We study the optimal pricing strategies of a monopolist selling a divisible good (service) to consumers that are embedded in a social network. A key feature of our model is that consumers experience a (positive) local network effect. In particular, each consumer's usage level depends directly on the usage of her neighbors in the social network structure. Thus, the monopolist's optimal pricing strategy may involve offering discounts to certain agents, who have a central position in the underlying network. Our results can be summarized as follows. First, we consider a setting where the monopolist can offer individualized prices and derive an explicit characterization of the optimal price for each consumer as a function of her network position. In particular, we show that it is optimal for the monopolist to charge each agent a price that is proportional to her Bonacich centrality in the social network. In the second part of the paper, we discuss the optimal strategy of a monopolist that can only choose a single uniform price for the good and derive an algorithm polynomial in the number of agents to compute such a price. Thirdly, we assume that the monopolist can offer the good in two prices, full and discounted, and study the problem of determining which set of consumers should be given the discount. We show that the problem is NP-hard, however we provide an explicit characterization of the set of agents that should be offered the discounted price. Next, we describe an approximation algorithm for finding the optimal set of agents. We show that if the profit is nonnegative under any feasible price allocation, the algorithm guarantees at least 88 % of the optimal profit. Finally, we highlight the value of network information by comparing the profits of a monopolist that does not take into account the network effects when choosing her pricing policy to those of a monopolist that uses this information optimally.

In this paper we introduce a novel flow representation for finite games in strategic form. This representation allows us to develop a canonical direct sum decomposition of an arbitrary game into three components, which we refer to as the potential, harmonic and nonstrategic components. We analyze natural classes of games that are induced by this decomposition, and in particular, focus on games with no harmonic component and games with no potential component. We show that the first class corresponds to the well-known potential games. We refer to the second class of games as harmonic games, and study the structural and equilibrium properties of this new class of games.Intuitively, the potential component of a game captures interactions that can equivalently be represented as a common interest game, while the harmonic part represents the conflicts between the interests of the players. We make this intuition precise, by studying the properties of these two classes, and show that indeed they have quite distinct and remarkable characteristics. For instance, while finite potential games always have pure Nash equilibria, harmonic games generically never do. Moreover, we show that the nonstrategic component does not affect the equilibria of a game, but plays a fundamental role in their efficiency properties, thus decoupling the location of equilibria and their payoff-related properties. Exploiting the properties of the decomposition framework, we obtain explicit expressions for the projections of games onto the subspaces of potential and harmonic games. This enables an extension of the properties of potential and harmonic games to "nearby" games. We exemplify this point by showing that the set of approximate equilibria of an arbitrary game can be characterized through the equilibria of its projection onto the set of potential games.

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Motivated by the prevalence of ride-sharing platforms, in “Spatial Pricing in Ride-Sharing Networks,” Bimpikis, Candogan, and Saban explore the impact of the demand pattern for rides across a network’s locations on a platform’s optimal pricing and compensation policy, profits, and consumer surplus. They explicitly account for the pricing problem’s spatial dimension and the fact that the drivers endogenously determine whether and where to provide service. Their first contribution is to develop a tractable model to study a platform operating on a network of locations that may differ in both the size of their potential demand and the destination preferences of riders. Second, they provide a characterization of the platform’s optimal policy and identify “balancedness” of the demand pattern as a property that captures the profit potential of a given network. Finally, they discuss the benefits and limitations of a number of alternative pricing and compensation schemes.

