On the complexity of strong Nash equilibrium: Hard-to-solve instances and smoothed complexity
aa r X i v : . [ c s . G T ] A p r On the complexity of strong Nash equilibrium:Hard–to–solve instances and smoothed complexity
Nicola Gatti and
Marco Rocco
Politecnico di MilanoPiazza Leonardo da Vinci 32Milano, Italy { ngatti, mrocco } @elet.polimi.it Tuomas Sandholm
Carnegie MellonComputer Science Department5000 Forbes AvenuePittsburgh, PA 15213, [email protected]
Abstract
The computational characterization of game–theoretic solution concepts is a central topic in arti-ficial intelligence, with the aim of developing com-putationally efficient tools for finding optimal waysto behave in strategic interactions. The central so-lution concept in game theory is
Nash equilibrium (NE). However, it fails to capture the possibilitythat agents can form coalitions (even in the 2–agentcase).
Strong Nash equilibrium (SNE) refines NEto this setting. It is known that finding an SNE is
N P –complete when the number of agents is con-stant. This hardness is solely due to the existenceof mixed–strategy SNEs, given that the problem ofenumerating all pure–strategy SNEs is trivially in P . Our central result is that, in order for a gameto have at least one non–pure–strategy SNE, theagents’ payoffs restricted to the agents’ supportsmust, in the case of 2 agents, lie on the same line,and, in the case of n agents, lie on an ( n − –dimensional hyperplane. Leveraging this result, weprovide two contributions. First, we develop worst–case instances for support–enumeration algorithms.These instances have only one SNE and the supportsize can be chosen to be of any size—in particular,arbitrarily large. Second, we prove that, unlike NE,finding an SNE is in smoothed polynomial time:generic game instances (i.e., all instances exceptknife–edge cases) have only pure–strategy SNEs. The computational characterization of game–theoretic solu-tion concepts is a central topic in artificial intelligence, withthe aim of developing computationally efficient tools for find-ing optimal ways to behave in strategic interactions. Thecentral solution concept provided by game theory is
Nashequilibrium (NE). Every finite game admits at least one NEin mixed strategies. Computer scientists have characterizedthe complexity of NE finding and provided a number of al-gorithms. Finding an NE of a strategic–form (aka normal-form) game is
PPAD –complete [Daskalakis et al. , 2006]even with just two agents [Chen et al. , 2009]. Such games arecalled bimatrix games. Although
PPAD ⊆ N P , it is gen- erally believed that
PPAD 6 = P and therefore that there notexists any polynomial–time algorithm to find an NE unless P = N P . Furthermore, bimatrix games do not have a fullypolynomial–time approximation scheme unless
PPAD ⊆P [Chen et al. , 2009] and finding an NE in bimatrix games isnot in smoothed– P unless PPAD ⊆ RP [Chen et al. , 2006]and, therefore, by definition of smoothed complexity, gameinstances remain hard even if subjected to small pertur-bations. Instances [Savani and von Stengel, 2006] that re-quire exponential time when solved with a number ofalgorithms [Lemke and Howson, 1964; Porter et al. , 2009]are known. (However, these instances are unstable:they become easy when small perturbations are ap-plied [Gatti et al. , 2012].)NE captures the situation in which no agent can gainmore by unilaterally changing her strategy. When agentscan form coalitions and change their strategies multilaterallyin a coordinated way, the most natural solution concept is strong Nash equilibrium (SNE) [Aumann, 1960]. An SNEis a strategy profile from which no coalition can deviate in away that benefits each of the deviators. SNE has significantlydifferent properties than NE. An SNE is not assured to exist.Finding an SNE (determining if one exists) is
N P –completewhen the number of agents is constant;
N P –hardness wasproven in [Conitzer and Sandholm, 2008] and membershipin
