{"title": "Computing Nash Equilibria in Generalized Interdependent Security Games", "book": "Advances in Neural Information Processing Systems", "page_first": 2735, "page_last": 2743, "abstract": "We study the computational complexity of computing Nash equilibria in generalized interdependent-security (IDS) games. Like traditional IDS games, originally introduced by economists and risk-assessment experts Heal and Kunreuther about a decade ago, generalized IDS games model agents\u2019 voluntary investment decisions when facing potential direct risk and transfer risk exposure from other agents. A distinct feature of generalized IDS games, however, is that full investment can reduce transfer risk. As a result, depending on the transfer-risk reduction level, generalized IDS games may exhibit strategic complementarity (SC) or strategic substitutability (SS). We consider three variants of generalized IDS games in which players exhibit only SC, only SS, and both SC+SS. We show that determining whether there is a pure-strategy Nash equilibrium (PSNE) in SC+SS-type games is NP-complete, while computing a single PSNE in SC-type games takes worst-case polynomial time. As for the problem of computing all mixed-strategy Nash equilibria (MSNE) efficiently, we produce a partial characterization. Whenever each agent in the game is indiscriminate in terms of the transfer-risk exposure to the other agents, a case that Kearns and Ortiz originally studied in the context of traditional IDS games in their NIPS 2003 paper, we can compute all MSNE that satisfy some ordering constraints in polynomial time in all three game variants. Yet, there is a computational barrier in the general (transfer) case: we show that the computational problem is as hard as the Pure-Nash-Extension problem, also originally introduced by Kearns and Ortiz, and that it is NP complete for all three variants. Finally, we experimentally examine and discuss the practical impact that the additional protection from transfer risk allowed in generalized IDS games has on MSNE by solving several randomly-generated instances of SC+SS-type games with graph structures taken from several real-world datasets.", "full_text": "Computing Nash Equilibria in Generalized\n\nInterdependent Security Games\n\nHau Chan\n\nLuis E. Ortiz\n\nDepartment of Computer Science, Stony Brook University\n\n{hauchan,leortiz}@cs.stonybrook.edu\n\nAbstract\n\nWe study the computational complexity of computing Nash equilibria in gener-\nalized interdependent-security (IDS) games. Like traditional IDS games, origi-\nnally introduced by economists and risk-assessment experts Heal and Kunreuther\nabout a decade ago, generalized IDS games model agents\u2019 voluntary investment\ndecisions when facing potential direct risk and transfer-risk exposure from other\nagents. A distinct feature of generalized IDS games, however, is that full invest-\nment can reduce transfer risk. As a result, depending on the transfer-risk reduc-\ntion level, generalized IDS games may exhibit strategic complementarity (SC)\nor strategic substitutability (SS). We consider three variants of generalized IDS\ngames in which players exhibit only SC, only SS, and both SC+SS. We show that\ndetermining whether there is a pure-strategy Nash equilibrium (PSNE) in SC+SS-\ntype games is NP-complete, while computing a single PSNE in SC-type games\ntakes worst-case polynomial time. As for the problem of computing all mixed-\nstrategy Nash equilibria (MSNE) ef\ufb01ciently, we produce a partial characterization.\nWhenever each agent in the game is indiscriminate in terms of the transfer-risk ex-\nposure to the other agents, a case that Kearns and Ortiz originally studied in the\ncontext of traditional IDS games in their NIPS 2003 paper, we can compute all\nMSNE that satisfy some ordering constraints in polynomial time in all three game\nvariants. Yet, there is a computational barrier in the general (transfer) case: we\nshow that the computational problem is as hard as the Pure-Nash-Extension prob-\nlem, also originally introduced by Kearns and Ortiz, and that it is NP-complete\nfor all three variants. Finally, we experimentally examine and discuss the practi-\ncal impact that the additional protection from transfer risk allowed in generalized\nIDS games has on MSNE by solving several randomly-generated instances of\nSC+SS-type games with graph structures taken from several real-world datasets.\n\n1\n\nIntroduction\n\nInterdependent Security (IDS) games [1] model the interaction among multiple agents where each\nagent chooses whether to invest in some form of security to prevent a potential loss based on both\ndirect and indirect (transfer) risks. In this context, an agent\u2019s direct risk is that which is not the result\nof the other agents\u2019 decisions, while indirect (transfer) risk is that which does.