This paper provides threshold policies with tight guarantees for online selection with convex cost (OSCC). In OSCC, a seller wants to sell some asset to a sequence of buyers with the goal of maximizing her profit. The seller can produce additional units of the asset, but at non-decreasing marginal costs. At each time, a buyer arrives and offers a price. The seller must make an immediate and irrevocable decision in terms of whether to accept the offer and produce/sell one unit of the asset to this buyer. The goal is to develop an online algorithm that selects a subset of buyers to maximize the seller’s profit, namely, the total selling revenue minus the total production cost. Our main result is the development of a class of simple threshold policies that are logistically simple and easy to implement, but have provable optimality guarantees among all deterministic algorithms. We also derive a lower bound on competitive ratios of randomized algorithms and prove that the competitive ratio of our threshold policy asymptotically converges to this lower bound when the total production output is sufficiently large. Our results generalize and unify various online search, pricing, and auction problems, and provide a new perspective on the impact of non-decreasing marginal costs on real-world online resource allocation problems.

We study the formation of stable outcomes via simple dynamics in cardinal hedonic games, where the valuations of agents change over time depending on the history of the coalition formation process. Specifically, we analyze situations where members of a coalition decrease their valuation for a leaving agent ( resentment ) or increase their valuation for a joining agent ( appreciation ). We show a series of convergence results for dynamics for resentful or appreciative agents which do not hold for classic dynamics. In particular, resentment turns out to be a strong stability-driving force. We complement our theoretical analysis with simulations that shed some light on the average running time of the dynamics and on the structure of the produced outcomes. From an algorithmic perspective, we obtain general hardness results for determining the fastest convergence time, results which also carry over to classic dynamics under static valuation functions. Finally, we explore a related model for preference updates where resentment is expressed by the deviator. We find a more nuanced picture, but still a broad possibility of convergence.

This paper uses reinforcement learning (RL) to approximate the policy rules of banks participating in a high-value payment system (HVPS). The objective of the RL agents is to learn a policy function for the choice of amount of liquidity provided to the system at the beginning of the day and the rate at which to pay intraday payments. Individual choices have complex strategic effects precluding a closed form solution of the optimal policy, except in simple cases. We show that in a stylized two-agent setting, RL agents learn the optimal policy that minimizes the cost of processing their individual payments—without complete knowledge of the environment. We further demonstrate that in more complex settings, both agents learn to reduce the cost of processing their payments and effectively respond to liquidity-delay trade-off. Our results show the potential of RL to solve liquidity management problems in HVPS and provide new tools to assist policymakers in their mandates of ensuring safety and improving the efficiency of payment systems.

Most online platforms learn from interactions with users, and engage in exploration : making potentially suboptimal choices in order to acquire new information. We study the interplay between exploration and competition : how such platforms balance the exploration for learning and competition for users. We consider a stylized duopoly in which two firms face the same multi-armed bandit problem. Users arrive one by one and choose between the two firms, so that each firm makes progress on its bandit problem only if it is chosen. We study whether competition incentivizes the adoption of better algorithms. We find that stark competition disincentivizes exploration, leading to low welfare. However, weaker competition incentivizes better exploration algorithms and increases welfare. We investigate two channels for weakening the competition: stochastic user choice models and a first-mover advantage. Our findings speak to the competition-innovation relationship and the first-mover advantage in the digital economy.

The airport problem is a classical and well-known model of fair cost-sharing for a single facility among multiple agents. This paper extends it to a more general setting involving multiple facilities. Specifically, in our model, each agent selects a facility and shares the cost with the other agents using the same facility. This scenario frequently occurs in sharing economies, such as sharing subscription costs for a multi-user license or taxi fares among customers traveling to potentially different destinations along a route. Our model can be viewed as a coalition formation game with size constraints, based on the fair cost-sharing of the airport problem. We refer to our model as a fair ride allocation on a line . We incorporate Nash stability and envy-freeness as criteria for solution concepts in our setting. We show that if a feasible allocation exists, a Nash stable feasible allocation that minimizes the social cost of agents exists and can be computed efficiently. Regarding envy-freeness, we provide several structural properties of envy-free allocations. Based on these properties, we design efficient algorithms for finding an envy-free allocation when either (1) the number of facilities, (2) the number of agent types, or (3) the capacity of facilities, is small. Moreover, we show that a consecutive envy-free allocation can be computed in polynomial time. On the negative front, we show NP-completeness of determining the existence of an allocation under two relaxed envy-free concepts.

We evaluate the fairness of a rule allocating among agents with equal rights by computing their utility for a hypothetical worst-case share, which only depends on their own valuation and the number of agents. For indivisible goods, Budish proposed the maximin share : the least utility of a bundle in the best partition of the objects; unfortunately maximin share is not always satisfiable. Earlier, Hill proposed the worst maximin share over all utilities with the same largest possible single-object value. More conservative than maximin share, it is guaranteed to be satisfiable for any possible profile of utilities and its computation is elementary, whereas it is NP-hard to compute maximin share. We apply Hill’s approach to the allocation of indivisible bads (objects with disutilities) and compute in closed form the worst-case minimax share for a given value of the worst single bad. We show that the worst-case minimax share is close to the original minimax share and that its monotonic closure is the best guaranteed share for all allocation instances with a common upper bound on the value of the worst single bad.