## Information Design for Bayesian Networks

### Project description

Selfish behavior degrades the performance of traffic networks. A popular model to study these effects is the traffic model of Wardrop. Here, the road network is modeled as a directed graph where each edge of the graph corresponds to a road segment. The travel times in the graph are modeled as cost functions on the edges where the cost of an edge is a function of the total flow on that edge.

We study a generalization of the model where the travel time of each edge also depends on a state of the world $\theta\in \Theta$ that is unknown to the traffic participants. The traffic participants share a common belief regarding the state of the world. In real-world traffic networks, the state $\theta$ may model certain weather or traffic scenarios.

We put ourselves in the shoes of a benevolent mediator who knows the true state of the world before any flow is routed. In practice, this corresponds to a traffic service provider before the rush hour.
The mediator may strategically reveal information about the state of the world regarding the true state of the world in order to induce a good traffic equilibrium, i.e., an equilibrium with a low overall travel time. In order to obtain a good equilibrium it may be beneficial for the system designer to not reveal the full information about the state of the world. Instead, it may be good to only partially reveal the knowledge in order to have more control on the user behavior.
We study the mathematical problem of computing the optimal information structures that induce best-possible equilibria in these settings.

*Explanation of the figures:*
We assume that both scenarios occur with a 50% chance.
Our goal is to find a convex decomposition of the prior 50% which as the least cost.
The different signals are given on the x-axis: signaling x corresponds to signaling that scenario 2 will occur with x% probability.
The cost of each signal is given by the function $C^*(\lambda)$.
The graph shows that signaling 25% in 2 out of 3 times and signaling 100% otherwise is the optimal solution.