Strategyproof Mechanisms for Competitive Influence in Networks

Abstract

Motivated by applications to word-of-mouth advertising, we consider a game-theoretic scenario in which competing advertisers want to target initial adopters in a social network. Each advertiser wishes to maximize the resulting cascade of influence, modeled by a general network diffusion process. However, competition between products may adversely impact the rate of adoption for any given firm. The resulting framework gives rise to complex preferences that depend on the specifics of the stochastic diffusion model and the network topology. We study this model from the perspective of a central mechanism, such as a social networking platform, that can optimize seed placement as a service for the advertisers. We ask: given the reported budgets of the competing firms, how should a mechanism choose seeds to maximize overall efficiency? Beyond the algorithmic problem, competition raises issues of strategic behaviour: rational agents should be incentivized to truthfully report their advertising budget. For a general class of influence spread models, we show that when there are two players, the social welfare can be $\frac{e}{e-1}$-approximated by a polynomial-time strategyproof mechanism. Our mechanism uses a dynamic programming procedure to randomize the order in which advertisers are allocated seeds according to a greedy method. For three or more players, we demonstrate that under an additional assumption (satisfied by many existing models of influence spread) there exists a simpler strategyproof $\frac{e}{e-1}$-approximation mechanism; notably, this natural greedy mechanism is not necessarily strategyproof when there are only two players.

Appendices

Appendix 1: Relation with Other Diffusion Models

In our results, we have made a number of modelling assumptions about agent utilities and social welfare. To some extent, we can argue that these assumptions may be necessary to be able to obtain truthfulness and constant approximation on the social welfare. Furthermore, we now provide some background on the relevance of our assumptions to the existing work on influence diffusion in social networks, which served as the running example throughout the paper.

1.1 Non-decreasing and Submodular Utilities and Social Welfare

To the best of our knowledge, in order to establish a constant approximation on the social welfare, all of the known models in competitive and non-competitive diffusion assume that the overall expected spread is a non-decreasing and submodular function with respect to the set of initial adopters. A main part of the seminal work by Kempe et al. [6] is the proof that the expected spread of two models of non-competitive diffusion process is indeed non-decreasing and submodular. This was later extended to more general processes in [7]. In the case of the competitive influence spread models in [10–12], it is shown that a player’s expected spread is a non-decreasing and submodular function of his initial set of nodes, while fixing the competitors’ allocations of nodes. This also implies that the total influence spread is a non-decreasing and submodular. Without any assumption on the nature of the social welfare function, it is NP-hard to obtain any non trivial approximation on the social welfare even for a single player.

1.2 Adverse Competition

In the initial adoption of (say) a technology, a competitor can indirectly benefit from competition so as to insure widespread adoption of the technology. However, once a technology is established (e.g., cell phone usage), the issue of influence spread amongst competitors should satisfy adverse competition. The same can be said for selecting a candidate in a political election. We also note that the previous competitive spread models [10–12] mentioned above also satisfy adverse competition. In its generality, the Goyal and Kearns model need not satisfy this assumption, but in order to obtain their positive result on the price of anarchy, they adopt a similar restriction (namely, that the adoption function at every node satisfies the condition that a player’s probability of influencing an adjacent node cannot decrease in the absense of other players competing).

Furthermore, a simple example shows that the assumption of adverse competition is necessary for truthfulness. Consider the following two-player setting. The ground set is composed of two items: $u_1$, which contributes a value of 1 to the receiving player and a value of N to her competitor (who did not receive $u_1$), and item $u_2$ which gives both players a value of 1. Now, consider the outcome of any mechanism when the bid profile is (1, 1). Without loss of generality, one player, say player A, will receive $u_1$, while the other player will get $u_2$. The valuations would therefore be 2 and $N+1$ for players A and B, respectively. In that case, player A would prefer to lower her bid to 0, which would guarantee her a valuation of N (player B would have to get $u_1$, as otherwise the approximation ratio of the social welfare is unbounded as N grows). We conclude that unless the competition assumption holds, no strategyproof mechanism can, in general, obtain a bounded approximation ratio to the optimal social welfare. Although the example refers to deterministic allocations, the same argument can be made for randomized allocations.

