I see you’ve edited the post to use my solution. Cool! But I think this remark of yours no longer applies:
Of course, it’s possible for two members of the triumvirate to create a second duumvirate that will profit from the hapless third member. Feel free to add whatever political metaphor you think this fits.
Let’s say the duumvirate chooses (red, blue) and the triumvirate chooses (red, blue, blue). Now who’s the hapless third member of the triumvirate that’s being preyed upon? Feel free to add a metaphor :-)
About politics, it’s interesting that the version with picking mates feels more “cutthroat” than the version with choosing colors. I guess the reason is that in the mate-picking game, the gains are spread out and the losses are concentrated (a Nash equilibrium has only r losers who pay everyone else), while in the color-choosing game the losses are spread out more (a Nash equilibrium has r(q+2) losers), which makes it feel more “safe” to a risk-averse person.
You could make it even more “safe” by designating only one of the least popular colors as winning (e.g. the lowest numbered one), so nobody has to feel less fortunate than the majority. Or you could make it more “cutthroat” and designate only one of the most popular colors as losing, which would ensure that losses can be concentrated to a minority. I think both would work for any n≥5, like your original result.
That said, I think your result was already enough to make the fraction of winners arbitrarily large or small as n grows, so this is just a fun distraction :-)
Of course, it’s possible for two members of the triumvirate to create a second duumvirate that will profit from the hapless third member. Feel free to add whatever political metaphor you think this fits.
That still applies. The duumvriate chooses (red, blue) and split the loss and gain. Two of the ex-triumvirate also choose (red, blue) and agree to split the loss and gain between themselves only. The last member to the triumvirate is now left out in the cold: whatever they choose, they lose, and they take the whole loss, while both other pairs gain.
I see you’ve edited the post to use my solution. Cool! But I think this remark of yours no longer applies:
Let’s say the duumvirate chooses (red, blue) and the triumvirate chooses (red, blue, blue). Now who’s the hapless third member of the triumvirate that’s being preyed upon? Feel free to add a metaphor :-)
About politics, it’s interesting that the version with picking mates feels more “cutthroat” than the version with choosing colors. I guess the reason is that in the mate-picking game, the gains are spread out and the losses are concentrated (a Nash equilibrium has only r losers who pay everyone else), while in the color-choosing game the losses are spread out more (a Nash equilibrium has r(q+2) losers), which makes it feel more “safe” to a risk-averse person.
You could make it even more “safe” by designating only one of the least popular colors as winning (e.g. the lowest numbered one), so nobody has to feel less fortunate than the majority. Or you could make it more “cutthroat” and designate only one of the most popular colors as losing, which would ensure that losses can be concentrated to a minority. I think both would work for any n≥5, like your original result.
That said, I think your result was already enough to make the fraction of winners arbitrarily large or small as n grows, so this is just a fun distraction :-)
That still applies. The duumvriate chooses (red, blue) and split the loss and gain. Two of the ex-triumvirate also choose (red, blue) and agree to split the loss and gain between themselves only. The last member to the triumvirate is now left out in the cold: whatever they choose, they lose, and they take the whole loss, while both other pairs gain.