Optimal Targeting in Super-Modular Games
OOptimal targeting in super-modular games ∗ Giacomo Como † , Stephane Durand ‡ , and Fabio Fagnani § Department of Mathematical Sciences “G.L. Lagrange”,Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Torino,Italy.September 22, 2020
Abstract
We study an optimal targeting problem for super-modular games withbinary actions and finitely many players. The considered problem consistsin the selection of a subset of players of minimum size such that, when theactions of these players are forced to a controlled value while the others areleft to repeatedly play a best response action, the system will converge tothe greatest Nash equilibrium of the game. Our main contributions consistin showing that the problem is NP-complete and in proposing an efficientiterative algorithm with provable convergence properties for its solution.We discuss in detail the special case of network coordination games and itsrelation with the notion of cohesiveness. Finally, we show with simulationsthe strength of our approach with respect to naive heuristics based onclassical network centrality measures.
In a game with multiple Nash equilibria, what is the minimum number of playersto target in order to force the system to move from an original Nash equilibrium A to a desired Nash equilibrium B ? This paper deals with such a problem forthe class of super-modular games with binary actions and where the two Nashequilibria A and B are, respectively, the least and the greatest in the game.Our contribution is twofold: we show that the problem is NP-complete and wepropose the design of an iterative algorithm for an efficient solution. ∗ This work was partially supported by MIUR grant Dipartimenti di Eccellenza 2018–2022[CUP: E11G18000350001], the Swedish Research Council, and by the Compagnia di San Paolo. † Email: [email protected]. G. Como is also with the Department of AutomaticControl, Lund University, BOX 118, SE-22100, Lund, Sweden. ‡ Email: [email protected] § Email: [email protected] a r X i v : . [ c s . G T ] S e p he considered problem can be framed in the more general setting of study-ing minimal intervention strategies needed to drive a multi-agent system gov-erned by agents’ myopic utility maximization to a desired configuration. Inapplications where the goal is to achieve a social optimum, such interventionsare often modeled as perturbations of the utility functions that lead to a modifi-cation of the Nash equilibria of the game. This viewpoint is natural for instancein analyzing the effect of taxes or subsidies in economic models or prices and tollsin transportation systems. More recently, a similar approach has been proposedin the context of network quadratic games [6] to model incentive interventionsfor instance in school and economic systems.A different viewpoint, that is the one considered in this paper, is that ofindividuating a subset of nodes (hopefully small) that if suitably controlled willlead the entire system to the desired equilibrium. The minimum cardinality ofthis set can also be interpreted as a measure of resilience of the system’s equilib-rium: the larger it is, the more energy is needed by an external intervention todestabilize it. In the context of binary actions { , } considered in this paper,the control action simply amounts to force the set of chosen players, originallyplaying action 0 state, to play action 1. This well models situations where action1 indicates the use of a certain technology or the adoption of a new product andthe control action corresponds for instance to a marketing intervention where,at the targeted individuals, a certain item is offered for free.Super-modular games have received a great deal of attention in the recentyears as the basic way to model strategic complementarity effects [11]. Itsvariegate applications include modeling of social and economic behaviors likeadoption of a new technology, participation to an event, provision of a publicgood effort. They are typically endowed with multiple Nash equilibria thatadmit a Pareto ordering and the problem of the minimal effort needed to pushthe system from a lower to a greater equilibrium is natural and relevant in allthese applicative contexts.A fundamental example of super-modular games is that of coordinationgames over networks. The binary coordination game is analyzed in detail in[16] where the key concept of cohesiveness of a set of players is introduced andthen used in order to characterize all Nash equilibria. Moreover, the question ifan initial seed of influenced players (that maintain action 1 in all circumstances)is capable of propagating to the all network is addressed in the same paper andan equivalent characterization of this spreading phenomenon is also expressedin terms of cohesiveness.This contagion phenomenon is the content of our analysis in the more generalframework of super-modular games. A subset of nodes from which propagationis successful is called a sufficient control sets and our goal is to find such setsof minimum possible cardinality. We notice that the condition proposed in [16]is computationally quite demanding and in practice it cannot be used directlyto solve the optimization problem even for medium size games. Indeed, even todetermine if a single set is a sufficient control set, it requires a number of checkgrowing exponentially in the cardinality of the complement of such set.The complementary problem of understanding (for binary coordination games)2hat is the maximum possible spreading of the state 1 starting from an initialseed of a given number k of targeted players, was studied in a seminal paperby [12]. While their problem and ours are related, they are independent, in thesense that solving one does not provide a solution of the other. Another pointworth stressing is that, in their setting, [12] consider players equipped with ran-dom independent activation thresholds and chose to optimize the expected sizeof the maximum spreading. They prove that such functional is sub-modularand then they design a greedy algorithm for obtaining sub-optimal solutions.The randomness that they introduce is actually crucial in their approach, asthe functional considered is not sub-modular for deterministic choices of thresh-olds. This lack of sub-modularity is actually a key feature of coordination gameswhere the utility functions present a threshold behavior and make it unfeasibleto try to approximate our targeting problem by iteratively adding target nodesin a greedy way.A targeting intervention problem, related to the one studied in [12], is con-sidered in [7]. There, the authors consider the problem of a firm that sells a goodto a set of individuals organized through a social network. The firm, in orderto maximize its profit, chooses a set of individuals on which to concentrate itsadvertising efforts