Uniform topologies on types


 Tracy Roberts
 1 years ago
 Views:
Transcription
1 Theoretcal Economcs 5 (00), / Unform topologes on types YChun Chen Department of Economcs, Natonal Unversty of Sngapore Alfredo D Tllo IGIER and Department of Economcs, Unverstà Lug Boccon Eduardo Fangold Department of Economcs, Yale Unversty Syang Xong Department of Economcs, Rce Unversty We study the robustness of nterm correlated ratonalzablty to perturbatons of hgherorder belefs. We ntroduce a new metrc topology on the unversal type space,called unformweak topology, under whch two types are close f they have smlar frstorder belefs, attach smlar probabltes to other players havng smlar frstorder belefs, and so on, where the degree of smlarty s unform over the levels of the belef herarchy. Ths topology generalzes the now classc noton of proxmty to common knowledge based on common pbelefs (Monderer and Samet 989). We show that convergence n the unformweak topology mples convergence n the unformstrategc topology (Dekel et al. 006). Moreover, when the lmt s a fnte type, unformweak convergence s also a necessary condton for convergence n the strategc topology. Fnally, we show that the set of fnte types s nowhere dense under the unform strategc topology. Thus, our results shed lght on the connecton between smlarty of belefs and smlarty of behavors n games. Keywords. Ratonalzablty, ncomplete nformaton, hgherorder belefs, strategc topology, Electronc Mal game. JEL classfcaton. C70, C7. YChun Chen: Alfredo D Tllo: Eduardo Fangold: Syang Xong: We are very grateful to a coedtor and three anonymous referees for ther comments and suggestons, whch greatly mproved ths paper. We also thank Perpaolo Battgall, Martn Crpps, Edde Dekel, Jeffrey C. Ely, Amanda Fredenberg, Drew Fudenberg, Qngmn Lu, George J. Malath, Stephen Morrs, Marcn Pesk, Dov Samet, Marcano Snscalch, Aaron Sojourner, Tomasz Strzaleck, Satoru Takahash, Jonathan Wensten, and Muhamet Yldz for ther nsghtful comments. Chen and Xong gratefully acknowledge fnancal support from the NSF (Grant SES ) and the Northwestern Unversty Economc Theory Center. Copyrght 00 YChun Chen, Alfredo D Tllo, Eduardo Fangold, and Syang Xong. Lcensed under the Creatve Commons AttrbutonNonCommercal Lcense 3.0. Avalable at DOI: 0.398/TE46
2 446 Chen, D Tllo, Fangold, and Xong Theoretcal Economcs 5 (00). Introducton The Bayesan analyss of ncomplete nformaton games requres the specfcaton of a type space, whch s a representaton of the players uncertanty about fundamentals, ther uncertanty about the other players uncertanty about fundamentals, and so on, ad nfntum. Thus the strategc outcomes of a Bayesan game may depend on entre nfnte herarches of belefs. Crtcally, n some games ths dependence can be very senstve at the tals of the herarches, so that a mspecfcaton of hgherorder belefs, even at arbtrarly hgh orders, can have a large mpact on the predctons of strategc behavor, as shown by the Electronc Mal game of Rubnsten (989). As a matter of fact, ths phenomenon s not specal to the EMal game. Recently, Wensten and Yldz (007) have shown that n any game satsfyng a certan payoff rchness condton, f a player has multple actons that are consstent wth nterm correlated ratonalzablty the soluton concept that embodes common knowledge of ratonalty then any of these actons can be made unquely ratonalzable by sutably perturbng the player s hgherorder belefs at any arbtrarly hgh order. Ths phenomenon rases a conceptual ssue: f predctons of strategc behavor are not robust to mspecfcaton of hgherorder belefs, then the common practce n appled analyss of modelng uncertanty usng small type spaces often fnte may gve rse to spurous predctons. A natural approach to study ths robustness problem s topologcal. Consder the correspondence that maps each type of player nto hs set of nterm correlated ratonalzable (ICR) actons. The fraglty of strategc behavor dentfed by Rubnsten (989) and Wensten and Yldz (007) can be recast as a certan knd of dscontnuty of the ICR correspondence n the product topology over herarches of belefs,.e., the topology of weak convergence of korder belefs, for each k. Whle n every game the ICR correspondence s upper hemcontnuous n the product topology, lower hemcontnuty can fal even for the strct ICR correspondence a refnement of ICR that requres the ncentve constrants to hold wth strct nequalty. Strctness rules out ncentves that hnge on a knfe edge, whch can always be destroyed by sutably perturbng the payoffs of the game. Indeed, nonstrct soluton concepts are known to fal lower hemcontnuty n other contexts, e.g., n complete nformaton games, Nash equlbrum, and, n fact, even bestreply correspondences fal to be lower hemcontnuous wth respect to payoff perturbatons. By contrast, the strct Nash equlbrum and the strct bestreply correspondences are lower hemcontnuous. It s, therefore, surprsng that ths form of contnuty breaks down when t comes to perturbatons of hgherorder belefs. There exst, of course, fner topologes under whch the ICR correspondence s upper hemcontnuous and the strct ICR correspondence s lower hemcontnuous n all games. The coarsest such topology s the strategc topology ntroduced by Dekel et al. (006); t embodes the mnmum restrctons on the class of admssble perturbatons of hgherorder belefs necessary to render ratonalzable behavor contnuous. Thus See Dekel et al. (007, Proposton ) and Battgall et al. (008, Theorem4). Here, the noton of strctness s actually qute strong: the slack n the ncentve constrants s requred to be bounded away from zero unformly on a bestreply set. Despte ths, the strct ICR correspondence fals to be lower hemcontnuous n the product topology.
3 Theoretcal Economcs 5 (00) Unform topologes on types 447 the strategc topology gves a tght measure of the robustness of strategc behavor: f the analyst consders any larger set of perturbatons, he s bound to make a nonrobust predcton n some game. Gven ths sgnfcance, we beleve the strategc topology deserves closer examnaton. Indeed, Dekel et al. (006) only defne t mplctly n terms of proxmty of behavor n games, as opposed to explctly usng some noton of proxmty of probablty measures. Ths leaves open the mportant queston as to what proxmty n the strategc topology means n terms of the belefs of the players. To address ths queston, we ntroduce a new metrc topology on types, called unformweak topology, under whch a sequence of types (t n ) n converges to a type t f the korder belef of t n weakly converges to that of t and the rate of convergence s unform over k. More precsely, for each k, we consder the Prohorov metrc, d k,overkorder belefs a standard metrc that metrzes the topology of weak convergence of probablty measures and then defne the unformweak topology as the topology of convergence n the metrc d UW sup k d k. Our frst man result, Theorem, s that convergence n the unformweak topology mples convergence n the unformstrategc topology. The latter, also ntroduced by Dekel et al. (006), s the coarsest topology on types under whch the ICR correspondence s upper hemcontnuous and the strct ICR correspondence s lower hemcontnuous, where the contnuty s now requred to hold unformly across all games. 3 In partcular, Theorem mples that convergence n the unformweak topology s a suffcent condton for convergence n the strategc topology. The unformweak topology s nterestng n ts own rght, as t generalzes the classc noton of approxmate common knowledge due to Monderer and Samet (989). Gven a payoffrelevant parameter θ, say that a type of a player has common pbelef n θ f he assgns probablty no smaller than p to θ, assgns probablty no smaller than p to the event that θ obtans and the other players assgn probablty no smaller than p to θ,and so forth, ad nfntum. A sequence of types (t n ) n has asymptotc common certanty of θ f for every p<, t n has common pbelef n θ for all n large enough. Monderer and Samet (989) use ths noton of proxmty to common knowledge to study the robustness of Nash equlbrum to small amounts of ncomplete nformaton. Although they focus on an ex ante noton of robustness and consder only common pror perturbatons, ther man result has the followng counterpart n our nterm, noncommon pror, nonequlbrum framework. If a sequence of types (t n ) n has asymptotc common certanty of θ, then, for every game, every acton that s strctly nterm correlated ratonalzable when θ s common certanty remans nterm correlated ratonalzable for type t n, for all n large enough. It turns out that asymptotc common certanty of θ s equvalent to unformweak convergence to the type that has common certanty of θ (.e., common belef). Thus, our Theorem s a generalzaton of Monderer and Samet s (989) man result to envronments where the lmt game has ncomplete nformaton. 3 See Secton 3 for the precse defnton of the modulus of contnuty on whch the unformty s based.
