Braess's Paradox

A city has two routes from source $S$ to destination $T$: Top route $S \to A \to T$: link $SA$ is congestion-dependent; link $AT$ has a fixed travel time. Bottom route $S \to B \to T$: link $SB$ has a fixed travel time; link $BT$ is congestion-dependent. The network is symmetric. A city planner proposes adding a new shortcut link $A \to B$ with near-zero travel time, creating a third route $S \to A \to B \to T$. To her surprise, adding the shortcut makes everyone’s travel time longer at the selfish-routing Nash equilibrium. Without the shortcut Both routes are symmetric. In equilibrium, traffic splits evenly. If $N/2$ drivers use each route and the congested links have delay $\alpha \cdot n$ (where $n$ is the number of cars): $$t_{\text{top}} = \frac{N}{2}\alpha + c = t_{\text{bottom}}$$ With the shortcut $A \to B$ Each driver thinks, “Link $AB$ is free; I can use $SA$, slip across to $B$, then take $BT$ instead of the slow constant link $AT$.” All $N$ drivers make this choice. The…