How Bad are Selfish Investments in Network Security?

Size: px
Start display at page:

Download "How Bad are Selfish Investments in Network Security?"

Transcription

1 1 How Bad are Selfsh Investments n Networ Securty? Lbn Jang, Venat Anantharam and Jean Walrand EECS Department, Unversty of Calforna, Bereley Abstract Internet securty does not only depend on the securty-related nvestments of ndvdual users, but also on how these users affect each other. In a non-cooperatve envronment, each user chooses a level of nvestment to mnmze hs own securty rs plus the cost of nvestment. Not surprsngly, ths selfsh behavor often results n undesrable securty degradaton of the overall system. In ths paper, (1) we frst characterze the prce of anarchy (POA) of networ securty under two models: an Effectve-nvestment model, and a Bad-traffc model. We gve nsght on how the POA depends on the networ topology, ndvdual users cost functons, and ther mutual nfluence. We also ntroduce the concept of weghted POA to bound the regon of all feasble payoffs. (2) In a repeated game, on the other hand, users have more ncentve to cooperate for ther long term nterests. We consder the socally best outcome that can be supported by the repeated game, and gve a rato between ths outcome and the socal optmum. (3) Next, we compare the benefts of mprovng securty technology or mprovng ncentves, and show that mprovng technology alone may not offset the effcency loss due to the lac of ncentves. (4) Fnally, we characterze the performance of correlated equlbrum (CE) n the securty game. Although the paper focuses on Internet securty, many results are generally applcable to games wth postve externaltes. Index Terms Internet securty, game theory, prce of anarchy, repeated game, correlated equlbrum, postve externalty I. INTRODUCTION Securty n a communcaton networ depends not only on the securty nvestment made by ndvdual users, but also on the nterdependency among them. If a careless user puts n lttle effort n protectng hs computer system, then t s easy for vruses to nfect ths computer and through t contnue to nfect others. On the contrary, f a user nvests more to protect hmself, then other users wll also beneft snce the chance of contagous nfecton s reduced. Defne each user s strategy as hs nvestment level, then each user s nvestment has a postve externalty on other users. Users n the Internet are heterogeneous. They have dfferent valuatons of securty and dfferent unt cost of nvestment. For example, government and commercal webstes usually prortze ther securty, snce securty breaches would lead to large fnancal losses or other consequences. They are also more wllng and effcent n mplementng securty measures. On the other hand, an ordnary computer user may care less about securty, and also may be less effcent n mprovng t due to the lac of awareness and expertse. There are many Ths wor s supported by the Natonal Scence Foundaton under Grant NeTS-FIND : Maret Enablng Networ Archtecture other users lyng between these two categores. If users are selfsh, some of them may choose to nvest more, whereas others may choose to free rde, that s, gven that the securty level s already good thans to the nvestment of others, such users mae no nvestment to save cost. However, f every user tends to rely on others, the resultng outcome may be far worse for all users. Ths s the free rdng problem n game theory as studed n, for example, [1]. Besdes user preferences, the networ topology, whch descrbes the (logcal) nterdependent relatonshp among dfferent users, s also mportant. For example, assume that n a local networ, user A drectly connected to the Internet. All other users are connected to A and exchange a large amount of traffc wth A. Intutvely, the securty level of A s partcularly mportant for the local networ snce A has the largest nfluence on other users. If A has a low valuaton of hs own securty, then t wll nvest lttle and the whole networ suffers. How the networ topology affects the effcency of selfsh nvestment n networ securty wll be one of our focuses. In ths paper, we study how networ topology, users preference and ther mutual nfluence affect networ securty n a non-cooperatve settng. In a one-shot game (.e., strategcform game), we derve the Prce of Anarchy (POA) [2] as a functon of the above factors. Here, POA s defned as the worst-case rato between the socal cost at a Nash Equlbrum (NE) and Socal Optmum (SO). Furthermore, we ntroduce the concept of Weghted-POA to bound the regons of all possble vectors of payoffs. In a repeated game, users have more ncentve to cooperate for ther long-term nterest. We study the socally best equlbrum n the repeated game, and compare t to the Socal Optmum. Next, we compare the benefts of mprovng securty technology or mprovng ncentves, and show that mprovng technology alone may not offset the effcency loss due to the lac of ncentves. Fnally, we consder the performance of correlated equlbrum (CE) (a more general noton than NE) n the securty game and characterze the best and worst CE s. Interestngly, some performance bounds of CE concde wth the POA of NE. A. Related Wors Varan studed the networ securty problem usng game theory n [1]. There, the effort of each user (or player) s assumed to be equally mportant to all other users, and the

2 2 networ topology s not taen nto account. Also, [1] s not focused on the effcency analyss (.e., POA). Prce of Anarchy (POA) [2], measurng the performance of the worst-case equlbrum compared to the Socal Optmum, has been studed n varous games n recent years, most of them wth negatve externalty. Roughgarden et al. shows that the POA s generally unbounded n the selfsh routng game [3], [4], where each user chooses some ln(s) to send hs traffc n order to mnmze hs congeston delay. Ozdaglar et al. derved the POA n a prce competton game n [5] and [6], where a number of networ servce provders choose ther prces to attract users and maxmze ther own revenues. In [7], Johar et al. studed the resource allocaton game, where each user bds for the resource to maxmze hs payoff, and showed that the POA s 3/4 assumng concave utlty functons. In all the above games, there s negatve externalty among the players: for example n the selfsh routng game, f a user sends hs traffc through a ln, other users sharng that ln wll suffer larger delays. On the contrary, n the networ securty game we study here, f a user ncreases hs nvestment, the securty level of other users wll mprove. In ths sense, t falls nto the category of games wth postve externaltes. Therefore, many results n ths paper may be applcable to other smlar scenaros. For example, assume that a number of servce provders (SP) buld networs whch are nterconnected. If a SP nvests to upgrade her own networ, the performance of the whole networ mproves and may brng more revenue to all SP s. In [8], Aspnes et al. formulated an noculaton game and studed ts POA. There, each player n the networ decdes whether to nstall ant-vrus software to avod nfecton. Dfferent from our wor, [8] has assumed bnary decsons and the same cost functon for all players. II. PRICE OF ANARCHY (POA) IN THE STRATEGIC-FORM GAME Assume there are n players. The securty nvestment (or effort, we use them nterchangeably) of player s x 0. Ths ncludes both money (e.g., for purchasng antvrus software) and tme/energy (e.g., for system scannng, patchng). So ths s not a one-tme nvestment. The cost per unt of nvestment s c > 0. Denote f (x) as player s securty rs : the loss due to attacs or vrus nfectons from the networ, where x s the vector of nvestments by all players. f (x) s decreasng n each x j (thus reflectng postve externalty) and non-negatve. We assume that t s convex and dfferentable, and that f (x = 0) > 0 s fnte. Then the cost functon of player s g (x) := f (x) + c x (1) Note that the functon f ( ) s generally dfferent for dfferent players. In a Nash game, player chooses hs nvestment x 0 to mnmze g (x). Frst, we prove n Appendx A1 that Proposton 1: There exsts some pure-strategy Nash Equlbrum (NE) n ths game. In ths paper we consder pure-strategy NE. Denote x as the vector of nvestments at some NE, and x as the vector of nvestments at Socal Optmum (SO). Also denote the unt cost vector c = (c 1,c 2,...,c n ) T. We am to fnd the POA, Q, whch upper-bounds ρ( x), where ρ( x) := G( x) G = g ( x) g (x ) s the rato between the socal cost at the NE x and at the socal optmum. For convenence, sometmes we smply wrte ρ( x) as ρ f there s no confuson. Before gettng to the dervaton, we llustrate the POA n a smple example. Assume there are 2 players, wth ther nvestments denoted as x 1 0 and x 2 0. The cost functon s g (x) = f(y) + x, = 1,2, where f(y) s the securty rs of both players, and y = x 1 + x 2 s the total nvestment. Assume that f(y) s non-negatve, decreasng, convex, and satsfes f(y) 0 when y. The socal cost s G(x) = g 1 (x) + g 2 (x) = 2 f(y) + y. Fg f (y) A NE ȳ 2*f (y) C B SO y D POA n a smple example y = x 1 + x 2 At a NE x, g( x) = f ( x 1 + x 2 ) + 1 = 0, = 1,2. Denote ȳ = x 1 + x 2, then f (ȳ) = 1. Ths s shown n Fg 1. Then, the socal cost Ḡ = 2 f(ȳ) + ȳ. Note that ȳ ( f (z))dz = f(ȳ) f( ) = f(ȳ) (snce f(y) 0 as y ), therefore n Fg 1, 2 f(ȳ) s the area B + C + D, and Ḡ s equal to the area of A + (B + C + D). At SO (Socal Optmum), on the other hand, the total nvestment y satsfes 2f (y ) = 1. Usng a smlar argument as before, G = 2f(y )+y s equal to the area of (A+B)+D. Then, the rato Ḡ/G = [A+(B+C+D)]/[(A+B)+D] (B + C)/B 2. We wll show later that ths upper bound s tght. So the POA s 2. Now we analyze the POA wth the general cost functon (1). In some sense, t s a generalzaton of the above example. Lemma 1: For any NE x, ρ( x) satsfes ρ( x) max{1,max {( f ( x) f ( x) x )/c }} (2) Note that ( x ) s the margnal beneft to the securty of all users by ncreasng x at the NE; whereas c s the margnal cost of ncreasng x. The second term n the RHS (rght-hand-sde) of (2) s the maxmal rato between these two.

