[cmajgimh] Round-robin payoffs

Adjust the prize structure of a round robin tournament.  A player receives prize money in proportion (or some formula) of how much they affected who ultimately wins first place. 

Consider the most extreme scenario where the first place and second place both win a lot of games, with first place barely coming out ahead.  The difference between first and second is in fact due to in large part an upset loss of the second place player to the last place player.

The last place player will earn a payoff, despite coming in last, for his or her win being one which decisively affected who would win first place.  The first place player would also, naturally, win a lot of prize money: if he or she had lost any of his or her games, then he or she would have been edged out for first, so he played a large number of important games.

The second place player might win some prize money, though there's definitely a feel of "no prize for second place".  The second player earns prize money only to the extent that his or her victories prevented the third (fourth, etc.) place winner from being first.

We would like to avoid incentives for players to form deals to throw games.  I'm not sure if this is possible under this system.

It seems that the first place winner wins more if he or she barely defeats second place, so there is incentive to lose if he or she is very far ahead.  In order to avoid this, maybe first place wins a fixed constant prize.  I think the second place player might have incentive to throw the last round game first place is out of reach and if offered a portion of the payoff that would be given to the opponent.  Therefore, the tournament payoff formula needs to incorporate time.

Morozevich, Mexico City. Also Grischuk the same round!