N P in [Gatti et al. , 2013]. Unlike for NE, the literaturehas very few algorithms for SNE. There are algorithms forfinding pure–strategy SNEs for specific classes of games,e.g., congestion games [Holzman and Law-Yone, 1997;Hayrapetyan et al. , 2006; Rozenfeld and Tennenholtz, 2006;Hoefer and Skopalik, 2010], connec-tion games [Epstein et al. , 2007], maxcutgames [Gourv`es and Monnot, 2009], and continuousgames [Nessah and Tian, 2012]. However, the hardnessin the general setting is due to the existence of mixed–strategy SNEs, given that the pure–strategy ones can befound in polynomial time by combining support enu-meration and verification. The only prior SNE–findingalgorithm that works also for mixed strategies is onlyfor 2–agent games [Gatti et al. , 2013]. Its applicationto instances from the ubiquitous NE benchmark testbed,GAMUT [Nudelman et al. , 2004], shows that these instanceseither admit pure SNEs or do not admit any SNE andtherefore that new benchmark testbeds for SNE–findinglgorithms are needed.In this paper, we provide the following main contributions. • We show that, if there is a mixed–strategy SNE, then thepayoffs restricted to the actions in the support must sat-isfy specific conditions. For example, in 2–agent games,they must lie on the same line in agents’ utilities space. • We show how to generate 2–agent games with m actionsper agent, only one SNE, and any desired number of ac-tions { , . . . , m } in the support of each agent’s mixedstrategy. These are the worst–case instances for support–enumeration algorithms, requiring time O (4 m ) . • We show that, for any number of agents, finding anSNE is in smoothed– P , thus admitting a deterministicsupport–enumeration algorithm with smoothed polyno-mial running time. In the generic case (i.e., in all exceptknife–edge cases), all SNEs are pure. A strategic–form game is a tuple ( N, A, U ) where [Shoham and Leyton-Brown, 2008]: • N = { , . . . , n } is the set of agents (we denote by i ageneric agent), • A = { A , . . . , A n } is the set of agents’ actions and A i is the set of agent i ’s actions (we denote a generic actionby a , and by m i the number of actions in A i ), • U = { U . . . , U n } is the set of agents’ utility arrayswhere U i ( a , . . . , a n ) is agent i ’s utility when the agentsplay actions a , . . . , a n .We denote by x i ( a i ) the probability with which agent i playsaction a i ∈ A i and by x i the vector of probabilities x i ( a i ) ofagent i . We denote by ∆ i the space of well–defined probabil-ity vectors over A i . We denote by S i the support of agent i ,that is, the set of actions played with positive probability, andby S the support profile ( S , . . . , S n ) .The most central solution concept in game theory is NE. Astrategy profile x = ( x , . . . , x n ) is an NE if, for each i ∈ N , x Ti U i Q j = i x − j ≥ x ′ Ti U i Q j = i x − j for every x ′ i ∈ ∆ i .Every finite game admits at least one NE in mixed strategies.The problem of finding an NE can be expressed as: v i − X a − i ∈ A − i U i ( a i , a − i ) · Y j ∈ N : j = i x j ( a j ) ≥ ∀ i ∈ N, a i ∈ A i (1) x i ( a i ) · v i − X a − i ∈ A − i U i ( a i , a − i ) ·· Y j ∈ N : j = i x j ( a j ) ! = 0 ∀ i ∈ N, a i ∈ A i (2) x i ( a i ) ≥ ∀ i ∈ N, a i ∈ A i (3) X ai ∈ Ai x i ( a i ) = 1 ∀ i ∈ N (4) Here v i is the expected utility of agent i . Constraints (1) forcethe expected utility v i to be non–smaller than the expectedutility given by every action a i available to agent i . Con-straints (2) force the expected utility v i of agent i to be equalto the expected utility given by every action a i that is playedwith positive probability by agent i . Constraints (3) forceeach probability x i ( a i ) to be nonnegative. Constraints (4)force each probability vector x i to sum to one. An SNE [Aumann, 1960] strengthens the NE concept byrequiring the strategy profile to be resilient also to multilateraldeviations by any coalition of agents. That is, in an SNE nocoalition of agents can deviate in a way that strictly increasesthe expected utility of each member of the coalition, againkeeping the strategies of the agents outside the coalition fixed.So, an SNE combines two notions: an SNE is an NE and itis weakly Pareto efficient over the space of all the strategyprofiles for each possible coalition. Unlike an NE, an SNE isnot assured to exist even in mixed strategies.Multi–objective programming provides Pareto optimal-ity conditions [Miettinen, 1999], based on Karush–Kuhn–Tucker (KKT) results. These conditions are: X i λ i · ∇ f i ( z ) + X j µ j · ∇ g j ( z ) + X k ν k · ∇ h k ( z ) = (5) µ j · g j ( z ) = 0 ∀ j (6) λ i , µ j ≥ ∀ i, j (7) X i λ i = 1 (8) where f i ( z ) are the objective functions to minimize, g j ( z ) are inequality constraints as g j ( z ) ≤ , and h k are equalityconstraints as h k ( z ) = 0 . The λ i , µ j , ν k are called KKT mul-tipliers; λ i is the weight of