Modelling Information Warfare as a Game

Transcription

1 Modellng Informaton Warfare as a ame Jorma Jormakka and Jarmo V. E. Mölsä Department of echnology Natonal Defence College P.O. ox 7 FI-0086 Helsnk Fnland E-mal: orma.ormakka@ml.f Communcatons Laboratory Helsnk Unversty of echnology P.O. ox 3000 FI-005 HU Fnland E-mal: armo.molsa@tkk.f Abstract ame theory s one of the possble ways to study nformaton warfare wth mathematcal models. hs paper presents four example games whch llustrate the dfferent requrements for an effectve playng strategy n nformaton warfare. hese games study how a bold playng strategy can lead to domnaton how a mxed playng strategy can reduce domnaton how t can be useful to play a domnatng strategy only part of the tme and how excessve domnaton can lead to rebels where all playng partes lose. hs paper also descrbes meta-strateges whose goal s to modfy the perceved costs and condtons of a game. hs knd of percepton management s closely related to the Observe-Orent-Decde- Act (OODA) loop. Keywords: Informaton warfare game theory OODA-loop. Introducton hs paper presents four example games for modellng nformaton warfare. Wth the help of game theory the possblty for achevng and mantanng a domnatng poston n dfferent knds of nformaton warfare cases s nvestgated. In general the purpose of mathematcal modellng of nformaton warfare s to produce a quanttatve evaluaton of an attack or a defence strategy. hs paper concentrates on fndng quanttatve answers n descrbng the most effectve strateges for certan nformaton warfare scenaros. In nformaton warfare the fundamental weapon and target s nformaton whle the man goal s nformaton superorty (Hutchnson and Warren 00). An mportant ssue here s the possblty to modfy the enemy s percepton of costs and condtons related to warfare. hese percepton management ssues are closely related to the Observe-Orent-Decde-Act (OODA) loop whch descrbes the decson process and ts dependence on tmely and correct observatons (oyd 996). he OODA-loop emphaszes the orentaton phase whch s responsble for analysng and nterpretng the dfferent observatons to fnd out the most useful decson. he structure of ths paper s the followng. Frst an overvew of game theory s gven. hen some general observatons about modellng nformaton warfare as a game are descrbed. he frst game n ths paper shows the beneft of bold strateges n asymmetrc warfare. he second game ndcates the benefts of usng mxed defence strateges for mtgatng the domnance of a stronger party. he thrd game analyses the optmal dsturbance strategy when a domnator wants to prevent the usage of a communcaton network. he fourth game Journal of Informaton Warfare (005) 4 (): 5 ISSN prnt/issn onlne

2 Modellng Informaton Warfare as a ame ponts out that excessve domnance can result n rebels whch cause massve costs to all playng partes. All these four games are dependent on correct perceptons of costs and other condtons of the game. he next secton descrbes how meta-strateges can be used to modfy the costs and condtons perceved by an enemy. Fnally some related work s descrbed and the conclusons are gven. An Overvew of ame heory hs secton descrbes the most mportant ssues n game theory to make t easer to understand the games defned later n ths paper. For an extensve ntroducton to game theory see for example bbons (99). ame theory treats mult-person decson problems as games where each player chooses such decsons whch result n the best possble payoff for hmself/herself regardng the most probable decsons of the other players. All players are thus expected to be ratonal.e. they want to maxmze payoffs. It s not possble to predct the outcome of a play f some players are rratonal. Actually t should be common knowledge that all the players are ratonal: all the players know that all the players are ratonal and that all the players know that all the players know that all the players are ratonal and so on. he (normal-form) representaton of an n -player game specfes the players of the game the strateges for each player and the payoff functon for each player: = S K S ; u K u }. { n n Here S denotes the set of strateges avalable to the player n. A strategy s S s one complete plan of acton for a game. he payoff for the player s defned by the functon u s K s ) where s n s the strategy chosen by the player. ( n ames are generally dvded nto statc and dynamc games. A statc game s a smultaneousmove game and all players choose ther strateges smultaneously wthout knowng what strateges other players have chosen. A dynamc game s a sequental-move game and players choose ther strateges n sequence. Statc games can be combned n a dynamc fashon and a repeated game s one example of ths. ames can have dfferent propertes. In a cooperatve game players try to maxmze the ont payoff but n a non-cooperatve game all players concentrate only on maxmzng ther own payoff. In warfare however there s no cooperaton. In a game of complete nformaton the payoff functon of all players s common knowledge but n a game of ncomplete nformaton at least one player s not sure about the payoff functon of another player. In a game of perfect nformaton a player wth the move knows the full hstory of the game played so far (.e. what strateges other players have chosen) but n a game of mperfect nformaton at least one player wth the move does not know the full hstory of the game. he strateges n S are the pure strateges for the player. If player has M dfferent pure strateges ( S = s K s } ) then a mxed strategy for player s a probablty dstrbuton { M p = ( p K pm ) where 0 p m for m = K M and p + L + pm =. A mxed strategy ndcates one player s uncertanty about what another player wll do. A pure strategy s m can be represented as a mxed strategy by settng p m to (and the remanng terms n the probablty dstrbuton beng 0). Journal of Informaton Warfare 3