4 448 Chen, D Tllo, Fangold, and Xong Theoretcal Economcs 5 (00) An mportant corollary of Theorem s that the strategc, unformstrategc, and product topologes generate the same σalgebra. 4 Indeed, a fundamental result of Mertens and Zamr (985), whch s the Bayesan foundaton of Harsany s ( ) model of types, s that the space of herarches of belefs, called the unversal type space, exhausts all the relevant uncertanty of the players when endowed wth the product σ algebra. It s reassurng to know that ths unversalty property remans vald when the players can reason about any strategc event. 5 Our second man result, Theorem, s that unformweak convergence s also a necessary condton for strategc convergence when the lmt s a fnte type,.e., a type belongng to a fnte type space. Indeed, for any fnte type t and for any sequence of (possbly nfnte) types (t n ) n that fals to converge to t unformweakly, we construct a game n whch an acton s strctly nterm correlated ratonalzable for t, but not nterm correlated ratonalzable for t n, nfntely often along the sequence. 6 Thus, the unformweak topology fully characterzes the strategc topology around fnte types. Moreover, the assumpton that the lmt s a fnte type cannot be dspensed wth. Under the unformweak topology, the unversal type space s not separable,.e.,t does not contan a countable dense subset; by contrast, Dekel et al. (006) show that a countable set of fnte types s dense under the strategc topology. 7 Ths mples the exstence of nfnte types to whch unformweak convergence s not a necessary condton for strategc convergence. (We explctly construct such an example n Secton 4.) Whle ths fact mposes a natural lmt to our analyss, fnte type spaces play a promnent role n both appled and theoretcal work, so t s mportant to know that our suffcent condton for strategc convergence s also necessary n ths case. Fnte types are also the focus of our thrd man result, Theorem 3. We show that, under the unformstrategc topology, the set of fnte types s nowhere dense,.e.,ts closure has an empty nteror. To understand the conceptual mplcatons of ths result, recall that Dekel et al. (006) demonstrate the denseness of fnte types under the nonunform verson of the strategc topology. 8 Arguably, ths result provdes a compellng justfcaton for why t mght be wthout loss of generalty to model uncertanty wth fnte type spaces: Irrespectve of how large the true type space T s, for any gven game there s always a fnte type space T wth the property that the predctons of strategc behavor 4 Ths s because unformweak balls are countable ntersectons of fnteorder cylnders and the strategc topologes are sandwched between the unformweak and the product topologes, by Theorem. 5 Morrs (00, Secton 4.) rases the queston of whether the Mertens Zamr constructon s stll meanngful when strategc topologes are assumed. 6 Ths complements the man result of Wensten and Yldz (007), who fx a game (satsfyng a payoffrchnessassumpton) andafntetype t, andthen construct a sequence of types convergng to t n the product topology such that the behavor of t s bounded away from the behavor of all types n the sequence. By way of contrast, we fx a sequence of types thatfalstoconvergetoafntetypet n the unformweak topology and then construct a game for whch the behavor of t s bounded away from the behavor of the types n the sequence nfntely often. 7 Whle Dekel et al. (006) state only the weaker result that the set of all fnte types s dense n the strategc topology, ther proof actually establshes the stronger result above. 8 Mertens and Zamr (985) prove the denseness of fnte types under the product topology. Dekel et al. (006) argue that ths result does not provde a sound justfcaton for restrctng attenton to fnte types, for strategc behavor s not contnuous n the product topology.
5 Theoretcal Economcs 5 (00) Unform topologes on types 449 based on T are arbtrarly close to those based on T. Our nowhere denseness result mples that such fnte type space T cannot be chosen ndependently of the game. Ths s partcularly relevant for envronments such as those of mechansm desgn, where the game both payoffs and acton sets s not a pror fxed. More generally, our result mples that the unformstrategc topology s strctly fner than the strategc topology. Thus, whle a pror these two notons of strategc contnuty seem equally compellng, assumng one or the other can have a large mpact on the ensung theory. The exercse n ths paper s smlar n sprt to that of Monderer and Samet (996) and Kaj and Morrs (998), who, lke us, consder perturbatons of ncomplete nformaton games. These papers provde belefbased characterzatons of strategc topologes for Bayesan Nash equlbrum n countable partton models à la Aumann (976). However, snce both of these papers assume a common pror and adopt an ex ante approach, whle we adopt an nterm approach wthout mposng a common pror, t s dffcult to establsh a precse connecton. 9 Another mportant dfference between ther approachandourssnthedstnctpayoffrelevance constrants adopted: we fx the set of payoffrelevant states, so our games cannot have payoffs that depend drectly on players hgherorder belefs; Monderer and Samet (996) andkaj and Morrs (998) have no such payoffrelevance constrant. The connecton between unform and strategc topologes frst appears n Morrs (00), who studes a specal class of games, called hgherorder expectaton (HOE), games, and shows that the topology of unform convergence of hgherorder terated expectatons s equvalent to the coarsest topology under whch a certan noton of strct ICR correspondence dfferent from the one we consder s lower hemcontnuous n every game of the HOE class. 0 Compared to the unformweak topology, the topology of unform convergence of terated expectatons s nether fner nor coarser, even around fnte types. We further elaborate on ths relatonshp n Secton 5. Ths paper s also related to contemporaneous work by Ely and Pęsk (008). Followng ther termnology, a type t s crtcal f, under the product topology, the strct ICR correspondence s dscontnuous at t n some game. Ely and Pęsk (008) provde an nsghtful characterzaton of crtcal types n terms of a common belef property: a type s crtcal f and only f, for some p>0, t has common pbelef n some closed (n product topology) proper subset of the unversal type space. Conceptually, ths result shows that the usual type spaces that appear n applcatons consst almost entrely of crtcal types, as these type spaces typcally embody nontrval common belef assumptons. For nstance, all fnte types are crtcal and so are almost all types belongng to a common 9 Monderer and Samet (996) fx the common pror and consder proxmty of nformaton parttons, whereas Kaj and Morrs (998) vary the common pror on a fxed nformaton structure. For ths reason, the precse connecton between these papers s already unclear. 0 Morrs (00) defnes hs strategc topology for HOE games usng a dstance that makes no reference to ICR. But, as we clamed above, t can be shown that hs strategc topology concdes wth the coarsest topology under whch a certan noton of strct ICR correspondence s contnuous n every HOE game. The noton of strctness mplct n Morrs (00) analyss, unlke ours, does not requre the slack n the ncentve constrants to be unform. Moreover, they show that under the product topology the regular types,.e., those types whch are not crtcal, form a resdual subsetoftheunversaltypespace astandardtopologcalnoton ofa generc set.
