Attacking Power Indices by Manipulating Player Reliability

03/04/2019 ∙ by Gabriel Istrate, et al. ∙ Association for Computing Machinery 0

We investigate the manipulation of power indices in TU-cooperative games by stimulating (subject to a budget constraint) changes in the propensity of other players to participate to the game. We display several algorithms that show that the problem is often tractable for so-called network centrality games and influence attribution games, as well as an example when optimal manipulation is intractable, even though computing power indices is feasible.

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1 Introduction

Control is a fundamental but difficult issue in multi-agent systems. A multi-agent society may be difficult to control due to the concurrence of several factors, that may interact and drive the dynamics in complex, unpredictable ways. Some of these factors could include uncertainty about agent involvement [1], coalition formation [2], the rules [3], the environment [4], about rewards [5], the presence (or lack) of synergies between players [6], etc.

A common type of control is manipulation111We use the word with its wider, commonsense meaning, rather than the specialized one from voting theory [7]. Our usage encompasses both strategic behavior by an agent or coalition (voting theory ”manipulation”) and interventions by a chair or outside agent (such as control and bribery in voting [8]). We assume, however, that all such interventions are costly., which often aims to change the power (index) of a given player by means of interventions in the settings or the dynamics of the agent society. Many types of manipulation have been considered in the literature, often in a computational social choice context. They include identity [9], cloning [10] and quota [11] manipulation in voting games, collusion and mergers [12], sybil attacks [13] and, finally, multi-mode attacks [14], just to name a few.

We contribute to this direction by studying yet another natural mechanism for manipulation: changing the propensity of players to participate to the game. This type of manipulation is quite frequent in real-life situations, a central example being voting - while parties cannot control with absolute certainty voter turnout on election day, they may employ tactics that aim to mobilize their supporters and deter participation of their opponents’ voters222Such scenarios are best modeled as multichoice voting games [15]. However, since such games are multi-cooperative (rather than cooperative) games [16], they fall outside of the scope of the present work, and will be dealt with in a subsequent paper.. Manipulation could be performed by a centralized actor (like in the voting example), or by a coalition of players [17], strategically modifying their behavior (in our case their reliabilities) in response to a perceived dominance of a player whose power index they wish to decrease.

The main impetus for our work was [18], where a model of strategic manipulation of player reliabilities was first investigated. Bachrach et al. [18] considered max games

. In these games each player possesses a weight; the value of a coalition is the maximum weight of a component of the coalition. They proved a ”no sabotage theorem” for (the reliability extension of) max-games with a common failure probability. They remarked that manipulating player reliabilities can be studied in principle for all coalitional games, and asked for further investigations of this problem, in settings similar to the one we consider, i.e. under costly player manipulation. Given the negative results for max-games

[18] and the fact that computing power indices is often intractable [19], we concentrate mostly on proving positive results, showing that there exist natural scenarios where optimal attacks on power indices by manipulating players’ reliabilities are easy to compute (and interesting). We hope that these positive results will encourage renewed interest (and research) on the scope and limits of reliability manipulation.

Contributions and outline In Section 2 we begin by informally stating the problem and justifying our choice of the two classes of coalitional games studied in this paper: network centrality games [20, 21, 22, 23] and credit attribution games [24, 25]. Even though credit attribution games may seem to be somewhat exotic/of limited use, their importance extends well-beyond scientometry: they were, in fact, anticipated, as hypergraph games (see [19] Section 3). The two games we consider from this class, full credit and full obligation games, are natural examples of read-once marginal contribution (MC) nets [26]. Full credit games are equivalent to the subclass of basic MC-nets [27] whose rules are conjunctions of positive variables; full obligation games correspond to generalized MC-nets whose rules consist of disjunctions of positive variables. Full obligation games can simulate induced subgraph games [19]. Full credit games capture an important subclass of coalitional skill games (CSG) [28, 29], that of CSG games with tasks consisting of a single skill.

Section 3 contains technical details and precise specifications of the models we investigate. We deal with two types of attacks: (node) removal, where we are allowed to remove (decrease to zero the reliability of) certain nodes, and fractional attacks, where reliability probabilities can be altered continuously.