Except for special classes of games, there is no systematic framework for analyzing the dynamical properties of multi-agent strategic interactions. Potential games are one such special but restrictive class of games that allow for tractable dynamic analysis. Intuitively, games that are "close" to a potential game should share similar properties. In this paper, we formalize and develop this idea by quantifying to what extent the dynamic features of potential games extend to "near-potential" games.We study convergence of three commonly studied classes of adaptive dynamics: discretetime better/best response, logit response, and discrete-time fictitious play dynamics. For better/best response dynamics, we focus on the evolution of the sequence of pure strategy profiles and show that this sequence converges to a (pure) approximate equilibrium set, whose size is a function of the "distance" from a close potential game. We then study logit response dynamics parametrized by a smoothing parameter that determines the frequency with which the best response strategy is played. Our analysis uses a Markov chain representation for the evolution of pure strategy profiles. We provide a characterization of the stationary distribution of this Markov chain in terms of the distance of the game from a close potential game and the corresponding potential function. We further show that the stochastically stable strategy profiles (defined as those that have positive probability under the stationary distribution in the limit as the smoothing parameter goes to 0) are pure approximate equilibria. Finally, we turn attention to fictitious play, and establish that in near-potential games, the sequence of empirical frequencies of player actions converges to a neighborhood of (mixed) equilibria of the game, where the size of the neighborhood increases with distance of the game to a potential game. Thus, our results suggest that games that are close to a potential game inherit the dynamical properties of potential games. Since a close potential game to a given game can be found by solving a convex optimization problem, our approach also provides a systematic framework for studying convergence behavior of adaptive learning dynamics in arbitrary finite strategic form games.

We show that in zero-sum polymatrix games, a multiplayer generalization of two-person zerosum games, Nash equilibria can be found efficiently with linear programming. We also show that the set of coarse correlated equilibria collapses to the set of Nash equilibria. In contrast, other important properties of two-person zero-sum games are not preserved: Nash equilibrium payoffs need not be unique, and Nash equilibrium strategies need not be exchangeable or max-min.

Data on population movements can be helpful in designing targeted policy responses to curb epidemic spread. However, it is not clear how to exactly leverage such data and how valuable they might be for the control of epidemics. To explore these questions, we study a spatial epidemic model that explicitly accounts for population movements and propose an optimization framework for obtaining targeted policies that restrict economic activity in different neighborhoods of a city at different levels. We focus on COVID-19 and calibrate our model using the mobile phone data that capture individuals’ movements within New York City (NYC). We use these data to illustrate that targeting can allow for substantially higher employment levels than uniform (city-wide) policies when applied to reduce infections across a region of focus. In our NYC example (which focuses on the control of the disease in April 2020), our main model illustrates that appropriate targeting achieves a reduction in infections in all neighborhoods while resuming 23.1%–42.4% of the baseline nonteleworkable employment level. By contrast, uniform restriction policies that achieve the same policy goal permit 3.92–6.25 times less nonteleworkable employment. Our optimization framework demonstrates the potential of targeting to limit the economic costs of unemployment while curbing the spread of an epidemic. This paper was accepted by Carri Chan, healthcare management.

We propose a spatial epidemic spread model to study the Covid-19 epidemic. In our model, a city consists of multiple neighborhoods, each of which has five disease compartments (susceptible/exposed/infected clinical/infected subclinical/recovered). Due to the movement of individuals across neighborhoods (e.g., commuting to work), the infections in one neighborhood can trigger infections in others. We consider the problem of a planner who reduces the economic activity in a targeted way to curb the spread of the epidemic. We focus both on the regime with a small number of infections and the regime with a large number of infections, and provide a framework for obtaining the policies that induce the lowest economic costs.We use the available data on individuals' movements, level of economic activity in different neighborhoods, and the state of the epidemic to apply our framework to the control of the epidemic in NYC. Our results indicate that targeted closures can achieve the same policy goals at substantially lower economic losses than city-wide closure policies. In addition, to curb the spread of the epidemic in NYC, coordination with other counties is paramount. Finally, the optimal policy (under different scenarios) promotes some level of economic activity in Midtown Manhattan locations (due to their economic importance) while imposing closures in many other neighborhoods in the city (to curb the spread of the disease). Contrary to what might be intuitively expected, and due to the spatial aspect of the epidemic spread, neighborhoods with higher level of infections should not necessarily be the ones exposed to the most stringent economic closure measures.