or other marketing strategies relative to that specific good.The role of the social network is either of propagating information (in a gossippairwise style) regarding the good so to push other people to buy it, or ratherto model a positive externality effect where the utility of an individual to buythat product depends on the number of neighbors already using it. This secondinstance is particularly related to the problem studied in [12] with the impor-tant difference that here authors model the network in a mean field fashion onlyconsidering the degree distribution.A different targeting intervention problem is studied in [1] where authorsconsider network quadratic games and individuate the k most influential playersby studying how the aggregate output decreases when this set of players isremoved from the network.The general problem of determining the best set of nodes to exert the mosteffective control in a networked system has recently appeared in other contexts.In [13, 8, 17] this is studied in the context of controllability problems for generallinear network systems. In [24], [22], [9] authors focus on the problem of theoptimal position of stubborn influencers in voter models or in linear opiniondynamics.Our main contribution is twofold. First, we prove that the proposed prob-lem is NP-complete, reducing it to the well known 3-SAT problem. Second,we design an iterative randomized search algorithm with provable convergenceproperties towards sufficient control sets of minimum cardinality. The core ofthe algorithm is a time-reversible Markov chain over the family of all sufficientcontrol sets that starts with the full set, moves through all of them in an ergodicway, and concentrates its mass on those of minimum cardinality. A preliminary version of the second part of our results for the special case of networkcoordination games and not containing any complexity analysis were presented at the 21st x , y , z . We de-fine the binary vectors δ i : ( δ i ) i = 1 and ( δ i ) j = 0 for every j (cid:54) = i . For a subset S ⊆ { , . . . , n } , we put S = (cid:80) i ∈S δ i . Every x in { , } n can be written as x = S for some S ⊆ { , . . . , n } . We use the notation for the all-1 vector. We consider finite strategic form games with set of players V = { , . . . , n } whereby each player i choses her action x i from a binary set A = { , } . Let X = A n denote the (strategy) profile space, whose elements x will be referredto as (strategy) profiles. We shall consider the standard partial order on thestrategy profile space X , given by x ≤ y ⇐⇒ x i ≤ y i , ∀ i ∈ V . (1)As customary, given a strategy profile x in X and a player i , we indicate with x − i the strategy profile of all players but i . Each player i is endowed with autility function u i : X → R , so that u i ( x ) = u i ( x i , x − i )denotes the utility of player i when she plays action x i while the rest of theplayers’ strategy profile is x − i . A game will be formally identified by the triple( V , A , { u i } ).The best response for a player i in V is captured by the set-valued function B i ( x − i ) = argmax a ∈A u i ( a, x − i ) , while the set of pure strategy Nash equilibria is formally defined by N = { x ∈ X | x i ∈ B i ( x − i ) ∀ i ∈ V} . Throughout the paper, we shall consider games satisfying the following in-creasing difference property [14].
IFAC World Congress and published in its proceedings [4]. ssumption 1. For every player i in V and every two strategy profiles x , y in X such that x − i ≥ y − i , u i (1 , x − i ) − u i (0 , x − i ) ≥ u i (1 , y − i ) − u i (0 , y − i ) . (2)Assumption 1 states that the marginal utility of increasing player i ’s actionfrom x i = 0 to x i = 1 is a non-decreasing function of the strategy profile x − i of all the other players. For finite games, as is our case, such increasingdifference property is equivalent to super-modularity [20, 23, 21]. For this reason,we will refer to a game ( V , A , { u i } ) satisfying (1) as to a finite super-modulargame. In the economic literature, these are also referred to as games of strategiccomplements [15].A standard result for super-modular games ensures that their set of purestrategy Nash equilibria is always nonempty and there exist a minimal and amaximal Nash equilibria with respect to the partial order (1). Throughout thepaper, we shall assume that such minimal and maximal pure strategy Nashequilibria are the all-0 profile and, respectively, the all-1 profile . This as-sumption implies no effective loss of generality since the presence of playersthat maintain a strict preference for action 0 or action 1 independently from theactions played by the other players can be easily integrated in our frameworkby suitably modifying the other players’ utilities.In this paper, we study the problem of finding subsets of players S ⊆ V ofminimal cardinality for which there exists an improvement path from S to thewhole player set V . This is formalized by the following definitions. Definition 1.
For a finite game with binary actions ( V , A , { u i } ) , a sequence ofstrategy profiles ( x k ) k =0 ,...,m is an improvement path from the set S ⊆ V to theset
T ⊆ V if1. x = S , x m = T
2. for every k = 0 , . . . , m − there exists i k in V \ S such that • x k +1 − i k = x k − i k and x k +1 i k (cid:54) = x ki k • u i k ( x k +1 ) ≥ u i k ( x k ) Definition 2 (Sufficient control set) . For a finite game with binary actions ( V , A , { u i } ) , • S ⊆ V is a sufficient control set if there exists an improvement path from S to V . • A sufficient control set
S ⊆ V is optimal if there exists no sufficient controlset of strictly smaller cardinality. Notice that sufficient control sets always exist, as the whole set of players V trivially is a sufficient control set. Our objective is to find optimal sufficientcontrol sets. 5 key fact is that, in dealing with the concept of sufficient control set, itis not restrictive to consider exclusively improvement paths where all actionchanges are from 0 to 1. Such improvement paths are formally defined below. Definition 3 (Monotone Improvement path) . For a finite game with binaryactions ( V , A , { u i } ) , an improvement path ( x k ) k =0 ,...,m from the set S ⊆ V tothe set
T ⊆ V is called monotone if there exists a sequence of distinct players i k in T \ S for k = 0 , . . . , m − such that x k +1 = x k + δ i k for k = 0 , . . . , m − .Remark . Notice that a monotone improvement path from S to T is completelyspecified by the sequence of players i k in T \ S , k = 1 , . . . , m , which are sequen-tially changing their actions from 0 to 1. Observe that T \ S = { i , . . . , i m } and thus the path length m = |T \ S| coincides with the difference between thecardinality of the arrival set T and the one of the departure set S .The following result formalizes our previous claim. Lemma 1.
