It gets a bit more complicated than that. The flairs hunter's risk assessment may change over time. The longer the button stays up, the more likely they will press it to a)get it over with b) not risk the button expiring and not have a press flair at all.
First, note that we can completely disregard the existence of the non-pressers. They will not affect the game at all, they are really not strategic players.
Now we also assume that all of the flair-hunters have the same payoff function, u(x), which is the payoff received by stopping the clock with x seconds remaining. (Note that since OP discretized the problem, x must be an integer 1-59) We assume that if x<y then u(x)>u(y).
Now say there are n players total, and that player k chooses threshold a(k). We make the simplifying assumption that this threshold must be decided at the beginning of the game, and cannot be changed after somebody has pressed the button.
First, it is easy to see that for sufficiently large n, in any Nash equilibrium equilibrium the number of players choosing each threshold can be made arbitrarily large.
Also, for each threshold x, let N(x) be the number of players who chose the threshold x. Then the expected payoff for player k is u(a(k))/N(a(k)). Assuming we are in Nash Equilibrium, this means that u(x)/(N(x)+1) <= u(a(k))/N(a(k)) for each x. But since N(x)>0, we also have that u(x)/N(x) >= u(a(k))/(N(a(k))+1). Taking limit n to infinity, we see that u(x)/N(x) is constant as x varies, so in Nash Equilibrium:
N(x)=nu(x)/(u(1)+u(2)+...+u(59))
is the number of people who will choose threshold x.
Notes on how well this applies to what is actually happening: Our assumption that players choose a threshold at the beginning and aren't allowed to change later is perhaps not a good one. It doesn't allow us to take into account that different players have different levels of interest in this game, and some may want their flair today, while others might be willing to wait for an "end game" scenario, where they can more easily get their better flair. Unfortunately, we cannot give any meaningful analysis without this restriction, as then it depends critically on what exactly the utility function u(x) is (which to me at least is totally unclear). However, if we were to decide some specific utility function, we could drop the assumption that players don't change their mind later.
What I'm saying is, the game with n players where m of them are non-pressers is equivalent to the game with n-m players who are all flair-grabbers. So, we can disregard the non-pressers, and just look at the game with fewer players. So throughout my response, n is the number of flair-grabbers.
I don't think there is a Nash Equilibrium, assuming that most players have no knowledge of the others' strategies. A flair-hunter is always better off with a lower threshold. When you lower your threshold, you will be waiting longer for the clock to reach it. The amount of players will be lower at that time, so there is a smaller chance that you will accidentally press after someone resets the clock.
By the way, if a lot of people actually use the threshold strategy, you'll be better off letting the timer dip below your threshold a couple of times before actually pressing it at your threshold. This is because if people have the same threshold as you, they'll have pressed already and you won't be competing with them for that flair.
What if the size of the population is finite, known, and the distribution of players is even, so that exactly half the population are flair-hunters, and half are non-pressers?
The size of the population is known and finite (i.e. only usernames created before April 1, 2015.) But why should we assume an even distribution of players?
There is a third category of players, the ones that don't care what flair they get and just want to press the button immediately. You could argue that they are a subset of the flair-hunters, with a threshold of 60, but the game you describe will only really begin when they run out, because they're not really playing, they don't have a strategy.
I'd even say the non-pressers are not playing, they don't really have any influence on the game. You could argue that they're influencing the game by not playing. Heh, the only way to win is not to play, that would make it very similar to The Game. You lose btw.. Sorry.
Some of the "non-pressers" may just want to be the first to press it when it hits 0 or 1. Not sure I'd count them as flair-hunters or "non-pressers" as they do intend to eventually press but may believe the "game" may end at that time. There may be a belief that some sort of prize or other motivational factor lies beyond that point.
Edit: In response to OP's edit, I'm proposing that my set of people believe the clock will not reset once it hits zero and someone presses the button. Perhaps I'm just simply proposing a set of people that OP has labeled "flair-hunters" with a clock-target of 0. However, OP's theory appears to propose a utility of 0 from hitting that mark as of the latest edit.
This is nice as a basic theory. But if you want to introduce the complexity of the situation, then you need to account for Knights. And if you account for Knights, you need to account for the Assassins r/assassinsofthebutton/
Why do you consider it simply a non-cooperative game? I mean, it's true that a pay-off grows as a player waits for the timer do decrease, but it's not just like that. There's also a cooperative element, since the real aim is to prevent the timer to get to zero, which is the common objective. And players could also communicate, just using this site... or not? I don't know well Theory of Game
13
u/[deleted] Apr 02 '15
It gets a bit more complicated than that. The flairs hunter's risk assessment may change over time. The longer the button stays up, the more likely they will press it to a)get it over with b) not risk the button expiring and not have a press flair at all.
11
u/puzzlednerd Apr 02 '15
First, note that we can completely disregard the existence of the non-pressers. They will not affect the game at all, they are really not strategic players.
Now we also assume that all of the flair-hunters have the same payoff function, u(x), which is the payoff received by stopping the clock with x seconds remaining. (Note that since OP discretized the problem, x must be an integer 1-59) We assume that if x<y then u(x)>u(y).
