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.

Stratergy

Being the Game Theoretic guy that I am, I thought I might do a little tribute to some great lesser-known strategic games out there.
Risk, Chess and the classics are obvious choices, but these are some of the more recent or marginalized games.

Bear in mind, this is not necessarilly strategy games, but real games of strategy.

1. Illuminati. Easily Steve Jackson's best game. The rules, while a little tricky at first, are straightforward once underway. Lots of underhanded wheeling and dealing can go on, each different illuminati you can play has different and preferred tactics, and the game can go on for quite a while. You can spend lots of hours on this game, and be amused the whole time, thanks in part to Jackson's tounge-in-cheek game design. A great game all around. I highly recommend picking it up.

2. Settlers of Catan. A relative newcomer, this game is HUGE here at Waterloo. It seemed to be almost singly responsible for most of my residence to give up video games in exchange for board games for a long time. The rules are very starightforward, the game is very efficient and fast paced, and there are countless strategies and permutations of the game. Plus there's always the hilarity that ensues from asking "Got wood for sheep?"

3. Imperialsim. A turn based computer strategy game where you play as a great power, and you have to turn it into the dominant force in the world, whether through conquest or diplomacy. The trick is in managing your resources and securing trade routes. Unlike many nation-competing strategy games out there, this is one where it is possible(though difficult) to win without firingt a single shot. If you are so far ahead of the rest of the world in technology, economy and infrastructure, then the AI tends to leave you alone. This game is incredibly addicitive. Hard to find, though.

4. Fluxx. The game with the ever changing rules. This is a card game where you don't so much act until you fulfill the requirements to win, you change around the requirements until you win. A very different kind of strategy paradigm. The best moves are those where you change the rules during your turn to put you far ahead, then change them at the end of your turn to make things horrible for your opponents.

Any other lesser know great strategy games out there? I'm always on the lookout.