In a finite super-modular game with binary actions ( V , A , { u i } ) , S ⊆V is a sufficient control set if and only if there exists a monotone improvementpath from S to V .Proof. Clearly, if there exists a monotone improvement path from S to V , then S is a sufficient control set.Conversely, if S is a sufficient control set, then there exists a (not necessarilymonotone) improvement path ( y k ) k =0 ,...,T in X from S to V . For every player i in V \ S , define k ( i ) = min { k = 1 , . . . , T | y k = y k − + δ i } that is the first time that player i changes her action from 0 to 1 along the path( y k ) k =0 ,...,T . Now, let m = n − |S| and order the players in V \ S as i , . . . , i m in such a way that k ( i ) < k ( i ) < · · · < k ( i m ). Then, for every h = 0 , , . . . , m ,define x h = S + h (cid:88) j =1 δ i j and notice that x h − ≥ y k ( i h ) − . Using the increasing difference property wenow obtain that u i h ( x h ) − u i h ( x h − ) ≥ u i h ( y k ( i h ) ) − u i h ( y k ( i h ) − ) ≥ , for every h = 1 , . . . , m . This shows that ( x k ) k =0 ,...,m is an improvement pathfrom S to V . By construction, this improvement path is also monotone, thusproving the claim.This new characterization of sufficient control sets, allows for proving thefollowing intuitive fact. Proposition 1 (monotonicity for inclusion) . In a finite super-modular gamewith binary actions ( V , A , { u i } ) , if S ⊆ V is a sufficient control set then every
T ⊆ V such that
S ⊆ T is also a sufficient control set. roof. Assume that S is a sufficient control set and let T ⊇ S . Because ofLemma 1, there exists a monotone improvement path ( x k ) k =0 ,...,m from S to V . Consider the associated sequence of players ( i k ) for k = 1 , . . . , m suchthat x k +1 − x k = δ i k for each k . Consider now the subsequence of points i k , i k , . . . , i k m (cid:48) that are in V \ T and put y h = max { T , x k h } for h = 1 , . . . , m (cid:48) .By construction, we have that y h ≥ x k h +1 − . By the increasing differenceproperty (2) and the fact that ( x k ) k =0 ,...,m is a monotone improvement pathfrom S it follows that, for every h , putting i = i k h +1 , u i ( y h +1 ) − u i ( y h ) = u i (1 , y h − i ) − u i (0 , y h − i ) ≥ u i (1 , x k h +1 − − i ) − u i (0 , x k h +1 − − i )= u i ( x k h +1 ) − u i ( x k h +1 − ) ≥ y k ) k =0 ,...,m (cid:48) is a monotone improvement path from T to V . Remark . The notion of sufficient control set introduced in Definition 2 canbe reinterpreted in terms of the asynchronous best response dynamics. Givena subset
S ⊆ V , consider the Markov chain X t on the strategy profile space X whose transitions are described as follows. At every discrete time, a player,among those in V \ S , is chosen uniformly at random and updates her playedaction choosing uniformly at random among the actions of her current bestresponse to the other players’ strategy profile. Notice that the existence of animprovement path from S to V is equivalent to say that, for every initial state X such that X i = 1 for all i in S , the Markov chain X t will reach the all-1profile in finite time with positive probability.Actually, more is true. Consider any strategy profile x in X such that X i = 1for all i in S , equivalently such that x = S (cid:48) for some superset S (cid:48) ⊇ S . If S isa sufficient control set, it follows from Proposition 1 that also S (cid:48) is a sufficient.This implies that there exists a monotone improvement path from S (cid:48) to V andthus X t will also reach from x in finite time with positive probability. If the all-1 strategy profile is a strict Nash equilibrium (in the sense that all players have,in that profile, a best response consisting of the singleton 1) then this argumentproves that S ⊆ V is a sufficient control set if and only if the correspondingMarkov chain X t is absorbed in in finite time with probability one. In themore general case, if there are players for which 0 and 1 are always indifferentindependently from the behavior of the other players, then the condition on theMarkov chain is replaced by the existence of a set of profiles containing onwhich the Markov chain X t gets trapped in finite time with probability one andwithin such set it moves ergodically. A notable example of super-modular games with binary actions is that of net-work coordination games. In this section, after reviewing this class of games, we7tudy the optimal targeting problem for two special instances. We first studycoordination games on arbitrary undirected networks where the players havehomogeneous thresholds characterizing their best responses and we highlight,for this case, the connection of our problem with the notion of cohesiveness [16].The second case we consider is that of coordination games on a complete graphwith heterogeneous thresholds for which, we show that the optimal targetingproblem admits a relatively simple analytical solution (c.f. [10, 18]).Let G = ( V , E , W ) be a finite weighted directed graph, whereby V is the setof nodes, E ⊆ V × V is the set of directed links, and W in R V×V + is the weightmatrix, such that W ij > i, j ) in E directed fromits tail node i to its head node j . A positive entry W ij of the weight matrix W represents the weight of the link ( i, j ). Let w i = (cid:80) j (cid:54) = i W ij denote the out-degreeof a node i in V . We shall assume that G contains no self-loops, equivalently,that the diagonal elements of the weight matrix W are all zero, and no sinks,i.e., that w i > i in V . We shall refer to the graph G as simple if W ij = W ji in { , } (in this case W is completely determined by E ).A network coordination game on a graph G = ( V , E , W ) is a game ( V , A , { u i } )with binary action set A = { , } and utilities u i ( x ) = (cid:88) j (cid:54) = i W ij ((1 − x i )(1 − x j ) + x i x j ) + c i x i , (3)where the constant c i in [ − w i , w i ] models a possible bias of player i towardsaction 0 (if c i <
0) or action 1 (if c i > B i ( x − i ) = { } if w i (cid:80) j (cid:54) = i W ij x j < θ i { , } if w i (cid:80) j (cid:54) = i W ij x j = θ i { } if w i (cid:80) j (cid:54) = i W ij x j > θ i (4)where θ i = w i − c i w i (5)is the threshold of player i in V . In the special case when the graph is simpleand c i = 0 (so that the threshold is θ i = 1 /
2) for every player i in V , this is alsoknown as the majority game . In this subsection, we focus on the special case when the players all have the samethreshold θ i = θ in [0 , S ⊆ V is called α - cohesive in a graph G if (cid:88) j ∈S W ij ≥ αw i , ∀ i ∈ S , (6)8igure 1: An optimal sufficient control set for the complete graphFor a simple graph, the above means that every node in S has at least a fraction α of its neighbors within S (equivalently, at most a fraction 1 − α outside S ).Considerations in [16] and in [11] yield the following characterization of sufficientcontrol sets. Define a subset S ⊆ V uniformly no more than θ -cohesive if nosubset S (cid:48) ⊆ S is θ (cid:48) -cohesive for some θ (cid:48) > θ . The following is a consequence ofthis definition and explicitly proven in [11] (see Proposition 4 therein). Proposition 2.
