Mechanisms and allocations with positive network externalities

Abstract

With the advent of social networks such as Facebook and LinkedIn, and online offers/deals web sites, network externalties raise the possibility of marketing and advertising to users based on influence they derive from their neighbors in such networks. Indeed, a user's knowledge of which of his neighbors "liked" the product, changes his valuation for the product. Much of the work on the mechanism design under network externalities has addressed the setting when there is only one product. We consider a more natural setting when there are multiple competing products, and each node in the network is a unit-demand agent.We first consider the problem of welfare maximization under various different types of externality functions. Specifically we get a O(log n log(nm)) approximation for concave externality functions, a 2 O(d) -approximation for convex externality functions that are bounded above by a polynomial of degree d, and we give a O(log 3 n)-approximation when the externality function is submodular. Our techniques involve formulating non-trivial linear relaxations in each case, and developing novel rounding schemes that yield bounds vastly superior to those obtainable by directly applying results from combinatorial welfare maximization.We then consider the problem of Nash equilibrium where each node in the network is a player whose strategy space corresponds to selecting an item. We develop tight characterization of the conditions under which a Nash equilibrium exists in this game. Lastly, we consider the question of pricing and revenue optimization when the users in the network are selfish agents, and their private information is the vector of valuations for different items. We show that for single parameter settings (when an agents's intrinsic valuation for every item can be described using one parameter), all our approximation results for welfare maximization extend to revenue maximization. For the multi-parameter setting, we design an O(1)-approximate revenue optimal mechanism for IID agents, when the action of a single agent does not affect the externality enjoyed by remaining agents.

Welfare Maximization with Friends-of-Friends Network Externalities

“…Then we give a 5/18·(1−1/e) ≈ 0.17-approximation algorithm for 2-hop step function externalities (Theorem 6). Our technique also leads to a combinatorial (1 − 1/e)/2 ≈ 0.3-approximation algorithm for 1-hop step function externalities (Theorem 7), improving the approximation ratio of the LP-based algorithm in [7].…”

“…As discussed earlier the prior work that is most closely related to our work is the work by Bhalgat et al [7], that studies a similar setting where multiple different items are assigned to unit-demand agents with 1-hop externalities.…”

“…There is a large body of work on mechanisms and pricing strategies for this problem with a single [1,3,5,6,9,10,16,19,21,28] or multiple items [2,7,12,18,24,[30][31][32] when the valuation function of a user depends solely on the 1-hop neighborhood of a user and the user itself. All this work assumes that there is an infinite supply of items (of each type if there are different items) and the users have unit-demand, that is, they want to buy only one item.…”

“…The prior work that is most closely related to our work is the work by Bhalgat et al [7], where they study the multi-item setting with 1-hop externality functions and give approximation algorithms based on LP-relaxations and randomized rounding for different classes of externality functions. For linear externalities they give a 1/64-approximation algorithm and for step function externalities they get an approximation ratio of (1 − 1/e)/16 ≈ 0.04.…”

“…(2) On the positive side we present (i) an O( √ n)-approximation algorithm for general concave externality functions, (ii) an O(log m)-approximation algorithm for linear externality functions, and (iii) a 5 18 (1 − 1/e)-approximation algorithm for 2-hop step function externalities. We also improve the result from [7] for 1-hop step function externalities by giving a …”