6 450 Chen, D Tllo, Fangold, and Xong Theoretcal Economcs 5 (00) pror type space. Thus ElyandPęsk s (008) resulttellsus when basedon thecommon belefs of the players there wll be some game and some productconvergent sequence along whch strategc behavor s dscontnuous, whereas we dentfy a condton for an arbtrary sequence to dsplay contnuous strategc behavor n all games. The rest of the paper s organzed as follows. Secton ntroduces the standard model of herarches of belefs and type spaces, and revews the soluton concept of ICR. Secton 3 revews the strategc and unformstrategc topologes of Dekel et al. (006), ntroduces the unformweak topology, and presents our two man results concernng the relatonshp between these topologes (Theorems and ). Secton 4 examnes the nongenercty of fnte types under the unformstrategc and unformweak topologes, and presents the nowhere denseness result (Theorem 3). Secton 5 dscusses the relaton wth some other topologes. Secton 6 concludes wth some open questons for future research.. Prelmnares Throughout the paper, we fx a twoplayer set I and a fnte set of payoffrelevant states wth at least two elements. Gven a player I, wewrte to desgnate the other player n I. All topologcal spaces, when vewed as measurable spaces, are endowed wth ther Borel σalgebra. For a topologcal space S, wewrte (S) to desgnate the space of probablty measures over S equpped wth the topology of weak convergence. Unless explctly noted, all product spaces are endowed wth the product topology and subspaces are endowed wth the relatve topology.. Herarches of belefs and types Our formulaton of ncomplete nformaton follows Mertens and Zamr (985). 3 Defne X 0 =,andx = X 0 (X 0 ),and,foreachk, defne recursvely { } k X k = (θ μ μ k ) X 0 (X l ) : marg X l μ l = μ l l = k l= By vrtue of the above coherency condton on margnal dstrbutons, each element of X k s determned by ts frst and last coordnates, so we can dentfy X k wth (X k ). For each I and k, welett k = (X k ) desgnate the space of korder belefs of player,sothatt k = ( T k ). ThespaceT of herarches of belefs of player s { } T = (μ k ) k (X k ) : marg X k μ k = μ k k k We restrct attenton to twoplayer games for ease of notaton. Our results reman vald wth any fnte number of players. 3 An alternatve, equvalent formulaton s found n Brandenburger and Dekel (993).
7 Theoretcal Economcs 5 (00) Unform topologes on types 45 Snce s fnte, T s a compact metrzable space. Moreover, there s a unque mappng μ : T ( T ) that s belef preservng,.e., for all t = (t t ) T and k, μ (t )θ (π k ) (E)]=t k+ θ E] for all θ and measurable E T k where π k s the natural projecton of T onto T k. Furthermore, the mappng μ s a homeomorphsm, so to save on notaton, we dentfy each herarchy of belef t T wth ts correspondng belef μ (t ) over T. Smlarly, for each t T,wewrtet k T k nstead of the more cumbersome π k(t ). Herarches of belefs can be mplctly represented usng a type space,.e., a tuple (T φ ) I,whereeachT s a Polsh space of types and each φ : T ( T ) s a measurable functon. Indeed, every type t T s mapped nto a herarchy of belefs ν (t ) = (ν k(t )) k n a natural way: ν (t ) = marg φ (t ) and, for k, ν k (t )θ E]=φ (t )θ (ν k ) (E)] for all θ and measurable E T k Thetypespace(T μ ) I s called the unversal type space, snce for every type space (T φ ) I there s a unque belefpreservng mappng from T nto T,namelythemappng ν above. 4 When the mappngs (ν ) I are njectve, the type space (T φ ) I s called nonredundant. In ths case, (ν ) I are measurable embeddngs onto ther mages (ν (T )) I, whch are measurable and can be vewed as a nonredundant type space, snce we have μ (ν (t )) ν (T )]= for all I and t T.Conversely,any(T ) I such that T T and μ (t ) T ]= for all I and t T can be vewed as a nonredundant type space.. Bayesan games and nterm correlated ratonalzablty A game s a tuple G = (A g ) I,whereA s a fnte set of actons for player and g : A A M M] s hs payoff functon, wth M>0 an arbtrary bound on payoffs that we fx throughout. 5 We wrte G to denote the set of all games and, for each nteger m, wewrteg m for the set of games wth A m for all I. The soluton concept of nterm correlated ratonalzablty (ICR) was ntroduced n Dekel et al. (007). Gven a γ R, atypespace(t φ ) I,andagameG, for each player I, nteger k 0, andtypet T,weletR k (t G γ) A desgnate the set of korder γratonalzable actons of t. These sets are defned as: R 0 (t G γ)= A and recursvely for each nteger k, R k (t G γ) s the set of all actons a A for whch there s a conjecture,.e., a measurable functon σ : T (A ) such that 6 supp σ (θ t ) R k (t G γ) (θ t ) T () 4 To say that ν s belefpreservng means that μ (ν (t ))θ E]=φ (t )θ (ν ) (E)] for all θ and measurable E T. 5 We wll also denote by g the payoff functon n the mxed extenson of G,wrtngg (α α θ)wth the obvous meanng for any α (A ) and α (A ). 6 Relaxng condton () by requrng t to hold only for φ (t )almost every (θ t ) would not alter the defnton of ratonalzablty. Indeed, any conjecture that has a (k )order ratonalzable support φ (t ) almost everywhere can be changed nto one that yelds the same expected payoff and satsfes the condton
8 45 Chen, D Tllo, Fangold, and Xong Theoretcal Economcs 5 (00) and for all a A, T g (a σ (θ t ) θ) g (a σ (θ t ) θ) ] φ (t )(dθ dt ) γ () For future reference, a conjecture σ : T (A ) that satsfes the former condton wll be called a (k )order γratonalzable conjecture. Thesetofγratonalzable actons of type t s then defned as R (t G γ)= k R k (t G γ) Fnally, followng Ely and Pęsk (008), an acton a A s strctly nterm correlated γ ratonalzable for type t and we wrte a R (t G γ)f a R (t G γ ) for some γ <γ. As shown n Dekel et al. (007), R (t G γ) s nonempty for every game G, typet and γ 0. 7 Interm correlated ratonalzablty has a characterzaton n terms of bestreply sets. A par of measurable functons ς : T A, I, hastheγbestreply property f for each I and t T,eachactona ς (t ) s a γbest reply for t to a conjecture σ : T (A ) wth supp σ (θ t ) ς (t ) (θ t ) T If (ς ) I has the γbestreply property, then ς (t ) R (t G γ)for all I and t T.As shown n Dekel et al. (007), the par (R ( G γ)) I s the maxmal par of correspondences wth the γbestreply property. Ths means there s no other par (ς ) I wth the γbestreply property such that R (t G γ) ς (t ) for each I and t T,wth strct ncluson for some I and t T. Therefore, an acton s γratonalzable for a type t f and only f t s a γbest reply to a γratonalzable conjecture,.e., a conjecture σ : T (A ) such that supp σ (θ t ) R (t G γ) (θ t ) T Dekel et al. (007)also show that the set of γratonalzable actons of a type s determned by the nduced herarchy of belefs. Indeed, for any k, any two types (possbly belongng to dfferent type spaces) mappng nto the same korder belef must have the same set of korder γratonalzable actons. Ths has two mplcatons. Frst, for nterm correlated ratonalzablty, t s wthout loss of generalty to dentfy types wth ther correspondng herarches. Thus, n what follows we restrct attenton to type spaces (T ) I wth T T and t T ]= for all I and t T. 8 Accordngly, we take the unversal type space T to be the doman of the correspondence R ( G γ): T A.Second, everywhere. Ths s possble because the correspondence R k s upper hemcontnuous, and hence t admts a measurable selecton by the Kuratowsk Ryll Nardzewsk selecton theorem (see, e.g., Alprants and Border 999). 7 Note that for γ< M,wehaveR (t G γ)=, and for γ>m we have R (t G γ)= A. 8 Recall that we dentfy each type t T wth hs belef μ (t ) ( T ).