S ⊆ V is a sufficient control set for the network coordinationgame where all players have threshold θ if and only if V \S is uniformly no morethan (1 − θ ) -cohesive. This reformulation of the concept of sufficient control set is of limited interestfrom the computational point of view. Indeed, checking that the set
V \ S isuniformly no more than (1 − θ )-cohesive involves an analysis of all possiblesubsets of V \ S . Nevertheless, this characterization can be used to analyzespecific cases.Below we present examples of sufficient control sets for the special case ofthe majority game for specific simple connected graphs.
Example 1.
In this example the game we are considering is always the majoritygame (namely the network coordination game where all players have threshold / ) on a simple and connected graph G = ( V , E , W ) . • Let G be the complete graph with n nodes. Then, all subsets consistingof (cid:98) n/ (cid:99) nodes are optimal sufficient control sets. This is because everysubset of (cid:100) n/ (cid:101) or fewer nodes is not θ -cohesive for any θ > / . This saysthat every subset of (cid:100) n/ (cid:101) nodes is uniformly no more than / -cohesiveand result follows from Proposition 2. Optimality follows directly from thefact that smaller subsets are never sufficient control sets. • Let G be a simple connected graph where every node has degree at most . Then, every set consisting of a single node is a sufficient control set(and is automatically optimal). To see this, notice that every strict subsetof players S (cid:40) V must possess a node i in S with W ij > for some j in V \ S (otherwise the graph would not be connected). This implies that theset S cannot be θ -cohesive for any θ > / . We can then conclude as inthe previous item. An instance is depicted in Figure 2. • Let G be a tree. Then, the set of the leaf nodes is always a sufficientcontrol set. Indeed let S be any subset of the nodes not containing leaves and consider a path (a walk with no repeated nodes) of maximum length allconsisting of nodes in S , say ( i , . . . , i l ) . Notice that i can not have otherneighbors in S otherwise the path could be extendable. On the other hand,since i is not a leaf in the tree, it must have degree at least , namely, atleast one neighbor outside of S . This implies that S can not be θ -cohesivefor θ > / . We conclude using again Proposition 2. In general, such setsare not optimal. Indeed, the argument above shows that also the set ofnodes that are neighbors of the leaves is a sufficient control set, typicallyof smaller cardinality than the set of leaves. An example is reported inFigure 3. • Let G be the d -dimensional grid graph having node set V and link set E respectively given by V = { , . . . , k − } d , E = (cid:26) ( a , b ) ∈ V × V | (cid:88) kh =1 | a h − b h | = 1 (cid:27) . Put S l = { ( a , . . . , a d ) ∈ V | (cid:80) a i = l } . We claim that S k − is a sufficientcontrol set. To see this, notice that any a in S l has exactly d neighborsin S l +1 if l < k − . Similarly, any a in S l has exactly d neighbors in S l − if l > k − . Considering that the degree of every node is at most d in G , a simple induction argument then allows to construct a monotoneimprovement path from S k − to the whole of V . It can be checked directlythat this control set is optimal for d = 1 and d = 2 , while is not for d ≥ . n of players and we expect the same to hold in very well connected graphs asfor instance random Erdos-Renji graphs. This conjecture is corroborated bynumerical simulations presented in Section 6. In contrast, for more loosely con-nected graphs (trees, grids), the size of optimal sufficient control sets scales asa negligible fraction of the size n . In this subsection, we focus on network coordination games on the completegraph, whereby W ij = 1 for every i (cid:54) = j in V . In contrast with the previoussubsection, we shall allow for full heterogeneity of the players’ thresholds, thatin this case are given by θ i = n − c i − n − , i ∈ V . Our results show that optimal sufficient control sets can be completely charac-terized in terms of the threshold distribution function F ( z ) = 1 n |{ i ∈ V : θ i ≤ z }| , z ∈ [0 , . First, we have the following technical result.
Lemma 2.
Consider a heterogeneous network coordination game on the com-plete graph with threshold distribution F ( z ) . Then, ∅ is a sufficient control setif and only if F ( z ) ≥ z , ∀ z ∈ [0 , . (7) Proof.
