Throw the Fight.

A staple of sports movies and conspiracies, paying off someone to give up a competition has its own interesting game theoretic psychology.

For simplicity, I will consider a 2-person competition, from the eyes of one competitor, ie, the one getting bribed.

Game: Player P is in a 2-person competition(boxing, chess, whatever) where he has a percentage chance of winning equal to w, the prize for winning being worth c, the prize for losing being 0(for simplicity).

P is approached by some people who have vested interests in P losing. They offer P a reward of b for losing on purpose.

P now has a choice to make, and so we will look at a couple cases depending on P's attitudes to determine whether or not P should take this offer.

CASE 1: P cares only for his own utility.
In this case, it's a fairly simple matter of comapring possibilities.
If he denies the bribe, his expected utility is w*c. If he takes the bribe,his utility is b.
If b>=w*c, then it is clear P should take the bribe, because while c by itself may be greater than b, there is still the chance that P will lose anyway, getting a utility of zero.
If b<(w*c), then it is simply a matter of how much of a gambler P is. If w is greater than 1/2, then it is wise to decline the bribe, since then he would have a good chance to win c. If w is less than 1/2, then it is wiser to take the bribe, since the sure prize of b is better than a slim chance of winning c.
But, as I said, it depends on how much of a gambler P is.

CASE 2: P discounts utility for pride.
IE, P has a distaste for the dishonesty of the bribe. She would prefer to win or lose by her own power.
There is a discount factor d, where d is P's "pride". It is added when she refuses, and subtracted when she accepts the bribe.
So, her expected utility for not taking the bribe increases is w*(c+d) + (1-w)d = (w*c)+d.
Her utility for taking the bribe is b-d.
If b>=(w*c)+2d, then b-d>=(w*c)+d and we are in a similar situation to above. This represents the "everyone has their price" situation. P would be hard pressed to refuse such an offer, since her "pride" has been factored into the bribe.
If (w*c)+d <= b < (w*c)+2d, then b-d<(w*c)+d, but it's still not a clear cut decision, because (w*c)+d is still only an expected utility. "That's pride fuckin' wit' you" as Marcellus Wallace from Pulp Fiction would say. There's no clear decision to make here, unfortunately, and such decisions are tests of one's character.

All this is not even taking into account possible legal repercussions of the bribe, but I think this is a decent enough simulation with out it. Pride in a competition is a tricky thing, and even if you are unscrupulous and selfish, there are still insufficient prices.

Until next time, cheers.