3 3 Proof: At NE, { f( x) = c f x > 0 f ( x) c f x = 0 By defnton, ρ( x) = G( x) G = f ( x) + c T x f (x ) + c T x Snce f ( ) s convex for all. Then f ( x) f (x )+( x x ) T f ( x). So ρ ( x x ) T f ( x) + c T x + f (x ) f (x ) + c T x = x T f ( x) + x T [c + f ( x)] + f (x ) f (x ) + c T x Note that x T [c + f ( x)] = x [c + f ( x) ] There are two possbltes for every player : (a) If x = 0, then x [c + f ( x) ] = 0. (b) If x > 0, then f( x) = c. Snce f ( x) 0 for all, then f ( x) c, so x [c + f ( x) ] 0. As a result, (3) ρ( x) x T f ( x) + f (x ) f (x ) + c T x (4) () If x = 0 for all, then the RHS s 1, so ρ( x) 1. Snce ρ cannot be smaller than 1, we have ρ = 1. () If not all x = 0, then ct x > 0. Note that the RHS of (4) s not less than 1, by the defnton of ρ( x). So, f we subtract f (x ) (non-negatve) from both the numerator and the denomnator, the resultng rato upper-bounds the RHS. That s, ρ( x) x T f ( x) c T x f ( x) max {( f ( x) x )/c } where x s the th element of the vector f ( x). Combnng case () and (), the proof s completed. In the followng, we gve two models of the networ securty game. Each model defnes a concrete form of f ( ). They are formulated to capture the ey parameters of the system whle beng amenable to mathematcal analyss. A. Effectve-nvestment ( EI ) model Generalzng [1], we consder an Effectve-nvestment (EI) model. In ths model, the securty rs of player depends on an effectve nvestment, whch we assume s a lnear combnaton of the nvestments of hmself and other players. Specfcally, let p ( n j=1 α jz j ) be the probablty that player s nfected by a vrus (or suffers an attac), gven the amount of efforts every player puts n. The effort of player j, z j, s weghted by α j, reflectng the mportance of player j to player. Let v be the cost of player f he suffers an attac; and c be the cost per unt of effort by player. Then, the total cost of player s g (z) = v p ( n j=1 α jz j ) + c z. For convenence, we normalze the expresson n the followng way. Let the normalzed effort be x := c z,. Then g (x) = v p ( n α j = v p ( α c c j x j ) + x n j=1 β jx j ) + x j=1 where β j := c α j α c j (so β = 1). We call β j the relatve mportance of player j to player. Defne the functon V (y) = v p ( α c y), where y s a dummy varable. Then g (x) = f (x) + x, where f (x) = V ( n j=1 β jx j ) (5) Assume that p ( ) s decreasng, non-negatve, convex and dfferentable. Then V ( ) also has these propertes. Proposton 2: In the EI model defned above, ρ max {1 + : β }. Furthermore, the bound s tght. Proof: Let x be some NE. Denote h := f ( x). Then the th element of h h V (Èn j=1 βj xj) x = = β V ( n j=1 β j x j ) V (Èn j=1 βj xj) From (3), we have = β V ( n j=1 β j x j ) = V ( n j=1 β j x j ) 1. So h β. Plug ths nto (2), we obtan an upper bound of ρ: ρ max{1,max { h }} Q := max {1 + β } (6) : whch completes the proof. (6) gves some nterestng nsght nto the game. Snce β s player s relatve mportance to player, then 1 + : β = β s player s relatve mportance to the socety. (6) shows that the POA s bounded by the maxmal socal mportance among the players. Interestngly, the bound does not depend on the specfc form of V ( ) as long as t s convex, decreasng and non-negatve. It also provdes a smple way to compute POA under the model. We defne a dependency graph as n Fg. 2, where each vertex stands for a player, and there s a drected edge from to f β > 0. In Fg. 2, player 3 has the hghest socal mportance, and ρ 1 + ( ) = 3.2. In another specal case, f for each par (,), ether β = 1 or β = 0, then the POA s bounded by the maxmum out-degree of the graph plus 1. If all players are equally mportant to each other,.e., β = 1,,, then ρ n (.e., POA s the number of players). Ths also explans why the POA s 2 n the example consdered n Fg 1. The followng s a worst case scenaro that shows the bound s tght. Assume there are n players, n 2. β = 1,,; and for all, V (y ) = [(1 ǫ)(1 y )] +, where [ ] + means postve part, y = n j=1 β jx j = n j=1 x j, ǫ > 0 but s very small. 1 Gven x = 0, g (x) = [(1 ǫ)(1 x )] + +x = (1 ǫ)+ ǫ x when x 1, so the best response for player s to let 1 Although V (y ) s not dfferentable at y = 1, t can be approxmated by a dfferentable functon arbtrarly closely, such as the result of the example s not affected.

4 Fg. 2. Dependency Graph and the Prce of Anarchy (In ths fgure, ρ 1 + ( ) = 3.2) x = 0. Therefore, x = 0, s a NE, and the resultng socal cost G( x) = [V (0) + x ] = (1 ǫ)n. Snce the socal cost s G(x) = n [(1 ǫ)(1 x )] + + x, the socal optmum s attaned when x = 1 (snce n(1 ǫ) > 1). Then, G(x ) = 1. Therefore ρ = (1 ǫ)n n when ǫ 0. When ǫ = 0, x = 0, s stll a NE. In that case ρ = n. B. Bad-traffc ( BT ) Model Next, we consder a model whch s based on the amount of bad traffc (e.g., traffc that causes vrus nfecton) from one player to another. Let r be the total rate of traffc from to. How much traffc n r wll do harm to player depends on the nvestments of both and. So denote φ, (x,x ) as the probablty that player s traffc does harm to player. Clearly φ, (, ) s a non-negatve, decreasng functon. We also assume t s convex and dfferentable. Then, the rate at whch player s nfected by the traffc from player s r φ, (x,x ). Let v be player s loss when t s nfected by a vrus, then g (x) = f (x)+x, where the nvestment x has been normalzed such that ts coeffcent (the unt cost) s 1, and f (x) = v r φ, (x,x ) If the frewall of each player s symmetrc (.e., t treats the ncomng and outgong traffc n the same way), then t s reasonable to assume that φ, (x,x ) = φ, (x,x ). v Proposton 3: In the BT model, ρ 1+max r j (,j): j v jr j. The bound s also tght. Proof: Let h := f ( x) for some NE x. Then the j-th element h j = = j We have q j := = f ( x) = j v r j φ j, ( x j, x ) f ( x) j f j( x) = f ( x) f j( x) + v j j v r j φ j,( x j, x ) j v jr j φ j,( x j, x ) j r j φ,j ( x, x j ) j v φ j,( x j, x ) r j v j j r j φ,j( x, xj) v r j max : j v j r j where the 3rd equalty holds because φ,j (x,x j ) = φ j, (x j,x ) by assumpton. From (3), we now that fj( x) 1. So h j = (1 + q j ) f j( x) v r j (1 + max ) : j v j r j Accordng to (2), t follows that v r j ρ max{1,max{ h j }} Q := 1 + max (7) j (,j): j v j r j whch completes the proof. Note that v r j s the damage to player caused by player j f player s nfected by all the traffc sent by j, and v j r j s the damage to player j caused by player f player j s nfected by all the traffc sent by. Therefore, (7) means that the POA s upper-bounded by the maxmum mbalance of the networ. As a specal case, f each par of the networ s balanced,.e., v r j = v j r j,,j, then ρ 2! To show the bound s tght, we can use a smlar example as n secton II-A. Let there be two players, and assume v 1 r 21 = v 1 r 12 = 1; φ 1,2 (x 1,x 2 ) = (1 ǫ)(1 x 1 x 2 ) +. Then t becomes the same as the prevous example when n = 2. Therefore ρ 2 as ǫ 0. And ρ = 2 when ǫ = 0. Note that when the networ becomes larger, the mbalance between a certan par of players becomes less mportant. Thus ρ may be much less than the worst case bound n large networs due to the averagng effect. III. BOUNDING THE PAYOFF REGIONS USING WEIGHTED POA So far, the research on POA n varous games has largely focused on the worst-case rato between the socal cost (or welfare) acheved at the Nash Equlbra and Socal Optmum. Gven one of them, the range of the other s bounded. However, ths s only one-dmensonal nformaton. In any mult-player game, the players payoffs form a vector whch s multdmensonal. Suppose that a NE payoff vector s nown, t would be nterestng to characterze or bound the regon of all feasble vectors of ndvdual payoffs, sometmes even wthout nowng the exact cost functons. Ths regon gves much more nformaton than solely the socal optmum, because t characterzes the tradeoff between effcency and farness among dfferent players. Conversely, gven any feasble payoff vector, t s also nterestng to bound the regon of the possble payoff vectors at all Nash Equlbra. We show that ths can be done by generalzng POA to the concept of Weghted POA, Q w, whch s an upper bound of ρ w ( x), where ρ w ( x) := G w( x) G w = w g ( x) w g (x w) Here, w R n ++ s a weght vector, x s the vector of nvestments at a NE of the orgnal game; whereas x w mnmzes a weghted socal cost G w (x) := w g (x). To obtan Q w, consder a modfed game where the cost functon of player s ĝ (x) := ˆf (x) + ĉ x = w g (x) = w f (x) + w c x