We start with a general consideration that will be used to prove bothimplications. Fix
S ⊆ V and let x = S . Put n = |S| . It follows from (4) thatfor every player i such that x i = 0, it holds B i ( x − i ) = { } if only if θ i > n n − . (8)11uppose ∅ is not a sufficient control set and let S (cid:40) V be a set of maximumcardinality for which there exists a monotone improvement set from ∅ to S . Put x = S and n = |S| ≤ n −
1. It follows from previous consideration that allplayers i such that x i = 0, have a threshold θ i satisfying (8). Therefore, n − n ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) i ∈ V : θ i > n n − (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) = n (cid:18) − F (cid:18) n n − (cid:19)(cid:19) By dividing both sides by n and rearranging terms, we obtain F (cid:18) n n − (cid:19) ≤ n n < n n − . (9)This implies that (7) does not hold true.Suppose instead that (7) does not hold true and let z in [0 ,
1] be such that F ( z ) < z . By the way F is defined, there exists n in { , , . . . , n − } such that F ( z ) = n /n . Observe that n = nF ( z ) < zn implies that n ≤ zn − n n − ≤ zn − n − ≤ z . Then, by monotonicity of the threshold distribution function we get F (cid:18) n n − (cid:19) ≤ F ( z ) = n n . (10)Let S be a set consisting of n players with the least possible threshold andlet x = S . It then follows from (10) that each player i playing x i = 0 hasthreshold satisfying (8) and hence, as observed at the beginning of this proof,it is such that B i ( x − i ) = { } . This implies that there cannot be a monotoneimprovement path from S to V . Consequently S is not a sufficient control setand neither is ∅ by Proposition 1.As an application of Lemma 2 we obtain the following characterization ofthe optimal sufficient control sets for heterogeneous network coordination gameson the complete graph. Proposition 3.
Consider a heterogeneous network coordination game on thecomplete graph with threshold distribution F ( z ) . Then, the minimal size of asufficient control set is M = (cid:24) n · sup ≤ z ≤ [ z − F ( z )] + (cid:25) . In particular, every S consisting of M players i in V with the M largest thresh-olds θ i gives an optimal sufficient control set.Proof. First observe that a subset of players
S ⊆ V is a sufficient control set forthe network coordination game with utilities (3) if and only if ∅ is a sufficientcontrol set for the modified network coordination game with utilities u i ( x ) = (cid:26) (cid:80) j (cid:54) = i ((1 − x i )(1 − x j ) + x i x j ) + ( n − x i if i ∈ S (cid:80) j (cid:54) = i ((1 − x i )(1 − x j ) + x i x j ) + c i x i if i ∈ V \ S , (11)12hereby all the players i in S have modified threshold θ i = 0 and the rest of theplayers j in V \ S have the same threshold θ j = θ j . Let F ( z ) be the thresholddistribution function of this modified game and observe that0 ≤ F ( z ) − F ( z ) ≤ |S| /n , ∀ z ∈ [0 , . (12)We now show that any subset S ⊆ V such that |S| < M can not be asufficient control set. If M = 0 there is nothing to prove. Assume now that M ≥ n · sup ≤ z (cid:48) ≤ [ z (cid:48) − F ( z (cid:48) )] + = M − (cid:15) (13)for some (cid:15) >
0. Notice that, since sup ≤ z (cid:48) ≤ [ z (cid:48) − F ( z (cid:48) )] + >
0, we have thatsup ≤ z (cid:48) ≤ [ z (cid:48) − F ( z (cid:48) )] + = sup ≤ z (cid:48) ≤ [ z (cid:48) − F ( z (cid:48) )] (14)If |S| < M , then (12), (13), and (14) imply that, for every z in [0 , ≤ n (cid:0) F ( z ) − F ( z ) (cid:1) ≤ |S| ≤ M − n · sup ≤ z (cid:48) ≤ [ z (cid:48) − F ( z (cid:48) )] − ε , This yields z − F ( z ) ≥ z − F ( z ) − sup ≤ z (cid:48) ≤ [ z (cid:48) − F ( z (cid:48) )] + ε/n , ∀ z ∈ [0 , . and taking the sup on both sides, we finally obtainsup ≤ z ≤ (cid:2) z − F ( z ) (cid:3) ≥ ε/n > ∅ is not a sufficient control set for the modifiednetwork coordination game with utilities (11) and, consequently, S is not asufficient control set for the original game.To complete the proof, we now consider a set S of M players with the highestthresholds. In this case, F ( z ) = min { , F ( z ) + M/n } ≥ min (cid:26) , F ( z ) + sup ≤ z (cid:48) ≤ [ z (cid:48) − F ( z (cid:48) )] + (cid:27) ≥ z , for every z in [0 , ∅ is a sufficient controlset for the modified network coordination game with utilities (11), thus showingthat S is a sufficient control set for the original game. In this section, we study the complexity of finding sufficient control sets forarbitrary super-modular games and prove that it is an NP-complete problem[19, Section 7.4].Formally, given a binary super-modular game and a positive integer n wedefine SCS to be the logical proposition ”there exists a sufficient control set ofsize less then or equal to s for the game”.13 heorem 1. The problem
SCS is NP-complete.
In order to prove Theorem 1, we will first show that
SCS belongs to thecomplexity class NP (c.f., [19, Definition 7.19]) and then that it is NP-hard.
Lemma 3.
The problem
SCS belongs to NP.Proof.