\nLet us be more concrete and consider an application of IDS games. Imagine that you are an owner\nof an apartment. One day, there was a \ufb01re alarm in the apartment complex. Luckily, it was nothing\nmajor: nobody got hurt. As a result, you realize that your apartment can be easily burnt down\nbecause you do not have any \ufb01re extinguishing mechanism such as a sprinkler system. However, as\nyou wonder about the cost and the effectiveness of the \ufb01re extinguishing mechanism, you notice that\nthe \ufb01re extinguishing mechanism can only protect your apartment if a small \ufb01re originates in your\napartment. If a \ufb01re originates in the \ufb02oor below, or above, or even the apartment adjacent to yours,\nthen you are out of luck: by the time the \ufb01re gets to your apartment, the \ufb01re would be \ufb01erce enough\n\n1\n\n\f\u03b1 \u223c N (0.4, 0.2)\n\n\u03b1 \u223c N (0.6, 0.2)\n\n\u03b1 \u223c N (0.8, 0.2)\n\nFigure 1: \u03b1-IDS Game of Zachary Karate Club at a Nash Equilibrium. Legend: Square \u2261 SC player,\nCircle \u2261 SS player, Colored \u2261 Invest, and Non-Colored \u2261 No Invest\n\nGame type\n\n(n SC players)\n\nSC\n\nSS\n\n(n SS players)\n\nSC + SS\n\n(nsc + nss = n)\n\nTable 1: Complexity of \u03b1-IDS Games\n\nOne PSNE\n\nAlways Exists\n\nO(n2)\n\nMaybe Not Exist\n\nAll MSNE\n\nUniform Transfers (UT)\n\nO(n4)\n\nUT wrt Ordering 1\n\nPure-Nash Extension\n\nNP-Complete\n\nNP-complete\n\nO(n4)\n\nUT wrt Ordering 1\nscn4\nO(n4\nss)\n\nss + n3\n\nscn3\n\nalready. You realize that if other apartment owners invest in the \ufb01re extinguishing mechanism, the\nlikelihood of their \ufb01res reaching you decreases drastically. As a result, you debate whether or not\nto invest in the \ufb01re extinguishing mechanism given whether or not the other owners invest in the\n\ufb01re extinguishing mechanism. Indeed, making things more interesting, you are not the only one\ngoing through this decision process; assuming that everybody is concerned about their safety in the\napartment complex, everybody in the apartment complex wants to decide on whether or not to invest\nin the \ufb01re extinguishing mechanism given the individual decision of other owners.\nTo be more speci\ufb01c, in the IDS games, the agents are the apartment owners, each apartment owner\nneeds to make a decision as to whether or not to invest in the \ufb01re extinguishing mechanism based on\ncost, potential loss, as well as the direct and indirect (transfer) risks. The direct risk here is the chance\nthat an agent will start a \ufb01re (e.g., forgetting to turn off gas burners or overloading electrical outlets).\nThe transfer risk here is the chance that a \ufb01re from somebody else\u2019s (unprotected) apartment will\nspread to other apartments. Moreover, transfer risk comes from the direct neighbors and cannot be\nre-transferred. For example, if a \ufb01re from your neighbors is transferred to you, then, in this model,\nthis \ufb01re cannot be re-transferred to your neighbors. Of course, IDS games can be used to model\nother practical real-world situations such as airline security [2], vaccination [3], and cargo shipment\n[4]. See Laszka et al. [5] for a survey on IDS games.\nNote that in the apartment complex example, the \ufb01re extinguishing mechanism does not protect an\nagent from \ufb01res that originate from other apartments. In this work, we consider a more general,\nand possibly also more realistic, framework of IDS games where investment can partially protect\nthe indirect risk (i.e., investment in the \ufb01re extinguishing mechanism can partially extinguish some\n\ufb01res that originate from others). To distinguish the naming scheme, we will call these generalized\nIDS games as \u03b1-IDS games where \u03b1 is a vector of probabilities, one for each agent, specifying the\nprobability that the transfer risk will not be protected by the investment. In other words, agent i\u2019s\ninvestment can reduce indirect risk by probability (1-\u03b1i). Given an \u03b1, the players can be partitioned\ninto two types: the SC type and the SS type. The SC players behave strategic complementarily:\nthey invest if suf\ufb01ciently many people invest. On the other hand, the SS players behave strategic\nsubstitutability: they do not invest if too many people invest.\nAs a preview of how the \u03b1 can affect the number of SC and SS players and Nash equilibria, which is\nthe solution concept used here (formally de\ufb01ned in the next section), Figure 1 presents the result of\nour simulation of an instance of SC+SS \u03b1-IDS games using the Zachary Karate Club network [6].