3 Modellng Informaton Warfare as a ame he expected payoff v for player n an n -player statc game = S K S ; u K u } when player ( n ) plays the mxed strategy p s the followng: where () s { n n M M n n v ( p K p ) n L = pkm ( ) k u s m K s nm () n m = mn = k= M s the number of pure strateges avalable to the player. For a two-player case M = M ( p p ) p m p m u ( s m s m ) m = m = v. () When modellng decson problems wth game theory t s often possble to dentfy one or more equlbrum condtons whch defne a strategcally stable predcton of the outcome of a play. In an equlbrum condton no player has nterest to change hs decson because ths would only result n a lower beneft for the player changng hs/her decson. In an n -player statc game = S K S ; u K u } of complete nformaton the mxed strateges { n n ( p K p ) are a Nash equlbrum f for each player n mxed strateges specfed for the other players. In other words max v ( p p K p p p+ K p n ) p s player s best response to the p solves for any player n. A Nash equlbrum provdes a local maxmum value for v n. Sngle-player strategy changes are thus not useful at a Nash equlbrum because any two Nash equlbrums (provded they exst) must dffer n the strateges of at least two players. So to change the outcome of a game from one Nash equlbrum to another at least two players must change ther strateges at the same tme. It has been shown that there exsts at least one Nash equlbrum possbly nvolvng mxed strateges for any normal-form statc game wth a fnte number of players and strateges. If a game has exactly one Nash equlbrum t s the unque soluton to the game. ame theory cannot necessarly predct the outcome of a game f there are more than one Nash equlbrums for the game. Especally when a game has multple Nash equlbrums wth conflctng payoffs ths game can have an outcome whch s not a Nash equlbrum. Credblty s a central ssue n dynamc sequental-move games. Only credble threats or promses can have an effect on how a game proceeds. In the context of warfare we wll concentrate on threats nstead of promses. In a two-player game one player can threat to change hs/her strategy f the other player does not act as requred by the stronger player. If the threat s not accepted the play wll end up n another Nash equlbrum whch provdes a worse outcome for the player not acceptng the threat. Credble threats about future behavour can thus have an nfluence on current behavour. he concept of a Nash equlbrum descrbes the possble outcomes of a statc game of complete nformaton but there are stronger equlbrum concepts for more complcated (rcher) types of games. For example a dynamc game of complete nformaton may have many Nash equlbrums but only sub-game perfect Nash equlbrums do not nvolve noncredble threats or promses. An example of a dynamc game s a repeated game ( ) where a statc game of complete nformaton (the stage game) s repeated tmes. he outcome of a sngle stage of ths 4 Journal of Informaton Warfare

