X Truthful Mechanisms for Competing Submodular Processes

Motivated by models of competitive influence spread in networks, we study mechanisms for allocating nodes to self-interested agents with negative externalities. For example, a social network provider may wish to allow advertisers to provide special offers to influential individuals. The advertisers benefit in that product adoption may spread through the network. However, a competing product may adversely impact the rate of adoption for any given advertiser. More generally, there is a ground set U of elements (e.g. nodes representing individuals in a social network) and rational agents (e.g. advertisers) who vie for elements of this set. A valid allocation is a collection of subsets of U – one for each agent. In our applications, the utility of each agent is non-decreasing in its own allocation and non-increasing in its opponents’ allocation, and the sum of utilities (the social welfare) is a non-decreasing submodular function. This framework captures many natural models of influence spread, and more generally (set-)monotone processes with negative externalities in which only the total welfare is required to be submodular. Much of the prior work in modelling competitive influence spread assumes that the players themselves will choose which network nodes to target, and study equilibria of the resulting game. However, in many applications, agents do not have this level of control: instead, the owner of the network chooses the allocation. We therefore consider a mechanism that receives as input a demand from each player, representing the number of elements to allocate, and returns sets respecting those demands with the goal of maximizing social welfare. The introduction of competition raises issues of strategies, where rational agents may strategically underreport their demands. The complicating factor is that while the social welfare function is submodular, the expected utility for each individual agent may undergo complex interactions due to externalities. In a broad class of such scenarios, the social welfare can be O(1) approximated by an iterative (but not necessarily strategyproof) algorithm that selects an (arbitrary) agent in each iteration and then greedily allocates a node to that agent. Under some natural assumptions about the social welfare and the players’ utility functions, we show that a variant of this algorithm with a randomized ordering is strategyproof when there are at least three players. Interestingly, this approach fails to be strategyproof when there are only two advertisers. In the case of two players, we recursively construct a randomized mechanism that is strategyproof and maintains the social welfare approximation of the greedy algorithm. Furthermore, this method also works when the extra assumptions about the social welfare and the players’ utilities are lifted.