1.3 Mechanism and Agent Indifference

In both the Wave Propagation model and the Distance-Based model presented in [10], the propagation of influence upholds both the mechanism and agent indifference properties. Goyal and Kearns [13] assume that the switching function that determines the probability that a node will adopt some technology is a function of the fraction of influenced neighbours (regardless of their assumed technologies). This immediately implies MeI, as these switching functions do not depend on the technologies involved. However, we note that whether or not the model satisfies the AgI assumption depends on the manner by which the nodes select specific technologies. The functions that determine this are termed selection functions in [13]. For example, suppose $\alpha _j$ is the fraction of nodes infected by technology j. If the switching function determining the probability of adopting technology i is $s(i) = \frac{\alpha _i}{\sum _j \alpha _j}$, then the model satisfies AgI. On the other hand, if $s(i) = \frac{\alpha ^2_i}{\sum _j \alpha ^2_j}$, then the model would not satisfy AgI when there are more than 2 technologies.

1.4 Multiset Allocations and Disjointness

Our model assumes that each agent can be allocated a node at most once, and indeed most influence models assume that a node is allocated at most once in any initial allocation. However, we can extend our model to allow a node to be allocated multiple times to the same agent, as in (for example) the model of Goyal and Kearns [13]. To implement such an extension, we can simply consider a modified problem instance with many identical copies of a given node, treating each copy as a distinct element, and then proceed as though each element can be allocated at most once. The output of Algorithm 1 for two players and Algorithm 2 for more than two players would then be a profile of multisets with regard to the original network model. We note, however, that for the case of Algorithm 2, the MeI and AgI definitions effectively imply that if multisets are permitted, then non-disjoint allocations must be permitted as well, as the conditions cannot distinguish between an element being allocated to one agent twice or to two different agents.

1.5 Generality of the Model

A few words are in order about the generality of the model of diffusion under which we prove that Algorithm 1 is strategyproof and provides a $\frac{e}{e-1}$-approximation. As noted, with the exception of the OR model, the analysis in previous competitive influence models assumes anonymous agents. Our general model does not require anonymity and hence we can accommodate agent specific edge weights (e.g. in determining the probability that influence is spread across an edge, or for determining whether the weighted sum of influenced neighbors crosses a given threshold of adoption). Our model also notably allows agent-independent node weights, for determining the value of an influenced node. Moreover, our abstract model does not specify any particular influence spread process, so long as the social welfare function is monotone submodular and each player’s payoff is monotonically non-decreasing in his own set and non-increasing in the allocations to other players. In particular, our framework can be used to model probabilistic cascades as well as submodular threshold models.

Appendix 2: Counterexamples When There are Two Agents (Extended Discussion)

The locally greedy algorithm is defined over an arbitrary permutation of the allocation turns. At the core of our work, we seek to carefully construct such orderings in a manner that induces strategyproofness. We demonstrate that this algorithm due to Nemhauser et al [9] (see also Goundan and Schultz [17]) is not, in general, strategyproof for some natural methods for choosing the ordering of the allocation between two agents.

To clarify the context when there are only two agents, we refer to them as agent A and agent B and their utilites as $f_A$ and $f_B$ respectively. We give examples of a set U and functions $f_A$ and $f_B$ (satisfying the conditions of our model) such that natural greedy algorithms for choosing sets S and T result in non-monotonicities. Our examples will all easily extend to the case of $k > 2$ agents (but not satisfying agent indifference).

1.1 The OR Model

We will consider examples of a special case of the OR model for influence spread, as studied in [12]. Let $G=(V,E)$ be a graph with fractional edge-weights $p: E \rightarrow [0,1]$, vertex weights $w_v$ for each $v \in V$, and sets $I_A,I_B \subseteq V$ of “initial adopters” allocated to each player. We use vertex weights for clarity in our examples; in Appendix “Counterexamples with Unweighted Nodes” we show how to modify the examples given in this section to be unweighted. We emphasize that all “infected” nodes (including any initially selected) contribute their weight to the expected social welfare and individual values of the players. The process unfolds in discrete steps. For each $u_A \in I_A$ and $v_A$ such that $(u_A,v_A) \in E$, $u_A$, once infected, will have a single chance to “infect” $v_A$ with probability $p(u_A,v_A)$. Define the same, single-step process for the nodes in $I_B$, and let $O_A$ and $O_B$ be the nodes infected by nodes in $I_A$ and $I_B$, respectively. Note that the infection process defined for each individual player is an instance of the Independent Cascade model as studied by Kempe et al. [6]. Finally, nodes that are contained in $O_A \backslash O_B$ will be assigned to player A, nodes in $O_B \backslash O_A$ will be assigned to B, and any nodes in $O_A \cap O_B$ will be assigned to one player or the other by flipping a fair coin.