4 Modellng Informaton Warfare as a ame dynamc game can however dffer from the Nash equlbrums of the basc statc stage game. If the stage game has multple Nash equlbrums a sub-game perfect outcome of (t) at a sngle stage t t < s not necessarly a Nash equlbrum of when credble threats about the future behavour are nvolved. Informaton Warfare as a ame A natural settng for modellng nformaton warfare s to have a game wth two players: an attacker and a defender. All players are expected to be ratonal here. he payoff for an attacker s the damage to a vctm. As a startng pont for the forthcomng games we wll frst gve some possble scenaros for nformaton warfare:. An army wth hgh technology attacks an opponent s Command Control Communcaton and Intellgence (C 3 I) system and tres to dsable t at the begnnng of a strategc strke. In addton to hacker warfare methods ths knd of an attack can nclude many other methods lke physcal destructon dsablng of sensors electronc warfare and so on. he goal of all of these attack methods s to cause loss of nformaton and delays n the OODA-loop whch causes wrong decsons to be taken (Kopp 00).. A group of attackers carres out a massve attack aganst a crtcal nformaton technology nfrastructure of a socety wth easly avalable tools such as Denal of Servce (DoS) tools vruses worms roans and password crackers. he goal s to produce chaos economc losses and psychologcal effects. 3. A group of terrorsts carres out a targeted well-planned and coordnated attack usng precson weapons such as new vruses worms and DoS tools. he attack ams n brngng down mportant organzatons and sabotagng busness operatons. hs knd of attack needs nformaton of the target before the attack can be made and the tme of the attack s lkely to be carefully chosen. he goal of a targeted attack s to cause economc losses to nfluence wllngness to fght or to support other actons lke terrorst operatons. 4. A group of attackers carres out long-term nformaton warfare to cause economc losses and slow down techncal development. All these scenaros nvolve more than one manfestatons of nformaton warfare: commandand-control (C ) ntellgence-based electronc psychologcal hacker economc and cyber warfare (Lbck 995). Each game here concentrates on a sngle manfestaton of nformaton warfare. Attack and defence strateges must not be too unspecfc. Only one strategy should be possble to play at a tme and there should be a lmted amount of clearly dfferent strateges. Otherwse the outcome of a game would be unknown f several strateges can be played at the same tme. he players of the games n ths paper can only choose one strategy at a tme. errorst ame: old Strategy Can Result n Domnaton he frst game n ths paper s the terrorst game. errorsm can be combned wth many nformaton warfare methods. he man contrbuton of ths game s to show that n a game wth more than one conflctng Nash equlbrums can only end up n domnaton. In asymmetrc warfare at least one of the players s expected to be n a weaker poston than the other players. Journal of Informaton Warfare 5

5 Modellng Informaton Warfare as a ame he terrorst game s a two-player statc game of complete nformaton where both players are ratonal. errorsts () capture hostages and threaten to blow up the hostages f the requrements of the terrorsts are not accepted. he government () proposes that terrorsts should surrender and be put to al. oth players have two strateges π and π. he strategy π means acceptng what the other player suggests: terrorsts surrender or the government accepts the requrements (e.g. pays the ransom). he strategy π means reectng what the other player suggests (terrorsts kll the hostages or the government reects to negotate). he payoffs are the followng: If both players play π both players get -: accepts requrements surrenders and s put to al but gets beneft from the accepted requrements. If both players play π both players get -0: reects klls hostages and terrorsts themselves. If player plays π and player plays π gets -5 and gets 0: reects surrenders and s put to al. If player plays π and player plays π gets and gets -5: gets free wth accepted requrements hostages released. Let us assume plays the mxed strategy ( pπ( p ) π ) where 0 p. hs means that plays π wth the probablty of p and π wth the probablty of ( p ). plays the mxed strategy ( pπ ( p) π ) where 0 p. he expected payoff v for s accordng to () v = p p 0( p )( p ) 5p ( p ) + ( p ) p = p ( 5 7p ) 0 + p and the expected payoff v for s v = p p 0( p )( p ) + 0p ( p ) 5( p ) p = p ( 5 6p ) 0 + 0p. ( he Nash equlbrum ponts p p ) for ths game are calculated by analysng the bestresponse correspondences p ( p ) (the value of p whch maxmzes v ( p ) ) and p ( p ) (the value of p whch maxmzes v ( p ) ). hese correspondences descrbe how the own optmal mxed strategy selecton probablty depends on the opponents probablty. p = 0 p = ): Frst we dentfy the pure strategy Nash equlbrums ( { } p ( 0) = and p ( ) = 0: Nash equlbrum pont s ( 0) p ( ) = 0 and p ( 0) = : Nash equlbrum pont s ( 0) he possble Nash equlbrums nvolvng mxed strateges can be found by dfferentatng the payoff functons: v 5 = 5 7p = 0 p = p 7 v p 5 = 5 6 p = 0 p =. 6 6 Journal of Informaton Warfare