5 5 Note that n ths game, the NE strateges are the same as the orgnal game: gven any x, player s best response remans the same (snce hs cost functon s only multpled by a constant). So the two games are strategcally equvalent, and thus have the same NE s. As a result, the weghted POA Q w of the orgnal game s exactly the POA n the modfed game (Note the defnton of x w). Applyng (2) to the modfed game, we have ρ w ( x) max{1,max {( = max{1, max {( ˆf ( x) x )/ĉ }} w f ( x) x )/(w c )}}(8) Then, one can easly obtan the weghted POA for the two models n the last secton. Proposton 4: In the EI model, ρ w Q w := max {1 + : w β w } (9) In the BT model, w v r j ρ w Q w := 1 + max (10) (,j): j w j v j r j Snce ρ w ( x) = Gw( x) G =È w g( x) w È w g(x w ) Q w, we have w g (x w) w g ( x)/q w. Notce that x w mnmzes G w (x) = w g (x), so for any feasble x, w g (x) w g (x w) w g ( x)/q w Then we have Proposton 5: Gven any NE payoff vector ḡ, then any feasble payoff vector g must be wthn the regon B := {g w T g w T ḡ/q w, w R n ++} Conversely, gven any feasble payoff vector g, any possble NE payoff vector ḡ s n the regon B := {ḡ w T ḡ w T g Q w, w R n ++} In other words, the Pareto fronter of B lower-bounds the Pareto fronter of the feasble regon of g. (A smlar statement can be sad for B.) As an llustratng example, consder the EI model, where the cost functon of player s n the form of g (x) = V ( n j=1 β jx j )+x. Assume there are two players n the game, and β 11 = β 22 = 1, β 12 = β 21 = 0.2. Also assume that g (x) = (1 2 j=1 β jx ) + +x, for = 1,2. It s easy to verfy that x = 0, = 1,2 s a NE, and g 1 ( x) = g 2 ( x) = 1. One can further fnd that the boundary (Pareto fronter) of the feasble payoff regon n ths example s composed of the two axes and the followng lne segments (the computaton s omtted): { g2 = 5 (g ) g 1 [0, 5 6 ] g 2 = 0.2 (g ) g 1 [0,5] whch s the dashed lne n Fg. 3. By Proposton 5, for every weght vector w, there s a straght lne that lower-bounds the feasble payoff regon. After plottng the lower bounds for many dfferent w s, we obtan a bound for the feasble payoff regon (Fg 3). Note that the bound only depends on the coeffcents β j s, but not the specfc form of V 1 ( ) and V 2 ( ). We see that the feasble regon s ndeed wthn the bound g (x,x ) Fg An NE Feasble regon g 1 (x 1,x 2 ) Boundng the feasble regon usng weghted POA IV. REPEATED GAME Unle the strategc-form game, n repeated games the players have more ncentves to cooperate for ther long term nterests. In ths secton we consder the performance gan provded by the repeated game of selfsh nvestments n securty. The Fol Theorem [9] provdes a Subgame Perfect Equlbrum (SPE) n a repeated game wth dscounted costs when the dscount factor suffcently close to 1, to support any cost vector that s Pareto-domnated by the reservaton cost vector g. The th element of g, g, s defned as g := mn x 0 g (x) gven that x j = 0, j and we denote x as a mnmzer. g = g (x = x,x = 0) s the mnmal cost achevable by player when other players are punshng hm by mang mnmal nvestments 0. Wthout loss of generalty, we assume that g (x) = f (x)+ x, nstead of g (x) = f (x)+c x n (1). Ths can be done by normalzng the nvestment and re-defnng the functon f (x). For smplcty, we mae some addtonal assumptons n ths secton: 1) f (x) (and g (x)) s strctly convex n x f x = 0. So x s unque. g 2) (0) < 0 for all. So, x > 0. 3) For each player, f (x) s strctly decreasng wth x j for some j. That s, postve externalty exsts. By assumpton 2 and 3, we have g (x) < g (x = x,x = 0) = g,. Therefore g(x) < g s feasble. A Performance Bound of the best SPE Accordng to the Fol Theorem [9], any feasble vector g < g can be supported by a SPE. So the set of SPE s qute large n general. By negotatng wth each other, the players can

6 6 agree on some SPE. In ths secton, we are nterested n the performance of the socally best SPE that can be supported, that s, the SPE wth the mnmum socal cost (denoted as G E ). Such a SPE s optmal for the socety, provded that t s also ratonal for ndvdual players. We wll compare t to the socal optmum by consderng the performance rato γ = G E /G, where G s the optmal socal cost, and G E = nf x 0 g (x) s.t. g (x) < g, Snce g ( ) s convex by assumpton, due to contnuty, G E = mn x 0 g (x) s.t. g (x) g, (11) (12) where g (x) g s the ratonalty constrant for each player. Denote by x E a soluton of (12). Then g (x E ) = G E. Recall that g (x) = f (x) + x, where the nvestment x has been normalzed such that ts coeffcent (unt cost) s 1. Then, to solve (12), we form a partal Lagrangan L(x,λ ) := g (x) + λ [g (x) g ] = (1 + λ )g (x) λ g and pose the problem max λ 0 mn x 0 L(x,λ ). Let λ be the vector of dual varables when the problem s solved (.e., when the optmal soluton x E s reached). Then dfferentatng L(x,λ ) n terms of x, we have the optmalty condton { (1 + λ )[ f (x E) ] = 1 + λ f x E, > 0 (1 + λ )[ f (x E) (13) ] 1 + λ f x E, = 0 Proposton 6: The performance rato γ s upper-bounded by γ = G E /G max {1 + λ }. (The proof s gven n Appendx A2.) Ths result can be understood as follows: f λ = 0 for all, then all the ncentve-compatblty constrants are not actve at the optmal pont of (12). So, ndvdual ratonalty s not a constranng factor for achevng the socal optmum. In ths case, γ = 1, meanng that the best SPE acheves the socal optmal. But f λ > 0 for some, the ndvdual ratonalty of player prevent the system from achevng socal optmum. Larger λ leads to a poorer performance bound on the best SPE relatve to SO. Proposton 6 gves an upper bound on γ assumng the general cost functon g (x) = f (x) + x. Although t s applcable to the two specfc models ntroduced before, t s not explctly related to the networ parameters. In the followng, we gve an explct bound for the EI model. Proposton 7: In the EI model where g (x) = V ( n j=1 β jx j ) + x, γ s bounded by γ mn{max,j, β β j,q} where Q = max {1 + : β }. The part γ Q s straghtforward: snce the set of SPE ncludes all NE s, the best SPE must be better than the worst NE. The other part s derved from Proposton 6 (ts proof s ncluded n Appendx A3). β Note that the nequalty γ max,j, β j may not gve a tght bound, especally when β j s very small for some j,. But n the followng smple example, t s tght and shows that the best SPE acheves the socal optmum. Assume n players, and β j = 1,,j. Then, the POA n the strategc-form game s ρ Q = n accordng to (6). In the repeated game, β however, the performance rato γ max m,j,m β jm = 1 (.e., socal optmum s acheved). Ths llustrates the performance gan resultng from the repeated game. It should be noted that, however, although repeated games can provde much better performance, they usually requre more communcaton and coordnaton among the players than strategc-form games. V. IMPROVEMENT OF TECHNOLOGY Recall that the general cost functon of player s g (x) = f (x) + x. (14). Now assume that the securty technology has mproved. We would le to study how effectve s technology mprovement compared to the mprovement of ncentves. Assume that the new cost functon of player s g (x) = f (a x) + x,a > 1. (15) Ths means that the effectveness of the nvestment vector x has mproved by a tmes (.e., the rs decreases faster wth x than before). Equvalently, f we defne x = a x, then (15) s g (x) = f (x )+x /a, whch means a decrease of unt cost f we regard x as the nvestment. Proposton 8: Denote by G the optmal socal cost wth cost functons (14), and by G the optmal socal cost wth cost functons (15). Then, G G G /a. That s, the optmal socal cost decreases but cannot decrease more than a tmes. Proof: Frst, for all x, g (x) g (x). Therefore G G. Let the optmal nvestment vector wth the mproved cost functons be x. We have g (a x ) = f (a x )+a x. Also, g ( x ) = f (a x )+ x. Then, a g ( x ) = a f (a x )+a x g (a x ), because f ( ) s non-negatve and a > 1. Therefore, we have a g ( x ) = a G G(a x ) G(x ) = G, snce x mnmzes G(x) = g (x). Ths completes the proof. Here we have seen that the optmal socal cost (after technology mproved a tmes) s at least a fracton of 1/a of the socal optmum before. On the other hand, we have the followng about the POA after technology mprovement. Proposton 9: The POA of the networ securty game wth mproved technology (.e., cost functon (15)) does not change n the EI model and the BT model. (That s, the expressons of POA are the same as those gven n Proposton 2 and 3.) Proof: The POA n the EI model only depends on the values of β j s, whch does not change wth the new cost functons. To see ths, note that g (x) = f (a x) + x = V (a β j x j ) + x. j

