That’s basically what I was aiming at, but that game isn’t symmetric according to the definition (eg if player 3 chooses player 2, then a permutation of 1 and 2 should end up with them choosing the old player 1). That’s because there’s no “internal” permutation within the action (so a_p(1) means “the action player p(1) had chosen”, not “the action player p(1) had chosen, updated according to permutation p if the action involve pointing at specific other players” .
My clunky setup essentially tries to do your idea without needing to have actions that point at other players.
I see. Yeah, it seems easy to prove that my game can’t be made symmetric by your definition. Players 1 and 2 can’t pair off, because there’s nothing stopping player 3 from submitting the same action as player 2.
Here’s another simple example that seems to work under your definition. Each of five people must pick either red or blue. That divides them into two unequal groups. Then everyone in the bigger group pays a dollar to everyone in the smaller group, unless there’s only one group, in which case nothing happens. The winning strategy for a team of two is to pick different colors.
You can generalize it as well, by making people choose one of m colors and then having everyone in the smallest group receive a dollar from everyone else. If the number of people isn’t divisible by m, then a team of m can win by choosing different colors.
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.
It’s interesting that the example with picking mates feels very political, with people callously discarding others to profit from them, but the example with colors doesn’t feel political at all. The last person to pick a color isn’t any worse off than the majority. I guess concentrating gains and spreading losses is more merciful than concentrating losses and spreading gains.
I see. Yeah, it seems easy to prove that my game can’t be made symmetric by your definition. (If player 1 and player 2′s mutual pairing must happen independently from their identities and also from the actions of everyone else, there’s nothing stopping player 3 from submitting the same action as player 2.) I wonder if the definition I was implicitly using is studied anywhere...
That’s basically what I was aiming at, but that game isn’t symmetric according to the definition (eg if player 3 chooses player 2, then a permutation of 1 and 2 should end up with them choosing the old player 1). That’s because there’s no “internal” permutation within the action (so a_p(1) means “the action player p(1) had chosen”, not “the action player p(1) had chosen, updated according to permutation p if the action involve pointing at specific other players” .
My clunky setup essentially tries to do your idea without needing to have actions that point at other players.
I see. Yeah, it seems easy to prove that my game can’t be made symmetric by your definition. Players 1 and 2 can’t pair off, because there’s nothing stopping player 3 from submitting the same action as player 2.
Here’s another simple example that seems to work under your definition. Each of five people must pick either red or blue. That divides them into two unequal groups. Then everyone in the bigger group pays a dollar to everyone in the smaller group, unless there’s only one group, in which case nothing happens. The winning strategy for a team of two is to pick different colors.
You can generalize it as well, by making people choose one of m colors and then having everyone in the smallest group receive a dollar from everyone else. If the number of people isn’t divisible by m, then a team of m can win by choosing different colors.
I think that works, and is much simpler.
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.
It’s interesting that the example with picking mates feels very political, with people callously discarding others to profit from them, but the example with colors doesn’t feel political at all. The last person to pick a color isn’t any worse off than the majority. I guess concentrating gains and spreading losses is more merciful than concentrating losses and spreading gains.
I see. Yeah, it seems easy to prove that my game can’t be made symmetric by your definition. (If player 1 and player 2′s mutual pairing must happen independently from their identities and also from the actions of everyone else, there’s nothing stopping player 3 from submitting the same action as player 2.) I wonder if the definition I was implicitly using is studied anywhere...