objective function f i , µ j is theweight of constraint g j , and h k is the weight of constraint h k .The KKT conditions (5)–(8) are necessary conditions forlocal Pareto efficiency [Miettinen, 1999]. For games, we canmap these conditions to the case of Pareto efficiency for asingle coalition C ⊆ N as follows: • f i is agent i ’s expected utility multiplied by ‘ − ’ (giventhat in KKT the objective f i is to minimize); • g j is a constraint of the form − x i ( a i ) ≤ ; • h k is a constraint of the form P a i ∈ A i x i ( a i ) − . We study the properties of mixed–strategy SNEs. We focuson the basic case of 2–agent games and we discuss how thereasoning can be extended to games with 3 or more agents.We denote by P mix and by P cor the sets of points in theagents’ utility spaces E [ U ] × E [ U ] that are on the Paretofrontier when the agents play mixed and correlated strategies,respectively. Obviously, points in P cor non–strictly Paretodominate points in P mix , given that mixed strategies consti-tute a subset of correlated strategies. In addition, we denoteby P mix ( S ) and P cor ( S ) the Pareto frontiers in mixed andcorrelated strategies, respectively, when the game is restrictedto the sets of actions in support profile S . Theorem 3.1
Consider a non–degenerate 2–agent gamewith two actions per agent. If there is a mixed–strategy SNE,then P mix = P cor .Proof . We can write down the game as follows: agent 2 a g e n t a a a p , q p , q a p , q p , q By assumption: • There is a mixed–strategy NE. Therefore: x ( a ) · p + x ( a ) · p = x ( a ) · p + x ( a ) · p (9) x ( a ) · q + x ( a ) · q = x ( a ) · q + x ( a ) · q (10) The NE is on P mix , being an SNE. Therefore, the KKTconditions are satisfied (since they are necessary condi-tions for local weak Pareto efficiency): − λ · ( x ( a ) · p + x ( a ) · p ) −− λ · ( x ( a ) · q + x ( a ) · q ) = ν (11) − λ · ( x ( a ) · p + x ( a ) · p ) −− λ · ( x ( a ) · q + x ( a ) · q ) = ν (12) − λ · ( x ( a ) · p + x ( a ) · p ) −− λ · ( x ( a ) · q + x ( a ) · q ) = ν (13) − λ · ( x ( a ) · p + x ( a ) · p ) −− λ · ( x ( a ) · q + x ( a ) · q ) = ν (14) By combining (9), (10) with (11)–(14), we obtain x ( a ) · q + x ( a ) · q = x ( a ) · q + x ( a ) · q (15) x ( a ) · p + x ( a ) · p = x ( a ) · p + x ( a ) · p (16) By trivial mathematics, we can rewrite (9), (10), (15), (16) as x ( a ) · ( p − p ) = x ( a ) · ( p − p ) x ( a ) · ( q − q ) = x ( a ) · ( q − q ) x ( a ) · ( q − q ) = x ( a ) · ( q − q ) x ( a ) · ( p − p ) = x ( a ) · ( p − p ) We can safely assume p = p , p = p , p = p , p = p ,and the analogous inequalities for agent , since this assump-tion excludes only degenerate games. Indeed, if p = p ,then, by the above conditions, we have p = p , and there-fore actions a and a are the same; if p = p , then, by theabove conditions, we have p = p , and therefore, in orderto have a mixed–strategy NE, we need p = p = p = p .The same reasoning holds for agent . Thus, we derive thefollowing conditions: p − p p − p = q − q q − q p − p p − p = q − q q − q We can give a simple geometric interpretation of the aboveconditions. Call R i = ( p i , q i ) . Each R i is a point in the space E [ U ] × E [ U ] . The above conditions state that: • R R is parallel to R R , • R R is parallel to R R ,and therefore R , R , R , R are the vertices of a parallelo-gram, see Fig. 1( a ). Given that • a mixed–strategy NE is strictly inside the parallelogram(it being the convex (non–degenerate) combination ofthe vertices), see Fig. 1( a ), and that • it must be on a Pareto efficient edge (since, if it isstrictly inside the parallelogram—as in Fig. 1( a )—thenit is Pareto dominated by some point on some edge),we have that R , R , R , R must be aligned according to aline of the form E [ U ] + φ · E [ U ] = const with φ ∈ ( − , ,see, e.g., Fig. 1( b ). Thus, the combination of R , R , R , R through every mixed–strategy profile lies on the line con-necting the two extreme vertices; for example, in Fig. 1( b )the extreme vertices are R and R . It trivially follows that P mix = P cor . (cid:3) The proof of the above theorem provides necessary condi-tions for a game to admit a mixed–strategy SNE. These con-ditions are not sufficient. Indeed, we can show that only somealignments of R , R , R , R can lead to an SNE: b bbbb R R R R NE E [ U ] E [ U ] ( a ) bbbb b SNE R R R R E [ U ] E [ U ] ( b ) Figure 1: Examples used in the proof of Theorem 3.1.