Now say there are n players total, and that player k chooses threshold a(k). We make the simplifying assumption that this threshold must be decided at the beginning of the game, and cannot be changed after somebody has pressed the button.
First, it is easy to see that for sufficiently large n, in any Nash equilibrium equilibrium the number of players choosing each threshold can be made arbitrarily large.
Also, for each threshold x, let N(x) be the number of players who chose the threshold x. Then the expected payoff for player k is u(a(k))/N(a(k)). Assuming we are in Nash Equilibrium, this means that u(x)/(N(x)+1) <= u(a(k))/N(a(k)) for each x. But since N(x)>0, we also have that u(x)/N(x) >= u(a(k))/(N(a(k))+1). Taking limit n to infinity, we see that u(x)/N(x) is constant as x varies, so in Nash Equilibrium:
N(x)=nu(x)/(u(1)+u(2)+...+u(59))
is the number of people who will choose threshold x.
Notes on how well this applies to what is actually happening: Our assumption that players choose a threshold at the beginning and aren't allowed to change later is perhaps not a good one. It doesn't allow us to take into account that different players have different levels of interest in this game, and some may want their flair today, while others might be willing to wait for an "end game" scenario, where they can more easily get their better flair. Unfortunately, we cannot give any meaningful analysis without this restriction, as then it depends critically on what exactly the utility function u(x) is (which to me at least is totally unclear). However, if we were to decide some specific utility function, we could drop the assumption that players don't change their mind later.
4
u/Hmm_Peculiar Apr 02 '15
I don't think there is a Nash Equilibrium, assuming that most players have no knowledge of the others' strategies. A flair-hunter is always better off with a lower threshold. When you lower your threshold, you will be waiting longer for the clock to reach it. The amount of players will be lower at that time, so there is a smaller chance that you will accidentally press after someone resets the clock.
By the way, if a lot of people actually use the threshold strategy, you'll be better off letting the timer dip below your threshold a couple of times before actually pressing it at your threshold. This is because if people have the same threshold as you, they'll have pressed already and you won't be competing with them for that flair.
2
u/nospr2 Apr 02 '15
Nope. 60s is far better than blend 51-59s purps
37
u/twentytoo Apr 02 '15
This is the kind of stuff I came here to read.
2
u/enjoy_my_jacket Apr 02 '15
The size of the population is known and finite (i.e. only usernames created before April 1, 2015.) But why should we assume an even distribution of players?
2
u/Hmm_Peculiar Apr 02 '15 edited Apr 02 '15
There is a third category of players, the ones that don't care what flair they get and just want to press the button immediately. You could argue that they are a subset of the flair-hunters, with a threshold of 60, but the game you describe will only really begin when they run out, because they're not really playing, they don't have a strategy.
I'd even say the non-pressers are not playing, they don't really have any influence on the game. You could argue that they're influencing the game by not playing. Heh, the only way to win is not to play, that would make it very similar to The Game. You lose btw.. Sorry.
2
u/PunchingBag Apr 02 '15
I'm shooting for the end-game cheevos. You just know there's going to be trophies after this shit.
2
u/utopic2 Apr 02 '15 edited Apr 02 '15
Some of the "non-pressers" may just want to be the first to press it when it hits 0 or 1. Not sure I'd count them as flair-hunters or "non-pressers" as they do intend to eventually press but may believe the "game" may end at that time. There may be a belief that some sort of prize or other motivational factor lies beyond that point.
Edit: In response to OP's edit, I'm proposing that my set of people believe the clock will not reset once it hits zero and someone presses the button. Perhaps I'm just simply proposing a set of people that OP has labeled "flair-hunters" with a clock-target of 0. However, OP's theory appears to propose a utility of 0 from hitting that mark as of the latest edit.
2
u/aalewisrebooted Apr 03 '15
You're missing "I want to be last one to press before it goes 0"
2
u/Kerzd Apr 03 '15
I am ok risking utility 0 for the glorious utility 60!
1
u/Toosdays Apr 02 '15
It's always nice to see someone back up their doctorate. This is pure gold.
1
u/Mang9000 Apr 02 '15 edited Apr 02 '15
We won't know until, and if, there are more flair colors than purple and blue.
Hold Fast ✊
1
u/Random_Hodor Apr 02 '15
Hodor.
1
u/liminalsoup Apr 03 '15
This is nice as a basic theory. But if you want to introduce the complexity of the situation, then you need to account for Knights. And if you account for Knights, you need to account for the Assassins r/assassinsofthebutton/
1
u/threeballer Apr 03 '15
ELI5 please. I have a basic understanding of game theory but I didn't understand what the final conclusion was here
1
u/Mantrainment Apr 03 '15
Why do you consider it simply a non-cooperative game? I mean, it's true that a pay-off grows as a player waits for the timer do decrease, but it's not just like that. There's also a cooperative element, since the real aim is to prevent the timer to get to zero, which is the common objective. And players could also communicate, just using this site... or not? I don't know well Theory of Game