I declare that

introducing bribes would not change how the sport is performed!

This is as a result of

there’ll all the time be two or extra gamers who would profit from a bribe, but the participant being bribed may settle for any and all of them. The bribers, recognizing this, select to not bribe in any respect, as a result of if one supplied $$x$, then one other would supply $$x+epsilon$, and so forth, till all bribers supplied $$100$. At this level, the participant being bribed has no extra incentive to behave a sure approach than if no cash was supplied.

To see this, let’s perceive how a sport with gamers A, B, C, D, and E performs out:

For A to have any likelihood of profitable, they need to select a quantity with two extra numbers on both aspect. This permits 3, 4, 5, 6, 7, or 8, however it is going to develop into obvious that they like one of many numbers within the center, 5 or 6. Assume WLOG they select 5.

B picks the opposite quantity to set their likelihood of profitable equal to A’s; in our instance, they decide 6.

To have any likelihood of profitable, C should decide adjoining to A or B. Suppose they decide 4.

Now comes the fascinating half: D and E are assured to lose, so their solely supply of earnings is bribes. Everyone’s actions will now play out primarily based on the next info:

A wins if and provided that D and E decide from reverse sides of A.
B wins if and provided that D and E decide from B’s aspect of A.
C wins if and provided that D and E decide from C’s aspect of A.

Therefore, A doesn’t bribe D in any respect, as a result of that will threat D being counter-bribed; as a substitute, they will all the time wait to bribe E to choose reverse to D. However, B and C every want D on their aspect to win, however this runs into the runaway bribing scenario, so neither bribe D in any respect. Thus, D picks randomly and their anticipated revenue is $$0$.

Suppose WLOG D chooses from B’s aspect. This means A and B every want E on their aspect to win. So once more, neither of them bribe E; E picks randomly, and their anticipated revenue is $$0$.

Thus, by the symmetry of their selections, A and B every have the identical likelihood of profitable. C’s likelihood of profitable is $frac37cdotfrac26=frac17$, so A’s and B’s possibilities every develop into $frac37$.

This provides the next approximate anticipated income:

A: $$42.86$.
B: $$42.86$.
C: $$14.29$.
D: $$00.00$.
E: $$00.00$.