\nThe nodes are the players, and the edge between nodes u and v represents the potential transfers\nfrom u to v and v to u. As we increase \u03b1\u2019s value, the number of SC players increases while the\n\n2\n\n123456789111213141820223231102829331734242630252715161921231234567891112131418202232311028293317342426302527151619212312345678911121314182022323110282933173424263025271516192123\fnumber of SS players decreases. Interestingly, almost all of the SC players invest, and all of the SS\nplayers are \u201cfree riding\u201d as they do not invest at the NE.\nOur goal here is to understand the behavior of the players in \u03b1-IDS games. Achieving this goal will\ndepend on the type of players, as characterized by the \u03b1, and our ability to ef\ufb01ciently compute NE,\namong other things. While Heal and Kunreuther [1] and Chan et al. [7] previously proposed similar\nmodels, we are unaware of any work on computing NE in \u03b1-IDS games and analyzing agents\u2019\nequilibrium behavior. The closest work to ours is Kearns and Ortiz [8], where they consider the\nstandard/traditional IDS model in which one cannot protect against the indirect risk (i.e., \u03b1 \u2261 1).\nIn particular, we study the computational aspects of computing NE of \u03b1-IDS games in cases of\nall game players being (1) SC, (2) SS, and (3) both SC and SS. Our contributions, summarized in\nTable 1, follow.\n\n\u2022 We show that determining whether there is a PSNE in (3) is NP-complete. However, there\nis a polynomial-time algorithm to compute a PSNE for (1). We identify some instances for\n(2) where PSNE does and does not exist.\n\u2022 We study the instances of \u03b1-IDS games where we can compute all NE. We show that\nif the transfer probabilities are uniform (independent of the destination), then there is a\npolynomial-time algorithm to compute all NE in case (1). Cases (2) and (3) may still take\nexponential time to compute all NE. However, based on some ordering constraints, we are\nable to ef\ufb01ciently compute all NE that satisfy the ordering constraints.\n\u2022 We consider the general-transfer case and show that the pure-Nash-extension problem [8],\nwhich, roughly, is the problem of determining whether there is a PSNE consistent with\nsome partial assignments of actions to some players, is NP-complete for cases (1), (2), and\n(3). This implies that computing all NE is likely as hard.\n\u2022 We perform experiments on several randomly-generated instances of SC+SS \u03b1-IDS games\nusing various real-world graph structures to show \u03b1\u2019s effect on the number of SC and SS\nplayers and on the NE of the games .\n\n2 \u03b1-IDS games: preliminaries, model de\ufb01nition, and solution concepts\n\nIn this section, we borrow de\ufb01nitions and notations of (graphical) IDS games from Kearns et al.\n[9], Kearns and Ortiz [8], and Chan et al. [7]. In an \u03b1-IDS game, we have an underlying (directed)\ngraph G = (V, E) where V = {1, 2, ..., n} represents the n players and E = {(i, j)|qij > 0} such\nthat qij is the transfer probability that player i will transfer the bad event to player j. As such, we\nde\ufb01ne Pa(i) and Ch(i) as the set of parents and children of player i in G, respectively.\nIn an \u03b1-IDS game, each player i has to make a decision as to whether or not to invest in protection.\nTherefore, the action or pure-strategy of player i is binary, denoted here by ai, with ai = 1 if i\ndecides to invest and ai = 0 otherwise. We denote the joint-action or joint-pure-strategy of all\nplayers by the vector a \u2261 (a1, . . . , an). For convenience, we denote by a\u2212i all components of a\nexcept that for player i. Similarly, given S \u2282 V , we denote by aS and a\u2212S all components of a\ncorresponding to players in S and V \u2212 S, respectively. We also use the notation a \u2261 (ai, a\u2212i) \u2261\n(aS, a\u2212S) when clear from context.\nIn addition, in an \u03b1-IDS game, there is a cost of investment Ci and loss Li associated with the bad\nevent occurring, either through direct or indirect (transfered) contamination. For convenience, we\ndenote the cost-to-loss ratio of player i by Ri \u2261 Ci/Li. We can parametrize the direct risk as pi,\nthe probability that player i will experience the bad event from direct contamination.\nSpeci\ufb01c to \u03b1-IDS games, the parameter \u03b1i denotes the probability of ineffectiveness of full invest-\nment in security (i.e., ai = 1) against player i\u2019s transfer risk. Said differently, the parameter \u03b1i mod-\nels the degree to which investment in security can potentially reduce player i\u2019s transfer risk. Player\nj\u2208Pa(i)[1 \u2212 (1 \u2212 aj)qji],\nis a function of joint-actions of Pa(i) because of the potential overall transfer probability (and thus\nrisk) from Pa(i) to i given Pa(i)\u2019s actions. One can think of the function si as the transfer-safety\nfunction of player i. The expression of si makes explicit the implicit assumption that the transfers\nof the bad event are independent. Putting the above together, the cost function of player i is\nMi(ai, aP a(i)) \u2261ai[Ci + \u03b1iri(aP a(i))Li] + (1 \u2212 ai)[pi + (1 \u2212 pi)ri(a\u2212i)]Li .