7 7 Defne the functon Ṽ(y) = V (a y),, where y s a dummy varable, then g (x) = Ṽ( j β jx j )+x, where Ṽ( ) s stll convex, decreasng and non-negatve. So the β j values do not change. By Proposton 2, the POA remans the same. In the BT model, defne φ, (x,x ) := φ, (a x,a x ), then φ, (x,x ) s stll non-negatve, decreasng and convex, and φ, (x,x ) = φ, (x,x ). So by Proposton 3, the POA has the same expresson as before. To compare the effect of ncentve mprovement and technology mprovement, consder the followng two optons to mprove the networ securty. 1) Wth the current technology, deploy proper ncentvzng mechansms (.e., stc and carrot ) to acheve the socal optmum. 2) All players upgrade to the new technology, wthout solvng the ncentve problem. Wth opton 1, the resultng socal cost s G. Wth opton 2, the socal cost s G( x NE ), where G( ) = g ( ) s the socal cost functon after technology mprovement, wth g ( ) defned n (15), and x NE s a NE n the new game. Defne ρ( x NE ) := G( x NE )/ G, then the rato between the socal costs wth opton 2 and opton 1 s G( x NE )/G = ρ( x NE ) G /G ρ( x NE )/a where the last step follows from Proposton 8. Also, by Proposton 9, n the EI or BT model, ρ( x NE ) s equal to the POA shown n Prop. 2 and 3 n the worst case. For example, assume the EI model wth β j = 1,,j. Then n the worst case, ρ( x NE ) = n. When the number of players n s large, G( x NE )/G may be much larger than 1. From ths dscusson, we see that the technology mprovement may not offset the negatve effect of the lac of ncentves, and solvng the ncentve problem may be more mportant than merely countng on new technologes. VI. CORRELATED EQUILIBRIUM (CE) Correlated equlbrum (CE) [10] s a more general noton of equlbrum whch ncludes the set of NE. In ths secton we consder the performance bounds of CE. Conceptually, one may thn of a CE as beng mplemented wth the help of a medator [11]. Let µ be a probablty dstrbuton over the strategy profles x. Frst the medator selects a strategy profle x wth probablty µ(x). Then the medator confdentally recommends to each player the component x n ths strategy profle. Each player s free to choose whether to obey the medator s recommendatons. µ s a CE ff t would be a Nash equlbrum for all players to obey the medator s recommendatons. Note that gven a recommended x, player only nows µ(x x ) (.e., the condtonal dstrbuton of other players recommended strateges gven x ). Then n a CE, x should be a best response to the randomzed strateges of other players wth dstrbuton µ(x x ). CE can also be mplemented wth a pre-play meetng of the players [9], where they decde the CE µ they wll play. Later they use a devce whch generates strategy profles x wth the dstrbuton µ and separately tells the th component, x, to player. Interestngly, CE can also arse from smple and natural dynamcs (wthout coordnaton va a medator or a preplay meetng). References [12] and [13] showed that n an nfnte repeated game, f each player observes the hstory of other players actons, and decdes hs acton n each perod based on a regret-mnmzng crteron, then the emprcal frequency of the players actons converge to some CE. In these dynamcs, each player does not need to now other players cost functons, but only ther prevous actons [12][13]. (Specfcally n the networ securty game, observng the actons of hs neghbors s suffcent.) Ths s very natural snce n practce, dfferent players tend to adjust ther nvestments based on ther observaton of others nvestments. For smplcty, n ths paper we focus on CE whose support s on a dscrete set of strategy profles. We call such a CE a dscrete CE. More formally, µ s a dscrete CE ff (1) t s a CE; and (2) the dstrbuton µ only assgns postve probabltes to x S µ, where S µ, the support of the dstrbuton µ, s a dscrete set of strategy profles. That s, S µ = {x R n +, = 1,2,...,M µ }, where x denotes a strategy profle, M µ < s the cardnalty of S µ and x S µ µ(x) = 1. (But the strategy set of each player s stll R +.) Dscrete CE exsts n the securty game snce a pure-strategy NE s clearly a dscrete CE, and pure-strategy NE exsts (Proposton 1). Also, any convex combnaton of multple pure-strategy NE s s a dscrete CE. (An example of dscrete CE whch s not a pure-strategy NE or a convex combnaton of pure-strategy NE s s gven n Appendx A3 of [16], due to the lmt of space.) We frst wrte down the condtons for a dscrete CE wth the general cost functon g (x) = f (x) + x,. (16) If µ s a dscrete CE, then for any x wth a postve margnal probablty (.e., (x, x ) S µ for some x ), x s a best response to the condtonal dstrbuton µ(x x ),.e., x arg mn x R + x [f (x,x )+x ]µ(x x ). (Recall that player can choose hs nvestment from R +.) Snce the objectve functon n the rght-hand-sde s convex and dfferentable n x, the frst-order condton s { f (x,x ) x µ(x x ) + 1 = 0 f x > 0 f (x,x ) (17) x µ(x x ) f x = 0 where f (x,x ) x µ(x x ) can also be smply wrtten as E µ ( f(x,x ) x ). A. How good can a CE get? The frst queston we would le to understand s: does there always exst a CE that acheves the socal optmum (SO) n the securty game? The answer s generally not. If a CE acheves SO, then the CE should have probablty 1 on the set of x that mnmzes the socal cost. For convenence, assume there s a unque x that mnmzes the socal cost. In other words, each tme, the medator chooses x and recommends x to player. If x > 0, then t satsfes f (x ) = 1

8 8 Snce f (x ) f(x ), we have g(x ) = f(x ) If the nequalty s strct, then player has ncentve to nvest less than x. Therefore n general, CE cannot acheve SO n ths game. But, a CE can be better than all NE s n ths game. Due to the lmt of space, an example s gven n Appendx A3 of [16]. The example s dfferent n nature from that n [10] snce each player can choose hs nvestment from R +. B. The worst-case dscrete CE As mentoned before, CE can result from smple and natural dynamcs n an nfntely repeated game wthout coordnaton. But le NE s, the resultng CE may not be effcent. In ths secton, we consder the POA of dscrete CE, whch s defned as the performance rato of the worst dscrete CE compared to the SO. In the EI model and BT model, we show that the POA of dscrete CE s dentcal to that of pure-strategy NE derved before, although the set of dscrete CE s s larger than the set of pure-strategy NE s n general. Frst, the followng lemma can be vewed as a generalzaton of Lemma 1. Lemma 2: Wth the general cost functon (16), the POA of dscrete CE, denoted as ρ CE, satsfes ρ CE max{max{1,max [E µ( µ C D f (x) x )]}} where C D s the set of dscrete CE s, the dstrbuton µ defnes a dscrete CE, and the expectaton s taen over the dstrbuton µ. Although the dstrbuton µ seems qute complcated, the proof of Lemma 2 (shown n Appendx A4) s smlar to that of Lemma 1. Proposton 10: In the EI model and the BT model, the POA of dscrete CE s the same as the POA of pure-strategy NE. That s, n the EI model, ρ CE max {1 + β }, : and n the BT model, v r j ρ CE (1 + max ). (,j): j v j r j The proof s ncluded n Appendx A5. VII. CONCLUSIONS We have studed the equlbrum performance of the networ securty game. Our model explctly consdered the networ topology, players dfferent cost functons, and ther relatve mportance to each other. We showed that n the strategcform game, the POA can be very large and tends to ncrease wth the networ sze, and the dependency and mbalance among the players. Ths ndcates severe effcency problems n selfsh nvestment. Not surprsngly, the best equlbrum n the repeated games usually gves much better performance, and t s possble to acheve socal optmum f that does not conflct wth ndvdual nterests. Implementng the strateges supportng an SPE n a repeated game, however, needs more communcatons and cooperaton among the players. We have compared the benefts of mprovng securty technology and mprovng ncentves. In partcular, we show that the POA of pure-strategy NE s nvarant wth the mprovement of technology, under the EI model and the BT model. So, mprovng technology alone may not offset the effcency loss due to the lac of ncentves. Fnally, we have studed the performance of correlated equlbrum (CE). We have shown that although CE cannot acheve SO n general, t can be much better than all pure-strategy NE s. In terms of the worst-case bounds, the POA s of dscrete CE are the same as the POA s of pure-strategy NE under the EI model and the BT model. Gven that the POA s large n many scenaros, a natural queston s how to desgn mechansms to mprove the nvestment ncentves for better networ securty. Ths has not been a focus of ths paper, and we would le to study t more n the future. Possble remedes for the problem nclude new protocols, prcng mechansms, regulatons and cyber-nsurance. For example, a conceptually smple scheme wth a regulator s called due care (see, for example, [1]). In ths scheme, each player s requred to nvest no less than x, the nvestment n the socally optmal confguraton. Otherwse, he s punshed accordng to the negatve effect he causes to other players. Although ths scheme can n prncple acheve the socal optmum, t s not easy to mplement n practce. Frstly, the optmal level of nvestment by each user s not easy to now unless a large amount of networ nformaton s collected. Secondly, to enforce the scheme, the regulator needs to montor the players actual nvestments, whch causes prvacy concerns. In the future, we would le to further explore effectve and practcal schemes to mprove the effcency of securty nvestments. REFERENCES [1] H. R. Varan, System Relablty and Free Rdng, Worshop on Economcs and Informaton Securty, [2] E. Koutsoupas, C. H. Papadmtrou, Worst-case equlbra, Annual Symposum on Theoretcal Aspects of Computer Scence, [3] T. Roughgarden, É Tardos, How bad s selfsh routng, Journal of the ACM, [4] T. Roughgarden, The prce of anarchy s ndependent of the networ topology, Proceedngs of the thry-fourth annual ACM symposum on Theory of computng, 2002, pp [5] D. Acemoglu and A. Ozdaglar, Competton and Effcency n Congested Marets, Mathematcs of Operatons Research, [6] A. Ozdaglar, Prce Competton wth Elastc Traffc, LIDS report, MIT, [7] R. Johar and J.N. Tstsls, Effcency loss n a networ resource allocaton game, Mathematcs of Operatons Research, 29(3): , [8] J. Aspnes, K. Chang, A. Yampolsy, Inoculaton Strateges for Vctms of Vruses and the Sum-of-Squares Partton Problem, Proceedngs of the sxteenth annual ACM-SIAM symposum on Dscrete algorthms, pp , [9] D. Fudenberg, J. Trole, Game Theory, MIT Press, Cambrdge, [10] R. J. Aumann, Subjectvty and Correlaton n Randomzed strateges, Journal of Mathematcal Economcs, 1:67-96, [11] R. B. Myerson, Dual Reducton and Elementary Games, Games and Economc Behavor, vol. 21, no. 1-2, pp , [12] D. Foster, R. Vohra, Calbrated Learnng and Correlated Equlbrum, Games and Economc Behavor, 21:40-55, [13] G. Stoltz, G. Lugos, Learnng Correlated Equlbra n Games wth Compact Sets of Strateges, Games and Economc Behavor, vol. 59, no. 1, pp , Aprl [14] J. B. Rosen, Exstence and Unqueness of Equlbrum Ponts for Concave N-Person Games, Econometrca, 33, , July 1965.

