# define hξ (s) ∈ {0, 1, 2} as the number of individuals using segment ξ.

define hξ (s) ∈ {0, 1, 2} as the number of individuals using segment ξ..

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: ABCDA. There are two agents,

1 and 2the first living in city and the second in city BIndividual 1 wants to go

to city C, whereas individual 2 wants to reach city DThe cost of travel depends

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

with cξ (k) denoting the cost of travel segment ξ ∈ ” ≡ {ABBC,CDDA} when

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

= (s1s2)define hξ (s) ∈ {012} as the number of individuals using segment ξ. Moreover, define the function ϒ → R as follows:

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

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define hξ (s) ∈ {0, 1, 2} as the number of individuals using segment ξ.

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