\n\ni\u2019s transfer-risk function ri(aPa(i)) \u2261 1 \u2212 si(aPa(i)), where si(aPa(i)) \u2261(cid:81)\n\n3\n\n\f\uf8f1\uf8f2\uf8f30,\n\nBRsc\n\ni (aPa(i)) \u2261\n\n\u2206sc\n\u2206sc\n1,\n[0, 1], \u2206sc\n\ni > si(aPa(i)),\ni < si(aPa(i)),\ni = si(aPa(i)) .\n\nNote that the safety function describes the situation where a player j can only be \u201crisky\u201d to player\ni if and only if j does not invest in protection. We assume, without loss of generality (wlog), that\nCi (cid:28) Li, or equivalently, that Ri (cid:28) 1; otherwise, not investing would be a dominant strategy.\nWhile a syntactically minor addition to the traditional IDS model, the parameter \u03b1 introduces a\nmajor semantic difference and an additional complexity over the traditional model. The semantic\ndifference is perhaps clearer from examining the best response of the players: player i invests if\nCi + \u03b1iri(aPa(i))Li < [pi + (1 \u2212 pi)ri(aPa(i))]Li \u21d4 Ri \u2212 pi < (1 \u2212 pi \u2212 \u03b1i)ri(aPa(i)) .\n\nThe expression (1\u2212 pi \u2212 \u03b1i) is positive when \u03b1i < 1\u2212 pi and negative when \u03b1i > 1\u2212 pi. The best\nresponse condition \ufb02ips when the expression is negative. (When \u03b1i = 1 \u2212 pi, player i\u2019s investment\ndecision simpli\ufb01es because the player\u2019s internal risk fully determines the optimal choice.)\nIn fact, the parameter \u03b1 induces a partition of the set of players based on whether the corresponding\n\u03b1i value is higher or lower than 1 \u2212 pi. We will call the set of players with \u03b1i > 1 \u2212 pi the\nset of strategic complementarity (SC) players. SC players exhibit as optimal behavior that their\npreference for investing increases as more players invest: they are \u201cfollowers.\u201d The set of players\nwith \u03b1i < 1 \u2212 pi is the set of strategic substitutability (SS) players.\nIn this case, SS players\u2019\npreference for investing decreases as more players invest: they are \u201cfree riders.\u201d\nFor all i \u2208 SC, let \u2206sc\nresponse correspondence for player i \u2208 SC as\n\ni , for i \u2208 SS. We can de\ufb01ne the best-\n\ni \u2261 1 \u2212 Ri\u2212pi\n1\u2212pi\u2212\u03b1i\n\n; similarly for \u2206ss\n\ni\n\nfor player i \u2208 SS is similar, except that we replace \u2206sc\n\nThe best-response correspondence BRss\ni by\ni and \u201creverse\u201d the strict inequalities above. We use the best-response correspondence to de\ufb01ne\n\u2206ss\nNE (i.e., both PSNE and MSNE). We introduce randomized strategies: in a joint-mixed-strategy x \u2208\n[0, 1]n, each component xi corresponds to player i\u2019s probability of invest (i.e. P r(ai = 1) = xi).\nPlayer i\u2019s decision depends on expected cost, and, with abuse of notation, we denote it by Mi(x).\nDe\ufb01nition A joint-action a \u2208 {0, 1}n is a pure-strategy Nash equilibrium (PSNE) of an IDS game\nif ai \u2208 BRi(aPa(i)) for each player i. Replacing a with a joint mixed-strategy x \u2208 [0, 1]n in the\nequilibrium condition and the respective functions it depends on leads to the condition for x being a\nmixed-strategy Nash equilibrium (MSNE). Note that the set of PSNE \u2282 MSNE. Hence, we use NE\nand MSNE interchangably.\n\nFor general (and graphical) games, determining the existence of PSNE is NP-complete [10]. MSNE\nalways exist [11], but computing a MSNE is PPAD-complete [12\u201314].\n\n3 Computational results for \u03b1-IDS games\n\nIn this section, we present and discuss the\nresults of our computational study of \u03b1-IDS\ngames. We begin by considering the problem\nof computing PSNE, then moving to the more\ngeneral problem of computing MSNE.\n\n3.1 Finding a PSNE in \u03b1-IDS games\n\nIn this subsection, we look at the complexity\nof determining a PSNE in \u03b1-IDS games, and\n\ufb01nding it if one exists. Our \ufb01rst result follows.\n\nFigure 2: 3-SAT-induced \u03b1-IDS game graph\n\nTheorem 1 Determining whether there is a PSNE in n-player SC+SS \u03b1-IDS games is NP-complete.\n\nProof (Sketch) We are going to reduce an instance of a 3-SAT variant into our problem. Each clause\nof the 3-SAT variant contains either only negated variables or only un-negated variables [15]. We\n\n4\n\n\fi > (1 \u2212 q).\n\ni > (1 \u2212 q)3.\n\nhave an SC player for each clause and two SS players for each variable. The clause players invest\nif there exists a neighbor (its literal) that invests. For each variable vi, we introduce two players\nvi and \u00afvi with preference for mutually opposite actions. They invest if there exists a neighbor\n(its clause and \u00afvi) that does not invest. Figure 2 depicts the basic structure of the game. Nodes\nat the botton-row of the graph correspond to a variable, where the un-negated-variables-clauses\nand negated-variables-clauses are connected to their corresponding un-negated-variable and negated\nvariable with bidirectional transfer probability q.