In Section 4 we give closed-form formulas for the Shapley values of the reliability extensions of network centrality and credit allocation games. Next we particularize these results to centrality games on specific network: first we show that no removal attack is beneficial; as for fractional attacks, we show that in the complete graph or when attacking the center of the star graph , a greedy approach works: one should increase the reliabilities of neighbors of the attacked node, in descending order of baseline reliabilities. When attacking a non-center player in the result is similar, with the important exception that increasing the reliability of the center should precede all other moves. In contrast, the situation for the cycle graph is more complex, involving all distance-two neighbors of the attacked node. A simple characterization is provided for the optimum as the best of four fixed “greedy” solutions. This characterization allows the determination of the optimum for all combinations of reliability values and budget.333The precise formula for the optimum is cumbersome, hence deferred to the full version. An interesting, and unintuitive, qualitative feature of the result is that in the optimal attack a non-neighbor of the attacked node could be targeted before some of the direct neighbors of the attacked node.

In Section 6 we analyze full credit and full obligation games. Although these two games have the same Shapley value [25], we show that they behave very differently with respect to attacks: removal attacks are not beneficial for full credit games, NP-hard for full obligation games. Fractional attacks also behave differently, modifying probabilities in opposite directions. In a particular setting which includes the case of induced subgraph games we obtain greedy algorithms for both games, derived from expressing the problems as fractional knapsack problems. The determining quantities for the attack orders are (two different) cost-benefit measures.

2 Problem Statement and Choice of Games

The power index attack problem, the main problem of interest in this paper, has the following simple informal statement: we consider the reliability extension of a cooperative game. We are given a positive budget and are allowed to modify reliabilities of all nodes, other than the targeted player , as long as the total cost incurred is at most . Which nodes should we target, and how should we change their reliabilities, in order to decrease as much as possible the Shapley value of node ?

A variant of the previous problem, called the pairwise power index attack problem and motivated by Example 7 below, is the following: we are given not one but two players . The goal is to decrease as much (within the budget) the Shapley value of , while not affecting at all the Shapley value of . This restriction makes some nodes exempt from attacks: we are not allowed to change the reliabilities of players who contribute to the Shapley value of .

Choice of games The problems described above could be investigated in all classes of TU-cooperative games, or compact representation frameworks. However, we feel that the most compelling cases are those where the computation of power indices, e.g. the Shapley value, of (the reliability extensions) of our games is tractable444This requirement disqualifies many natural candidate games such as weighted voting games [19, 30], as well as most subclasses of coalitional skill games [31]. In other words the intractability of manipulating a power index should not be a consequence of our inability to compute these indices. In particular, we are interested in scenarios where computing power indices is easy, but computing an optimal attack on them is hard. Theorem 7 below provides such an example.

The appeal of studying attacks on node centrality in social networks is quite self-evident: game-theoretic concepts such as those considered in [20, 21, 22, 23] formalize appealing notions of leadership in social situations. They have been proposed as tools for identifying key actors, with applications e.g. to terrorist networks [32, 33]. In such a setting, a direct (physical) attack on a leading node may be infeasible. Instead, one could attempt to indirectly affect its status (centrality), by incentivizing some of its peers.

Relevant examples of targeting nodes in order to affect power indices arose (implicitly) in even earlier work [24], that attempted to develop coalitional models of credit allocation in scientific work. The following is a version of the example in [24]:

Example 1.

Two scientists are compared with respect to their publication record555We do not condone and caution against the real-life use of such crude quantitative metrics for tasks like the one described in this example or our models.. All their papers have exactly one co-author. Figure 1 displays this information as a graph, listing for each author pair, the number of publications they have authored and the number of citations. If using the Hirsch index, it would seem that candidate has a better track record than candidate . But if we discard publications both of them have co-written with “famous scientist ” (that is, remove and its publications from consideration), then their relative ranking would be reversed.

