce-valued and ncreasng n the strong set order. Frst, I develop some needed machnery that apples to any fxed 25

26 dmenson m {1,..., k} of the acton space. Let BR m (t ) { a m L m : ( a m, a m) BR (t ) for some a m L m }. Frst pont: revealed preference. Gven that a ( ), a ( ) are best response strateges, revealed preference mples that a m (α) BR m (t ) for all types t who play an acton wth a m (α) as ts mth coordnate n ether strategy. Ths ncludes all types t nt ( S am (α) (a ( )) ) nt ( S am (α) (a ( )) ) where S am (α) (a ( )) [A (a ( ); m, a m (α)), A (a ( ); m, a m (α) + 1)] S am (α) (a ( )) [A (a ( ); m, a m (α)), A (a ( ); m, a m (α) + 1)]. S am (α) (a ( )) s the closure of the order nterval of types who play an acton wth mth coordnate a m (α) n the strategy a ( ). Smlarly, S am (α) (a ( )) contans types who play an acton wth mth coordnate a m (α) n the strategy a ( ). Defne shorthand H m [ˆt, 1 ] ( S am (α) (a ( )) S am (α) (a ( )) ) L m [ 0, ˆt ] ( S a m (α) (a ( )) S am (α) (a ( )) ). H m (L m ) s mnemonc for types that are Hgher (Lower) than ˆt that play an acton equal to a m (α) on the mth dmenson n ether strategy a ( ) or a ( ). (L m should not be confused wth the acton lattce L = Π k m=1l m.) Note that these sets are closed and that all types t n the nteror of H m L m have a best response acton whose mth coordnate equals a m (α). Second pont: reduce to 1/2-1/2 convex combnatons. The set of types t D such that a (t, α) = a m (α) s the nteror of the nterval S am (α) (â ( ; α)) αs am (α) (a ( )) + (1 α)s am (α) (a ( )) where ths s the usual convex combnaton of sets. In partcular, for any such type t, the acton a (t ; α) = a (t ; α) for all α n a neghborhood of α. Thus, I only need to prove that a (t ; α) BR (t ) for α belongng to a dense subset of [0, 1]. By an nducton argument, therefore, t suffces to prove that a (t ; 1/2) BR (t ) (.e. for α = 1/2). Thrd pont: some type has best response acton whose mth coordnate equals a m (1/2). Snce ˆt / D, one of the ntervals S am (1/2) (a ( )), S am (1/2) (a ( )) must have non-empty nteror. Thus, there must be some type t so that ether a m (t ) = a m (1/2) or a m (t ) = a m (1/2), mplyng that a m (1/2) BR m (t ). 26

27 Fourth pont: m propertes. Snce ˆt / D, ˆt nt ( S am (α) (a ( ; 1/2)) ) where S am (1/2) (a ( ; 1/2)) was defned n the frst pont. Thus, also has non-empty nteror. Defne W H m ( 2ˆt L m) m max W ˆt. In words, m s the maxmum length y such that ˆt y L m and ˆt +y H m. Key propertes of m nclude: 1. m > 0: Follows from the fact that W has non-empty nteror and mn W ˆt. (Ths fact wll be used n Part 2 when I argue that types ˆt m + ε and ˆt + m ε have a best response acton wth mth coordnate equal to a m (1/2).) 2. It can not be that both ˆt m = max L m and ˆt + m = mn H m : Otherwse, by defnton of m, one of the sets L m, H m must be a sngleton and ˆt / nt ( S am (α) (a ( ; 1/2)) ), a contradcton. (For example, f L m = { t m } and ˆt + m = mn H m, then ˆt = mn S am (α) (a ( ; 1/2)).) 3. max { a m (t ), a m (t ) } a m (1/2) for all t < ˆt m : Ths and the next facts follow mmedately from the defnton of m. 4. mn { a m (t ), a m (t ) } a m (1/2) for all t < ˆt + m. 5. max { a m (t ), a m (t ) } a m (1/2) for all t > ˆt m. 6. mn { a m (t ), a m (t ) } a m (1/2) for all t > ˆt + m. By property 2, ether ˆt m +ε L m or ˆt + m ε H m for small enough ε. Ths mples that ether max{a m mn{a m (ˆt m + ε ), a m (ˆt m + ε ) } = a m (1/2) or (ˆt + m ε ), a m (ˆt + m ε ) } = a m (1/2). 27