6 Modellng Informaton Warfare as a ame 5 5 he Nash equlbrums are thus the followng ponts: ( p p) (0)(0)( ). hese 6 7 ponts reflect the ntersecton ponts of the best-response correspondences shown n the Fg.. he payoffs ( v 5 0 v ) at the equlbrum ponts are: ( 50)( 5)( ). 7 6 Fgure : he ntersectons of the best-response correspondences are the Nash equlbrums. he two Nash equlbrum ponts wth pure strateges ( π π ) and ( π π ) gve the best payoffs for v and v respectvely. hus the thrd equlbrum pont s not n nterest for nether player. As such however these results do not yet provde any unque soluton to ths statc game. A bold strategy (as defned n gamblng theory n Dubns 965) can however result n a unque soluton n the long term when the statc terrorst game s repeated. Let us assume that s bold and always plays π. states that t wll not negotate wth. Player may not beleve that wll play boldly and may try π for fntely many tmes but f stcks on to playng π wll eventually fnsh wth a fnte negatve gan and wll have to start playng π n order to mnmze the losses. hs famlar real-lfe game can only end up n domnaton where accepts that always plays π and wll accept losng on ths game or n blowng up the hostages and terrorsts. hen ratonal player must always play π. A bold ratonal player always wns over the less bold ratonal player n the long term when the terrorst game s repeated. hs can be formulated as the followng Proposton. Proposton. Let = { S K S n; u K un} be a non-cooperatve n-player statc game whch s repeated. S s the set of strateges and u s the payoff functon of the player. Let the set E of Nash equlbrums satsfy the followng two crtera:. here are players and and two equlbrum ponts x y E such that u ( x) u ( z) and u ( y) u ( z) for any z E. hus there s no unque soluton to ths statc game because t s not possble for players and to acheve the best possble payoff at the same tme. hs s a typcal case for example n asymmetrc warfare.. Let the strategy of the player n x be x and the strategy of the player n y be y. Let z be the outcome of the statc game. If z E and the strategy of the player s x Journal of Informaton Warfare 7

7 Modellng Informaton Warfare as a ame then z = x and f the strategy of the player s y z = y. In other words f the game ends up n a Nash equlbrum then only ether of the players and wll get the maxmum payoff. he repeated game can only end up to the player domnatng the player or the player domnatng the player or both and beng domnated by some player k or to a pont outsde E. A pont outsde E wll result for example f players and both nsst on playng ther best-response strategy. Proof. hs proposton does not say anythng that s not already known from Nash equlbrum ponts but t s gven because of the conclusons we can draw from t. If the soluton s not one of the equlbrum ponts some player has a reason to change hs strategy. hus the players try to get to a soluton whch s n the set E. As the player s ratonal and the game s non-cooperatve he wll try to get the best gan and force others to play x = ( x... x n ). He can only play x but the only equlbrum pont where hs strategy s x s x. hus the other ratonal players can only play the strategy they play n x. If the player succeeds he wll domnate over. If some other player succeeds he wll domnate over. he last possblty s that the result s not an equlbrum pont whch happens for example f nether player nor player accept domnaton. In ths case player wll choose x and player wll choose y but due to conflctng Nash equlbrums ths combnaton wll result n an outcome whch s not n the set E. Domnaton strategy seems acceptable n the repeated terrorst game but n general the acceptablty of the soluton depends on what sde one s. he cause for asymmetrc warfare s often domnaton n the frst place. Domnaton strateges are not necessarly stable. When the cost of acceptng domnaton s on the same range as the estmated cost of fghtng aganst t we should expect a crss to appear. Evldoer ame: Mxed Defence Strateges Can Reduce Domnaton he second game n ths paper s the evldoer game. In ths game an evl hacker tres to attack and compromse a vctm network or host. hs can be seen as a relevant (sub)goal for many forms of nformaton warfare especally for C hacker and cyber warfare. he man contrbuton of ths game s to show that usng mxed defence strateges can result n better tolerance aganst domnatve attack strateges when no generc defence strategy s avalable. he evldoer game s a statc game wth complete nformaton where both players are ratonal. here are two players an attacker A and a vctm. he goal of the attacker s to cause damage to the vctm by crashng hosts nstallng malcous software gettng remote access wth root-prvleges causng Denal-of-Servce and so on. o acheve the goal the attack must fnd a prmary attack whch cannot be drectly detected (e.g. no fngerprnt avalable for a new explot). he attacker has two strateges π A and π A where π A means overloadng the vctm wth many secondary attacks at the same tme wth the prmary attack to cause delays n the OODA-loop of the vctm and make the detecton of the prmary attack more dffcult. he strategy π A means tryng only the prmary attack. he vctm also has two strateges π and π where π means detectng and alertng on all suspcous network 8 Journal of Informaton Warfare