The authors of [24] attempted to use the Shapley value of a game based on the Hirsch index for credit allocation. An ulterior, more general and cleaner game-theoretic approach is  [25]. The author defines several credit allocation games, and uses their (identical) Shapley values as a measure of individual publication record. Slightly modified versions of this measure have (regrettably) actually been used in some countries to set minimum publication thresholds for access and promotion to academic positions, e.g. the minimal standards in Romania.

In such a context one could naturally ask the following question: what are the top coauthors that account for most of a scientists’ publication record? When using the game-theoretic framework for scientific credit from [25], this is equivalent to finding the coauthors whose removal (together with the joint papers) causes the scientist’s’ Shapley value to decrease the most.

Collaborations may, however, be genuinely productive or just bring to one of the scientists the benefits of association with leading scientists666One could argue, of course, that such an association itself reflects positively on the scientist. But the opposite argument, that prestige drives scientific inequality, has recently been substantiated by real data [34] and is, at the very least, hard to ignore.. The Shapley value approach of [25] does not distinguish between these two scenarios, as it gives equal credit to all authors of a joint paper, irrespective of ”leadership status”. Recent work, e.g. Hirsch’s alpha index [35], has attempted to quantify ”scientific leadership”. It is possible to define a measure based on the reliability extension of credit allocation games that factors out the ”well connectedness” of an individual from its score777The measure computes appropriate values of reliability probabilities, the lower the probability the more of a ”scientific leader” a coauthor is; we are currently investigating the practicality of such an approach.. Given such a measure, the previous question, that of finding the top- co-authors is still interesting, as it identifies the most (genuinely) fruitful collaborations of a given author, irrespective of status. This is modeled by the power index attack problem in credit allocation games.

Figure 1: The scenario (from [24]) in Example 7

3 Technical Details

We will be working in the framework of Algorithmic Cooperative Game Theory, see

[36] for a readable introduction.

We will make use of notation as a shorthand for . Given graph and vertex , we will denote by the set of neighbors of and by . Given , we denote by the set of nodes such that there exists , . We generalize the setting above to the case when is a weighted graph, i.e. there exists a weight function . Given set and integer we define , the ball of radius around , to be the set . We may omit from this notation when it is simply the graph distance in . Also, given ”cutoff” distance we define .

We will deal with cooperative games with transferable utility, that is pairs where is a set of players and is a value function under the partial sets of . If is a set of players, is the value that players in coalition can guarantee for themselves irrespective of the other players’ participation.

Although we could prove similar results for other power indices, e.g. the Banzhaf value, in this paper we restrict ourselves to the Shapley value. This index measures the portion of the grand coalition value that a given player could fairly request for itself. It has the formula [36] , and is the set of permutation.

We are concerned with two classes of cooperative games. The first one arose from efforts to define game-theoretic notions of network centrality [20, 21, 22, 23]. We define these games as follows:

  • Game is specified by its value function .

  • Given integer , game is specified by its value function .

  • In game graph is weighted. We are also given a positive ”cutoff distance”

    . We give the characteristic function

    by

A second class of games, related to the example in [24] is that of influence-attribution games, formally defined by Karpov [25]. A credit-attribution game is formalized by a set of authors and a set of publications . Each paper is naturally endowed with a set of authors and a quality score . In real-life scenarios the quality measure could be 1 (i.e. we simply count papers), a score based on the ranking of the venue the paper was published in, the number of its citations, or even some iterative, PageRank-like variant of the above measures.

  • The full credit game is specified by its value function which is simply the sum of weights of papers whose authors’ list contains at least one member from .

  • The full obligation game is specified by its value function which is the sum of weights of papers whose authors are all members of .

Denote by the set of papers of , and by the set of co-authors of , i.e. the set of players for which there exists a . If denote by the joint contribution of .

Reliability extension and attack models We will be working within the framework of reliability extension of games, first defined in [1] and further investigated in [18]. The reliability extension of cooperative game with parameters is the cooperative game with ,

A useful result about these quantities is:

Claim 1.