9 Theoretcal Economcs 5 (00) Unform topologes on types 453 to establsh whether an acton s korder γratonalzable for a type t,wecanrestrct attenton to (k )order γratonalzable conjectures σ, whch are measurable wth respect to (k )order belefs. 9 Fnally, the followng result shows that, smlar to ratonalzablty n complete nformaton games, nterm correlated ratonalzablty has a characterzaton n terms of terated domnance, where the noton of domnance now becomes an nterm one. Proposton. Fx γ and a game G = (A g ) I.Foreachk,player I,typet T, and acton a A, we have a R k (t G γ)f and only f, for each α (A \{a }),there exsts a measurable σ : T (A ) wth such that supp σ (θ t ) R k (t G γ) (θ t ) T (3) T g (a σ (θ t ) θ) g (α σ (θ t ) θ) ] t (dθ dt ) γ The proof of ths proposton, relegated to the Appendx, uses a separaton argument analogous to that whch establshes the equvalence between strctly domnated and never bestreply strateges n complete nformaton games. Here, too, the usefulness of the result comes from the fact that to check whether an acton s ratonalzable for a type, we are able to reverse the order of quantfers and seek a possbly dfferent conjecture for each possble (mxed) devaton. 3. Topologes on types The strategc (or smply S) topology ntroduced n Dekel et al. (006) s the coarsest topology on the unversal type space T under whch the ICR correspondence s upper hemcontnuous and the strct ICR correspondence s lower hemcontnuous n all games. More explctly, followng a formulaton due to Elyand Pęsk (008), the S topologys the topology generated by the collecton of all sets of the form {t T : a / R (t G γ)} and {t T : a R (t G γ)} where G = (A g ) I, a A,andγ R. 0 The S topology on T s metrzable by the dstance d S, defned as follows. For each game G = (A g ) I,actona A,andtypet T,let h (t a G)= nf{γ : a R (t G γ)} 9 Ths means that σ (θ s ) = σ (θ t ) for all θ and all types s t wth the same (k )order belefs. 0 The strategc topology can be gven an equvalent defnton that makes no drect reference to γ ratonalzablty for γ 0. Indeed, by Ely and Pęsk (008, Lemma 4), a subbass of the strategc topology s the collecton of all sets of the form {t : a / R(t G 0)} and {t : a R (t G 0)}. Dekel et al. (006) defne the S topology drectly usng the dstance d S, rather than usng the topologcal defnton above.
10 454 Chen, D Tllo, Fangold, and Xong Theoretcal Economcs 5 (00) Then, for each s and t T, d S (s t ) = m m sup G=(A g ) I G m max a A h (s a G) h (t a G) In terms of convergence of sequences, Dekel et al. (006) show that for every t T and every sequence (t n ) n n T,wehaved S (t n t ) 0 f, and only f, for every game G = (A g ) I,actona A,andγ R, the followng upper hemcontnuty (u.h.c.) and lower hemcontnuty (l.h.c.) propertes hold: For every sequence γ n γ, and for some sequence γ n γ, a R (t n G γ n ) n a R (t G γ) (u.h.c.) a R (t G γ) a R (t n G γ n ) n (l.h.c.) Dekel et al. (006) also ntroduce the unformstrategc (US) topology, whch strengthens the defnton of the strategc topology by requrng the convergence to be unform over all games. More precsely, the US topology s the topology of convergence under the metrc d US, whch s defned as d US (t s ) = sup G=(A g ) I G max h (t a G) h (s a G) a A Ths unformty renders the US topology partcularly relevant for envronments where the game both payoffs and acton sets s not fxed a pror, such as n a mechansm desgn envronment. We now ntroduce a metrc topology on types, whch we call unformweak (UW) topology, under whch two types of player are close f they have smlar frstorder belefs, attach smlar probabltes to other players havng smlar frstorder belefs, and so on, where the degree of smlarty s unform over the levels of the belef herarchy. Thus, unlke the S and US topologes, whch are behavorbased, the UW topology s a belefbased topology,.e., a metrc topology defned explctly n terms of proxmty of herarches of belefs. The two man results of ths secton, Theorems and below, establsh a connecton between these behavor and belefbased topologes. Before we present the formal defnton of the UW topology, recall that for a complete separable metrc space (S d), the topology of weak convergence on (S) s metrzable by the Prohorov dstance ρ, defnedas ρ(μ μ ) = nf{δ>0:μ(e) μ (E δ ) + δ for each measurable E S} μ μ (S) where E δ ={s S :nf s S d(s s )<δ}. generated by the dstance The UW topology s the metrc topology on T d UW (s t ) = sup d k (s t ) k s t T where d 0 s the dscrete metrc on and recursvely for k, d k s the Prohorov dstance on ( T k ) nduced by the metrc max{d 0 d k } on T k.
11 Theoretcal Economcs 5 (00) Unform topologes on types 455 In the remander of Secton 3 we explore the relatonshp between the UW topology and the S and US topologes. Frst, we show that the UW topology s fner than the US topology (Theorem ). Second, we prove a partal converse, namely that around fnte types,.e., types belongng to a fnte type space, the S topology (and hence also the US topology) s fner than the UW topology (Theorem ). 3. UW convergence mples US convergence Theorem. For each player I and for all types s t T, d US (s t ) 4Md UW (s t ) Thus the UW topology s fner than the US topology. Ths theorem s a drect mplcaton of the followng proposton. Proposton. Fx a game G, γ 0 and δ>0. Foreachntegerk, d k (s t )<δ R k (t G γ) R k (s G γ+ 4Mδ) I s t T The man challenge n provng ths result s due to the fact that (k )order ratonalzable conjectures σ : T (A ) need not be contnuous under the topology of weak convergence of (k )order belefs. Ths mples that, keepng the conjecture fxed, the ncentve constrants of player for korder γratonalzablty (cf. ()) may be dscontnuous n hs type under the topology of weak convergence of korder belefs. Our proof overcomes ths ssue by endowng closeby types wth smlar, but not dentcal, conjectures. Indeed, the characterzaton of ICR from Proposton mples that for a gven acton a A and a gven mxed devaton α (A ),therealwaysexsts a (k )order ratonalzable conjecture that s optmal to γratonalze a aganst α at order k. Followng ths observaton, n our proof we endow type t wth an optmal conjecture for γratonalzablty and endow type s wth an optmal conjecture for (γ + 4Mδ)ratonalzablty. Usng these optmal conjectures, we then prove, usng an ntegratonbyparts type argument, that every acton that s korder γratonalzable for t remans korder (γ + 4Mδ)ratonalzable for s. Proof of Proposton. Fx a game G = (A g ) I, γ 0 and δ>0. The proof s by nducton on k. For k =, lets and t T be such that d (s t )<δ. Fx an arbtrary a R (t G γ) and let us show that a R (s G γ + 4Mδ) usng Proposton. Fx α (A \{a }) and let σ : (A ) beaconjecturesuchthat 3 ( g (a σ (θ) θ) g (α σ (θ) θ) ) t θ] γ (4) θ To be precse, when we say that σ s an optmal conjecture to γratonalze a aganst α at order k, we mean that σ s a (k )order γratonalzable conjecture that satsfes the followng property: for any type t, the expected payoff dfference between a and α for type t s at least γ under some (k )order γratonalzable conjecture f and only f ths expected payoff dfference s at least γ under σ. 3 Recall that t desgnates the frstorder belef of type t.