We show that, given an instance of a finite binary-action super-modulargame and a witness consisting in subset of players
S ⊆ V , checking if S is asufficient control set can be done in a time growing proportionally to the squareof n − s , where n = |V| and s = |S| . In fact, this can be achieved by an iterativealgorithm that starts with time index t = 0 and profile x (0) = S and thenproceeds as follows. If there exists at least one player i in V such that x i ( t ) = 0 , ∈ B i ( x − i ( t )) , (15)then arbitrarily chose one such player i , increase the time index t by one unit anddefine the new profile x ( t ) with x i ( t ) = 1 and x − i ( t ) = x − i ( t − i satisfying (15) exists, then halt and return the current value of thetime index t . Since, by Proposition 1, every superset of a sufficient control set isitself a sufficient control set, we have that S is a sufficient control set if and onlyif the algorithm defined above terminates with t = n − s . Clearly, the numberof steps of the algorithm is at most n − s and at the t -th step, it is necessaryto compute the best responses of at most n − t players, so that the algorithmeffectively requires at most (cid:80) n − s − t =0 ( n − t ) = ( n − s )( n − s + 1) / SCS is NP-hard by showing that the 3-SAT problem[19, Ch. 7.2] can be reduced, in polynomial time, to a particular instance of
SCS . Consider any instance I = ( X, C ) of the 3-SAT problem, consisting of aset of variables X = { x , x · · · x s − } and clauses C = { c , c , · · · c m } , such thatin every clause in C exactly three, possibly negated, variables from X appear.Then, we associate to I a simple graph G I = ( V I , E I ) of order |V I | = 2 s + 5 m and size |E I | = s + 8 m as follows. The node set V I is the union of the followingsix disjoint sets of nodes: • A set W = { w , w , . . . , w m } , whose elements correspond each to a clausein C ; • A set Y = { y , y , . . . , y s − } , whose elements correspond each to a variablein I , with the interpretation that y i encodes x i if x i is true; • A set ¯ Y = { ¯ y , ¯ y , . . . , ¯ y s − } , whose elements correspond each to a variablein I , with the interpretation that ¯ y i encodes x i if x i is false; • A single node z , whose role will be to break possible ties; • Two sets of leaves L and M , of cardinality |L| = 3 m and |M| = m + 1.141)(1) (3) (3)(4)(2)(2) Y ¯ Y W z L M
Links in E I only connect pairs of nodes belonging to different sets and in par-ticular:(1) A node w j in W is connected to a node y i in Y if and only if the variable x i appears in the clause c j and to a node ¯ y i in ¯ Y if and only if the variable¯ x i appears in the clause c j ;(2) For each clause containing the variable x i , node y i in Y is connected to adifferent node in L , and for each clause containing the variable x i , node y i in Y is connected to a different node in L , in such a way that the elementsof L are each connected to exactly one element either of Y or of Y ;(3) The node z is connected to every element of W and of M ;(4) For every i = 1 , . . . , s −
1, node y i is connected to the corresponding node¯ y i .There is a total of 3 m links of type (1), 3 m links of type (2), 2 m + 1 links oftype (3), and s − L and M all have degree 1, nodesin W all have degree 4, node z has degree 2 m + 1, while the degree of a node y i in Y (respectively y i in Y ) is 1 plus twice the number of clauses the variable x i (respectively, x i ) appears in.Now, we shall consider the majority game on the graph G I , whereby eachplayer in V I has action set { , } and the utility of player i is equal to the numberof her neighbors that play the same action as her. We then ask the question ”isthere a sufficient control set of size less then or equal to s for this game?” Wewill now show that the answer to this question is true if and only if the instanceof 3-SAT is satisfiable. Lemma 4.
Let I = ( X, C ) be an instance of the 3-SAT problem, and let G I =( V I , E I ) be the simple graph defined above. If I is satisfiable with a solution x ∗ in { , } s − , then S = { z } ∪ { y i : x ∗ i = 1 } ∪ { y i : x ∗ i = 0 } is a sufficient control set of size s for the majority game on G I .Proof. Since I is satisfied by x ∗ , for every clause c j in C there exists i in { , . . . , s − } such that either x i appears in c j and x ∗ i = 1 or x i appears in15 j and x ∗ i = 1. Thus, in the graph G I , all clause-related nodes in W have atleast one neighbor in ( Y ∪Y ) ∩S . Since they are all connected to z in S also, andhave all degree 4 in G I , this implies that there exists a monotone improvementpath from S to S ∪ W .Now, consider a variable x i in X and let m i be the number of clauses itappears in. Then, notice that, if the corresponding node y i in Y does notbelong to S , it necessarily has one neighbor in S ( y i ) as well as m i neighborsin W (those corresponding to the clauses it belongs to). Since its degree in G I is exactly 2 m i + 1, this implies that S ∪ W ∪ Y can be reached by a monotoneimprovement path from
S ∪ W , hence from S . Analogously, one proves that S ∪ W ∪ Y ∪ Y can be reached by a monotone improvement path from S .Finally, since every remaining node in L ∪ M is of degree one and connectedto a node in
Y ∪ Y ∪ { z } , we get that the monotone improvement path from S can be extended to reach the whole node set V I , thus proving that S is asufficient control set.We will now show that the converse of Lemma 4 holds true. Lemma 5.