9 9 [15] S. Boyd and L. Vandenberghe, Convex Optmzaton, Cambrdge Unversty Press, [16] L. Jang, V. Anantharam, J. Walrand, How Bad are Selfsh Investments n Networ Securty? Techncal Report, UC Bereley, Dec URL: html A1. Proof of Proposton 1 APPENDIX Consder player s set of best responses, BR (x ), to x 0. Defne x,max := [f (0) + ǫ]/c where ǫ > 0, then due to convexty of f (x) n x, we have f (x = 0,x ) f (x = x,max,x ) x,max ( f (x,max,x ) ) = f (0) + ǫ c ( f (x,max,x ) ). Snce f (x = 0,x ) f (0), and f (x = x,max,x ) 0, t follows that f (0) f (0) + ǫ c ( f (x,max,x ) ) whch means that f(x,max,x ) + c > 0. So, BR (x ) [0,x,max ]. Let x max = max x,max. Consder a modfed game where the strategy set of each player s restrcted to [0,x max ]. Snce the set s compact and convex, and the cost functon s convex, therefore ths s a convex game and has some pure-strategy NE [14], denoted as x. Gven x, x s also a best response n the strategy set [0, ), because the best response cannot be larger than x max as shown above. Therefore, x s also a pure-strategy NE n the orgnal game. A2. Proof of Proposton 6 Consder the followng convex optmzaton problem parametrzed by t = (t 1,t 2,...,t n ), wth optmal value V (t): V (t) = mn x 0 g (x) s.t. g (x) t, (18) When t = g, t s the same as problem (12) that gves the socal cost of the best SPE; when t = g, t gves the same soluton as the Socal Optmum. Accordng to the theory of convex optmzaton ([15], page 250), the value functon V (t) s convex n t. Therefore, V (g) V (g ) V (g)(g g ) Also, V (g) = λ, where λ s the vector of dual varables when the problem wth t = g s solved. So, Then G E = V (g) V (g ) + λ T (g g) = G + λ T (g g) G + λ T g γ = G E G whch completes the proof. 1 + λt g 1 T g max {1 + λ } A3. Proof of Proposton 7 It s useful to frst gve a setch of the proof before gong to the detals. Roughly, the KKT condton [15] (for the best SPE), as n equaton (13), s (1 + λ )[ f (x E) ] = 1 + λ, (except for some corner cases whch wll be taen care of by Lemma 4). Wthout consderng the corner cases, we have the followng by nequalty (19): γ max,j 1 + λ = max 1 + λ j,j max { f (x E ) / f (x E ) },j, (1 + λ )[ f (x E) ] (1 + λ )[ f (x E) ] whch s Proposton 11. Then by pluggng n f ( ) of the EI model, Proposton 7 mmedately follows. Now we begn the detaled proof. As assumed n secton 4, g(x) < g s feasble. Lemma 3: If g(x) < g s feasble, then at the optmal soluton of problem (12), at least one dual varable s 0. That s, 0 such that λ 0 = 0. Proof: Suppose λ > 0,. Then all constrants n (12) are actve. As a result, G E = g. Snce x such that g(x) < g, then for ths x, g (x) < g. x s a feasble pont for (12), so G E g (x) < g, whch contradcts G E = g. From Proposton 6, we need to bound max {1+λ }. Snce 1 + λ 1,, and 1 + λ 0 = 1 (by Lemma 3), t s easy to see that γ max {1 + λ 1 + λ } = max (19),j 1 + λ j Before movng to Proposton 11, we need another observaton: Lemma 4: If for some, (1 + λ )[ f (x E) ] < 1 + λ, then λ = 0. Proof: From (13), t follows that x E, = 0. Snce (1+ λ )[ f (x E) ] < 1 + λ, and every term on the left s nonnegatve, we have (1 + λ )[ f (x E ) ] < 1 + λ That s, f(xe) + 1 = g(xe) > 0. Snce f (x) s convex n x, and x E, = 0, then g (x,x E, ) g (x E,,x E, )+ g (x E ) (x 0) > g (x E ) where we have used the fact that x > 0. Note that g (x,x E, ) g (x,0 ) = g. Therefore, g (x E ) < g So λ = 0. Proposton 11: Wth the general cost functon g (x) = f (x) + x, γ s upper-bounded by γ mn{max { f (x E ) / f (x E ) },Q},j, where Q s the POA derved before for Nash Equlbra n the one-shot game (.e., ρ Q), and x E acheves the optmal socal cost n the set of SPE.

10 10 Proof: Frst of all, snce any NE s Pareto-domnated by g, the best SPE s at least as good as NE. So γ Q. Consder π,j := 1+λ 1+λ j. (a) If λ = 0, then π,j 1. (b) If λ,λ j > 0, then accordng to Lemma 4, we have (1 + λ )[ f (x E) ] = 1+λ and (1+λ )[ f (x E) ] = 1+λ j. Therefore π,j = (1 + λ )[ f (x E) ] (1 + λ )[ f (x E) ] max{ f (x E ) / f (x E ) } (c) If λ > 0 but λ j = 0, then from Lemma 4, (1 + λ )[ f (x E) ] = 1+λ and (1+λ )[ f (x E) ] 1+λ j. Therefore, π,j (1 + λ )[ f (x E) ] (1 + λ )[ f (x E) ] max{ f (x E ) / f (x E ) } Consderng the cases (a), (b) and (c), and from equaton (19), we have γ max π,j max { f (x E ) / f (x E ) },j,j, whch completes the proof. Proposton 11 apples to any game wth the cost functon g (x) = f (x)+x, where f (x) s non-negatve, decreasng n each x, and satsfes the assumpton (1)-(3) at the begnnng of secton 4. Ths ncludes the EI model and the BT model ntroduced before. It s not easy to fnd an explct form of the upper bound on γ n Proposton 11 for the BT model. However, for the EI model, we have the smple expresson shown n Proposton 7: γ mn{max,j, β β j,q} where Q = max {1 + : β }. Proof: The part γ Q s straghtforward: snce the set of SPE ncludes all NE s, the best SPE must be better than the worst NE. Also, snce f (x E) x = β V ( m β mx E,m ), and f (x E) x j = β j V we have γ max,j, β β j. ( m β mx E,m ), usng Proposton 11, Note that x T [1 + f (x)] = x [1 + have E[x (1+ So, f (x) ]. For every player, for each x wth postve probablty, there are two possbltes: (a) If x = 0, then x [1 + f (x) ] = 0, x; (b) If x > 0, then by (17), E( f(x) x ) = 1. Snce f (x) 0 for all, then E( f (x) x ) 1. Therefore for both (a) and (b), we f (x) ) x ] = x E[1+ f (x) x ] 0. = As a result, E{ [x (1 + E{E[x (1 + f (x) )]} f (x) ) x ]} 0. ρ(µ) E[x T f (x)] + f (x ) f (x ) + 1 T x. (20) Consder two cases: () If x = 0 for all, then the RHS s 1, so ρ(µ) 1. Snce ρ(µ) cannot be smaller than 1, we have ρ(µ) = 1. () If not all x = 0, then 1T x > 0. Note that the RHS of (20) s not less than 1, by the defnton of ρ(µ). So, f we subtract f (x ) (non-negatve) from both the numerator and the denomnator, the resultng rato upper-bounds the RHS. That s, ρ(µ) E[x T f (x)] 1 T x max {E( f ( x) f (x) x )} where x s the th element of the vector f ( x). Combnng cases () and (), we have ρ(µ) max{1,max E( f (x) x )}. Then, ρ CE s upper-bounded by max µ CD ρ(µ). A5. Proof of Proposton 10 A4. Proof of Lemma 2 Proof: The performance rato between the dscrete CE µ(x) and the socal optmal s ρ(µ) := G(µ) G = E[ (f (x) + x )] [f (x ) + x ] where the expectaton (and all other expectatons below) s taen over the dstrbuton µ. Snce f ( ) s convex for all. Then for any x, f (x) f (x ) + (x x ) T f (x). So ρ(µ) E[(x x ) T f (x) + 1 T x] + f (x ) f (x ) + 1 T x = E{ x T f (x) + x T [1 + f (x)]} + f (x ) f (x ) + 1 T x Proof: Snce µ s a dscrete CE, by (17), for any x wth postve probablty, E( f(x) x ) 1. Therefore E( f(x) ) 1. In the EI model, we have Therefore E( f (x) x f (x) ) = E( x = β [ f (x) ]. f (x) β ) β. So, ρ CE max {1 + : β }. In the BT model, smlar to the proof n Proposton 3, t s not dffcult to see that the followng holds for any x: [ : j f (x) ]/[ f j(x) v r j ] max. : j v j r j