8 Modellng Informaton Warfare as a ame traffc but some of ths traffc may have to allowed to pass through due to the hgh probablty of false alarms. π means detectng and blockng only the most mportant attacks. As the prmary attack wll succeed ntally the vctm may however be able to detect the attack afterwards wth a delay by payng attenton to the avalable alerts. he payoffs u A and u for the attacker and the vctm respectvely are the followng: u A ( π A π ) = 4and u ( π A π ) = 5 : the vctm does not detect the prmary attack at all because the vctm s overloaded by the large amount of alerts from the secondary attacks. u A ( π A π ) = 3and u ( π A π ) = : the vctm does not detect the prmary attack but some of the secondary attacks may cause alerts. hese alerts can be used as an ndcaton about a larger attack and sgns about ntrusons nto mportant hosts can be searched more carefully. here s a low probablty for dentfyng the successful prmary attack. ua( π A π ) = 0and u( π A π ) = 0 : the vctm detects all suspcous network traffc. As there s only a low volume of alerts the vctm has tme to analyse all data carefully and the ntruson s detected after a delay. u π π ) 30 and u π π ) = 0 : the vctm does not detect the prmary attack. A( A = ( A Let us assume that the attacker plays the mxed strategy ( pπ A ( p) π A) where 0 p and that the vctm plays the mxed strategy ( qπ ( q) π ) where 0 q. he expected payoff v A for A s accordng to () v A = p( 5q ) + 5 4q and the expected payoff v for s v = q( 5p) 3+ p. he best response correspondences p (q) and q ( p) for the attacker and the vctm respectvely have one ntersecton at p = /5 and q = /5 whch s the unque mxed strategy Nash equlbrum for ths evldoer game. he evldoer game has thus a unque outcome. hs result has the followng effect on the selecton of attack and defence strateges. If a specfc defence strategy s not generc but nstead works well only aganst a specfc attack strategy then changng the defence strategy more or less randomly results n better tolerance aganst the domnatve attack strateges. he attacker has thus more dffcultes n domnatng a vctm when the vctm s usng mxed defence strateges. In ths specfc evldoer game the attacker should overload the vctm n 40 % of the attacks and use only a sngle prmary attack mechansm n 60 % of the attacks. he vctm on the other hand should try to detect all possble suspcous network behavour 40 % of the tme and concentrate only on detectng and blockng the maor attacks 60 % of the tme. he mxed strategy Nash equlbrum reflects the mutual uncertanty about the other player s strategy. Vandal game: Domnaton Can Have a Lmted me Span he thrd game n ths paper s the vandal game. In ths game a vandal tres to dsturb the usage of a communcaton network for example by ntatng a floodng Denal of Servce (DoS) attack overloadng the target network or by ammng a wreless network. he man contrbuton of ths game s to show that domnaton can have a lmted tme span. If the vctm network s unusable for a too long tme legtmate users wll start usng another Journal of Informaton Warfare 9

9 Modellng Informaton Warfare as a ame network for communcaton purposes. hs new network s selected so that the vandal cannot dsturb t at least not wth the same attack methods. he vandal game s an n -player statc game of complete knowledge where all the players are ratonal. In ths game all the players are usng the servces of a communcaton network. he player V s the vandal who does not maxmze hs gan from havng good servce wth low costs but nstead maxmzes the harm to other users. he player L n s a legtmate user of the network. All legtmate users are dentcal. In total there are thus n players. All the players have two strateges π and π where π means usng the network and π means beng dle. If the vandal s usng the network then the network s expected to be overloaded and cannot be used for legtmate purposes. he payoffs for the vandal and the legtmate user are the followng: If V plays π V wll get 0. If V plays π V wll get m f there are m legtmate users playng π. If L n plays π If V plays π and If V plays π and L L L wll get 0. n plays π n plays π L wll get. L wll get -. Let us assume that the player V plays the mxed strategy ( pπ ( p) π ) and that the player L n plays a mxed strategy ( plπ( pl ) π ) where 0 p L. he expected payoff v V for the vandal s vv = ( n ) p p L and the expected payoff v L for the legtmate user L s v = p p + p ) p = p ( ). L L ( L L p We can concentrate on analysng the best-response correspondences between the vandal and only one legtmate user because all legtmate users are here expected to behave n a smlar fashon. hs results n a two-dmensonal presentaton for the best-response correspondences p ( pl) and p L ( p) and they do not have only a sngle ntersecton pont but nstead an nfnte set of ntersecton ponts because the correspondences overlap partally. he Nash equlbrums for ths vandal game are thus the followng: When p s n the range [..] p L s 0 and the expected payoff s 0 for all players. he soluton of the game s that the vandal cannot wn snce the other players stop usng the servce. Intally of course there wll be a loss for the legtmate users because network servce s not avalable but the legtmate users wll fnd another communcaton mechansm. When studed as a dynamc game the vandal should not always play π because legtmate users wll stop usng the communcaton network. If the vandal plays π only part of the tme (here less than 50% of the tme) he can gan. An mportant queston s how often to play π so that legtmate users wll not stop usng the communcaton network. 0 Journal of Informaton Warfare

