Doing Well by Doing Good: Sports Edition

New Zealand play Argentina in the Rubgy Championship again this weekend, so there will likely be more discussion of the role that the immediate past New Zealand coach, Graham Henry, has as a coaching advisor to the Argentinian team. When the two teams met a couple of weeks ago in New Zealand, the media commentary on this focused on patriotism (how dare Henry work for the opposition), industrial espionage (there should be a stand-down period between coaching stints for different teams, since Henry still has information about current NZ structures), and social policy (it is good for the game to provide help to the less-strong teams).

In contrast, no-one seems to have considered the possibility that Henry's assistance to Argentina could be good for New Zealand. File this one under Honours problem sets I would like to write if I were teaching an appropriate course. The model is as follows:

The Rugby Championship is a two-round, round robin tournament (home and away) involving four teams: New Zealand, Australia, South Africa and New Zealand. New Zealand is clearly currently the strongest of the four teams and Argentina the weakest. The disparity is not so great that any result is a foregone conclusion, but realistically the gap between New Zealand and Argentina is large. The biggest threat to New Zealand in the competition, then, is not the prospect that they might lose to Argentina. It is that they might split their two-game series with the other two teams, and have the competition decided by bonus points.

Helping out Argentina, therefore, would be to our advantage, if it raised the probability of their beating the other two teams by more than if it raised the probability of their beating New Zealand. I have no idea just what assumptions would be needed to make this model work, which is why I think it would make a great problem set. There might even be a letters-style publication here. I'll put co-authorship out for tender!
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