Post by ForgotToFlush on Dec 9, 2023 1:59:07 GMT
Navigating the Prisoner's Dilemma: Lessons from Tropicana 17
By ForgotToFlush
The Prisoner's Dilemma explained
The Prisoner's Dilemma is a very famous thought experiment, with some niche applications for the way we approach mafia games. In case you're unfamiliar with it, allow me to summarize:
Imagine there are two partner criminals, kept separately imprisoned, to be interrogated by the authorities. Each of the prisoners is presented with the same choice by their captors: give us information on your partner and you'll receive a lesser sentence, or, don't talk and we'll sentence you based on the information already available.
With each prisoner needing to make the decision, the scenarios can play out as follows:
1) Neither of them give up the other, leading to both of them to receive minimum sentences due to the lack of evidence.
2) One of them gives up the other, leading to the giver receiving a minimum sentence, while the givee receives a maximum sentence.
3) Both of them give up each other, and now the authorities have enough information to make sure they both receive maximum sentences. Nobody said the authorities were playing fair after all.
Here's how these scenarios play out in diagram form:
So, we can see there's a path for both prisoners to benefit, and for both prisoners to lose out, and that by being too aware of one or the other, they may be tempted to act in a way that is compromising to one or both of them. Hence the dilemma. But you may be wondering....
What does this have to do with Mafia?
Well, pretend it's a multiball game, with 1 scum player, versus another unaligned scum player, and then an amount of town where the exact number isn't too relevant. Pretend as well that a given scum player, has a good idea who the other scum is, and vice versa. Those unaligned scum players are our prisoners, and instead of giving up information, they're choosing whether to kill town, or to kill the other scum player. Now are options look like this:
1) Both scum players decide to kill town, and therefore they are able to advance their win conditions by two more town slots with that night phase.
2) One scum player decides to kill the other, whom kills a townie. They get closer to their win condition by one town slot, and they remove the scum player that would have impeded their win condition. Note that this isn't directly analogous to the prisoner's dilemma, and it isn't a unilateral benefit. While this erases their chances of being night-killed, they shrink the POE for the rest of townies, and they also remove the ability to have their win-condition advanced by the other scum player, who could have killed more townies.
3) The scum players kill each other. Townies go home happy.
So we see that there are some similarities, some differences, but overall the pattern of thought is clear. There are opportunities for cooperation and mutually assured destruction alike, and the logic that could lead you to or away from those outcomes is largely the same as the original Prisoner's Dilemma.
Until........
Tropicana 17: Expanding the Prisoner's Dilemma
Firstly, there's nothing specific to the Tropicana 17 setup that prompted this discovery, it just happened to be the game where it was noticed. As for what that discovery is, the astute among you may have noticed that Scum aren't exactly locked into the two options I outlined for them in the last section:
In Tropicana 17 the game went to a 2v1v1, and it looked like the standard Prisoner's Dilemma was going to play out. But litteleven, one of the scum players, changed the equation by adding a new element: idling his own kill. While we didn't get to see this play out in game, the implications are certainly exciting. What happens when players add idling to their list of options? Well we have the 3 standard scenarios of the Prisoner's Dilemma still available, but now we add:
4) Player A idles, Player B kills Player A. Player A is out of the game, but Player B doesn't benefit, other than removing their chances of being night-killed. Player B is still in the POE and they aren't necessarily closer to their win condition.
5) Player A idles, Player B kills a townie. Both players advance closer to their win conditions, but the idle from Player A means they're doing it less efficiently than in Scenario 1.
6) Both players idle. Neither player advances their win-condition. Both players are still in the POE.
Here's how the diagram looks now:
So going over the new scenarios, it's clear that both scum players idling is bad for both of them, despite both of them living through the night, as neither has been able to advance their win-condition.
When a player kills an idling scum, it's definitely a negative outcome for the idler, but the idle action means that the outcome for their killer isn't necessarily a strict positive, hence the question mark. Take the Tropicana situation for example. If litt dies while idling, the remaining scum is stuck in a 2v1. It's definitely better than a situation where they kill each other, but it isn't necessarily a winning position for the remaining scum.
When one player kills a townie while the other idles, it can be seen that both players are better off. However, this isn't necessarily a better situation than if they had both killed townies.
Takeaways
PS! plays a lot of multiball games, so I'm sure many of you have come across situations where the Prisoner's Dilemma was in effect, even if you didn't have explicit knowledge of what it is. You probably also had pre-conceived notions of the logical actions to take when it is in effect, so I'm curious to know, has idling ever been a consideration for you? How does it change the math for you?
We can see that the person idling seems to end up on the negative end of things more often from a pure numbers perspective. We can also see that there's strength in leveraging it as a spite play. Yes you may have killed me, but now you're in a situation where you aren't actually that much closer to your win-condition. Therefore, idling in this situation may be strong if you don't have a clear idea of who the last scum is. You stop yourself from misfiring on a townie and handing the game to the other scum this way.
For me, the strength of the Prisoner's Dilemma is in the ability to leverage the other scum's kill to advance your own win-condition. As long as you both keep killing town, you're both fine right? Well, if you're not playing for a draw, one of you has to die eventually. So I think there's further strategy in determining when you kill the other scum. Too soon means you don't get them clearing the player-list for you. Too late and, well they probably got you before you could get them. Where's the balance? Feel free to let me know how greedy you can get with this.
