Consider the following “congestion game,” taken from Monderer

and Shapley (1996a). There are four cities located around a lake with a single road

joining them in the following clockwise order: A–B–C–D–A. There are two agents,

1 and 2, the first living in city A and the second in city B. Individual 1 wants to go

to city C, whereas individual 2 wants to reach city D. The cost of travel depends

on “congestion,” i.e., how many individuals (one or two) use the same segment of

the road joining any two adjacent cities. Costs are additive across travel segments,

with cξ (k) denoting the cost of travel segment ξ ∈ ” ≡ {AB, BC,CD, DA} when

there are k individuals using it. Model the situation as a game where each player i has two possible strategies: “travel clockwise” or “travel counterclockwise.” For any strategy profile

s = (s1, s2), define hξ (s) ∈ {0, 1, 2} as the number of individuals using segment ξ. Moreover, define the function ϒ : S → R as follows:

where we make cξ (0) = 0. Prove that ϒ(·) is a potential for the game described.