I claim that
introducing bribes doesn't change how the game is played!
This is because
there will always be two or more players who would benefit from a bribe, yet the player being bribed could accept any and all of them. The bribers, recognizing this, choose not to bribe at all, because if one offered $\\\$x$, then another would offer $\\\$x+\epsilon$, and so on, until all bribers offered $\\\$100$. At this point, the player being bribed has no more incentive to behave a certain way than if no money was offered.
To see this, let's understand how a game with players A, B, C, D, and E plays out:
For A to have any chance of winning, they must choose a number with two more numbers on either side. This allows 3, 4, 5, 6, 7, or 8, but it will become apparent that they prefer one of the numbers in the middle, 5 or 6. Assume WLOG they choose 5.
B picks the other number to set their chance of winning equal to A's; in our example, they pick 6.
To have any chance of winning, C must pick adjacent to A or B. Suppose they pick 4.
Now comes the interesting part: D and E are guaranteed to lose, so their only source of income is bribes. Everyone's actions will now play out based on the following information:
A wins if and only if D and E pick from opposite sides of A.
B wins if and only if D and E pick from B's side of A.
C wins if and only if D and E pick from C's side of A.
Therefore, A does not bribe D at all, because that would risk D being counter-bribed; instead, they can always wait to bribe E to pick opposite to D. However, B and C each need D on their side to win, but this runs into the runaway bribing situation, so neither bribe D at all. Thus, D picks randomly and their expected profit is $\\\$0$.
Suppose WLOG D chooses from B's side. This means A and B each need E on their side to win. So again, neither of them bribe E; E picks randomly, and their expected profit is $\\\$0$.
Thus, by the symmetry of their choices, A and B each have the same chance of winning. C's chance of winning is $\frac37\cdot\frac26=\frac17$, so A's and B's chances each become $\frac37$.
This gives the following approximate expected profits: