Learning to Reach Agreement in a Continuous Ultimatum Game

It is well-known that acting in an individually rational manner, according to the principles of classical game theory, may lead to sub-optimal solutions in a class of problems named social dilemmas. In contrast, humans generally do not have much difficulty with social dilemmas, as they are able to balance personal benefit and group benefit. As agents in multi-agent systems are regularly confronted with social dilemmas, for instance in tasks such as resource allocation, these agents may benefit from the inclusion of mechanisms thought to facilitate human fairness. Although many of such mechanisms have already been implemented in a multi-agent systems context, their application is usually limited to rather abstract social dilemmas with a discrete set of available strategies (usually two). Given that many real-world examples of social dilemmas are actually continuous in nature, we extend this previous work to more general dilemmas, in which agents operate in a continuous strategy space. The social dilemma under study here is the well-known Ultimatum Game, in which an optimal solution is achieved if agents agree on a common strategy. We investigate whether a scale-free interaction network facilitates agents to reach agreement, especially in the presence of fixed-strategy agents that represent a desired (e.g. human) outcome. Moreover, we study the influence of rewiring in the interaction network. The agents are equipped with continuous-action learning automata and play a large number of random pairwise games in order to establish a common strategy. From our experiments, we may conclude that results obtained in discrete-strategy games can be generalized to continuous-strategy games to a certain extent: a scale-free interaction network structure allows agents to achieve agreement on a common strategy, and rewiring in the interaction network greatly enhances the agents ability to reach agreement. However, it also becomes clear that some alternative mechanisms, such as reputation and volunteering, have many subtleties involved and do not have convincing beneficial effects in the continuous case

Differentiable Game Mechanics

Deep learning is built on the foundational guarantee that gradient descent on an objective function converges to local minima. Unfortunately, this guarantee fails in settings, such as generative adversarial nets, that exhibit multiple interacting losses. The behavior of gradient-based methods in games is not well understood -- and is becoming increasingly important as adversarial and multi-objective architectures proliferate. In this paper, we develop new tools to understand and control the dynamics in n-player differentiable games. The key result is to decompose the game Jacobian into two components. The first, symmetric component, is related to potential games, which reduce to gradient descent on an implicit function. The second, antisymmetric component, relates to Hamiltonian games, a new class of games that obey a conservation law akin to conservation laws in classical mechanical systems. The decomposition motivates Symplectic Gradient Adjustment (SGA), a new algorithm for finding stable fixed points in differentiable games. Basic experiments show SGA is competitive with recently proposed algorithms for finding stable fixed points in GANs -- while at the same time being applicable to, and having guarantees in, much more general cases.Comment: JMLR 2019, journal version of arXiv:1802.0564