\nSetting the parameters of the clause players. Wlog, we can set the parameters to be identical\nfor all clause players i: \ufb01nd Ri > 0 and \u03b1i > 1 \u2212 pi such that (1 \u2212 q)2 > \u2206sc\nSetting the parameters of the variables players. Wlog, we can set the parameters to be identical\nfor all variable players i: \ufb01nd Ri > 0 and \u03b1i < 1 \u2212 pi such that 1 > \u2206ss\nWe now show that there exists a satis\ufb01able assignment if and only if there exists a PSNE.\nSatis\ufb01able assignment =\u21d2 PSNE. Suppose that we have a satis\ufb01able assignment of the variant\n3-SAT. This implies that every clause player is playing invest. Moreover, for each clause player,\nthere must be some corresponding variable players that play invest. Given a satis\ufb01able assignment,\nnegated and un-negated variable players cannot play the same action. One of them must be playing\ninvest and the other must be playing no-invest. The investing variable is best-responding because\nat least one of the players (namely its negation) is playing not invest. The not investing variable is\nbest-responding because all of its neighbors are investing. Hence, all the players are best-responding\nto each other and thus we have a PSNE.\nPSNE =\u21d2 satis\ufb01able assignment.\n(a) First we show that at every PSNE, all of the clause\nplayers must play invest. For the sake of contradiction, suppose that there is a PSNE in which there\nare some clause players that play no-invest. For the no-invest clause players, all of their variables\nmust play no-invest at PSNE. However, by the best-response conditions of the variable players, if\nthere exists a clause player that plays no-invest, then at least one of the variable players must play\ninvest, which contradicts the fact that we have a PSNE. (b) We now show that at every PSNE, the un-\nnegated variable player and the corresponding negated variable player must play different actions.\nSuppose that there is a PSNE, in which both of the players play the same action (i) no-invest or (ii)\ninvest. In the case of no-invest (i), by their best-response conditions (given that at every PSNE all\nclause players play invest), none of the variables are best-responding so one of them must switch\nfrom playing no-invest to invest. In the case of invest (ii), again by the best-response condition,\none of them must play no-invest. (c) Finally, we need to show that at every PSNE there must be\na variable player that makes every clause player play invest. To see this, note that, by the clause\u2019s\nbest-response condition, there must be at least one variable player playing invest. If there is a clause\nthat plays invest when none of its variable players play invest, then the clause player would not be\n(cid:117)(cid:116)\nbest-responding.\n\n3.1.1 SC \u03b1-IDS games\nWhat is the complexity of determining whether a PSNE exists in SC \u03b1-IDS games (i.e. \u03b1i > 1\u2212pi)?\nIt turns out that SC players have the characteristics of following the actions of other agents. If there\nare enough SC players who invest, then some remaining SC player(s) will follow suit. This is\nevident from the safety function and the best-response condition. Consider the dynamics in which\neverybody starts off with no-invest. If there are some players that are not best-responding, then their\nbest (dominant) strategy is to invest. We can safely change the actions of those players to invest.\nThen, for the remaining players, we continue to check to see if any of them is not best-responding.\nIf not, we have a PSNE, otherwise, we change the strategy of the not best-responding players to\ninvest. The process continues until we have reached a PSNE.\n\nTheorem 2 There is an O(n2)-time algorithm to compute a PSNE of any n-player SC \u03b1-IDS game.\n\nNote that once a player plays invest, other players will either stay no-invest or move to invest. The\nno-investing players do not affect the strategy of the players that already have decided to invest.\nPlayers that have decided to invest will continue to invest because only more players will invest.\n\n3.1.2 SS \u03b1-IDS games\n\nUnlike the SC case, an SS \u03b1-IDS game may not have a PSNE when n > 2.\n\n5\n\n\fi > (1 \u2212 qji) where j is\nProposition 1 Suppose we have an n-player SS \u03b1-IDS game with 1 > \u2206ss\nthe parent of i. (a) If the game graph is a directed tree, then the game has a PSNE. (b) If the game\ngraph is a a directed cycle, then the game has a PSNE if and only if n is even.