11 11 Then, f (x) v r j (1 + max )[ f j(x) ]. : j v j r j If µ s a dscrete CE, then E( fj(x) ) 1, j. Therefore E( f (x) ρ CE max E( j ) (1 + max : j v r j v jr j ). So, f (x) v r j ) (1 + max ). (,j): j v j r j PLACE PHOTO HERE Lbn Jang receved hs B.Eng. degree n Electronc Engneerng & Informaton Scence from the Unversty of Scence and Technology of Chna n 2003 and the M.Phl. degree n Informaton Engneerng from the Chnese Unversty of Hong Kong n 2005, and s currently worng toward the Ph.D. degree n the Department of Electrcal Engneerng & Computer Scence, Unversty of Calforna, Bereley. Hs research nterest ncludes wreless networs, game theory and networ economcs. PLACE PHOTO HERE Venat Anantharam s on the faculty of the EECS department at UC Bereley. He receved hs B.Tech n Electrcal Engneerng from the Indan Insttute of Technology, 1980, a M.S. n EE from UC Bereley, 1982, a M.A. n Mathematcs, UC Bereley, 1983, a C.Phl n Mathematcs, UC Bereley, 1984 and a Ph.D. n EE, UC Bereley, He s a corecpent of the 1998 Prze Paper award of the IEEE Informaton Theory Socety and a co-recpent of the 2000 Stephen O. Rce Prze Paper award of the IEEE Communcatons Theory Socety. He s a Fellow of the IEEE. Hs research nterest ncludes nformaton theory, communcatons and game theory. PLACE PHOTO HERE Jean Walrand receved hs Ph.D. n EECS from UC Bereley, where he has been a professor snce He s the author of An Introducton to Queueng Networs (Prentce Hall, 1988) and of Communcaton Networs: A Frst Course (2nd ed. McGraw- Hll,1998) and co-author of Hgh Performance Communcaton Networs (2nd ed, Morgan Kaufman, 2000). Prof. Walrand s a Fellow of the Belgan Amercan Educaton Foundaton and of the IEEE and a recpent of the Lanchester Prze and of the Stephen O. Rce Prze.

Recurrence. 1 Definitions and main statements

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 information

1 Approximation Algorithms

1 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 information

Luby s Alg. for Maximal Independent Sets using Pairwise Independence

Luby 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 information

A Lyapunov Optimization Approach to Repeated Stochastic Games

A 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://www-bcf.usc.edu/

More information

benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).

benefit 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 information

Extending Probabilistic Dynamic Epistemic Logic

Extending 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 σ-algebra: a set

More information

An Alternative Way to Measure Private Equity Performance

An 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 information

A Probabilistic Theory of Coherence

A 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 information

Power-of-Two Policies for Single- Warehouse Multi-Retailer Inventory Systems with Order Frequency Discounts

Power-of-Two Policies for Single- Warehouse Multi-Retailer Inventory Systems with Order Frequency Discounts Power-of-wo Polces for Sngle- Warehouse Mult-Retaler Inventory Systems wth Order Frequency Dscounts José A. Ventura Pennsylvana State Unversty (USA) Yale. Herer echnon Israel Insttute of echnology (Israel)

More information

Module 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur

Module 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 information

On the Interaction between Load Balancing and Speed Scaling

On the Interaction between Load Balancing and Speed Scaling On the Interacton between Load Balancng and Speed Scalng Ljun Chen and Na L Abstract Speed scalng has been wdely adopted n computer and communcaton systems, n partcular, to reduce energy consumpton. An

More information

When Network Effect Meets Congestion Effect: Leveraging Social Services for Wireless Services

When 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 information

Support Vector Machines

Support 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 information

How Sets of Coherent Probabilities May Serve as Models for Degrees of Incoherence

How 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 information

Institute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic

Institute 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 information

Answer: 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

Answer: 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 Multple-Choce Questons (30/00 ponts) Please, crcle the correct answer for each of the followng 0 multple-choce questons. For each queston, only one of the answers s correct.

More information

Activity Scheduling for Cost-Time Investment Optimization in Project Management

Activity Scheduling for Cost-Time Investment Optimization in Project Management PROJECT MANAGEMENT 4 th Internatonal Conference on Industral Engneerng and Industral Management XIV Congreso de Ingenería de Organzacón Donosta- San Sebastán, September 8 th -10 th 010 Actvty Schedulng

More information

Analyzing Self-Defense Investments in Internet Security Under Cyber-Insurance Coverage

Analyzing Self-Defense Investments in Internet Security Under Cyber-Insurance Coverage Analyzng Self-Defense Investments n Internet Securty Under Cyber-Insurance Coverage Ranjan Pal Department of Computer Scence Unv. of Southern Calforna Emal: rpal@usc.edu Leana Golubchk Department of Computer

More information

OPTIMAL INVESTMENT POLICIES FOR THE HORSE RACE MODEL. Thomas S. Ferguson and C. Zachary Gilstein UCLA and Bell Communications May 1985, revised 2004

OPTIMAL INVESTMENT POLICIES FOR THE HORSE RACE MODEL. Thomas S. Ferguson and C. Zachary Gilstein UCLA and Bell Communications May 1985, revised 2004 OPTIMAL INVESTMENT POLICIES FOR THE HORSE RACE MODEL Thomas S. Ferguson and C. Zachary Glsten UCLA and Bell Communcatons May 985, revsed 2004 Abstract. Optmal nvestment polces for maxmzng the expected

More information

On the Interaction between Load Balancing and Speed Scaling

On the Interaction between Load Balancing and Speed Scaling On the Interacton between Load Balancng and Speed Scalng Ljun Chen, Na L and Steven H. Low Engneerng & Appled Scence Dvson, Calforna Insttute of Technology, USA Abstract Speed scalng has been wdely adopted

More information

ANALYZING 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 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, 610-519-4390,

More information

Feature selection for intrusion detection. Slobodan Petrović NISlab, Gjøvik University College

Feature selection for intrusion detection. Slobodan Petrović NISlab, Gjøvik University College Feature selecton for ntruson detecton Slobodan Petrovć NISlab, Gjøvk Unversty College Contents The feature selecton problem Intruson detecton Traffc features relevant for IDS The CFS measure The mrmr measure

More information

The Development of Web Log Mining Based on Improve-K-Means Clustering Analysis

The Development of Web Log Mining Based on Improve-K-Means Clustering Analysis The Development of Web Log Mnng Based on Improve-K-Means Clusterng Analyss TngZhong Wang * College of Informaton Technology, Luoyang Normal Unversty, Luoyang, 471022, Chna wangtngzhong2@sna.cn Abstract.

More information

Combinatorial Agency of Threshold Functions

Combinatorial 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 information

Dynamic Pricing for Smart Grid with Reinforcement Learning

Dynamic Pricing for Smart Grid with Reinforcement Learning Dynamc Prcng for Smart Grd wth Renforcement Learnng Byung-Gook Km, Yu Zhang, Mhaela van der Schaar, and Jang-Won Lee Samsung Electroncs, Suwon, Korea Department of Electrcal Engneerng, UCLA, Los Angeles,

More information

8 Algorithm for Binary Searching in Trees

8 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 information

Efficient Project Portfolio as a tool for Enterprise Risk Management

Efficient 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 information

Fisher Markets and Convex Programs

Fisher 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 information

A Novel Auction Mechanism for Selling Time-Sensitive E-Services

A Novel Auction Mechanism for Selling Time-Sensitive E-Services A ovel Aucton Mechansm for Sellng Tme-Senstve E-Servces Juong-Sk Lee and Boleslaw K. Szymansk Optmaret Inc. and Department of Computer Scence Rensselaer Polytechnc Insttute 110 8 th Street, Troy, Y 12180,

More information

8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by

8.5 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 information

Lecture 3: Force of Interest, Real Interest Rate, Annuity

Lecture 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 annuty-mmedate, and ts present value Study annuty-due, and

More information

Chapter 4 ECONOMIC DISPATCH AND UNIT COMMITMENT

Chapter 4 ECONOMIC DISPATCH AND UNIT COMMITMENT Chapter 4 ECOOMIC DISATCH AD UIT COMMITMET ITRODUCTIO A power system has several power plants. Each power plant has several generatng unts. At any pont of tme, the total load n the system s met by the

More information

An Analysis of Central Processor Scheduling in Multiprogrammed Computer Systems

An Analysis of Central Processor Scheduling in Multiprogrammed Computer Systems STAN-CS-73-355 I SU-SE-73-013 An Analyss of Central Processor Schedulng n Multprogrammed Computer Systems (Dgest Edton) by Thomas G. Prce October 1972 Techncal Report No. 57 Reproducton n whole or n part

More information

Week 6 Market Failure due to Externalities

Week 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 information

On Competitive Nonlinear Pricing

On 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 information

Inequality and The Accounting Period. Quentin Wodon and Shlomo Yitzhaki. World Bank and Hebrew University. September 2001.

Inequality 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 information

Internet can be trusted and that there are no malicious elements propagating in the Internet. On the contrary, the

Internet can be trusted and that there are no malicious elements propagating in the Internet. On the contrary, the Prcng and Investments n Internet Securty 1 A Cyber-Insurance Perspectve Ranjan Pal, Student Member, IEEE, Leana Golubchk, Member, IEEE, arxv:submt/0209632 [cs.cr] 8 Mar 2011 Abstract Internet users such

More information

BERNSTEIN POLYNOMIALS

BERNSTEIN POLYNOMIALS On-Lne 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 information

Enabling P2P One-view Multi-party Video Conferencing

Enabling P2P One-view Multi-party Video Conferencing Enablng P2P One-vew Mult-party Vdeo Conferencng Yongxang Zhao, Yong Lu, Changja Chen, and JanYn Zhang Abstract Mult-Party Vdeo Conferencng (MPVC) facltates realtme group nteracton between users. Whle P2P

More information

1 Example 1: Axis-aligned rectangles

1 Example 1: Axis-aligned 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 information

Minimal Coding Network With Combinatorial Structure For Instantaneous Recovery From Edge Failures

Minimal Coding Network With Combinatorial Structure For Instantaneous Recovery From Edge Failures Mnmal Codng Network Wth Combnatoral Structure For Instantaneous Recovery From Edge Falures Ashly Joseph 1, Mr.M.Sadsh Sendl 2, Dr.S.Karthk 3 1 Fnal Year ME CSE Student Department of Computer Scence Engneerng

More information

The Stock Market Game and the Kelly-Nash Equilibrium

The Stock Market Game and the Kelly-Nash Equilibrium The Stock Market Game and the Kelly-Nash Equlbrum Carlos Alós-Ferrer, Ana B. Ana Department of Economcs, Unversty of Venna. Hohenstaufengasse 9, A-1010 Venna, Austra. July 2003 Abstract We formulate the

More information

Multiple-Period Attribution: Residuals and Compounding

Multiple-Period Attribution: Residuals and Compounding Multple-Perod Attrbuton: Resduals and Compoundng Our revewer gave these authors full marks for dealng wth an ssue that performance measurers and vendors often regard as propretary nformaton. In 1994, Dens

More information

Lecture 3: Annuity. Study annuities whose payments form a geometric progression or a arithmetic progression.