10 Modellng Informaton Warfare as a ame Rebel game: Extreme Domnaton Can Result n Rebellons Let us look at a game lke the terrorst game whch has several Nash equlbrum ponts and whch therefore leads to a domnatng soluton f played repeatedly wth a bold strategy. In ths rebel game the domnatng soluton s expected to cause extremely hgh costs to the weaker party. hese hgh costs wll eventually be seen unbearable by the weaker party whch wll eventually start to rebel. he man contrbuton of ths game s to show that t s useful for a domnatng party to prevent extremely hgh costs for the weaker party. Otherwse the weaker party may start to rebel whch wll result n substantal costs to the domnatng party. If a game wth hgh costs to the weaker party s played several tmes we do not know the outcome. he purpose of the rebel game s to suggest a way to treat such games and to propose a soluton concept. If the game s played up to some fnte fnshng tme and the players optmze ther strateges up to the fnshng tme then the game can be treated as a sngle step game wth all the outcomes up to the fnshng tme and wth all strateges whch are sequences of one step strateges. Such a game has at least one Nash equlbrum pont. hs however s not a realstc soluton snce the players cannot calculate strateges up to the fnshng tme and cannot assume the opponent s calculatng strateges up to the fnshng tme and playng ratonally unless the fnshng tme s only few steps ahead. For nstance f there are two players and each has two strateges n one step then after 5 steps there are two to power ffty outcomes. Usng a computer for numercal evaluaton of such a game approaches the computng lmt and memory demands are reached even earler. Analytc solutons can be used to gve probablty dstrbutons for the gans or losses of the players n n steps even for a large n provded that the model s suffcently smple. Let us brefly descrbe one such method the generatng functon method. Let users A and have strateges π A π A π π whch they play wth probabltes p A pa p p respectvely. Let s n qa q ra r denote the probablty of A havng cost r A and havng cost r after a sequence of moves where A has played π A q A tmes and has played π q tmes n n steps. he order of playng the strateges has an mportance to the costs and s n qa q ra r s the sum of probabltes over all possble orderngs of playng the strateges. here s the recurson equaton: s n qa q ra r pa p sn q A+ q+ ra+ fa ( ) r + f ( ) = = = (3) he functons f A ( ) and f ( ) are the losses of A and on one step. We can solve t ndvdually for A and. In the case of A the cost of s not of nterest thus we must solve an equaton of the general type s n q q r = sn q q r + pn q q + c q f r q n s K K K K K K K ( ) n q... q + c... f ( r q n) In the generatng functon method (Jormakka 003) we try to fnd a functon = n( u K uk r) = sn q K q K r k= 0 n( u v) u wth some functons y y ( k n q... qk r) v x( k n q... qk r) x such that the recurson equaton s changed to the form. (4) Journal of Informaton Warfare