By ForgotToFlush
The Prisoner's Dilemma explained
The Prisoner's Dilemma is a very famous thought experiment, with some niche applications for the way we approach mafia games. In case you're unfamiliar with it, allow me to summarize:
Imagine there are two partner criminals, kept separately imprisoned, to be interrogated by the authorities. Each of the prisoners is presented with the same choice by their captors: give us information on your partner and you'll receive a lesser sentence, or, don't talk and we'll sentence you based on the information already available.
With each prisoner needing to make the decision, the scenarios can play out as follows:
1) Neither of them give up the other, leading to both of them to receive minimum sentences due to the lack of evidence.
2) One of them gives up the other, leading to the giver receiving a minimum sentence, while the givee receives a maximum sentence.
3) Both of them give up each other, and now the authorities have enough information to make sure they both receive maximum sentences. Nobody said the authorities were playing fair after all.
Here's how these scenarios play out in diagram form:
So, we can see there's a path for both prisoners to benefit, and for both prisoners to lose out, and that by being too aware of one or the other, they may be tempted to act in a way that is compromising to one or both of them. Hence the dilemma. But you may be wondering....
What does this have to do with Mafia?
Well, pretend it's a multiball game, with 1 scum player, versus another unaligned scum player, and then an amount of town where the exact number isn't too relevant. Pretend as well that a given scum player, has a good idea who the other scum is, and vice versa. Those unaligned scum players are our prisoners, and instead of giving up information, they're choosing whether to kill town, or to kill the other scum player. Now are options look like this:
1) Both scum players decide to kill town, and therefore they are able to advance their win conditions by two more town slots with that night phase.
2) One scum player decides to kill the other, whom kills a townie. They get closer to their win condition by one town slot, and they remove the scum player that would have impeded their win condition. Note that this isn't directly analogous to the prisoner's dilemma, and it isn't a unilateral benefit. While this erases their chances of being night-killed, they shrink the POE for the rest of townies, and they also remove the ability to have their win-condition advanced by the other scum player, who could have killed more townies.
3) The scum players kill each other. Townies go home happy.
So we see that there are some similarities, some differences, but overall the pattern of thought is clear. There are opportunities for cooperation and mutually assured destruction alike, and the logic that could lead you to or away from those outcomes is largely the same as the original Prisoner's Dilemma.
Until........
Tropicana 17: Expanding the Prisoner's Dilemma
Firstly, there's nothing specific to the Tropicana 17 setup that prompted this discovery, it just happened to be the game where it was noticed. As for what that discovery is, the astute among you may have noticed that Scum aren't exactly locked into the two options I outlined for them in the last section:
In Tropicana 17 the game went to a 2v1v1, and it looked like the standard Prisoner's Dilemma was going to play out. But litteleven, one of the scum players, changed the equation by adding a new element: idling his own kill. While we didn't get to see this play out in game, the implications are certainly exciting. What happens when players add idling to their list of options? Well we have the 3 standard scenarios of the Prisoner's Dilemma still available, but now we add:
4) Player A idles, Player B kills Player A. Player A is out of the game, but Player B doesn't benefit, other than removing their chances of being night-killed. Player B is still in the POE and they aren't necessarily closer to their win condition.
5) Player A idles, Player B kills a townie. Both players advance closer to their win conditions, but the idle from Player A means they're doing it less efficiently than in Scenario 1.
6) Both players idle. Neither player advances their win-condition. Both players are still in the POE.
Here's how the diagram looks now:
So going over the new scenarios, it's clear that both scum players idling is bad for both of them, despite both of them living through the night, as neither has been able to advance their win-condition.
When a player kills an idling scum, it's definitely a negative outcome for the idler, but the idle action means that the outcome for their killer isn't necessarily a strict positive, hence the question mark. Take the Tropicana situation for example. If litt dies while idling, the remaining scum is stuck in a 2v1. It's definitely better than a situation where they kill each other, but it isn't necessarily a winning position for the remaining scum.
When one player kills a townie while the other idles, it can be seen that both players are better off. However, this isn't necessarily a better situation than if they had both killed townies.
Takeaways
PS! plays a lot of multiball games, so I'm sure many of you have come across situations where the Prisoner's Dilemma was in effect, even if you didn't have explicit knowledge of what it is. You probably also had pre-conceived notions of the logical actions to take when it is in effect, so I'm curious to know, has idling ever been a consideration for you? How does it change the math for you?
We can see that the person idling seems to end up on the negative end of things more often from a pure numbers perspective. We can also see that there's strength in leveraging it as a spite play. Yes you may have killed me, but now you're in a situation where you aren't actually that much closer to your win-condition. Therefore, idling in this situation may be strong if you don't have a clear idea of who the last scum is. You stop yourself from misfiring on a townie and handing the game to the other scum this way.
For me, the strength of the Prisoner's Dilemma is in the ability to leverage the other scum's kill to advance your own win-condition. As long as you both keep killing town, you're both fine right? Well, if you're not playing for a draw, one of you has to die eventually. So I think there's further strategy in determining when you kill the other scum. Too soon means you don't get them clearing the player-list for you. Too late and, well they probably got you before you could get them. Where's the balance? Feel free to let me know how greedy you can get with this.