Let I = ( X, C ) be an instance of the 3-SAT problem, and let G I =( V I , E I ) be the simple graph defined above. If there is a sufficient control set S of size s for the majority game on G I , then I is solvable.Proof. We will first show that there exists a sufficient control set S (cid:48) of the samesize s containing z and exactly one node between y i and ¯ y i for 1 ≤ i ≤ s − i = 1 , . . . , s −
1, at least onenode among y i , ¯ y i , and the leaves in L connected to them must be in S for,otherwise, it is easy to check that no improvement path would ever be able toreach the pair { y i , ¯ y i } . Similarly, at least one element among z and the leavesin M must be in S .In case when neither y i nor ¯ y i belong to S , removing the leaf connected to themthat is in S and adding its sole neighbor (either y i or ¯ y i ) maintains the controlset sufficient and preserves its size. We construct S (cid:48) in this way replacing leaveswith variable nodes and finally applying the same substitution idea to includethe node z removing a leaf connected to it.Now observe that, because of the structure of the graph and since S (cid:48) containsno leaves in L ∪ M , in any monotone improvement path from S (cid:48) to V I , a nodein ( Y ∪ Y ) \ S (cid:48) can only appear after all nodes in W have already appeared.Since all nodes in W have degree 4, this says that each of them must have atleast two neighbors in S (cid:48) . This implies that every node in W must have at leastone neighbor in S (cid:48) \ { z } ⊆ Y ∪ Y .Consider now the candidate solution x ∗ in { , } s − that has x ∗ i = 1 if andonly if y i in S (cid:48) . Then, it follows from the argument above that for every clause c j there exists i in { , . . . , s − } such that either x i appears in c j and x ∗ i = 1or x i appears in c j and x ∗ i = 1. This proves that I is solvable.Lemma 4 and Lemma 5 thus show that starting from an instance of the3-SAT, we could build an instance of the SCS problem in polynomial time and16f polynomial size, whose answer is the same as that of the 3-SAT. This showsthat SCS is NP-hard. Together with Lemma 3, this implies that SCS is anNP-complete problem. The characterization of sufficient control sets through the concept of monotoneimprovement paths (Lemma 1) suggests the possibility that such sets may besearched for by starting from the all-1 profile and iteratively replacing 1’swith 0’s in the attempt to follow backwards a monotone improvement path. Inorder to capture this intuition, in this section we introduce a family of discrete-time Markov chains ( Z εt ) t ≥ on the strategy profile space X , parameterized bya scalar ε in [0 , < ε ≤
1, the Markov chain( Z εt ) t ≥ is time-reversible and that, as ε vanishes, its stationary distributionconcentrates on the family of optimal sufficient control sets.The dynamics of the Markov chain Z εt are described as follows: at everydiscrete time t = 0 , . . . , given that Z εt = z , a player i is chosen uniformly atrandom from the whole player set V . Then, if u i (1 , z − i ) < u i (0 , z − i ), the stateis not changed, i.e., Z εt +1 = z . Otherwise, if u i (1 , z − i ) ≥ u i (0 , z − i ), then if thecurrent action of player i is z i = 1 it is changed to 0 with probability 1, while ifher current action is z i = 0, it is changed to 1 with probability ε . The transitionprobabilities of this Markov chain are then given by P (cid:15) x , y = /n if y = x − δ i and u i ( y ) ≤ u i ( x ) ε/n if y = x + δ i and u i ( y ) ≥ u i ( x )0 if otherwise , (16)for every x , y in X .Notice that, for ε = 0, only transitions from 1 to 0 are allowed. In fact, inthis case, the Markov chain Z t has absorbing states. Specifically, let Z = { x ∈ X | P ( ∃ t ≥ Z t = x | Z = ) > } (17)be the set of all states that are reachable by the Markov chain Z t when startedfrom Z = and let Z ∞ = { x ∈ X | P ( ∃ t ≥ Z t = x ∀ t ≥ t | Z = ) > } (18)be the set of absorbing states reachable by Z t from Z = . We have thefollowing result. Proposition 4.
For a finite super-modular game with binary actions ( V , A , { u i } ) ,let Z and Z ∞ be defined as in (17) and (18) , respectively. Then,(i) S ⊆ V is a sufficient control set if and only if S in Z ;(ii) if S is a minimal sufficient control set then S in Z ∞ . roof. (i) By definition, x = S belongs to the set of reachable states Z if andonly if there exists a sequence of strategy profiles ( y k ) k =0 ,...,l , such that y = , y l = S , and y k = y k − − δ i k , u i k ( y k ) ≤ u i k ( y k − ) 1 ≤ k ≤ l . (19)Notice that (19) is equivalent to say that the reversed path ( x k ) k =0 ,...,l with x k = y l − k for 0 ≤ k ≤ l is a monotone improvement path from S to V . ByLemma 1, this is equivalent to say that S is a sufficient control set.(ii) If S is a minimal sufficient control set, we know from point (i) that thestrategy profile S belongs to the reachable set Z . Now if, by contradiction, S did not belong to the set of reachable absorbing states Z ∞ , then, from x = S ,the Markov chain Z t could reach, in one step, a different state x (cid:48) = S (cid:48) with S (cid:48) (cid:40) S , thus contradicting the minimality assumption on S .Point (i) of Proposition 4 implies that the problem of finding optimal suffi-cient control sets can be equivalently stated as the problem of finding strategyprofiles x in Z of minimal l -norm || x || = (cid:80) k x k , i.e., that S is an optimalsufficient control set if and only if S ∈ argmin x ∈Z || x || . Point (ii) implies that we can actually restrict the minimization above to the set Z ∞ of absorbing states of the Markov chain Z t that are reachable from the all-1strategy profile. However, as the example below shows, Z ∞ may contain profilescorresponding to sufficient control sets that are suboptimal and, possibly, noteven minimal. Example 2.
Consider the majority game on the ring graph with four nodes { , , , } . Then, z = (1 , , , in Z ∞ corresponds to the sufficient controlset S = { , } , but it is not minimal since { } is also a sufficient control set. As a consequence, by simply simulating the Markov chain Z t started from Z = , we are not guaranteed to reach an optimal sufficient control set. Toovercome this issue, we will instead use the Markov chain Z εt with ε >
0, which,as shown below, is time-reversible and ergodic on whole set Z of reachablestrategy profiles and, hence, it does not get trapped in non-optimal control sets,and at the same time has a stationary distribution concentrating on the set ofoptimal control sets as the parameter ε vanishes. Theorem 2.