\n\nProof (a) The root of the tree will always play no-invest while the immediate children of the root\nwill always play invest at a PSNE. Moreover, assigning the action invest or no-invest to any node\nthat has an odd or even (undirected) distance to the root, respectively, completes the PSNE.\n(b) For even n, an assignment in which any independent set of n\n2 players play invest form a PSNE.\nFor odd n, suppose there is a PSNE in which I players invest and N players do not invest, such\nthat I + N = n. The investing players must have I parents that do not invest and the non-investing\nplayers must have N parents that play invest. Moreover, I \u2264 N and N \u2264 I implies that I = N.\n(cid:117)(cid:116)\nHence, an odd n cycle cannot have a PSNE.\n\nWe leave the computational complexity of determining whether SS \u03b1-IDS games have PSNE open.\n\n3.2 Computing all NE in \u03b1-IDS games\n\nWe now study whether we can compute all MSNE of \u03b1-IDS games. We prove that we can compute\nall MSNE in polynomial time in the case of uniform-transfer SC \u03b1-IDS games, and a subset of all\nMSNE in the case of SS and SC+SS games. A uniform transfer \u03b1-IDS game is an \u03b1-IDS game\nwhere the transfer probability to another players from a particular player is the same regardless of\nthe destination. More formally, qij = \u03b4i for all players i and j (i (cid:54)= j). Hence, we have a complete\ngraph with bidirectional transfer probabilities. We can express the overall safety function given joint\ni=1[1\u2212(1\u2212xi)\u03b4i]. Now, we can determine the best response\ni (1 \u2212 (1 \u2212 ai)\u03b4i), for SC, relative to\n\nmixed-strategy x \u2208 [0, 1]n as s(x) =(cid:81)n\n\nof SC or SS player exactly based solely on the values of \u2206sc\ns(x); similarly for SS.\nWe assume, wlog, that for all players i, Ri > 0, \u03b4i > 0, pi > 0, and \u03b1i > 0. Given a joint mixed-\nstrategy x, we partition the players by type wrt x: let I \u2261 I(x) \u2261 {i | xi = 1}, N \u2261 N (x) \u2261\n{i | xi = 0}, and P \u2261 P (x) \u2261 {i | 0 < xi < 1} be the set of players that, wrt x, fully invest in\nprotection, do not invest in protection, and partially invest in protection, respectively.\n\n3.2.1 Uniform-transfer SC \u03b1-IDS games\n\nThe results of this section are non-trivial extensions of those of Kearns and Ortiz [8]. In particu-\nlar, we can construct a polynomial-time algorithm to compute all MSNE of a uniform-transfer SC\n\u03b1-IDS game, along the same lines of Kearns and Ortiz [8], by extending their Ordering Lemma\n(their Lemma 3) and Partial-Ordering Lemma (their Lemma 4). 1 Appendixes A.1 and B of the sup-\nplementary material contain our versions of the lemmas and detailed pseudocode for the algorithm,\nrespectively. A running-time analysis similar to that for traditional uniform-transfer IDS games done\nby Kearns and Ortiz [8] yields our next algorithmic result.\n\nTheorem 3 There exists an O(n4)-time algorithm to compute all MSNE of an uniform-transfer\nn-player SC \u03b1-IDS game.\n\nThe signi\ufb01cance of the theorem lies in its simplicity. That we can extend almost the same computa-\ntional results, and structural implications on the solution space, to a considerably more general, and\nperhaps even more realistic, model, via what in hindsight were simple adaptations, is positive.\n\n3.2.2 Uniform-transfer SS \u03b1-IDS games\n\nUnlike the SC case, the ordering we get for the SS case does not yield an analogous lemma. Never-\ntheless, it turns out that we can still determine the mixed strategies of the partially-investing players\nin P relative to a partition. The result is a Partial-Investment Lemma that is analogous to that\nof Kearns and Ortiz [8] for traditional IDS games. 2 For completeness, Appendix A.2 of the supple-\nmentary material formally states the lemma. We remind the reader that the signi\ufb01cance and strength\n\n1Take their Ri/pi\u2019s and replace them with our corresponding \u2206sc\n2Take their Lemma 4 and replace Ri/pi there by \u2206ss\n\ni \u2019s.\n\ni here, and replace the expression for V there by\n\nV \u2261 [maxk\u2208N (1 \u2212 \u03b4k)\u2206ss\n\ni ].\nk , mini\u2208I \u2206ss\n\n6\n\n\fof this non-trivial extension lies in its simplicity, and particularly when we note that the nature of\nthe SS case is the complete opposite of the version of IDS games studied by Kearns and Ortiz [8].