I would like the community's help with a game I am trying to solve.
In this instance of a congestion game we have five commuters $A$, $B$, $C$, $D$, $E$ $(n=5)$ and two allowable resources which are bus and car. These are the strategies of our players. $c_i$ is the cost of each commuter using their own car where $i ∈ \{A, B, C, D, E\}$ and they are fixed: $c_A = 1$, $c_B = 3$, $c_C = 5$, $c_D = 7$, $c_E = 9$. Commute time by bus depends on how many others take the bus and that can be seen below and the cost would be the same for each:
People Cost
1 10
2 8
3 6
4 4
5 2
I am trying to find Nash Equilibrium here and eliminate Dominated strategies but I am struggling with it. The first thing I did was create a matrix where you can see the costs where column is commuters taking the car and row is number of commuters taking the bus:
Can someone let me know if there's better way to create the matrix so I can showcase iterated elimination of dominated strategies and NE?
Many thanks
$A$ will go by car, since their car costs are lower than the bus cost can be.
That leaves $4$ people who might go by bus, so the bus will cost at least $4$, so $B$ will go by car.
And so on.