Recurrence. 1 Definitions and main statements
Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.
More informationExtending Probabilistic Dynamic Epistemic Logic
Extendng Probablstc Dynamc Epstemc Logc Joshua Sack May 29, 2008 Probablty Space Defnton A probablty space s a tuple (S, A, µ), where 1 S s a set called the sample space. 2 A P(S) s a σalgebra: a set
More informationA Probabilistic Theory of Coherence
A Probablstc Theory of Coherence BRANDEN FITELSON. The Coherence Measure C Let E be a set of n propostons E,..., E n. We seek a probablstc measure C(E) of the degree of coherence of E. Intutvely, we want
More information1 Example 1: Axisaligned rectangles
COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture # 6 Scrbe: Aaron Schld February 21, 2013 Last class, we dscussed an analogue for Occam s Razor for nfnte hypothess spaces that, n conjuncton
More informationEmbedding lattices in the Kleene degrees
F U N D A M E N T A MATHEMATICAE 62 (999) Embeddng lattces n the Kleene degrees by Hsato M u r a k (Nagoya) Abstract. Under ZFC+CH, we prove that some lattces whose cardnaltes do not exceed ℵ can be embedded
More informationREGULAR MULTILINEAR OPERATORS ON C(K) SPACES
REGULAR MULTILINEAR OPERATORS ON C(K) SPACES FERNANDO BOMBAL AND IGNACIO VILLANUEVA Abstract. The purpose of ths paper s to characterze the class of regular contnuous multlnear operators on a product of
More informationBERNSTEIN POLYNOMIALS
OnLne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful
More informationLuby s Alg. for Maximal Independent Sets using Pairwise Independence
Lecture Notes for Randomzed Algorthms Luby s Alg. for Maxmal Independent Sets usng Parwse Independence Last Updated by Erc Vgoda on February, 006 8. Maxmal Independent Sets For a graph G = (V, E), an ndependent
More informationOn Lockett pairs and Lockett conjecture for πsoluble Fitting classes
On Lockett pars and Lockett conjecture for πsoluble Fttng classes Lujn Zhu Department of Mathematcs, Yangzhou Unversty, Yangzhou 225002, P.R. Chna Emal: ljzhu@yzu.edu.cn Nanyng Yang School of Mathematcs
More informationPSYCHOLOGICAL RESEARCH (PYC 304C) Lecture 12
14 The Chsquared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304C) Lecture 1 If a normal varable X, havng mean µ and varance σ, s standardsed, the new varable Z has a mean 0 and varance 1. When ths standardsed
More informationOn Competitive Nonlinear Pricing
On Compettve Nonlnear Prcng Andrea Attar Thomas Marott Franços Salané February 27, 2013 Abstract A buyer of a dvsble good faces several dentcal sellers. The buyer s preferences are her prvate nformaton,
More informationModule 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..
More informationv a 1 b 1 i, a 2 b 2 i,..., a n b n i.
SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 455 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces we have studed thus far n the text are real vector spaces snce the scalars are
More information8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by
6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng
More informationSupport Vector Machines
Support Vector Machnes Max Wellng Department of Computer Scence Unversty of Toronto 10 Kng s College Road Toronto, M5S 3G5 Canada wellng@cs.toronto.edu Abstract Ths s a note to explan support vector machnes.
More informationHow Sets of Coherent Probabilities May Serve as Models for Degrees of Incoherence
1 st Internatonal Symposum on Imprecse Probabltes and Ther Applcatons, Ghent, Belgum, 29 June 2 July 1999 How Sets of Coherent Probabltes May Serve as Models for Degrees of Incoherence Mar J. Schervsh
More informationAn Alternative Way to Measure Private Equity Performance
An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate
More informationINTERPRETING TRUE ARITHMETIC IN THE LOCAL STRUCTURE OF THE ENUMERATION DEGREES.
INTERPRETING TRUE ARITHMETIC IN THE LOCAL STRUCTURE OF THE ENUMERATION DEGREES. HRISTO GANCHEV AND MARIYA SOSKOVA 1. Introducton Degree theory studes mathematcal structures, whch arse from a formal noton
More informationWe are now ready to answer the question: What are the possible cardinalities for finite fields?
Chapter 3 Fnte felds We have seen, n the prevous chapters, some examples of fnte felds. For example, the resdue class rng Z/pZ (when p s a prme) forms a feld wth p elements whch may be dentfed wth the
More informationNONCONSTANT SUM REDANDBLACK GAMES WITH BETDEPENDENT WIN PROBABILITY FUNCTION LAURA PONTIGGIA, University of the Sciences in Philadelphia
To appear n Journal o Appled Probablty June 2007 OCOSTAT SUM REDADBLACK GAMES WITH BETDEPEDET WI PROBABILITY FUCTIO LAURA POTIGGIA, Unversty o the Scences n Phladelpha Abstract In ths paper we nvestgate
More informationCombinatorial Agency of Threshold Functions
Combnatoral Agency of Threshold Functons Shal Jan Computer Scence Department Yale Unversty New Haven, CT 06520 shal.jan@yale.edu Davd C. Parkes School of Engneerng and Appled Scences Harvard Unversty Cambrdge,
More informationProductForm Stationary Distributions for Deficiency Zero Chemical Reaction Networks
Bulletn of Mathematcal Bology (21 DOI 1.17/s11538195174 ORIGINAL ARTICLE ProductForm Statonary Dstrbutons for Defcency Zero Chemcal Reacton Networks Davd F. Anderson, Gheorghe Cracun, Thomas G. Kurtz
More information1 Approximation Algorithms
CME 305: Dscrete Mathematcs and Algorthms 1 Approxmaton Algorthms In lght of the apparent ntractablty of the problems we beleve not to le n P, t makes sense to pursue deas other than complete solutons
More informationbenefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).
REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or
More information8 Algorithm for Binary Searching in Trees
8 Algorthm for Bnary Searchng n Trees In ths secton we present our algorthm for bnary searchng n trees. A crucal observaton employed by the algorthm s that ths problem can be effcently solved when the
More informationWhat is Candidate Sampling
What s Canddate Samplng Say we have a multclass or mult label problem where each tranng example ( x, T ) conssts of a context x a small (mult)set of target classes T out of a large unverse L of possble
More informationRing structure of splines on triangulations
www.oeaw.ac.at Rng structure of splnes on trangulatons N. Vllamzar RICAMReport 201448 www.rcam.oeaw.ac.at RING STRUCTURE OF SPLINES ON TRIANGULATIONS NELLY VILLAMIZAR Introducton For a trangulated regon
More informationGeneralizing the degree sequence problem
Mddlebury College March 2009 Arzona State Unversty Dscrete Mathematcs Semnar The degree sequence problem Problem: Gven an nteger sequence d = (d 1,...,d n ) determne f there exsts a graph G wth d as ts
More informationCan Auto Liability Insurance Purchases Signal Risk Attitude?
