Equilibrium computation in resource allocation games

We study the equilibrium computation problem for two classical resource allocation games: atomic splittable congestion games and multimarket Cournot oligopolies. For atomic splittable congestion games with singleton strategies and player-specific affine cost functions, we devise the first polynomial time algorithm computing a pure Nash equilibrium. Our algorithm is combinatorial and computes the exact equilibrium assuming rational input. The idea is to compute an equilibrium for an associated integrally-splittable singleton congestion game in which the players can only split their demands in integral multiples of a common packet size. While integral games have been considered in the literature before, no polynomial time algorithm computing an equilibrium was known. Also for this class, we devise the first polynomial time algorithm and use it as a building block for our main algorithm. We then develop a polynomial time computable transformation mapping a multimarket Cournot competition game with firm-specific affine price functions and quadratic costs to an associated atomic splittable congestion game as described above. The transformation preserves equilibria in either game and, thus, leads – via our first algorithm – to a polynomial time algorithm computing Cournot equilibria. Finally, our analysis for integrally-splittable games implies new bounds on the difference between real and integral Cournot equilibria. The bounds can be seen as a generalization of the recent bounds for single market oligopolies obtained by Todd (Math Op Res 41(3):1125–1134 2016, 10.1287/moor.2015.0771).

Path Planning Problems with Side Observations—When Colonels Play Hide-and-Seek

Resource allocation games such as the famous Colonel Blotto (CB) and Hide-and-Seek (HS) games are often used to model a large variety of practical problems, but only in their one-shot versions. Indeed, due to their extremely large strategy space, it remains an open question how one can efficiently learn in these games. In this work, we show that the online CB and HS games can be cast as path planning problems with side-observations (SOPPP): at each stage, a learner chooses a path on a directed acyclic graph and suffers the sum of losses that are adversarially assigned to the corresponding edges; and she then receives semi-bandit feedback with side-observations (i.e., she observes the losses on the chosen edges plus some others). We propose a novel algorithm, Exp3-OE, the first-of-its-kind with guaranteed efficient running time for SOPPP without requiring any auxiliary oracle. We provide an expected-regret bound of Exp3-OE in SOPPP matching the order of the best benchmark in the literature. Moreover, we introduce additional assumptions on the observability model under which we can further improve the regret bounds of Exp3-OE. We illustrate the benefit of using Exp3-OE in SOPPP by applying it to the online CB and HS games.

Reciprocity Among Preschoolers in Relation to Resource Allocation Toward Siblings, Friends, and Strangers

Children at age 6 years differentially treat kin, friends, and strangers in resource allocation games by being more altruistic toward kin. However, it is unknown how previous allocation experience as a recipient influences the potential kinship effect in subsequent resource allocations. The present study investigated how 4- to 6-year-old children allocated resources between themselves and a sibling, a friend, or a stranger in three allocation tasks after the recipient had previously shared or nonshared with the participant. Results showed that, when a share would induce cost on the self, 6-year-old children were likely to share with a sibling whether the sibling had previously shared or not, but they would share only with friends or strangers who had previously shared. When a share would induce no cost, participants across ages were likely to share with a recipient who had previously shared. When the decision option was between sharing equally and sharing altruistically, participants would allow the recipient to have more only when the recipient was a sibling or friend who had previously allocated altruistically. These findings suggest that kin altruism in resource allocation emerges at around 6 years of age and that reciprocity partly overrides and partly reinforces kin altruism.