Lecture 3: Annuity. Study annuities whose payments form a geometric progression or a arithmetic progression. Lecture 3: Annuty Goals: Learn contnuous annuty and perpetuty. Study annutes whose payments form a geometrc progresson or a arthmetc progresson. Dscuss yeld rates. Introduce Amortzaton Suggested Textbook

More information

Price Competition in an Oligopoly Market with Multiple IaaS Cloud Providers

Price Competition in an Oligopoly Market with Multiple IaaS Cloud Providers Prce Competton n an Olgopoly Market wth Multple IaaS Cloud Provders Yuan Feng, Baochun L, Bo L Department of Computng, Hong Kong Polytechnc Unversty Department of Electrcal and Computer Engneerng, Unversty

More information

Supply network formation as a biform game

Supply network formation as a biform game Supply network formaton as a bform game Jean-Claude Hennet*. Sona Mahjoub*,** * LSIS, CNRS-UMR 6168, Unversté Paul Cézanne, Faculté Sant Jérôme, Avenue Escadrlle Normande Némen, 13397 Marselle Cedex 20,

More information

Cautiousness and Measuring An Investor s Tendency to Buy Options

Cautiousness 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, Arrow-Pratt measure of rsk averson explans a ratonal nvestor s behavor n stock markets

More information

Multi-Resource Fair Allocation in Heterogeneous Cloud Computing Systems

Multi-Resource Fair Allocation in Heterogeneous Cloud Computing Systems 1 Mult-Resource 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 mult-resource allocaton

More information

Logistic Regression. Lecture 4: More classifiers and classes. Logistic regression. Adaboost. Optimization. Multiple class classification

Logistic 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 information

Feasibility of Using Discriminate Pricing Schemes for Energy Trading in Smart Grid

Feasibility of Using Discriminate Pricing Schemes for Energy Trading in Smart Grid Feasblty of Usng Dscrmnate Prcng Schemes for Energy Tradng n Smart Grd Wayes Tushar, Chau Yuen, Bo Cha, Davd B. Smth, and H. Vncent Poor Sngapore Unversty of Technology and Desgn, Sngapore 138682. Emal:

More information

x 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

x 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 information

Causal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting

Causal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting Causal, Explanatory Forecastng Assumes cause-and-effect relatonshp between system nputs and ts output Forecastng wth Regresson Analyss Rchard S. Barr Inputs System Cause + Effect Relatonshp The job of

More information

Abteilung für Stadt- und Regionalentwicklung Department of Urban and Regional Development

Abteilung 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 SRE-Dscusson

More information

Section 5.4 Annuities, Present Value, and Amortization

Section 5.4 Annuities, Present Value, and Amortization Secton 5.4 Annutes, Present Value, and Amortzaton Present Value In Secton 5.2, we saw that the present value of A dollars at nterest rate per perod for n perods s the amount that must be deposted today

More information

General Auction Mechanism for Search Advertising

General 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 information

Downlink Power Allocation for Multi-class. Wireless Systems

Downlink Power Allocation for Multi-class. Wireless Systems Downlnk Power Allocaton for Mult-class 1 Wreless Systems Jang-Won Lee, Rav R. Mazumdar, and Ness B. Shroff School of Electrcal and Computer Engneerng Purdue Unversty West Lafayette, IN 47907, USA {lee46,

More information

An Interest-Oriented Network Evolution Mechanism for Online Communities

An Interest-Oriented Network Evolution Mechanism for Online Communities An Interest-Orented 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 information

Joint Resource Allocation and Base-Station. Assignment for the Downlink in CDMA Networks

Joint Resource Allocation and Base-Station. Assignment for the Downlink in CDMA Networks Jont Resource Allocaton and Base-Staton 1 Assgnment for the Downlnk n CDMA Networks Jang Won Lee, Rav R. Mazumdar, and Ness B. Shroff School of Electrcal and Computer Engneerng Purdue Unversty West Lafayette,

More information

Pricing Model of Cloud Computing Service with Partial Multihoming

Pricing Model of Cloud Computing Service with Partial Multihoming Prcng Model of Cloud Computng Servce wth Partal Multhomng Zhang Ru 1 Tang Bng-yong 1 1.Glorous Sun School of Busness and Managment Donghua Unversty Shangha 251 Chna E-mal:ru528369@mal.dhu.edu.cn Abstract

More information

Dynamic Online-Advertising Auctions as Stochastic Scheduling

Dynamic Online-Advertising Auctions as Stochastic Scheduling Dynamc Onlne-Advertsng Auctons as Stochastc Schedulng Isha Menache and Asuman Ozdaglar Massachusetts Insttute of Technology {sha,asuman}@mt.edu R. Srkant Unversty of Illnos at Urbana-Champagn rsrkant@llnos.edu

More information

1. Fundamentals of probability theory 2. Emergence of communication traffic 3. Stochastic & Markovian Processes (SP & MP)

1. Fundamentals of probability theory 2. Emergence of communication traffic 3. Stochastic & Markovian Processes (SP & MP) 6.3 / -- Communcaton Networks II (Görg) SS20 -- www.comnets.un-bremen.de Communcaton Networks II Contents. Fundamentals of probablty theory 2. Emergence of communcaton traffc 3. Stochastc & Markovan Processes

More information

Data Broadcast on a Multi-System Heterogeneous Overlayed Wireless Network *

Data Broadcast on a Multi-System Heterogeneous Overlayed Wireless Network * JOURNAL OF INFORMATION SCIENCE AND ENGINEERING 24, 819-840 (2008) Data Broadcast on a Mult-System Heterogeneous Overlayed Wreless Network * Department of Computer Scence Natonal Chao Tung Unversty Hsnchu,

More information

Can Auto Liability Insurance Purchases Signal Risk Attitude?

Can Auto Liability Insurance Purchases Signal Risk Attitude? Internatonal Journal of Busness and Economcs, 2011, Vol. 10, No. 2, 159-164 Can Auto Lablty Insurance Purchases Sgnal Rsk Atttude? Chu-Shu L Department of Internatonal Busness, Asa Unversty, Tawan Sheng-Chang

More information

The Greedy Method. Introduction. 0/1 Knapsack Problem

The 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 information

Using Series to Analyze Financial Situations: Present Value

Using Series to Analyze Financial Situations: Present Value 2.8 Usng Seres to Analyze Fnancal Stuatons: Present Value In the prevous secton, you learned how to calculate the amount, or future value, of an ordnary smple annuty. The amount s the sum of the accumulated

More information

A hybrid global optimization algorithm based on parallel chaos optimization and outlook algorithm

A hybrid global optimization algorithm based on parallel chaos optimization and outlook algorithm Avalable onlne www.ocpr.com Journal of Chemcal and Pharmaceutcal Research, 2014, 6(7):1884-1889 Research Artcle ISSN : 0975-7384 CODEN(USA) : JCPRC5 A hybrd global optmzaton algorthm based on parallel

More information

Capacity Reservation for Time-Sensitive Service Providers: An Application in Seaport Management

Capacity Reservation for Time-Sensitive Service Providers: An Application in Seaport Management Capacty Reservaton for Tme-Senstve Servce Provders: An Applcaton n Seaport Management L. Jeff Hong Department of Industral Engneerng and Logstcs Management The Hong Kong Unversty of Scence and Technology

More information

Dominant Resource Fairness in Cloud Computing Systems with Heterogeneous Servers

Dominant Resource Fairness in Cloud Computing Systems with Heterogeneous Servers 1 Domnant Resource Farness n Cloud Computng Systems wth Heterogeneous Servers We Wang, Baochun L, Ben Lang Department of Electrcal and Computer Engneerng Unversty of Toronto arxv:138.83v1 [cs.dc] 1 Aug

More information

Approximation algorithms for allocation problems: Improving the factor of 1 1/e

Approximation 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 information

Research Article Enhanced Two-Step Method via Relaxed Order of α-satisfactory Degrees for Fuzzy Multiobjective Optimization

Research Article Enhanced Two-Step Method via Relaxed Order of α-satisfactory Degrees for Fuzzy Multiobjective Optimization Hndaw Publshng Corporaton Mathematcal Problems n Engneerng Artcle ID 867836 pages http://dxdoorg/055/204/867836 Research Artcle Enhanced Two-Step Method va Relaxed Order of α-satsfactory Degrees for Fuzzy

More information

Addendum to: Importing Skill-Biased Technology

Addendum to: Importing Skill-Biased Technology Addendum to: Importng Skll-Based Technology Arel Bursten UCLA and NBER Javer Cravno UCLA August 202 Jonathan Vogel Columba and NBER Abstract Ths Addendum derves the results dscussed n secton 3.3 of our