11 Modellng Informaton Warfare as a ame n( u v) = g ( n u v) + n ( u v) + n ( u v) (5) he functon n ( u ) descrbes the contrbuton of the boundares. Notce v that g ( n u v) s not always a functon can also be a lnear operator. he equaton for the generatng functon s solved and n ( u v) s expanded nto a power seres out of whch the probabltes are extracted. he form of the soluton for s n qa q ra r s typcally a partton wth some restrctons. Parttons are also slow to evaluate numercally but n many cases one can fnd lmts or bounds whch can be effectvely evaluated. We wll not go nto the mathematcal detals of the method here. he theory s explaned n (Jormakka 003). When the probabltes are solved we can draw a few conclusons. If a sngle step game wth domnaton s played n tmes the probablty of player's cost exceedng a gven lmt can be obtaned from the probabltes s n qa q ra r. Assumng that a player has set a maxmum lmt to tolerable losses ths gves the probablty of a player startng to play rratonally n the sngle step game for fghtng aganst an ntolerable stuaton. he game has been named the rebel game because there s an easy perodc soluton where A plays a domnatng strategy and causes losses of value R on each step. Assumng that has the lmt M for tolerable costs he s lkely to rebel after n=m/r steps. Assumng that ths rebellon ncdence levels the costs of and A to zero and A agan starts playng the domnatng strategy we have a perodc sequence of rebels. In the perodc soluton plays rratonally on some steps and therefore t s not a good ratonal soluton. We may have the followng ratonal solutons: ) he mmedate costs to are so hgh that he can play only one strategy n each step. hat s the mmedate cost of one step exceeds M. hen t s ratonal for to accept all the tme the domnaton strategy played by A. ) A does not play a domnaton strategy but nstead he plays a mxed far strategy. hen 's losses never grow to M and hs ratonal behavor on the sngle step game s also ratonal n the mult-step game. hs requres A to be cooperatve. 3) A and play strateges whch are almost ratonal on the sngle step game but stll never lead to ntolerable losses to ether player. hen the players are almost non-cooperatve. he thrd soluton seems ntutvely to be a possble soluton concept for ths knd of a game. he players are almost ratonal and non-cooperatve on each sngle step. hey do not optmze moves over any sequence of steps whch s realstc. her motvaton s to avod ntolerable losses and the strategy should guarantee t. A rather smple mathematcal model guaranteeng that the probablty of losses hgher than M are mpossble can be made by multplyng the probabltes by ( r M ) where =A and scalng the equaton correspondngly by c n so that probabltes sum to one: s n q = A q ra r c n ra M A )( r M ) pa p sn q A+ q + ra+ f A( ) r + f ( ) ( (6) here s a natural connecton wth the rebel game and asymmetrc nformaton warfare. In asymmetrc warfare one of the partes s often a group of people who feel for some reason havng suffered ntolerable damage and s ready for acts lookng rratonal f the whole hstory Journal of Informaton Warfare

12 Modellng Informaton Warfare as a ame s not consdered. Domnaton strategy s not the soluton as the rebel game shows. It seems mpossble to mpose so hgh costs on one step that the soluton ) be realstc. he soluton ) makes lttle sense to A. Probably balancng between the extremes as n 3) s the only practcal soluton. Appled n nformaton warfare we should conclude that nether hgh penaltes nor gong along wth demands of cyber warrors s a workng soluton. he goal should be to remove the possblty of ntolerable losses to ether sde. Mathematcal models typcally gve only nsght but n ths case there may be a chance for rough estmaton of the tme before rratonal playng behavor happens and how much weaker should a strategy be of a domnatng strategy n order to brng a stable soluton. Meta-Strateges: Modfyng Observatons and Orentaton of the Enemy A wdely used model for descrbng the decson processes n warfare s the OODA-loop (oyd 996). he one who s able to get nsde the opponent s OODA-loop wll control how fast the opponent s able to react to new stuatons and also what decsons the opponent s makng. hs s nformaton superorty and the essental goal for nformaton warfare. Here we wll shortly descrbe how t s possble to model the modfcaton of the opponent s OODA-loop as a game of nformaton warfare. All the complete-knowledge games descrbed so far rely on accurate knowledge of the payoff functons. In real-lfe any player must observe and make as realstc assumptons about these payoffs (costs) as possble. If the observatons about an opponent s costs are unrealstc a player can end up choosng a non-optmal strategy. In the OODA-loop language ths s expressed as makng a wrong decson. o combne game theory and the OODA-loop we can end up n a meta-strategy whch defnes how to affect the strategy selecton process (decson process) of the enemy. he prmary goal of ths knd of a meta-strategy s to make threats look credble.e. lure the opponent to make wrong observatons about the costs of the attacker. If an opponent beleves some threats from an attacker ths opponent wll play accordng to these false observatons and the attacker s allowed to enter at least partally nsde the OODA-loop of the opponent. As a result the qualty and speed of the opponent s decsons wll be decreased. In a complete knowledge game wth realstc payoff functons the possble results of an nformaton warfare game are mostly known n advance. Only choosng a sutable metastrategy wll make t possble to modfy these pre-determned results. Related work Hacker warfare whch s one manfestaton of nformaton warfare can be modelled by usng attack trees (Cohen 998; Schneer 000). Attack tree modellng s a method smlar to usng rsk trees n rsk assessment. An attack tree classfes dfferent attack types and defnes a rsk for each attack type to be the product of the assocated probablty and the cost of a successful attack. It seems however mpossble to assgn probabltes to varous attack types. It s naturally possble to estmate the probabltes of partcular attack types usng some large set of analysed cases but such probabltes do not predct the stuaton n specal cases. he success probablty of usng an explot s strongly tme dependent: a vulnerablty s frst known only to a few then publc scrpts explotng ths vulnerablty wll be avalable and fnally the assocated securty patch wll be nstalled on most vulnerable computers and the explot wll no longer be useful. hs tme dependency assocated wth lack of knowledge makes buldng and updatng a detaled attack tree practcally mpossble. hus an attack tree grows bg wth too many unknown probabltes lke n Cohen (998). Journal of Informaton Warfare 3