Internatonal Journal of Busness and Economcs, 2011, Vol. 10, No. 2, 159164 Can Auto Lablty Insurance Purchases Sgnal Rsk Atttude? ChuShu L Department of Internatonal Busness, Asa Unversty, Tawan ShengChang
More informationImplied (risk neutral) probabilities, betting odds and prediction markets
Impled (rsk neutral) probabltes, bettng odds and predcton markets Fabrzo Caccafesta (Unversty of Rome "Tor Vergata") ABSTRACT  We show that the well known euvalence between the "fundamental theorem of
More informationThe literature on manyserver approximations provides significant simplifications toward the optimal capacity
Publshed onlne ahead of prnt November 13, 2009 Copyrght: INFORMS holds copyrght to ths Artcles n Advance verson, whch s made avalable to nsttutonal subscrbers. The fle may not be posted on any other webste,
More informationgreatest common divisor
4. GCD 1 The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no
More informationAnswer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy
4.02 Quz Solutons Fall 2004 MultpleChoce Questons (30/00 ponts) Please, crcle the correct answer for each of the followng 0 multplechoce questons. For each queston, only one of the answers s correct.
More informationA Lyapunov Optimization Approach to Repeated Stochastic Games
PROC. ALLERTON CONFERENCE ON COMMUNICATION, CONTROL, AND COMPUTING, OCT. 2013 1 A Lyapunov Optmzaton Approach to Repeated Stochastc Games Mchael J. Neely Unversty of Southern Calforna http://wwwbcf.usc.edu/
More informationGeneral Auction Mechanism for Search Advertising
General Aucton Mechansm for Search Advertsng Gagan Aggarwal S. Muthukrshnan Dávd Pál Martn Pál Keywords game theory, onlne auctons, stable matchngs ABSTRACT Internet search advertsng s often sold by an
More informationFaculty Research Working Papers Series
Faculty Research Workng Papers Seres Mechansm Desgn wth Multdmensonal, Contnuous Types and Interdependent Valuatons Nolan Mller John F. Kennedy School of Government  Harvard Unversty John W. Pratt Harvard
More informationx f(x) 1 0.25 1 0.75 x 1 0 1 1 0.04 0.01 0.20 1 0.12 0.03 0.60
BIVARIATE DISTRIBUTIONS Let be a varable that assumes the values { 1,,..., n }. Then, a functon that epresses the relatve frequenc of these values s called a unvarate frequenc functon. It must be true
More informationThe Stock Market Game and the KellyNash Equilibrium
The Stock Market Game and the KellyNash Equlbrum Carlos AlósFerrer, Ana B. Ana Department of Economcs, Unversty of Venna. Hohenstaufengasse 9, A1010 Venna, Austra. July 2003 Abstract We formulate the
More informationThe descriptive complexity of the family of Banach spaces with the πproperty
Arab. J. Math. (2015) 4:35 39 DOI 10.1007/s4006501401163 Araban Journal of Mathematcs Ghadeer Ghawadrah The descrptve complexty of the famly of Banach spaces wth the πproperty Receved: 25 March 2014
More informationThe OC Curve of Attribute Acceptance Plans
The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4
More informationTHE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek
HE DISRIBUION OF LOAN PORFOLIO VALUE * Oldrch Alfons Vascek he amount of captal necessary to support a portfolo of debt securtes depends on the probablty dstrbuton of the portfolo loss. Consder a portfolo
More informationA Novel Methodology of Working Capital Management for Large. Public Constructions by Using Fuzzy Scurve Regression
Novel Methodology of Workng Captal Management for Large Publc Constructons by Usng Fuzzy Scurve Regresson ChengWu Chen, Morrs H. L. Wang and TngYa Hseh Department of Cvl Engneerng, Natonal Central Unversty,
More informationWeek 6 Market Failure due to Externalities
Week 6 Market Falure due to Externaltes 1. Externaltes n externalty exsts when the acton of one agent unavodably affects the welfare of another agent. The affected agent may be a consumer, gvng rse to
More informationInequality and The Accounting Period. Quentin Wodon and Shlomo Yitzhaki. World Bank and Hebrew University. September 2001.
Inequalty and The Accountng Perod Quentn Wodon and Shlomo Ytzha World Ban and Hebrew Unversty September Abstract Income nequalty typcally declnes wth the length of tme taen nto account for measurement.
More informationDEFINING %COMPLETE IN MICROSOFT PROJECT
CelersSystems DEFINING %COMPLETE IN MICROSOFT PROJECT PREPARED BY James E Aksel, PMP, PMISP, MVP For Addtonal Informaton about Earned Value Management Systems and reportng, please contact: CelersSystems,
More informationStability, observer design and control of networks using Lyapunov methods
Stablty, observer desgn and control of networks usng Lyapunov methods von Lars Naujok Dssertaton zur Erlangung des Grades enes Doktors der Naturwssenschaften  Dr. rer. nat.  Vorgelegt m Fachberech 3
More informationn + d + q = 24 and.05n +.1d +.25q = 2 { n + d + q = 24 (3) n + 2d + 5q = 40 (2)
MATH 16T Exam 1 : Part I (InClass) Solutons 1. (0 pts) A pggy bank contans 4 cons, all of whch are nckels (5 ), dmes (10 ) or quarters (5 ). The pggy bank also contans a con of each denomnaton. The total
More informationLogistic Regression. Lecture 4: More classifiers and classes. Logistic regression. Adaboost. Optimization. Multiple class classification
Lecture 4: More classfers and classes C4B Machne Learnng Hlary 20 A. Zsserman Logstc regresson Loss functons revsted Adaboost Loss functons revsted Optmzaton Multple class classfcaton Logstc Regresson
More informationToday s class. Chapter 13. Sources of uncertainty. Decision making with uncertainty
Today s class Probablty theory Bayesan nference From the ont dstrbuton Usng ndependence/factorng From sources of evdence Chapter 13 1 2 Sources of uncertanty Uncertan nputs Mssng data Nosy data Uncertan
More informationCHAPTER 7 VECTOR BUNDLES
CHAPTER 7 VECTOR BUNDLES We next begn addressng the queston: how do we assemble the tangent spaces at varous ponts of a manfold nto a coherent whole? In order to gude the decson, consder the case of U
More informationANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING
ANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING Matthew J. Lberatore, Department of Management and Operatons, Vllanova Unversty, Vllanova, PA 19085, 6105194390,
More informationThe Market Organism: Long Run Survival in Markets with Heterogeneous Traders
The Market Organsm: Long Run Survval n Markets wth Heterogeneous Traders Lawrence E. Blume Davd Easley SFI WORKING PAPER: 200804018 SFI Workng Papers contan accounts of scentfc work of the authors) and
More informationA Secure PasswordAuthenticated Key Agreement Using Smart Cards
A Secure PasswordAuthentcated Key Agreement Usng Smart Cards Ka Chan 1, WenChung Kuo 2 and JnChou Cheng 3 1 Department of Computer and Informaton Scence, R.O.C. Mltary Academy, Kaohsung 83059, Tawan,
More information1. Fundamentals of probability theory 2. Emergence of communication traffic 3. Stochastic & Markovian Processes (SP & MP)
6.3 /  Communcaton Networks II (Görg) SS20  www.comnets.unbremen.de Communcaton Networks II Contents. Fundamentals of probablty theory 2. Emergence of communcaton traffc 3. Stochastc & Markovan Processes
More informationBrigid Mullany, Ph.D University of North Carolina, Charlotte
Evaluaton And Comparson Of The Dfferent Standards Used To Defne The Postonal Accuracy And Repeatablty Of Numercally Controlled Machnng Center Axes Brgd Mullany, Ph.D Unversty of North Carolna, Charlotte
More informationThe Development of Web Log Mining Based on ImproveKMeans Clustering Analysis
The Development of Web Log Mnng Based on ImproveKMeans Clusterng Analyss TngZhong Wang * College of Informaton Technology, Luoyang Normal Unversty, Luoyang, 471022, Chna wangtngzhong2@sna.cn Abstract.