Let . We have

We will consider in the sequel the following two attack models:

  • fractional attack: In this type of attack every node different from the attacked node has a baseline reliability . We are allowed to manipulate the reliability of each such node by changing it from to an arbitrary value . To do so we will incur, however, a cost . We assume that cost function is defined and has an unique zero888There is no cost for keeping the baseline reliability. at , is decreasing and linear on and increasing and linear on (Figure 2). That is: for every player there exist values such that

  • removal attack: In this type of attack we are only allowed to change the reliability of any node (different from the targeted node ) from to . To do so will incur a cost .

A basis for fractional attacks The following simple result will be used to analyze fractional attacks in network centrality games:

Lemma 1 (”Improving Swaps”).

Let be an open set in , let and be an analytic function. Assume are indices such that Define , with

(1)

Then there exists such that function , is monotonically decreasing.

In other words, to minimize function one could decrease the variables with the largest partial derivative, while symmetrically increasing a smaller one.

Proof.

By the chain rule

Since is continuous, is strictly negative on some interval . The result follows. ∎

Figure 2: Shape of utility functions in fractional attacks.

4 Closed-form formulas

The basis for our manipulation of network centralities is the following characterization of the Shapley value of the reliability extension:

Theorem 1.

The Shapley values of the reliability extensions of network centrality games have the formulas:

As for credit atribution games, the corresponding result is

Theorem 2.

The Shapley values of the reliability extensions of with probabilities have the formulas

(2)

where is the set of coauthors of paper and , and

(3)

5 Attacking network centralities

The next result follows from Theorem 1 and Claim 1:

Corollary 1.

In the reliability extensions of the centrality games , the Shapley values of player are monotonically decreasing functions of distance-two neighbors’ reliabilities (and do not depend on other players).

Proof.

Deferred to the full version. ∎

The previous corollary shows that for network centrality games no removal attack is beneficial:

Theorem 3.

No removal attack on the centrality of a player in games can decrease its Shapley value.

Fractional attacks on specific networks Given that removal attacks are not beneficial, we now turn to fractional attacks. The objective of this section is to show that the analysis of optimal fractional attacks is often feasible. Since the graphs in this section are fairly symmetric, we will assume (for these examples) that the slopes of all utility curves are identical. That is, there exist positive constants such that if are different agents then and (though, of course, baseline probabilities and may differ). The graphs we are going to be concerned with are the complete graph , the star graph (where node 1 is either the center or an outer node) and the -cycle (Figure 3).

1

1

1

Figure 3: Target topologies for fractional attacks.

Note that, when or , pairwise Shapley value attacks are trivially impossible: indeed, these graphs have diameter at most two. Since all distance-two neighbors influence the Shapley value of a given player, all nodes are exempt from attacks.

On the other hand, for these topologies it turns out that the best attack on Shapley value of player is to increase the reliabilities of its neighbors in the descending order of their baseline reliabilities:

Theorem 4.

Let be either the complete graph with vertices. or the star graph with vertices centered at node . To optimally attack the centrality of in the reliability extension of use the following algorithm:

- Consider nodes in the decreasing order of their baseline reliabilities, breaking ties arbitrarily. - While the budget allows it, increase to one (if not already equal to 1) the probabilities , starting with and successively increasing . - If the budget no longer allows increasing to one, increase it as much as possible. - Leave all other probabilities to their baseline values.

If, on the other hand, centered, say, at node 2, to optimally attack the centrality of node , the algorithm changes as follows:

- Consider nodes in the following order: node 2, followed by nodes sorted in decreasing order of their baseline reliabilities , breaking ties arbitrarily. Denote the new order by . - Follow the previous greedy protocol, increasing baseline probabilities up to one (if allowed by the budget) according to the new ordering .

Similar statements hold for game , and for for large enough values of parameter .

In the previous examples the optimal attack involved a determined node targeting order, which privileged direct neighbors and could depend on baseline reliabilities but was independent of the value of the budget. None of this holds in general: as the next result shows, on graph the optimum can be computed by taking the best of four node targeting orders. The optimum may lack the two previously discussed properties of optimal orders:

  • in optimal attacks one should sometimes target a distance-two neighbor (3 or n-1) before targeting both of ’s neighbors (2 and , see Figure 3).

