[Tom]

Some very rough game theory!

Interesting game theory exercise actually, but only for the top players.

Div 1 Second Prize = Div 2 First Prize.

Probably poses a conundrum only for the top 3 players, assuming all have a roughly equal chance of beating one another.

If no players throw a game, one is guaranteed a chance at missing out on a computer. If one player throws a game, everyone gets a computer (and one player gets a great computer). If two players throw a game, one gets a great computer, one gets an okay computer and one misses out on a computer. If all three throw a game, two miss out on a computer.

Optimal result - three top players get together and one agrees to throw a game. That player is "insured" by being compensation (to some degree) by the lost opportunity to compete for the best computer. The two competing players need not be compensated because the result is determined by skill.

Solution (non-cooperation) - do not let any player know the results of the qualifying games until all players have finished their game. Top players must then deal with the conundrum that if everyone (or even two) of them throw a game, there is a significant risk (1 in 2 or 2 in 3) they will miss out on a computer entirely. If all players compete to the maximum of their ability, all three have a 2 in 3 chance of getting a computer! By preventing any player from knowing the results of qualifiers in advance, the top three players are better off (3/3 or 2/3 versus 1/3) playing to the best of their ability.

Solution (cooperation) - planned "throwing" of games do ensure top three players all get a computer. This collusion cannot be prevented and (while I am not advocating this), the top three players' interests are maximised by colluding such that one player agrees to throws a game and receive compensation from the other two.

In terms of how this applies outside the top 3, I think if you restrict information about results, this will also have a trickle down effect to weaker players. This is because it may in the interests of a strong (but not top 3) player to throw a game, to get the best shot at a notebook. This is especially so if the top 3 players are rational and all play to the best of their ability to ensure at least a 2/3 chance at getting a computer. The issue then is that the (lets say top 4) player will probably get the Div 2 computer. Accordingly, players below the top 4 should also play from the strongest division to try and get a Prize 3 or 4 in Div 1, knowing that the "rational" fourth best player will aim for Division 2, thus freeing up a prize in Div 1 for them!

The organisers would dramatically increase the odds of most players trying their best by increasing the number of (low value) prizes offered in Div 1. This would mean it would only be worthwhile for 1 or 2 good players to "throw" a game to have a shot at the Div 2 notebook. Once this player throws the game, all the weaker players might as well just try their best to try and get a low value Div 1 prize.

TL;DR

Best way to get legitimate play - do not let any players have access to game results until everyone has qualified! This will not prevent private arrangements between players (but these players run the risk of someone not "honouring" the bargain).

Based on the prize structure, the top 3 players have an incentive to play to the best of their ability, assuming none have access to information about the other players' results.

The "fourth best" player is best off "throwing" a game, no matter what.

All other decent players might as well play their best. The way to maximise this is to offer some additional "low value" prizes for 5th, 6th etc in Div 1.