Corollary 3.2
The only alignments of R , R , R , R thatcan lead to an SNE satisfy the following conditions: • R or R is one extreme, • moving from the previous extreme, the next vertex is R or R , • the next vertex is R or R , • R or R is the other extreme.Proof . We initially show that it is not possible to have amixed–strategy NE for alignments different from those con-sidered in the corollary. For reasons of space, we study asingle case, the proof in the other cases is similar. Con-sider, for contradiction, the case in which R is the extremewith the maximum E [ U ] , then the sequence is R , R , and R . It can be observed that the first action of the first agentdominates the second action because U ( R ) > U ( R ) and U ( R ) > U ( R ) and therefore the first agent willplay the first action with probability one. So, there is nomixed–strategy NE. Instead, for the alignments consideredin the corollary, dominance does not apply, as shown, e.g., inFig. 2. (cid:3) agent 2 a g e n t a a a , R ) 0 , R ) a , R ) 2 , R ) bbbb b SNE R R R R E [ U ] E [ U ] Figure 2: Example 2–agent game (left) and its Pareto frontier(right).We now extend the previous result to the setting in whicheach agent has m actions and | S | = | S | = 2 . Corollary 3.3
Consider a non–degenerate 2–agent gamewith m actions per agent. If there is a mixed–strategy SNEwith support sizes | S | = | S | = 2 , then P mix ( S , S ) = P cor ( S , S ) .Proof . We can split the NE constraints and KKT conditionsinto two groups: those generated considering deviations to-wards pure or mixed strategies over the supports S and S and those generated considering deviations towards pure ormixed strategies over actions off the supports S and S . Theconstraints belonging to the first group are the same as in thecase with two actions per agent considered in the proof ofTheorem 3.1. The second group overconstrains the problemand it is not necessary for the proof. Thus, restricting thegame to the actions in S and S , Theorem 3.1 holds andtherefore P mix ( S , S ) = P cor ( S , S ) , see, e.g., Fig. 3. (cid:3) gent 2 a a a a a g e n t a , , − , − − , − a , , − , − − , − a − , − − , − , , a − , − − , − , , b SNE E [ U ] E [ U ] Figure 3: Example 2–agent game (left) and its Pareto frontier(right).Next we study the case of arbitrary–sized (potentially de-generate) games admitting SNEs with full supports.
Theorem 3.4
Consider a 2–agent game with m and m . Ifthere is a mixed–strategy SNE with | S | = m and | S | = m , then P mix = P cor .Proof sketch . The proof is similar to the proof of Theo-rem 3.1. Define R i,j = ( U ( i, j ) , U ( i, j )) . By NE con-straints and KKT conditions, we have that • for all j , P a i x ( a i ) · R i,j = const (i.e., the convexcombinations of the elements of each column of the bi-matrix must have the same value), and • for all i , P a j x ( a j ) · R i,j = const (i.e., the convexcombinations of the elements of each column of the bi-matrix must have the same value),and we have that • each convex combination is strictly inside the polygonwhose vertices are points R i,j (because the combinationis not degenerate), and • the combination must be on a Pareto efficient edge (oth-erwise it would be Pareto dominated),and therefore all the vertices must be aligned. In this way, thecombination of all the points R i,j for every mixed strategyleads to a point that lies on the line connecting all the R i,j .Thus we have P mix = P cor . (cid:3) The extension of Corollary 3.2 to the case with support sizelarger than two is very involved and we omit it here due tolimited space. Instead, Corollary 3.3 easily extends to thecase in which the support per agent is larger than two (weomit the proof because it is the same as that of Corollary 3.3).
