In revenue maximization of selling a digital product in a social network, the utility of an agent is often considered to have two parts: a private valuation, and linearly additive influences from other agents. We study the incomplete information case where agents know a common distribution about others' private valuations, and make decisions simultaneously. The "rational behavior" of agents in this case is captured by the well-known Bayesian Nash equilibrium. Two challenging questions arise: how to compute an equilibrium and how to optimize a pricing strategy accordingly to maximize the revenue assuming agents follow the equilibrium? In this paper, we mainly focus on the natural model where the private valuation of each agent is sampled from a uniform distribution, which turns out to be already challenging. Our main result is a polynomial-time algorithm that can exactly compute the equilibrium and the optimal price, when pairwise influences are non-negative. If negative influences are allowed, computing any equilibrium even approximately is PPAD-hard. Our algorithm can also be used to design an FPTAS for optimizing discriminative price profile.
Recently, @cite_20 consider the multi-stage model that the seller sets different prices for each stage. In contrast to @cite_1 , within each stage, agents are perfectly rational'', which is characterized by the pessimistic equilibrium in our setting with complete information . As mentioned in @cite_20 , they did not consider the case where a rational agent may defer her decision to later stages in order to improve the utility.
@cite_1 @cite_20
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We consider the problem of a monopolist seller who wants to sell some items to a set of buyers. The buyers are strategic, unit-demand, and connected by a social network. Furthermore, the utility of a buyer is a decreasing function of the number of neighbors who do not own the item. In other words, they exhibit negative externalities, deriving utility from being unique in their purchases. In this model, any fixed setting of the price induces a sub-game on the buyers. We show that it is an exact potential game which admits multiple pure Nash Equilibria. A natural problem is to compute those pure Nash equilibria that raise the most and least revenue for the seller. These correspond respectively to the most optimistic and most pessimistic revenues that can be raised. We show that the revenues of both the best and worst equilibria are hard to approximate within sub-polynomial factors. Given this hardness, we consider a relaxed notion of pricing, where the price for the same item can vary within a constant factor for different buyers. We show a 4-approximation to the pessimistic revenue when the prices are relaxed by a factor of 4. The interesting aspect of this algorithm is that it uses a linear programming relaxation that only encodes part of the strategic behavior of the buyers in its constraints, and rounds this relaxation to obtain a starting configuration for performing relaxed Nash dynamics. Finally, for the maximum revenue Nash equilibrium, we show a 2-approximation for bipartite graphs (without price relaxation), and complement this result by showing that the problem is NP-Hard even on trees. We consider the problem of dividing indivisible goods fairly among n agents who have additive and submodular valuations for the goods. Our fairness guarantees are in terms of the maximin share, that is defined to be the maximum value that an agent can ensure for herself, if she were to partition the goods into n bundles, and then receive a minimum valued bundle. Since maximin fair allocations (i.e., allocations in which each agent gets at least her maximin share) do not always exist, prior work has focussed on approximation results that aim to find allocations in which the value of the bundle allocated to each agent is (multiplicatively) as close to her maximin share as possible. In particular, Procaccia and Wang (2014) along with (2015) have shown that under additive valuations a 2 3-approximate maximin fair allocation always exists and can be found in polynomial time. We complement these results by developing a simple and efficient algorithm that achieves the same approximation guarantee. Furthermore, we initiate the study of approximate maximin fair division under submodular valuations. Specifically, we show that when the valuations of the agents are nonnegative, monotone, and submodular, then a 1 10-approximate maximin fair allocation is guaranteed to exist. In fact, we show that such an allocation can be efficiently found by using a simple round-robin algorithm. A technical contribution of the paper is to analyze the performance of this combinatorial algorithm by employing the concept of multilinear extensions. How should one sell an item to a buyer whose value for the item will only be realized next week? E.g. consider selling a flight to some executive who may or may not have a meeting with a client next week. Suppose that both the seller and the buyer only know a distribution, F, from which the buyer's value, v, for the item will be drawn. One way the seller could go about this sale is to make a take-it-or-leave-it offer today. The offer reads "pay the expected value today to get the item next week". A risk-neutral buyer would find this offer attractive, hence the seller would extract the full surplus. The unsettling feature of the afore-described mechanism is that, for some realizations of v, the bidder ends up with negative utility. In particular, while our mechanism is interim Individually Rational (IR), it is not ex-post IR. How could we fix this? One way is to wait until next week when the value is realized and make a take-it-or-leave-it offer of the item at an optimal monopoly price. The new mechanism is clearly ex-post IR, but its revenue could be much smaller than that of the previous one. Still, this trivial mechanism extracts the best possible revenue among all ex-post IR mechanisms, as a simple argument can establish. However, this optimality argument fails when several items are to be sold over consecutive periods. In this paper, we provide a characterization of the revenue-optimal, ex-post IR, dynamic mechanism selling k items over k periods to a bidder whose values are independent. In particular, we optimize the seller's revenue subject to the following strong individual rationality condition: at each period, the stage utility of the agent, defined to be the surplus from that period's allocation minus the agent's payment, must be non-negative. In particular, the non-negativity of the stage utilities implies that, at the end of each period, the agent's realized utility from participating in the mechanism so far is non-negative. We provide extensions to multiple bidders and an infinite horizon with discount factors. In this paper we study multi issue alternating-offers bargaining in a perfect information finite horizon setting, we determine the pertinent subgame perfect equilibrium, and we provide an algorithm to compute it. The equilibrium is determined by making a novel use of backward induction together with convex programming techniques in multi issue settings. We show that the agents reach an agreement immediately and that such an agreement is Pareto efficient. Furthermore, we prove that, when the multi issue utility functions are linear, the problem of computing the equilibrium is tractable and the related complexity is polynomial with the number of issues and linear with the deadline of bargaining.