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Subject: Sabotage the first mission? A mathematical answer (5 Player Game, no plot cards) rss

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David Murray
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Introduction
I decided to weigh up "If at least one spy on the first mission, then exactly one sabotage card is played (always play red on the first mission)" against "Always play blue on the first mission".

The quick answer
The spies are better of always playing blue into the first mission, this gives a 3/5 chance of winning, whereas always play (exactly one) red gives a 2/5 chance of winning.

The bad news

Unfortunately, I had to make quite a few assumptions which means the mathematical answer is for a slightly different game, but I still think it applies to real life games.

Assumptions:

The resistance know the spies Strategy:

In the case of "Always play blue" the resistance know that's what the spies will do and in the case of "Always sabotage" the resistance know that's what the spies will do.

For the second mission onwards, spies always sabotage
In missions 2-5, if there is at least one spy on the mission there will be exactly one sabotage. On the 3 player missions I have no doubt spies should be sabotaged, the only debate is mission 3. If the score is 2-0 (either way) then spies should sabotage it. The only argument really is whether spies should sabotage if the score is 1-1, but for simplicity I've decided spies always sabotage in that situation, but I'll probably do the maths for this later.

"The 6th player":
Imagine a that there is a 6th player, he wants the resistance to win, but cannot be on any missions and doesn't know what teams each player is on. He chooses the team for each mission based on previous mission results and will choose the team most likely to be Resistance (actually there is pretty much always a tie, so choose randomly or arbitrarily between the best teams, it doesn't matter as long as a best team is chosen).
In real life resistance a resistance member will (should) always include themselves and a spy will (should) always include at least one spy, but I think if the resistance vote yes for a "best" team and no for anything "not best" then I think it's very similar to the above.

What we end up with is something close to real life, but not the same since it's very hard to take voting into account so it's best to assume everyone is voting and picking teams optimally, in which case I think it's equivalent to the 6th player scenario described above.

Conclusion
Spies always playing blue into M1 is better for the spies then spies always playing red into M1, however I suspect the optimal strategy is for spies to play red a certain % of the time, which is what I plan to investigate next.
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Travis Worthington
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david707 wrote:

For the second mission onwards, spies always sabotage


I think for accuracy you need to test scenarios where spy fails the first mission, but that same spy passes when chosen for the next mission - that should increase the win rate for spies failing the first mission with the contstraints you've chosen.

BUt as you stated in your conclusion the spies are always easier to catch if their behaviour follows a strict pattern. You should build the moel using stochastic techniques for missions 2-4 (mission 5 is of course always a fail).
 
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Christopher M
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The best strategy in this game is to not be predictable. But I still prefer failing the first mission.
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Agent J
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For the record, PBF 8-13, every time a spy was on the first mission. When the mission passed, the spies went on to lose. When the mission failed, the spies went on to win. Different players with different number of players in each.

In PBF14, there was no spy in the first mission. Also, it was a variant, so no useful information can be gleaned.
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David Murray
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If the score is 1-1, then the spies are actually better of always playing blue then always playing red. As someone said above, spies want to sabotage a certain % of the time and this applies to mission 1 and 3. (Spies should always sabotage 3 player missions right?). So I plan to say spies sabotage mission 1 if they are on it with probability A and mission 3 if they are on it and the score is 1-1 with probability B and then try to work out the A and B that give spies best chance of winning.
 
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Agent J
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PBF10 has a spy passing M2, which automatically got him on M4. He failed that one and M5 for the win, having robbed us of the only chance to get good voting results on M3.
 
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David Murray
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T Worthington wrote:
david707 wrote:

For the second mission onwards, spies always sabotage


I think for accuracy you need to test scenarios where spy fails the first mission, but that same spy passes when chosen for the next mission - that should increase the win rate for spies failing the first mission with the contstraints you've chosen.

BUt as you stated in your conclusion the spies are always easier to catch if their behaviour follows a strict pattern. You should build the moel using stochastic techniques for missions 2-4 (mission 5 is of course always a fail).