Corollary 3.5
Consider a 2–agent game with m and m . Ifthere is a mixed–strategy SNE with | S | = m and | S | = m , then P mix ( S , S ) = P cor ( S , S ) . We will now discuss how the above results extend tomore than two agents. For example, in the 3–agent set-ting, the vector of payoffs for each action profile is R i,j,k =( U ( i, j, k ) , U ( i, j, k ) , U ( i, j, k )) . The crucial result is thatnecessary conditions, generated for only the actions in thesupports, for mixed–strategy SNEs forced by NE constraintswith KKT conditions for all the coalitions (i.e., { , } , { , } , { , } , { , , } ) require that all the R i,j,k lie on a plane (with n –agent games, all the payoff vectors restricted on S must lieon an ( n − –dimensional hyperplane). Thus, we have: Theorem 3.6
Consider an n –agent game. If there is amixed–strategy SNE with support profile S then P mix ( S ) = P cor ( S ) . The above theorem provides only some necessary conditions,given that many other conditions, due to, e.g., dominance, arerequired to have mixed–strategy SNEs. Interestingly, we canshow that there are vectors of payoffs that satisfy all theseconditions. For example, the following 3–agent game has amixed–strategy SNE in which all the agents randomize withuniform probability over all their actions: agent 2 a a a a g e n t a , , , , , , a , , , , , , a , , , , , , Agent 3 plays action a agent 2 a a a a g e n t a , , , , , , a , , , , , , a , , , , , , Agent 3 plays action a agent 2 a a a a g e n t a , , , , , , a , , , , , , a , , , , , , Agent 3 plays action a We now leverage the results from the previous section to showthat it is possible to generate games, with m actions per agent,that have only one SNE, and the support size can be set any-where in the range { , . . . , ⌈ m ⌉} . Thus, given m , we cangenerate O (4 m ) different game instances, each with a differ-ent SNE. These game instances are the worst–case instancesfor support–enumeration algorithms, given that, for each pos-sible enumeration, it is possible to generate an instance thatrequires the algorithm to scan an exponential number of sup-ports.We first introduce some results that we subsequently ex-ploit to generate our hard instances. Corollary 4.1
Given an even number m , consider the fol-lowing two–agent game in which each agent has m = 2 m actions. U i = " U , i U , i U , i U , i where U j,ki are matrices m × m defined as follows: U , ( i, k ) = ( if i + j odd otherwise U , ( i, k ) = ( − m if i = j otherwise U , ( i, k ) = − mU , ( i, k ) = 0 U , ( i, k ) = ( if i + j even otherwise U , = U , U , = U , U , = U , This game has an SNE with support sizes | S | = | S | = m and no other SNE.Proof . Denote by A i the first m actions of agent i and by A i the second m actions of agent i . For clarity we split theproof into two parts, discussed in the following paragraphs,respectively. An example game is in Fig. 4. gent 2 a a a a a a a a a g e n t a , , , , − m, − m − m, − m, − m, a , , , , − m, − m, − m − m, − m, a , , , , − m, − m, − m, − m − m, a , , , , − m, − m, − m, − m, − m a − m, − m , − m , − m , − m , , , , a , − m − m, − m , − m , − m , , , , a , − m , − m − m, − m , − m , , , , a , − m , − m , − m − m, − m , , , , . . . . . . b SNE E [ U ] E [ U ] Figure 4: Example game that has an SNE with | S | = | S | = 4 = m (left), and its Pareto frontier (right). There is an SNE with | S | = | S | = m . The strategy pro-file in which each agent i plays all actions in A i with uniformprobability m is an SNE. All the actions in the supports, i.e., A i , provide utility , while all the actions off the supports,i.e., A i , provide utility · ( m − − mm = − m . Therefore, it isan NE. Furthermore, the outcomes with utilities (1 , and (0 , are Pareto efficient, and the others are (weakly) domi-nated. Thus the Pareto frontier is a line that connects (1 , to (0 , ; therefore the NE is Pareto efficient. So, it is an SNE. There is no other SNE . We focus on the strategy of agent (the same reasoning can be applied for agent ). Supposeagent adopts a different strategy than the above uniformstrategy over A . If agent randomizes over a strict subsetof A composed of only odd actions or only even actions,then agent has two best responses, one in A and one in A (each provides agent utility ). However, there is no NE ofthis form, given that, if agent randomizes over such best re-sponses, agent ’s best response would be to play some actionin A with a positive utility in place of a utility of − m givenby the actions in A . If agent randomizes over a strict subsetof A with even and odd actions, then agent has one or morebest responses in A that provide utility to agent . Alsoin this case, if agent randomizes over such best responses,agent would play some actions in A gaining in place of − m given by actions in A . If agent randomizes over lessthan m actions in A and A , then agent ’s best response isto play some action in A . Also in this case agent ’s bestresponse would be to play some action in A . Finally, whenboth agents play only actions in A i or play more than m ac-tions, we can have NEs, but these NEs are Pareto dominated(because they are randomizations over Pareto–dominated out-comes). (cid:3) Games with m odd are a simple variation w.r.t. the evencase (the proof is omitted being very similar): Corollary 4.2
Given an odd number m , consider the follow-ing two–agent game in which each agent has m = 2 m ac-tions. U i = " U , i U , i U , i U , i where U j,ki are matrices m × m defined as follows: • U , i , U , i , U , i are generated as in the case of m even, • U , i are composed of a submatrix ( m − × ( m − generated as in the case of m even and all the otherentries are . This game has an SNE with support sizes | S | = | S | = m and no other SNE. We are now ready to define our hard–to–solve instances.