More informationInstitute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic
Lagrange Multplers as Quanttatve Indcators n Economcs Ivan Mezník Insttute of Informatcs, Faculty of Busness and Management, Brno Unversty of TechnologCzech Republc Abstract The quanttatve role of Lagrange
More informationSubstitution Effects in Supply Chains with Asymmetric Information Distribution and Upstream Competition
Substtuton Effects n Supply Chans wth Asymmetrc Informaton Dstrbuton and Upstream Competton Jochen Schlapp, Mortz Fleschmann Department of Busness, Unversty of Mannhem, 68163 Mannhem, Germany, jschlapp@bwl.unmannhem.de,
More informationWhat should (public) health insurance cover?
Journal of Health Economcs 26 (27) 251 262 What should (publc) health nsurance cover? Mchael Hoel Department of Economcs, Unversty of Oslo, P.O. Box 195 Blndern, N317 Oslo, Norway Receved 29 Aprl 25;
More informationCautiousness and Measuring An Investor s Tendency to Buy Options
Cautousness and Measurng An Investor s Tendency to Buy Optons James Huang October 18, 2005 Abstract As s well known, ArrowPratt measure of rsk averson explans a ratonal nvestor s behavor n stock markets
More information6. EIGENVALUES AND EIGENVECTORS 3 = 3 2
EIGENVALUES AND EIGENVECTORS The Characterstc Polynomal If A s a square matrx and v s a nonzero vector such that Av v we say that v s an egenvector of A and s the correspondng egenvalue Av v Example :
More informationFisher Markets and Convex Programs
Fsher Markets and Convex Programs Nkhl R. Devanur 1 Introducton Convex programmng dualty s usually stated n ts most general form, wth convex objectve functons and convex constrants. (The book by Boyd and
More information2.4 Bivariate distributions
page 28 2.4 Bvarate dstrbutons 2.4.1 Defntons Let X and Y be dscrete r.v.s defned on the same probablty space (S, F, P). Instead of treatng them separately, t s often necessary to thnk of them actng together
More informationIDENTIFICATION AND CORRECTION OF A COMMON ERROR IN GENERAL ANNUITY CALCULATIONS
IDENTIFICATION AND CORRECTION OF A COMMON ERROR IN GENERAL ANNUITY CALCULATIONS Chrs Deeley* Last revsed: September 22, 200 * Chrs Deeley s a Senor Lecturer n the School of Accountng, Charles Sturt Unversty,
More informationComplete Fairness in Secure TwoParty Computation
Complete Farness n Secure TwoParty Computaton S. Dov Gordon Carmt Hazay Jonathan Katz Yehuda Lndell Abstract In the settng of secure twoparty computaton, two mutually dstrustng partes wsh to compute
More informationA Two Stage Stochastic Equilibrium Model for Electricity Markets with Two Way Contracts
A Two Stage Stochastc Equlbrum Model for Electrcty Markets wth Two Way Contracts Dal Zhang and Hufu Xu School of Mathematcs Unversty of Southampton Southampton SO17 1BJ, UK Yue Wu School of Management
More informationCOLLOQUIUM MATHEMATICUM
COLLOQUIUM MATHEMATICUM VOL. 74 997 NO. TRANSFERENCE THEORY ON HARDY AND SOBOLEV SPACES BY MARIA J. CARRO AND JAVIER SORIA (BARCELONA) We show that the transference method of Cofman and Wess can be extended
More informationMAPP. MERIS level 3 cloud and water vapour products. Issue: 1. Revision: 0. Date: 9.12.1998. Function Name Organisation Signature Date
Ttel: Project: Doc. No.: MERIS level 3 cloud and water vapour products MAPP MAPPATBDClWVL3 Issue: 1 Revson: 0 Date: 9.12.1998 Functon Name Organsaton Sgnature Date Author: Bennartz FUB Preusker FUB Schüller
More informationNonbinary Quantum ErrorCorrecting Codes from Algebraic Curves
Nonbnary Quantum ErrorCorrectng Codes from Algebrac Curves JonLark Km and Judy Walker Department of Mathematcs Unversty of NebraskaLncoln, Lncoln, NE 685880130 USA emal: {jlkm, jwalker}@math.unl.edu
More informationThe covariance is the two variable analog to the variance. The formula for the covariance between two variables is
Regresson Lectures So far we have talked only about statstcs that descrbe one varable. What we are gong to be dscussng for much of the remander of the course s relatonshps between two or more varables.
More informationMultiResource Fair Allocation in Heterogeneous Cloud Computing Systems
1 MultResource Far Allocaton n Heterogeneous Cloud Computng Systems We Wang, Student Member, IEEE, Ben Lang, Senor Member, IEEE, Baochun L, Senor Member, IEEE Abstract We study the multresource allocaton
More informationPowerofTwo Policies for Single Warehouse MultiRetailer Inventory Systems with Order Frequency Discounts
Powerofwo Polces for Sngle Warehouse MultRetaler Inventory Systems wth Order Frequency Dscounts José A. Ventura Pennsylvana State Unversty (USA) Yale. Herer echnon Israel Insttute of echnology (Israel)
More informationEfficient Project Portfolio as a tool for Enterprise Risk Management
Effcent Proect Portfolo as a tool for Enterprse Rsk Management Valentn O. Nkonov Ural State Techncal Unversty Growth Traectory Consultng Company January 5, 27 Effcent Proect Portfolo as a tool for Enterprse
More informationRole of Bargaining in Marketing Channel Games of Quality Choice and Profit Share
Workng Paper Seres FSWP 200702 Role of Barganng n Marketng Channel Games of Qualty Choce and Proft Share Phlppe Bontems* Toulouse School of Economcs (GREMAQ, INRA and IDEI) Trtha Dhar Sauder School of
More informationCS 2750 Machine Learning. Lecture 17a. Clustering. CS 2750 Machine Learning. Clustering
Lecture 7a Clusterng Mlos Hauskrecht mlos@cs.ptt.edu 539 Sennott Square Clusterng Groups together smlar nstances n the data sample Basc clusterng problem: dstrbute data nto k dfferent groups such that
More information1.1 The University may award Higher Doctorate degrees as specified from timetotime in UPR AS11 1.