In our examples, we consider two identical players each having utility equal to the weight of the final set of nodes assigned by the spread process. It can be easily verified that both the expected social welfare (total weight of influenced nodes) and the expected individual values (fixing the other player’s allocation) are submodular set-functions.

1.2 Deterministic Greedy Algorithms that are Not Strategyproof

We demonstrate that the more obvious deterministic orderings for the greedy algorithm fail. First, consider the “dictatorship” ordering, in which (without loss of generality by symmetry) player A is first allocated nodes according to his budget, and only then player B is allocated nodes. Our example showing non-truthfulness also applies to an ordering that would always select the player having the largest current unsatisfied budget breaking ties (again without loss of generality by symmetry) in favor of player A. Consider the graph depicted in Fig. 2a. When player A bids 1 and player B bids 1 as well, the algorithm will allocate $c_1$ to player A, as it contributes the maximal marginal gain to the social welfare, and will allocate $c_3$ to player B. The value of the allocation for player A is 2.

Fig. 2

Counter-examples for the mechanism under the deterministic dictatorship and Round Robin orderings. In both cases, we set the weights $w_{c_i}=0$ and $w_{u_i}=1$, for all $1 \le i \le 4$. Additionally, we let $0<\epsilon < \frac{1}{8}$. a The counter-example for the deterministic mechanism with a dictatorship ordering. The initial budget for both players is 1. b The counter-example for the deterministic algorithm under a Round Robin ordering. The initial budgets for players A and B are 1 and 2, respectively

However, notice that if player A increases its bid to 2, the mechanism will allocate nodes $c_1$ and $c_3$ to player A, and allocate $c_2$ to B. In this case player A receives an extra value of $\frac{1}{2}$ from node $c_3$, but the allocation of $c_2$ to B will “pollute” player A’s value from $c_1$: he will receive nodes $u_1$ and $u_2$ each with probability $\frac{1}{10} + \frac{1}{2}\cdot \frac{9}{10} = \frac{11}{20}$. Thus the total expected value for player A is only $\frac{16}{10}$, and hence the algorithm is non-monotone in the bid of player A.

Next, consider the Round Robin ordering, in which the mechanism alternates between allocating a node to player A and to player B. Our example here also applies to the case when the mechanism always chooses the player having the smallest current unsatisfied budget breaking ties in favor of player A. Consider the instance given in Fig. 2b. When the bids of players A and B are 1 and 2, respectively, the algorithm will first allocate $c_1$ to player A, and then it will subsequently allocate nodes $c_3$ and $c_4$ to player B, which results in a payoff of 1 for player A. If player A were to increases his bid to 2, then the mechanism would allocate nodes $c_1$ and $c_4$ to player A, and nodes $c_2$ and $c_3$ to player B, for a payoff of $3 \cdot \epsilon + 2 \cdot \epsilon + (1 - 2 \cdot \epsilon ) \cdot \frac{1}{2} = \frac{1}{2}+4 \cdot \epsilon < 1$ (since $0< \epsilon < \frac{1}{8}$). Therefore, the monotonicity is violated for the payoff to player A.

1.3 The Uniform Random Greedy Algorithm is Not Strategyproof

As we shall see in Sect. 4, for the case of $k>2$ agents in the restricted setting that assumes mechanism and agent indifference, a very simple mechanism admits a strategyproof mechanism that provides an $\frac{e}{e-1}$ approximation to the optimal social welfare. More specifically, we show that under these assumptions on the social welfare agent utilities, taking a uniformly random permutation over the allocation turns is a strategyproof algorithm. In contrast, for the case of $k=2$, and even with these additional restrictions (although the agent indifference assumption turns out to be vacuous in this case), the uniformly random mechanism is not strategyproof.

Consider the example given in Fig. 3. We note that for this example, Algorithm 2 in Sect. 4 is equivalent to first choosing a random order of allocation (e.g. choosing all possible permutations satisfying agent budgets with equal probability) and then allocating greedily. The greedy algorithm will allocate one of $c_2,c_3,c_4$ and $c_5$ to one of the players, then allocate $c_1$, and then any remaining nodes.