Definition 4.3 (Hard–to–solve instances)
Given m and m ,if m ∈ { , . . . , ⌊ m ⌋} , a hard–to–solve game instance is com-posed as follows: • a sub–bimatrix of size m × m built as described inCorollaries 4.1 and 4.2, • all the other entries are drawn from {− m . . . , } withuniform probability.If m = 1 , all the entries are drawn from {− m . . . , } withuniform probability except for a pure action profile in whichboth agents have a utility of . Theorem 4.4
Hard–to–solve instances admit only one SNE.Proof sketch . The SNE described in Corollaries 4.1 and 4.2is still an SNE because all the additional entries are smallerthan the expected utility of the SNE. Because these additionalentries are Pareto dominated, no additional SNE exists. (cid:3)
We leave open the extension to n –agent games. Worst–case complexity, being too pessimistic, is often a badindicator of the actual performance of an algorithm, andaverage–case complexity is difficult to determine. A newermetric of complexity, called smoothed complexity, has beengaining interest in recent years [Bl¨aser and Manthey, 2012].It studies how the introduction of small perturbations affectsthe worst–case complexity. There might be several models ofperturbations. By far the most common perturbation modelsare the uniform one and the Gaussian one.In the case of the SNE-finding problem, given a perturba-tion D σ of magnitude σ , these are defined as follows. • Uniform perturbation: for each agent i , every entry in U i is subjected to an additive perturbation [ − σ, + σ ] withuniform probability. • Gaussian perturbation: for each agent i , every entry in U i is subjected to an additive perturbation [ − z, + z ] withprobability σ √ π e −| U i ( j,k ) − z | /σ .Denote by ˜ U i the perturbed utility matrix.We will first present results for the 2–agent setting. In theend of this section we show the generalization to any numberof agents.The smoothed running time of an algorithm A given a per-turbation D σ is defined as smoothed– t A = E ˜ U , ˜ U ∼D σ [ t A ( ˜ U , ˜ U ) | U , U ] here t A ( ˜ U , ˜ U ) is the running time for the games instance ( ˜ U , ˜ U ) . An algorithm has smoothed polynomial time com-plexity if for all < σ < there are positive constants c, k , k such that: smoothed– t A = O ( c · m k · σ − k ) where m is the size of the game in terms of actions per agent.Basically, a problem is in smoothed– P if it admits a smoothedpolynomial time algorithm. Theorem 5.1
Finding an SNE is in smoothed– P .Proof . We provide Algorithm 1. It has three main parts.In the first part (Steps 1–3), the algorithm searches for apure–strategy SNE by enumerating all the pure–strategy pro-files and verifying whether each strategy profile is an SNE.The verification is accomplished by checking whether or notNE constraints are satisfied (this can be done in polynomialtime in m ) and by checking whether or not the strategy pro-file is on the Pareto frontier (this can be done in polynomialtime as shown in [Gatti et al. , 2013]). If an SNE is found, thealgorithm returns it. The maximum number of iterations inthe first part of the algorithm is m .In the second part (Steps 4–7), the algorithm verifieswhether there there are strategy profile of support sizes | S | + | S | = 3 such that P mix ( S , S ) = P cor ( S , S ) . This can beaccomplished in time O ( m ) by checking whether there is aline connecting at least three entries of all the sub–bimatricesof size × . In the affirmative case, the temporary variable temp is set true . Otherwise, temp is set false .In the third part (Steps 8–13), if temp is false , the algo-rithm returns non–existence , given that, by Theorem 3.4,there is no mixed–strategy SNE. Otherwise, the algorithmenumerates all supports to find mixed–strategy NEs, and foreach of them the algorithm verifies whether it is an SNE asdone in Steps 1–3. In the latter case, the algorithm can takeexponential time as discussed in the previous section.Thus, the running time of Algorithm 1 is super–polynomialonly if it needs to enumerate supports during Steps 8–13(this can take exponential time). This happens only when P mix ( S , S ) = P cor ( S , S ) for some S , S with | S | + | S | = 3 . Given that the perturbations D σ over all theentries of the utility matrices are independent and identi-cally distributed, the probability that the perturbed entriesare aligned as required by Corollary 3.5 to have SNEs with | S | = | S | > is zero. Therefore, the smoothed runningtime of Algorithm 1 is polynomial in m and independent of σ (for both uniform and Gaussian perturbations). (cid:3) The above result shows that, except for a space of the pa-rameters with zero measure, games admit only pure–strategySNEs and therefore that verifying the existence of an SNEand finding them is computationally easy. Interestingly, theinstability is due to the combination of NE constraints andPareto efficiency constraints. Indeed, both the problem offinding an NE and the problem of finding Pareto efficientstrategies are not sensitive to perturbations. Thus, while anapproximate NE can be found by perturbing a game and find-ing an NE of the perturbed game (exactly, an NE of an ǫ –perturbed game is an ǫ –NE of the original game), this is notthe case with SNE. Indeed, if a game admits only mixed–strategy SNEs, once perturbed, it does not admit any SNE. Algorithm 1
SNEfinding for all pure–strategy support profiles do if the support profile has an SNE then return the SNE4: temp ← false for all strategy profiles with | S | + | S | = 3 do if payoffs restricted to supports are aligned then temp ← true if not temp then return non–existence else for all support profiles do if the support profile has an SNE then return the SNE The above results extend to the setting with any (constant)number of agents. As discussed in Section 3, in order to havemixed–strategy SNEs with n agents, all the payoffs vectorsof the actions in the agents’ supports must lie on the same ( n − –hyperplane. Thus, if perturbed even with small per-turbations, the probability that the payoff vectors satisfy suchconstraints is zero. Therefore, generic game instances (i.e., allinstances except knife–edge cases) have only pure–strategySNEs (if any). Strong Nash equilibrium (SNE) is the most natural solutionconcept for games where agents can form coalitions, i.e., co-ordinate their strategies. An SNE is a strategy profile whereno coalition can deviate in a way that every one of the devi-ating agents (strictly) benefits. Given a finite game, a Nashequilibrium always exists, but an SNE might not exist. Forfinding an SNE, there has been a shortage of algorithms. Mostalgorithms only find pure–strategy SNEs in special gameclasses. A recent algorithm finds SNEs generally, but onlyin 2–agent games.Our central result is that, in order for a game to have atleast one mixed–strategy (i.e., non–pure–strategy) SNE, theagents’ payoffs restricted to the agents’ supports must, in thecase of two agents, lie on the same line, and, in the case of n agents, lie on an ( n − –dimensional hyperplane. Leverag-ing this result, we provided two contributions. First, we de-veloped worst–case game instances for support–enumerationalgorithms. These instances have only one SNE and the sup-port size can be chosen to be of any size—in particular, arbi-trarily large. In this way, for each possible enumeration, it ispossible to generate an instance that requires the algorithm toscan an exponential number of supports. Second, we provedthat, unlike Nash equilibrium, finding an SNE is in smoothedpolynomial time: generic game instances (i.e., all instancesexcept knife–edge cases) have only pure–strategy SNEs.In future research we plan to study the computational com-plexity of approximating SNE and to design algorithms todo so. We also plan to study computational issues related to strong correlated equilibrium . This concept should presentdifferent properties than SNE, e.g., the convexity of the Paretofrontier with this solution concept could make the computa-tion of an equilibrium easier and could make equilibria notsensitive to small perturbations. eferences [Aumann, 1960] R. Aumann. Acceptable points in games ofperfect information. PAC J MATH , 10:381–417, 1960.[Bl¨aser and Manthey, 2012] M. Bl¨aser and B. Manthey.Smoothed complexity theory.
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