For a finite super-modular game with binary actions ( V , A , { u i } ) ,let Z be defined as in (17) . Then, for ε > , the Markov chain Z εt with transitionprobabilities (16) (i) keeps the set Z invariant, namely, if Z ε belongs to Z , then Z εt belongs to Z for every t ≥ ;(ii) is time-reversible and ergodic on the set Z ; iii) has stationary probability µ ε x := 1 K ε ε || x || , x ∈ Z , (20) where K ε = (cid:80) x ∈Z ε || x || . In particular, µ ε converges to a probabilitymeasure µ concentrated on the set of profiles corresponding to optimalsufficient control sets as ε vanishes.Proof. (i) Let x in Z be strategy profile that is reachable from the all-1 profileby the Markov chain Z t and let y in X be a strategy profile such that P ε x , y > y belongs to Z . If y = x − δ i for some player i in V ,then it follows from (16) that 0 < P ε x , y = 1 /n and then P x , y = 1 /n >
0, thusimplying that the strategy profile y belongs to Z .On the other hand, if y = x + δ i for some player i in V , we argue as follows.Since x in Z is a strategy profile reachable by the Markov chain Z t from theall-1 profile, we can find a sequence of profiles ( x k ) k =0 ,...,l such that x = and x l = x and P x k − , x k > k = 1 , . . . , l . From (16), this is equivalent to x k = x k − − δ i k and u i k ( x k ) ≤ u i k ( x k − ) for some i k in V , for every k = 1 , . . . , l .Let s in { , . . . , l } be such that i s = i and consider the sequence ( z k ) k =0 ,...,l − such that z k = x k for k ≤ s − z k = x k +1 + δ i for k ≥ s . Notice that, for k ≥ s , z k = x k +1 + δ i = x k + δ i − δ i k +1 = z k − − δ i k +1 (21)Relation (21) and the super-modularity property (2) yield u i k +1 ( x k +1 ) ≤ u i k +1 ( x k ) ⇒ u i k +1 ( z k ) ≤ u i k +1 ( z k − )for every k ≥ s −
1. This implies that P z k − , z k > k = 1 , . . . , l − z l − = x l + δ i = y , this proves that the strategy profile y belongs to Z .(ii) Notice that ε || x || P ε x , y = ε || y || P ε y , x , (22)for every two strategy profiles x and y in X . This implies that the Markovchain Z εt is time-reversible with respect to the stationary distribution (20).Since the transitions that have positive probability for the Markov chain Z t have also positive probability for the Markov chain Z εt , we have that all profilesin Z can be reached from the all-1 profile by the Markov chain Z εt . Moreover,Equation (22) implies that a transition probability P ε x , y is positive if and onlyif the reverse transition P ε y , x is positive. This implies that is reachable fromany other profile in Z and thus we conclude that Z εt is ergodic on Z .(iii) Ergodicity and Equation (22) imply that, for every ε >
0, the uniquestationary distribution of the Markov chain Z εt on the set Z has the form (20).As ε vanishes, a direct check shows that the stationary distribution µ ε convergesto a uniform distribution on the set argmin x ∈Z || x || . Using Proposition 4, theset argmin x ∈Z || x || coincides with the set of optimal sufficient control sets, thuscompleting the proof. 19igure 5: Size of Control Sets for random graphs E ( n, p ) with p = 0 . p = 4 log nn (right) In this section, we briefly present some numerical simulations of the proposedalgorithm for the case of the majority game on Erd¨os-Renyi random graphs. TheErd¨os-Renyi graph E ( n, p ) is a random undirected graph with n nodes whereundirected links between pairs of nodes are present with probability p in [0 , n rangingup to 70 and two different scalings for the probability p . In the first case, weconsider a constant p = 0 . n , thus leadingto quite a densely connected graph. In contrast, in the second case, we choose p = 4 log nn , a choice leading to a more sparse graph that nevertheless remainsconnected with high probability as the graph order n grows large [5, Theorem2.8.1]. We run the randomized algorithm Z εt , with (cid:15) = 0 .
3, for a number ofsteps proportional to the square of the size of the graph (exactly 100 n ) andthe control set returned is the one of minimum cardinality during the walk. Forsmall values of n , an explicit comparison with the optimal solution, obtainedthrough exhaustive search, proves the efficiency of our approach. Simulationsare reported in Figure 5. In Figure 6 we have made a comparison with respect toa naive heuristics selecting the highest degree nodes. Specifically, for each valueof n , we have considered the highest degree nodes set of the same cardinalityas the one found by our algorithm and we have plotted the percentage of thegraph nodes that would turn to 1 using that specific control set. When n issufficiently large this percentage is around 30% and shows how the degree is notthe right property to look at in the optimization of these control sets. In this paper, we have studied a novel optimal targeting problem for super-modular games with binary action set and finitely many players. The consideredproblem consists in the selection of a subset of players of minimum size suchthat, when the actions of these players are forced to the value 1, there exists20igure 6: Coverage obtained by taking the k highest degree node, with k thesize of the set found by the algorithm for random graphs E ( n, p ) with p = 0 . p = 4 log nn (right)a monotone improvement path from the minimal to the maximal pure strategyNash equilibrium of the constrained super-modular game. Our main contribu-tions consist in: (i) showing that this is an NP-complete problem; (ii) proposinga computationally simple randomized algorithm that provably selects an opti-mal solution with high probability. Finally, we have presented some numericalsimulations for the case of the majority game on Erd¨os-Renyi random graphs.We have compared the performance of our algorithm with that of an exhaus-tive search (for small problem sizes) and that of a simple heuristic where targetplayers are those with the highest centrality in the graph. The first such com-parison validates our theoretical results. The second comparison shows that thecentrality-based heuristic performs as much as 70% worse than our algorithmin this problem, thus highlighting the relevance of our analysis.The problem studied in this paper can be considered a particular instanceof a control problem in a game-theoretic framework. Our results show how thestructure of the game, i.e., super-modularity, can be leveraged to get insightinto the solution of the control problem. Several directions for future researchcan be considered. For instance, in the context of super-modular games, naturalgeneralizations include the extension to non-binary action sets and the consid-eration of possibly more complex actions altering the utilities of the controlledplayers rather than directly forcing their action to a desired one. Our techniquesstrongly leverage on the super-modularity assumption. Extensions to more gen-eral classes of games are challenging and would likely require the developmentof different technical tools. References [1] C. Ballester, A. Calv´o-Armengol, and Y. Zenou. Who’s who in networks.wanted: The key player.
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