Fig. 3

The counterexample for the mechanism that allocated according to a random ordering of the turns ($0 <\epsilon \ll 1$). $w_{c_i}=\epsilon , i=1,\ldots ,5$, $w_{u_i}=1, i=1,2$

Let player A’s budget be 3 and player B’s budget be 1. In this case, with probability $\frac{1}{4}$, player B will be allocated $c_1$ (i.e. when B’s allocation occurs second), in which case player A’s expected value would be 1. Also, with probability $\frac{3}{4}$, player B will be allocated one of $\{c_2, c_3, c_4, c_5\}$, in which case player A’s expected outcome would be $\frac{1}{2} + \epsilon$. In total, player A’s expected payoff will be $\frac{5}{8} + \frac{3}{4}\epsilon$.

If player A were to increase his budget to 4, then with probability $\frac{1}{5}$ player B will be allocated $c_1$, in which case player A’s outcome will be 1. On the other hand, player A’s expected payoff will be $\frac{1}{2} + \epsilon$ if B is allocated one of $\{c_2, c_3, c_4, c_5\}$, which occurs with probability $\frac{4}{5}$. In total, player A’s expected outcome will be $\frac{3}{5} + \frac{4}{5}\epsilon < \frac{5}{8} + \frac{3}{4}\epsilon$, implying that this algorithm is non-monotone.

Appendix 3: Counterexamples with Unweighted Nodes

In Sect. “Counterexamples When There are Two Agents (Extended Discussion)” we constructed specific examples of influence spread instances for the OR model, to illustrate that simple greedy methods are not necessarily strategyproof for the case of two players. These examples used weighted nodes which our model allows. For the sake of completeness, we now show that these examples can be extended to the case of unweighted nodes.

We focus on the example from Sect.“The Uniform Random Greedy Algorithm is Not Strategyproof” to illustrate the idea; the other examples can be extended in a similar fashion. In that example there were nodes $u_1$ and $u_2$ of weight 1, and nodes $c_1, \cdots , c_5$ of weight 0. We modify the example as follows. We choose a sufficiently large integer $N > 1$ and a sufficiently small $\epsilon > 0$. We will replace node $u_1$ with a set S of N independent nodes. We replace the $\epsilon$-weighted edge from $c_1$ to $u_1$ with an $\epsilon$-weighted edge from $c_1$ to each node in T.

Similarly, we replace $u_2$ by a set T of N independent nodes. For each $c_i$, we replace the unit-weight edge from $c_i$ to $u_2$ with a unit weight edge from $c_i$ to each node in T.

In this example, if the sum of agent budgets is at most 5, the greedy algorithm will never allocate any nodes in S or T. The allocation and analysis then proceeds just as in Sect.“The Uniform Random Greedy Algorithm is Not Strategyproof”, to demonstrate that if agent B declares 1 then agent A would rather declare 3 than 4.

Appendix 4: Tightness of Approach: More than Two Players

The mechanism we construct in Sect. 3.2 is applicable to settings in which there are precisely two competing players, and our mechanism in Sect. 4 for more than three players requires the MeI and AgI assumptions. A natural open question is whether these results can be extended to the general case of three or more agents without the MeI and AgI restrictions. In this section we briefly describe the difficulty in applying our approach to settings with three players.

For the case of two players in Sect. 3.2, our mechanism was built from an initial greedy algorithm by randomizing over orderings under which to assign elements to players. Our construction is recursive: we demonstrated that if we can define the behaviour of a strategyproof mechanism for all possible budget declarations up to a total of at most t, then we can extend this to a strategyproof mechanism for all possible budget declarations that total at most $t+1$. A key observation that makes this extension possible is the direct relation between the utilities of the two players. This manifests itself in the cross monotonicities that we utilize in the inductive argument. In addition, the strategyproofness condition (i.e. agent monotonicity) can be equivalently re-expressed as a certain “budget competition” property: if one player increases his budget, then the expected utility for the other player cannot increase by more than the marginal gain the total welfare. In other words, for all $a + b \ge 1$, a strategyproof mechanism must satisfy $w^A(a,b) - w^A(a,b-1) \le \varDelta ^{\oplus B}(a,b)$ where $\varDelta ^{\oplus B}(a,b) = w(a,b) - w(a,b-1)$ and a similar consequence with regard to $\varDelta ^{\oplus A}(a,b)$.

Claim 5

(Equivalence of monotonicity and budget competition for two players)

• 1: $w^A(a,b) - w^A(a,b-1) \le w(a,b) - w(a,b-1)$ iff $w^B(a,b-1) \le w^B(a,b)$

• 2: $w^B(a,b) - w^B(a-1,b) \le w(a,b) - w(a-1,b)$ iff $w^A(a-1,b) \le w^A(a,b)$

Proof

We provide the proof for $\varDelta ^{\oplus B}(a,b)$.