13 Modellng Informaton Warfare as a ame In the area of network securty game theory has been used to model msbehavour and selfshness n ad hoc networks (Mchard 003; Urp 003). Penetraton nto a computer system has been studed wth the help of game theory n Sallhammar (004). Conclusons hs paper used game theory to model four dfferent nformaton warfare cases. he results from these games ndcated that dfferent strateges are effectve n achevng and mantanng a domnatng poston n the long term when a sngle step game s repeated many tmes. In the terrorst game whch has two conflctng results a bold strategy was requred to force an enemy to beleve that a player wll not accept any threats. In the evldoer game t was shown that mxed strateges can mtgate the domnatve poston of the attacker especally when any defence strategy s effectve only aganst a specfc attack strategy. Changng the defence strategy somehow randomly wll ncrease the probablty of mtgatng attacks. In the vandal game where there s no cost for carryng out a DoS attack an attacker should overload a network only part of the tme so that the defender wll not stop usng a network completely. Fnally the rebel game ndcated that causng excessve damages to a weaker party can result n rebellons where both partes wll suffer large damages. Mantanng a domnatng poston requres the stronger player to lmt the long term costs to the weaker party. If the weaker party feels that t has suffered too much there wll be lttle dfference how much t wll lose the next tme. Here understandng the hstory of the play s requred to treat actons as ratonal n the sense of game theory. In a dynamc game such as a repeated game only credble threats can have an effect on current behavour. hs fact can be exploted n meta-strateges whch try to modfy the opponent s observatons and perceptons of the payoff functons and strateges n the game. hs percepton management s closely related to the OODA-loop. Meta-strateges are one possblty for gettng nsde the opponent s OODA-loop and controllng the observatons and orentatons and the resultng decson process as a whole. References oyd J. R. (996). he Essence of Wnnng and Losng Presentaton sldes January 996 Cohen F. et al. (998). A Prelmnary Classfcaton Scheme for Informaton System hreats Attacks and Defenses: A Cause and Effect Model and Some Analyss based on hat Model Sanda Natonal Laboratores. Dubns L. and Savage L. (965). How to amble f You Must; Inequaltes for Stochastc Processes Mcraw-Hll. bbons R. (99). A Prmer n ame heory Pearson Educaton Essex UK. Hutchnson W. and Warren M. (00). Prncples of Informaton Warfare. Journal of Informaton Warfare (): -6. Jormakka J. (003). Combnatoral Decson heory wth Applcatons n Hacker Warfare Modellng echncal Report (preprnt) Department of echnology Natonal Defence College Helsnk Fnland. Publcaton Seres (). 4 Journal of Informaton Warfare

14 Modellng Informaton Warfare as a ame Kopp C. (00). Shannon Hypergames and Informaton Warfare. Journal of Informaton Warfare (): Lbck M. (995). What s Informaton Warfare? Natonal Defense Unversty. Mchard P. and Molva R. (003). A ame heoretcal Approach to Evaluate Cooperaton Enforcement Mechansms n Moble Ad hoc Networks. In: Proceedngs of the WOpt 03 workshop March 003 Sopha-Antpols France. Sallhammar K. and Knapskog S. J. (004). Usng ame heory n Stochastc Models for Quantfyng Securty. In Proceedngs of the Nnth Nordc Workshop on Secure I Systems November 004 Espoo Fnland. Schneer. (000). Secrets & Les Wley New York. Urp A. and onuccell M. and ordano S. (003). Modellng cooperaton n moble ad hoc networks: a formal descrpton of selfshness. In: Proceedngs of the WOpt 03 workshop March 003 Sopha-Antpols France. Journal of Informaton Warfare 5