\nIndeed, a naive way to compute all NE is to consider all of the possible combinations of players\ninto the investment, partial investment, and not investment sets and apply the Partial-Investment\nLemma alluded to in the previous paragraph to compute the mixed strategies. However, this would\ntake O(nss3nss\n) worst-case time to compute any equilibrium. So, how can we ef\ufb01ciently perform\nthis computation? As mentioned earlier, SS players are less likely to invest when there is a large\nnumber of players investing and have \u201copposite\u201d behavior as the SC players (i.e., the best response\nis \ufb02ipped). Hence, imposing a \u201c\ufb02ip\u201d ordering (Ordering 1) that is opposite of the SC case seems\nnatural. If we assume such a speci\ufb01c ordering of the players at equilibrium, then we can compute all\nNE consistent with that speci\ufb01c ordering ef\ufb01ciently, as we discuss earlier for the SC case. Mirroring\nthe SC \u03b1-IDS game, we settle for computing all NE that satisfy the following ordering.\nOrdering 1 For all i \u2208 I ss, j \u2208 P ss, and k \u2208 N ss,\n\n(1 \u2212 \u03b4k)\u2206ss\n(1 \u2212 \u03b4j)\u2206ss\n(1 \u2212 \u03b4k)\u2206ss\n\nk \u2264 (1 \u2212 \u03b4j)\u2206ss\nj \u2264 \u2206ss\nj \u2264 \u2206ss\nk \u2264 (1 \u2212 \u03b4i)\u2206ss\n\ni\n\nj < \u2206ss\nj\n\ni \u2264 \u2206ss\n\ni\n\nThe \ufb01rst and last set of inequalities (ignoring the middle one) follow from the consistency constraint\nimposed by the overall safety function. The middle set of inequalities restrict and reduce the number\nof possible NE con\ufb01gurations we need to check. It is possible that the (1\u2212 \u03b4k)\u2206ss\nj or\n(1\u2212 \u03b4k)\u2206ss\ni at an NE, but we do not consider those types of NE. Our hardness results\npresented in the upcoming Section 3.2.4 suggest that, in general, computing all MSNE without any\nof the constraints above is likely hard. (See Algorithm 2 of the supplementary material.)\n\nk > (1\u2212 \u03b4j)\u2206ss\n\nk > (1\u2212 \u03b4i)\u2206ss\n\nTheorem 4 There exists an O(n4)-time algorithm to compute all MSNE consistent with Ordering 1\nof an uniform-transfer n-player SS \u03b1-IDS game.\n\n3.2.3 Uniform-transfer SC+SS \u03b1-IDS games\n\nFor the uniform variant of the SC+SS \u03b1-IDS games, we could partition the players into either SC or\nSS and modify the respective algorithms to compute all NE. Unfortunately, this is computationally\ninfeasible because we can only compute all NE in polynomial time in the SC case. Again, if we settle\nfor computing all NE consistent with Ordering 1, then we can devise an ef\ufb01cient algorithm. From\nnow on, the fact that we are only considering NE consistent with Ordering 1 is implicit, unless noted\notherwise. The idea is to partition the players into a class of SC and a class of SS players. From\nthe characterizations stated earlier, it is clear that there are only a polynomial number of possible\npartitions we need to check for each class of players. Since the ordering results are based on the same\noverall safety function, the orderings of SC and SS players do not affect each other. Hence, wlog,\nstarting with the algorithm described earlier as a based routine for SC players, we do the following.\nFor each possible equilibrium con\ufb01guration of the SC players, we \ufb01rst run the algorithm described\nin the previous section for SS players and then test whether the resulting joint mixed-strategy is a\nNE. This guarantees that we check every possible equilibrium combination. A running-time analysis\nyields our next result.\n\nTheorem 5 There exists an O(n4\nOrdering 1 of an uniform-transfer n-player SC+SS \u03b1-IDS game, where n = nsc + nss.\n\nss)-time algorithm to compute all NE consistent with\n\nss + n3\n\nscn4\n\nscn3\n\n3.2.4 Computing all MSNE of arbitrary \u03b1-IDS games is intractable, in general\n\nIn this section, we prove that determining whether there exists a PSNE consistent with a partial-\nassignment of the actions to some players is NP-complete, even if the transfer probability takes only\ntwo values: \u03b4i \u2208 {0, q} for q \u2208 (0, 1).\nWe consider the pure-Nash-extension problem [8] for binary-action n-player games that takes as\ninput a description of the game and a partial assignment a \u2208 {0, 1,\u2217}n. We want to know whether\nthere is a complete assignment b \u2208 {0, 1}n consistent with a. Indeed, computing all NE is at least\nas dif\ufb01cult as the pure-Nash extension problem. Appendix C presents proofs of our next results.\n\n7\n\n\fTable 2: Level of Investment of SC+SS \u03b1-IDS Games at Nash Equilibrium\n\n\u03b1i \u223c N (0.4, 0.2)\n%SC Invest\n\n%SS Invest\n\n\u03b1i \u223c N (0.8, 0.2)\n%SC Invest\n\n%SS Invest\n\n\u03b1i \u2208 [0, 1]\n%SC Invest\n\n%SS Invest\n\n100.00\n100.00\n100.00\n97.76*\n97.46*\n95.97*\n\n21.37\n17.93\n15.47\n19.38*\n17.87*\n19.91*\n\nHigh Ci\nLi\nDatasets\nKarate Club\nLes Miserables\nCollege Football\nPower Grid\nWiki Vote\nEmail Enron\nLow Ci\nLi\nKarate Club\n41.34\nLes Miserables\n49.26\nCollege Football\n54.87\nPower Grid\n45.07**\nWiki Vote\n44.45**\nEmail Enron\n44.72**\n*=0.001-NE, **=0.005-NE, %SS (%SC) = Percentage of SS (SC) players, N (\u00b5, \u03c32) =normal distribution with mean \u00b5 and variance \u03c32\n\n100.00\n99.40\n100.00\n97.31**\n97.00**\n94.39**\n\u03b1i \u2208 [0, 1]\n100.00\n100.00\n100.00\n99.13**\n98.51**\n98.0**\n\n14.88\n14.84\n13.46\n15.90**\n14.75**\n16.84**\n\n49.64\n51.17\n60.42\n49.45*\n46.50*\n49.80**\n\n100.00\n100.00\n100.00\n99.13*\n98.30*\n97.96**\n\n100.00\n99.85\n100.00\n98.79*\n98.92*\n97.92*\n\n%SS\n76.18\n75.45\n75.65\n75.47\n75.55\n75.29\n\n99.41\n98.96\n98.87\n98.68\n98.62\n98.73\n\n%SS\n56.18\n55.06\n55.39\n55.01\n55.02\n54.78\n\n86.18\n85.71\n86.35\n85.20\n85.01\n84.94\n\n%SS\n12.35\n11.82\n11.57\n12.82\n12.78\n12.53\n\n60.59\n59.22\n61.48\n59.41\n59.89\n59.85\n\n0.00\n0.67\n0.00\n2.13*\n2.06*\n2.24*\n\n23.19\n28.34\n28.30\n28.66*\n27.54*\n29.32*\n\n100.00\n100.00\n100.00\n98.81*\n97.38*\n96.48*\n\n\u03b1i \u223c N (0.4, 0.2)\n\n\u03b1i \u223c N (0.8, 0.2)\n\nTheorem 6 The pure-Nash extension problem for n-player SC \u03b1-IDS games is NP-complete.\n\nA similar proof argument yields the following computational-complexity result.\n\nTheorem 7 The pure-Nash extension problem for n-player SS \u03b1-IDS games is NP-complete.\n\nCombining Theorems 6 and 7 yields the next corollary.\n\nCorollary 1 The pure-Nash extension problem for n-player SC+SS \u03b1-IDS games is NP-complete.\n\n4 Preliminary Experimental Results\n\n(i.e. pi +(cid:80)\n\nTo illustrate the impact of the \u03b1 parameter on \u03b1-IDS games, we perform experiments on randomly-\ngenerated instances of \u03b1-IDS games in which we compute a possibly approximate NE. Given \u0001 > 0,\nin an approximate \u0001-NE each individual\u2019s unilateral deviation cannot reduce the individual\u2019s ex-\npected cost by more than \u0001. The underlying structures of the instances use network graphs from\npublicly-available, real-world datasets [6, 16\u201320]. Appendix D of the supplementary material pro-\nvides more speci\ufb01c information on the size of the different graphs in the real-world dataset. The\nnumber of nodes/players ranges from 34 to \u2248 37K while the number of edges ranges from 78 to\n\u2248 368K. The table lists the graphs in increasing size (from top to bottom). To generate each instance\nwe generate (1) Ci/Li where Ci = 103\u2217(1+random(0, 1)) and Li = 104 (or Li = 104/3) to obtain\na low (high) cost-to-loss ratio and \u03b1i values as speci\ufb01ed in the experiments; (2) pi such that \u2206sc\ni or\nis [0, 1]; and (3) qji\u2019s consistent with probabilistic constraints relative to the other parameters\n\u2206ss\nj\u2208P a(i) qji \u2264 1). On each instance, we initialize the players\u2019 mixed strategies uniformly\ni\nat random and run a simple gradient-dynamics heuristic based on regret minimization [21\u201323] until\nwe reach an (\u0001) NE. In short, we update the strategies of all non-\u0001-best-responding players i at each\nround t according to x(t+1)\nPa(i))). Note that for \u0001-NE to be\nwell-de\ufb01ned, all Mis\u2019 values are normalized. Given that our main interest is to study the structural\nproperties of arbitrary \u03b1-IDS games, our hardness results of computing NE in such games justify\nthe use of a heuristic as we do here. (Kearns and Ortiz [8] and Chan et al. [7] also used a similar\nheuristic in their experiments.). Table 2 shows the average level of investment at NE over ten runs\non each graph instance. We observe that higher \u03b1 values generate more SC players, consistent with\nthe nature of the game instances. Almost all of the SC players invest while most of the SS players\ndo not invest, regardless of the number of players in the games and the \u03b1 values. This makes sense\nbecause of the nature of the SC and SS players. Going from high to low cost-to-loss ratio, we see\nthat the number of SS players and the percentage of SS players investing at a NE increase across\nall \u03b1 values. In both high and low cost-to-loss ratio cases, we see a similar behavior in which the\nmajority of the SS players do not invest (\u2248 50%).\nAcknowledgments\nThis material is based upon work supported by an NSF Graduate Research Fellowship (\ufb01rst author)\nand an NSF CAREER Award IIS-1054541 (second author).\n\ni \u2212 10\u00d7 (Mi(1, x(t)\n\nPa(i))\u2212 Mi(0, x(t)\n\ni \u2190 x(t)\n\n8\n\n\fReferences\n[1] Geoffrey Heal and Howard Kunreuther. Interdependent security: A general model. Working\n\nPaper 10706, National Bureau of Economic Research, August 2004.\n\n[2] Geoffrey Heal and Howard Kunreuther. IDS models of airline security. Journal of Con\ufb02ict\n\nResolution, 49(2):201\u2013217, April 2005.\n\n[3] Geoffrey Heal and Howard Kunreuther. The vaccination game. Working paper, Wharton Risk\n\nManagement and Decision Processes Center, January 2005.\n\n[4] Konstantinos Gkonis and Harilaos Psaraftis. Container transportation as an interdependent\n\nsecurity problem. 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