• $w^B(a,b-1) \le w^B(a,b)$ iff

• $w^A(a,b) + w^B(a,b-1) \le w^A(a,b) + w^B(a,b)$ iff

• $w^A(a,b) + w^B(a,b-1) + w^A(a,b-1) \le w^A(a,b) + w^B(a,b) + w^A(a,b-1)$ iff

• $w^A(a,b) + w(a,b-1) \le w(a,b) + w^A(a,b-1)$ iff

• $w^A(a,b) - w^A(a,b-1) \le w(a,b) - w(a,b-1)$

$\square$

A direct extension of our approach to three players would involve proving inductively that an allocation rule that satisfies these conditions for all budgets that total at most t can always be extended to handle budgets that total up to $t+1$. We now give an example to show that this is not the case, even when our underlying submodular function takes a very simple linear form.

Suppose we have three players A, B, and C, and suppose our ground set U contains a single element g of value; all other elements are worth nothing. The utility for each agent is 1 if their allocation contains g, otherwise their utility is 0. In this case, the locally greedy algorithm simply gives element g to the first player that is chosen for allocation; the remaining allocations have no effect on the utility of any player. Note then that the marginal gain in social welfare is 1 for the first allocation, and 0 for all subsequent allocations made by the greedy algorithm.

We now define the behaviour of a mechanism for all budget declarations totalling at most 2. Note that the relevant feature of this mechanism is the (possibly randomized) choice of which agent is first in the order presented to the greedy algorithm. We present this behaviour in the following table.

Budgets (a, b, c)	Player selected	Utilities $(w^A, w^B, w^C)$
(0, 0, 0)	N/A	(0, 0, 0)
(1, 0, 0)	A	(1, 0, 0)
(0, 1, 0)	B	(0, 1, 0)
(0, 0, 1)	C	(0, 0, 1)
(1, 1, 0)	A	(1, 0, 0)
(0, 1, 1)	B	(0, 1, 0)
(1, 0, 1)	C	(0, 0, 1)

We note that this mechanism (restricted to these type profiles) is strategyproof, satisfies the budget competition property, and also satisfies the cross-monotonicity properties (i.e. in the invariants of Theorem 1). However, we claim that no allocation on input (1, 1, 1) that obtains positive social welfare can maintain the budget competition property. To see this, note that the budget competition property would imply that $w^A(1,1,1) \le w^A(1,0,1) + \varDelta ^{\oplus B}(1,1,1) = w^A(1,0,1) = 0$. Similarly, we must have $w^B(1,1,1) = w^C(1,1,1) = 0$. Thus, in order to maintain the budget competition property, our mechanism would have to generate social welfare 0 on input (1, 1, 1), resulting in an unbounded approximation factor. We conclude that there is no way to extend this specific mechanism for budgets totalling at most 2 to a (strategyproof) mechanism for budgets totalling at most 3 while maintaining the constant approximation factor of the locally greedy algorithm.

Roughly speaking, the problem illustrated by this example is that the presence of more than two bidders means that a substantial increase in the utility gained by one player does not necessarily imply any limits in the utility of another specific player. This is in contrast to the case of two players, in which the utilities of the two players are more directly related. This fundamental difference seems to indicate that substantially different techniques will be required in order to construct strategyproof mechanisms with three or more players.

A different (and natural) approach would be to employ the solution for two players by grouping all but one player at a time, and running the mechanism for two players recursively. However, this method seems ineffective in our setting, as interdependencies between the players’ outcomes can introduce non-monotonicities. This brings into question whether or not the locally greedy method can be made strategyproof by some method of randomizing over the order in which allocations are made.

This “2 vs 3 barrier” is, of course, not unique to our problem. Many optimization problems (such as graph coloring) are easily solvable when the size parameter is $k = 2$ but become NP-hard when $k \ge 3$. Closer to our setting, the 2 vs 3 barrier has been discussed in recent papers concerning mechanism design without payments, such as in the Lu et al. [27] results for k-facility location. Additionally Ashlagi et al. discussed similar issues [28] in the context of mechanisms for kidney exchange. They show that for n points on the line, there is a deterministic (respectively, randomized) strategyproof mechanism for placing $k = 2$ facilities (so as to minimize the sum of distances to the nearest facility) with approximation ratio $n-2$ (respectively, 4) whereas for $k = 3$ facilities, they do not know if there is any bounded ratio for deterministic strategyproof mechanisms and the best known approximation for randomized strategyproof mechanisms is O(n).