X The concavity of atomic splittable congestion games with non-linear utility functions

Congestion games come up often in practice they represent our drives to work in the morning, the performance of factories under load, and the routing of packets through computer networks. In addition, all of these instances involve uncertainty. In the automobile traffic model, this uncertainty can come from the random assortment of drivers on the road at one time, but also the random interactions of wildlife on the road; the chances of accidents; the weather. In computer networks, links can become overloaded and see quality of service degrade, both wired and wireless. Agents can respond to this uncertainty in different ways. When trying to get to work in time for an important meeting, a commuter might adjust her route to take a perhaps slower, but safer route. If a company is relying on having components in their supply chain ready for a product launch, they may well try and split the load across multiple factories on the hope that at least one will be able to get things done in time (a diversification effect). We consider the atomic-splittable setting, where each player can divide up their flow among all possible paths. These models are useful for modeling computer networks, among others, where the individual units of flow are nearly infinitesimal, but players can control perhaps large chunks of the flow. Our players then can either be risk-averse in their total cost, across all flow, or risk averse in the cost that any particular unit or piece of flow experiences. A coalition of commuters (as considered in [Cominetti et al. 2009]) may be more concerned with their individual risk aversion. Packets from a streaming video may be useless after a certain time, while a site may be less concerned with the exact arrival of bits for a large file download. A baker delivering loaves of fresh bread might be concerned with the time that each loaf takes to get to market. On the other hand, a supply chain with non-biodegradable parts may be more concerned with the total cost across all goods produced (or, equally, the average cost). We model the non-linear utility over arrival times with increasing cost functions. We do this in two ways: with per-path cost functions, and with total delay cost functions. In the former, a player defines the cost of receiving flow at a time in the future and then tries to minimize that total cost. A loaf of fresh bread may be worth $4, while a loaf of