More information

J. Parallel Distrib. Comput.

J. Parallel Distrib. Comput. J. Parallel Dstrb. Comput. 71 (2011) 62 76 Contents lsts avalable at ScenceDrect J. Parallel Dstrb. Comput. journal homepage: www.elsever.com/locate/jpdc Optmzng server placement n dstrbuted systems n

More information

NON-CONSTANT SUM RED-AND-BLACK GAMES WITH BET-DEPENDENT WIN PROBABILITY FUNCTION LAURA PONTIGGIA, University of the Sciences in Philadelphia

NON-CONSTANT SUM RED-AND-BLACK GAMES WITH BET-DEPENDENT WIN PROBABILITY FUNCTION LAURA PONTIGGIA, University of the Sciences in Philadelphia To appear n Journal o Appled Probablty June 2007 O-COSTAT SUM RED-AD-BLACK GAMES WITH BET-DEPEDET WI PROBABILITY FUCTIO LAURA POTIGGIA, Unversty o the Scences n Phladelpha Abstract In ths paper we nvestgate

More information

On the Optimal Control of a Cascade of Hydro-Electric Power Stations

On the Optimal Control of a Cascade of Hydro-Electric Power Stations On the Optmal Control of a Cascade of Hydro-Electrc Power Statons M.C.M. Guedes a, A.F. Rbero a, G.V. Smrnov b and S. Vlela c a Department of Mathematcs, School of Scences, Unversty of Porto, Portugal;

More information

What is Candidate Sampling

What 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 information

THE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek

THE 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 information

Optimality in an Adverse Selection Insurance Economy. with Private Trading. November 2014

Optimality 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 information

Calculation of Sampling Weights

Calculation of Sampling Weights Perre Foy Statstcs Canada 4 Calculaton of Samplng Weghts 4.1 OVERVIEW The basc sample desgn used n TIMSS Populatons 1 and 2 was a two-stage stratfed cluster desgn. 1 The frst stage conssted of a sample

More information

The Application of Fractional Brownian Motion in Option Pricing

The Application of Fractional Brownian Motion in Option Pricing Vol. 0, No. (05), pp. 73-8 http://dx.do.org/0.457/jmue.05.0..6 The Applcaton of Fractonal Brownan Moton n Opton Prcng Qng-xn Zhou School of Basc Scence,arbn Unversty of Commerce,arbn zhouqngxn98@6.com

More information

AN APPROACH TO WIRELESS SCHEDULING CONSIDERING REVENUE AND USERS SATISFACTION

AN APPROACH TO WIRELESS SCHEDULING CONSIDERING REVENUE AND USERS SATISFACTION The Medterranean Journal of Computers and Networks, Vol. 2, No. 1, 2006 57 AN APPROACH TO WIRELESS SCHEDULING CONSIDERING REVENUE AND USERS SATISFACTION L. Bada 1,*, M. Zorz 2 1 Department of Engneerng,

More information

Distributed Optimal Contention Window Control for Elastic Traffic in Wireless LANs

Distributed Optimal Contention Window Control for Elastic Traffic in Wireless LANs Dstrbuted Optmal Contenton Wndow Control for Elastc Traffc n Wreless LANs Yalng Yang, Jun Wang and Robn Kravets Unversty of Illnos at Urbana-Champagn { yyang8, junwang3, rhk@cs.uuc.edu} Abstract Ths paper

More information

Coordinated Denial-of-Service Attacks in IEEE 802.22 Networks

Coordinated Denial-of-Service Attacks in IEEE 802.22 Networks Coordnated Denal-of-Servce Attacks n IEEE 82.22 Networks Y Tan Department of ECE Stevens Insttute of Technology Hoboken, NJ Emal: ytan@stevens.edu Shamk Sengupta Department of Math. & Comp. Sc. John Jay

More information

2.4 Bivariate distributions

2.4 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 information

Economic Models for Cloud Service Markets

Economic Models for Cloud Service Markets Economc Models for Cloud Servce Markets Ranjan Pal and Pan Hu 2 Unversty of Southern Calforna, USA, rpal@usc.edu 2 Deutsch Telekom Laboratores, Berln, Germany, pan.hu@telekom.de Abstract. Cloud computng

More information

A Secure Password-Authenticated Key Agreement Using Smart Cards

A Secure Password-Authenticated Key Agreement Using Smart Cards A Secure Password-Authentcated Key Agreement Usng Smart Cards Ka Chan 1, Wen-Chung Kuo 2 and Jn-Chou Cheng 3 1 Department of Computer and Informaton Scence, R.O.C. Mltary Academy, Kaohsung 83059, Tawan,

More information

2008/8. An integrated model for warehouse and inventory planning. Géraldine Strack and Yves Pochet

2008/8. An integrated model for warehouse and inventory planning. Géraldine Strack and Yves Pochet 2008/8 An ntegrated model for warehouse and nventory plannng Géraldne Strack and Yves Pochet CORE Voe du Roman Pays 34 B-1348 Louvan-la-Neuve, Belgum. Tel (32 10) 47 43 04 Fax (32 10) 47 43 01 E-mal: corestat-lbrary@uclouvan.be

More information

Project Networks With Mixed-Time Constraints

Project Networks With Mixed-Time Constraints Project Networs Wth Mxed-Tme 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 information

Section 5.3 Annuities, Future Value, and Sinking Funds

Section 5.3 Annuities, Future Value, and Sinking Funds Secton 5.3 Annutes, Future Value, and Snkng Funds Ordnary Annutes A sequence of equal payments made at equal perods of tme s called an annuty. The tme between payments s the payment perod, and the tme

More information

7.5. Present Value of an Annuity. Investigate

7.5. Present Value of an Annuity. Investigate 7.5 Present Value of an Annuty Owen and Anna are approachng retrement and are puttng ther fnances n order. They have worked hard and nvested ther earnngs so that they now have a large amount of money on

More information

Efficient Bandwidth Management in Broadband Wireless Access Systems Using CAC-based Dynamic Pricing

Efficient Bandwidth Management in Broadband Wireless Access Systems Using CAC-based Dynamic Pricing Effcent Bandwdth Management n Broadband Wreless Access Systems Usng CAC-based Dynamc Prcng Bader Al-Manthar, Ndal Nasser 2, Najah Abu Al 3, Hossam Hassanen Telecommuncatons Research Laboratory School of

More information

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12 14 The Ch-squared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304-C) 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 information

APPLICATION OF PROBE DATA COLLECTED VIA INFRARED BEACONS TO TRAFFIC MANEGEMENT

APPLICATION OF PROBE DATA COLLECTED VIA INFRARED BEACONS TO TRAFFIC MANEGEMENT APPLICATION OF PROBE DATA COLLECTED VIA INFRARED BEACONS TO TRAFFIC MANEGEMENT Toshhko Oda (1), Kochro Iwaoka (2) (1), (2) Infrastructure Systems Busness Unt, Panasonc System Networks Co., Ltd. Saedo-cho

More information

Robust Design of Public Storage Warehouses. Yeming (Yale) Gong EMLYON Business School

Robust Design of Public Storage Warehouses. Yeming (Yale) Gong EMLYON Business School Robust Desgn of Publc Storage Warehouses Yemng (Yale) Gong EMLYON Busness School Rene de Koster Rotterdam school of management, Erasmus Unversty Abstract We apply robust optmzaton and revenue management

More information

Forecasting the Direction and Strength of Stock Market Movement

Forecasting the Direction and Strength of Stock Market Movement Forecastng the Drecton and Strength of Stock Market Movement Jngwe Chen Mng Chen Nan Ye cjngwe@stanford.edu mchen5@stanford.edu nanye@stanford.edu Abstract - Stock market s one of the most complcated systems

More information

Enforcement in Private vs. Public Externalities

Enforcement in Private vs. Public Externalities Enforcement n Prvate vs. Publc Externaltes Hakan Inal Department of Economcs, School of Busness and L. Douglas Wlder School of Government and Publc Affars Vrgna Commonwealth Unversty hnal@vcu.edu November

More information

Dynamic Cost-Per-Action Mechanisms and Applications to Online Advertising

Dynamic Cost-Per-Action Mechanisms and Applications to Online Advertising Dynamc Cost-Per-Acton Mechansms and Applcatons to Onlne Advertsng Hamd Nazerzadeh Amn Saber Rakesh Vohra Abstract We examne the problem of allocatng a resource repeatedly over tme amongst a set of agents.

More information

ABSTRACT. Categories and Subject Descriptors. General Terms. Keywords

ABSTRACT. Categories and Subject Descriptors. General Terms. Keywords On Desgnng Incentve-Compatble Routng and Forwardng Protocols n Wreless Ad-Hoc Networks An Integrated Approach Usng Game Theoretcal and Cryptographc Technques Sheng Zhong L (Erran) L Yanbn Grace Lu Yang

More information

The OC Curve of Attribute Acceptance Plans

The 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 information

An MILP model for planning of batch plants operating in a campaign-mode

An MILP model for planning of batch plants operating in a campaign-mode An MILP model for plannng of batch plants operatng n a campagn-mode Yanna Fumero Insttuto de Desarrollo y Dseño CONICET UTN yfumero@santafe-concet.gov.ar Gabrela Corsano Insttuto de Desarrollo y Dseño

More information

Relay Secrecy in Wireless Networks with Eavesdropper

Relay Secrecy in Wireless Networks with Eavesdropper Relay Secrecy n Wreless Networks wth Eavesdropper Parvathnathan Venktasubramanam, Tng He and Lang Tong School of Electrcal and Computer Engneerng Cornell Unversty, Ithaca, NY 14853 Emal : {pv45, th255,

More information