  • the order (among the four) that characterizes the optimum may depend on the budget value as well. Formally:

Theorem 5.

Let

be the vectors

, , , , respectively. Let , be the configurations obtained by increasing in turn (as much as possible, subject to the budget ) the reliabilities of nodes in the order(s) specified by , respectively. Then

  • The best of is an optimal attack on the centrality of in game on the cycle graph .

  • There exist values of s.t. is optimal for all values of (by symmetry a similar statement holds for ).

  • There exist values of and an nonempty open interval for the budget such that is an optimum for all (by symmetry a similar statement holds for ).

6 Attacks in credit attribution

In this section we study removal attacks in credit attribution games. Interestingly, while the Shapley values have identical formulas in [25], the two games are not similar with respect to attacks. Indeed, similarly to the case of network centrality, we have:

Theorem 6.

No removal attack can decrease the Shapley value of a given player in a full credit attribution game.

Proof.

At first, this seems counterintuitive, as it would seem to contradict Example 7. The answer is that this example does not correspond to the full credit game, but to the full obligation one: in game a player does not lose credit for a paper due to removal of a coauthor; in fact its Shapley value will increase, since the credit for the paper divides among fewer coauthors. It is in where players may lose credit as a result of coauthor removal. ∎

This difference between and is evident with respect to attacks: As the next result shows, in full-obligation games, finding optimal removal attacks can simulate a well-known hard combinatorial problem:

Theorem 7.

The budgeted maximum coverage problem (which is NP-complete) reduces to minimizing the Shapley value of a given player in the full-obligation game (under removal attacks).

Proof.

Deferred to the full version. ∎

Fractional attacks The following is a simple consequence of the formulas in Theorem 2 and Claim 1 shows that optimal attacks are different in games and irrespective of the topology of the coauthorship hypergraph: in the first case we need to increase the reliability of ’s coauthors, in the other case we aim to decrease it:

Theorem 8.

In the reliability extensions of the credit allocation games the Shapley value of player is a decreasing (respectively increasing) function of coauthors’ reliabilities (and does not depend on other players).

Optimal attacks can be explicitly described in the particular scenario when, just as in Example 7, each paper has exactly two authors (a situation that corresponds, under the full obligation model, to induced subgraph games). It turns out that the relevant quantity is the ratio between the score of coauthors’ joint contribution with the attacked node and its marginal cost:

Theorem 9.

To optimally decrease the Shapley value of node in game in the two-author special case:

(a). Sort the coauthors of in the decreasing order of the fractions , breaking ties arbitrarily.

(b). While the budget allows it, for , increase to 1 the probability of the ’th most valuable coauthor.

(c). If the budget does not allow increasing the probability of the ’th coauthor up to 1, increase it as much as possible.

(d). Leave all other probabilities to their baseline values.

Corollary 2.

In the setting of Theorem 9, to optimally solve the pairwise Shapley value attack problem for , run the algorithm in the Theorem only on those that are coauthors of but not of .

As for game , the optimal attack is symmetric. Since we are decreasing probabilities, we will be using fractions instead:

Theorem 10.

To optimally decrease the Shapley value of node in the full obligation game in the two-author special case:

(a). Sort the coauthors of in the decreasing order of the fractions , breaking ties arbitrarily.

(b). While the budget allows it, for , decrease to 0 the probability of the ’th most valuable coauthor.

(c). If the budget does not allow decreasing the probability of the ’th coauthor up to 0, decrease it as much as possible.

(d). Leave all other probabilities to their baseline values.

Corollary 3.

In the setting of Theorem 10, to solve the pairwise Shapley value attack problem for players , run the algorithm in the Theorem only on those that are coauthors of but not of .

7 Proof Highlights

In this section we present some of the proofs of our results. Some other proofs are included in the Appendix, others are deferred to the full version of the paper, to be posted on arxiv.org:

7.1 Proof of Theorem 1

We prove the formula for the first game only. Similarly to [22], the proofs for the other two games are completely analogous, and deferred to the full version. Define, for ,

A simple case analysis proves that, for every , We therefore have

We now introduce two notations that will help us reinterpret the previous sum: given , denote by the set of nodes in that are alive under the reliability extension model. Also, given permutation and , denote by the element of that appears first in enumeration . With these notations

If then the conditional probability that is , given that is alive, is . We thus get the desired formula.