HIGHER DOCTORATE DEGREES SUMMARY OF PRINCIPAL CHANGES General changes None Secton 3.2 Refer to text (Amendments to verson 03.0, UPR AS02 are shown n talcs.) 1 INTRODUCTION 1.1 The Unversty may award Hgher
More informationA Constant Factor Approximation for the Single Sink Edge Installation Problem
A Constant Factor Approxmaton for the Sngle Snk Edge Installaton Problem Sudpto Guha Adam Meyerson Kamesh Munagala Abstract We present the frst constant approxmaton to the sngle snk buyatbulk network
More informationForecasting the Demand of Emergency Supplies: Based on the CBR Theory and BP Neural Network
700 Proceedngs of the 8th Internatonal Conference on Innovaton & Management Forecastng the Demand of Emergency Supples: Based on the CBR Theory and BP Neural Network Fu Deqang, Lu Yun, L Changbng School
More informationOptimality in an Adverse Selection Insurance Economy. with Private Trading. November 2014
Optmalty n an Adverse Selecton Insurance Economy wth Prvate Tradng November 2014 Pamela Labade 1 Abstract Prvate tradng n an adverse selecton nsurance economy creates a pecunary externalty through the
More informationAn Empirical Study of Search Engine Advertising Effectiveness
An Emprcal Study of Search Engne Advertsng Effectveness Sanjog Msra, Smon School of Busness Unversty of Rochester Edeal Pnker, Smon School of Busness Unversty of Rochester Alan RmmKaufman, RmmKaufman
More informationNPAR TESTS. OneSample ChiSquare Test. Cell Specification. Observed Frequencies 1O i 6. Expected Frequencies 1EXP i 6
PAR TESTS If a WEIGHT varable s specfed, t s used to replcate a case as many tmes as ndcated by the weght value rounded to the nearest nteger. If the workspace requrements are exceeded and samplng has
More informationLecture 3: Force of Interest, Real Interest Rate, Annuity
Lecture 3: Force of Interest, Real Interest Rate, Annuty Goals: Study contnuous compoundng and force of nterest Dscuss real nterest rate Learn annutymmedate, and ts present value Study annutydue, and
More informationThe Greedy Method. Introduction. 0/1 Knapsack Problem
The Greedy Method Introducton We have completed data structures. We now are gong to look at algorthm desgn methods. Often we are lookng at optmzaton problems whose performance s exponental. For an optmzaton
More informationPrice Impact Asymmetry of Block Trades: An Institutional Trading Explanation
Prce Impact Asymmetry of Block Trades: An Insttutonal Tradng Explanaton Gdeon Saar 1 Frst Draft: Aprl 1997 Current verson: October 1999 1 Stern School of Busness, New York Unversty, 44 West Fourth Street,
More informationAbteilung für Stadt und Regionalentwicklung Department of Urban and Regional Development
Abtelung für Stadt und Regonalentwcklung Department of Urban and Regonal Development Gunther Maer, Alexander Kaufmann The Development of Computer Networks Frst Results from a Mcroeconomc Model SREDscusson
More informationApproximation algorithms for allocation problems: Improving the factor of 1 1/e
Approxmaton algorthms for allocaton problems: Improvng the factor of 1 1/e Urel Fege Mcrosoft Research Redmond, WA 98052 urfege@mcrosoft.com Jan Vondrák Prnceton Unversty Prnceton, NJ 08540 jvondrak@gmal.com
More informationDynamic OnlineAdvertising Auctions as Stochastic Scheduling
Dynamc OnlneAdvertsng Auctons as Stochastc Schedulng Isha Menache and Asuman Ozdaglar Massachusetts Insttute of Technology {sha,asuman}@mt.edu R. Srkant Unversty of Illnos at UrbanaChampagn rsrkant@llnos.edu
More informationWhen Network Effect Meets Congestion Effect: Leveraging Social Services for Wireless Services
When Network Effect Meets Congeston Effect: Leveragng Socal Servces for Wreless Servces aowen Gong School of Electrcal, Computer and Energy Engeerng Arzona State Unversty Tempe, AZ 8587, USA xgong9@asuedu
More informationPassive Filters. References: Barbow (pp 265275), Hayes & Horowitz (pp 3260), Rizzoni (Chap. 6)
Passve Flters eferences: Barbow (pp 6575), Hayes & Horowtz (pp 360), zzon (Chap. 6) Frequencyselectve or flter crcuts pass to the output only those nput sgnals that are n a desred range of frequences (called
More informationOptimality in an Adverse Selection Insurance Economy. with Private Trading. April 2015
Optmalty n an Adverse Selecton Insurance Economy wth Prvate Tradng Aprl 2015 Pamela Labade 1 Abstract An externalty s created n an adverse selecton nsurance economy because of the nteracton between prvate
More informationInequity Aversion and Individual Behavior in Public Good Games: An Experimental Investigation
Dscusson Paper No. 07034 Inequty Averson and Indvdual Behavor n Publc Good Games: An Expermental Investgaton Astrd Dannenberg, Thomas Rechmann, Bodo Sturm, and Carsten Vogt Dscusson Paper No. 07034 Inequty
More informationLeveraged Firms, Patent Licensing, and Limited Liability
Leveraged Frms, Patent Lcensng, and Lmted Lablty KuangCheng Andy Wang Socal Scence Dvson Center for General Educaton Chang Gung Unversty and YJe Wang Department of Economcs Natonal Dong Hwa Unversty
More informationProject Networks With MixedTime Constraints
Project Networs Wth MxedTme Constrants L Caccetta and B Wattananon Western Australan Centre of Excellence n Industral Optmsaton (WACEIO) Curtn Unversty of Technology GPO Box U1987 Perth Western Australa
More informationJoe Pimbley, unpublished, 2005. Yield Curve Calculations
Joe Pmbley, unpublshed, 005. Yeld Curve Calculatons Background: Everythng s dscount factors Yeld curve calculatons nclude valuaton of forward rate agreements (FRAs), swaps, nterest rate optons, and forward
More informationNBER WORKING PAPER SERIES POLITICAL ECONOMY IN A CHANGING WORLD. Daron Acemoglu Georgy Egorov Konstantin Sonin
NBER WORKING PAPER SERIES POLITICAL ECONOMY IN A CHANGING WORLD Daron Acemoglu Georgy Egorov Konstantn Sonn Workng Paper 19158 http://www.nber.org/papers/w19158 NATIONAL BUREAU OF ECONOMIC RESEARCH 1050
More informationAn InterestOriented Network Evolution Mechanism for Online Communities
An InterestOrented Network Evoluton Mechansm for Onlne Communtes Cahong Sun and Xaopng Yang School of Informaton, Renmn Unversty of Chna, Bejng 100872, P.R. Chna {chsun,yang}@ruc.edu.cn Abstract. Onlne
More informationChapter 11 Practice Problems Answers
Chapter 11 Practce Problems Answers 1. Would you be more wllng to lend to a frend f she put all of her lfe savngs nto her busness than you would f she had not done so? Why? Ths problem s ntended to make
More informationStaff Paper. Farm Savings Accounts: Examining Income Variability, Eligibility, and Benefits. Brent Gloy, Eddy LaDue, and Charles Cuykendall
SP 200502 August 2005 Staff Paper Department of Appled Economcs and Management Cornell Unversty, Ithaca, New York 148537801 USA Farm Savngs Accounts: Examnng Income Varablty, Elgblty, and Benefts Brent
More informationLecture 7 March 20, 2002
MIT 8.996: Topc n TCS: Internet Research Problems Sprng 2002 Lecture 7 March 20, 2002 Lecturer: Bran Dean Global Load Balancng Scrbe: John Kogel, Ben Leong In today s lecture, we dscuss global load balancng
More informationOptimal resource capacity management for stochastic networks
Submtted for publcaton. Optmal resource capacty management for stochastc networks A.B. Deker H. Mlton Stewart School of ISyE, Georga Insttute of Technology, Atlanta, GA 30332, ton.deker@sye.gatech.edu
More information