7.2 Proof of Theorem 2

Denote, for a set of authors , by the set of papers with at least one author in . We decompose function as where

(4)

which means that we can decompose , and the Shapley value of decomposes as well and similarly for . On the other hand

Given set of authors,

Now is 1 if , 0 otherwise. For , . Otherwise

We can interpret this quantity as the probability that the live subset of does not cover , but is live and does. Applying this to the Shapley value we infer that is the probability that in a random permutation the live subset of does not cover , but is live and does.

Full credit model: There are permutations of indices in , each of them equally likely when is a random permutation in . Given subset , the probability that starts with followed by is . To make pivotal for paper , none of the agents in must be live. This happens with probability . Given the above argument, we have

(5)

which is what we had to prove.

Full obligation model: For to be pivotal for paper , and all its coauthors in must all be live, and all elements of must appear before in ordering . This happens with probability

7.3 Proof of Theorem 4

First of all, the following claim holds for all graphs :

Claim 2.

The minimum of function exists and is reached on some profile with .

Proof.

Function is continuous and the set is compact, so the minimum is reached. Assuming some , we could increase up to , reducing total cost. This does not increase (and perhaps further decreases) the Shapley value. ∎

Next, we (jointly) prove cases and with being a center, since the proofs are practically identical. The remaining case (, not a center) is deferred to the Appendix. We start with the following

Lemma 2.

For or , and any probability profile ,

Proof.

Deferred to the full version. ∎

We first prove that in the optimal solution on these graphs no two variables could assume equal values, unless both equal to the endpoints of their restricting intervals:

Lemma 3.

In the setting of Theorem 4, suppose is such there is are indices with . Then there exists such that for every , , , (where is defined as in equation (1)).

Proof.

Deferred to the full version. ∎

Now we prove:

Claim 3.

In the optimal solution there is at most one index with . In other words, in the optimal solution some probabilities are increased up to 1, some ae left unchanged to their baseline values, and at most one variable is increased to a value less than 1.

Proof.

Suppose there were two different indices . We must have , or, by Lemma 1, one could decrease the Shapley value by increasing the larger one and symmetrically decreasing the smaller one. But this is impossible, due to Lemma 3. ∎

Note that the greedy solution has the structure from Claim 3 and that any permutation of OPT on variables has the same Shapley value as OPT (since have this symmetry).

We compare the vectors , both sorted in decreasing order. Our goal is to show that these sorted versions are equal. Without loss of generality, we may assume that creates the same ordering on variables as the ’s (and ), when considered in decreasing sorted order (we break ties, if any, in the same way). Indeed, if there were indices such that but then, since , we could simply swap values and and obtain another legal, optimal solution.

If were different from , since Greedy increases the largest variables first, there must be variables such that , and . Since and have the same ordering of variables, we also must have in fact , i.e. . But then, using either Lemma 1 (if ) or Lemma 3 (otherwise) we could further improve by increasing and symmetrically decreasing , a contradiction.

7.4 Proof of Theorem 5

A simple computation shows that for

As does not influence any attack on itself, w.l.o.g. we will assume . We need to minimize the above quantity, subject to

We now prove a result somewhat similar to Claim 3. However, now we will only interdict certain patterns.

Claim 4.

In an optimal solution it is not possible that , when:

  • , (and, symmetrically, , ). In fact, in this case we have the stronger implication . Symetrically, .

  • , .

  • , (and, symmetrically, , .) In the case when we have the stronger implication . Symetrically, in the case when , .

Proof.

Suppose there were two such indices . We must also have , otherwise we could decrease the Shapley value using Lemma 1. We reason in all cases by contradiction:

a. We prove directly the stronger result. Suppose . We have . So we can apply Lemma 1 to and , further decreasing the Shapley value as we increase and decrease .

b. Equality of partial derivatives can be rewritten as