Stochastic stuff is about 1 year time in my degree, so maybe you want to take it from here? I'm relatively confident I could assign probabilities A and B where A is P(Spies sabotage first mission) and B is P(Spies sabotage 3rd mission if the score is 1-1) and then work out optimal value for A and B, I'm assuming spies should always sabotage missions with 3 people on, but I'm not certain.
 
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David Murray
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Jythier wrote:
PBF10 has a spy passing M2, which automatically got him on M4. He failed that one and M5 for the win, having robbed us of the only chance to get good voting results on M3.


Oh wow, I always thought spies should sabotage 3 player missions.
 
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Clyde W
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1awesomeguy wrote:
The best strategy in this game is to not be predictable. But I still prefer failing the first mission.
I too tend to prefer failing the first mission, unless I've just stated that I as a spy always fail the first mission, in which case sometimes I won't fail it to gain cred.

I especially like to get on the first mission with my spy buddy and coordinate the one fail.

I also enjoy failing the first and passing the second.

I also enjoy passing both the first and the second, and arguing after the third fails that there must've been a botched coordinate on M2.

The great thing is, all of these are entirely valid spy tactics.
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I think you're thinking flawedly, mostly because you're not thinking about how Resistance players really react to things and what kind of information they get from a pass, from a fail, and the information missed in between if there's no reason not to trust a certain team.
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Basically I don't have the mathematical knowledge to take this much further, so I'd be happy to hand it over to T Worthington if he wishes to find %'s for mission 2-4.
 
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david707 wrote:
Jythier wrote:
PBF10 has a spy passing M2, which automatically got him on M4. He failed that one and M5 for the win, having robbed us of the only chance to get good voting results on M3.


Oh wow, I always thought spies should sabotage 3 player missions.
Nah man, especially not in the 5p game.

It's fun to sabotage the first mission then pass the second. Try it sometime, you'll find you easily win, "because a spy would never do that." I've won this way many times.

The other nice thing about that play is that it gets you past the dreaded M3. You get to be on M3, as a spy.

In fact, David, you recently posted about a game when all 5 missions had a spy on them. There was a spy on all 5 missions of PBF10 as well, though not the SAME spy.
 
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David Murray
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clydeiii wrote:
david707 wrote:
Jythier wrote:
PBF10 has a spy passing M2, which automatically got him on M4. He failed that one and M5 for the win, having robbed us of the only chance to get good voting results on M3.


Oh wow, I always thought spies should sabotage 3 player missions.
Nah man, especially not in the 5p game.

It's fun to sabotage the first mission then pass the second. Try it sometime, you'll find you easily win, "because a spy would never do that." I've won this way many times.

The other nice thing about that play is that it gets you past the dreaded M3. You get to be on M3, as a spy.

In fact, David, you recently posted about a game when all 5 missions had a spy on them. There was a spy on all 5 missions of PBF10 as well, though not the SAME spy.


All 5 missions had both spies on them , but the point is the 3 player missions were sabotaged. I'm looking forward to trying the "play blue into a 3 player mission" tactic next time I'm a spy.
 
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Technically there were only 4 missions.
 
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Jythier wrote:
For the record, PBF 8-13, every time a spy was on the first mission. When the mission passed, the spies went on to lose. When the mission failed, the spies went on to win. Different players with different number of players in each.

In PBF14, there was no spy in the first mission. Also, it was a variant, so no useful information can be gleaned.


Was there a spy on the first mission in PBF16?
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Yeah, and can you tell me if there was a spy on M2 in PBF15? Come on Jyth, you know there was...be honest!
 
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david707 wrote:
Basically I don't have the mathematical knowledge to take this much further, so I'd be happy to hand it over to T Worthington if he wishes to find %'s for mission 2-4.


He probably has experimental/statistical data on this. He's the publisher.
 
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1awesomeguy wrote:
david707 wrote:
Basically I don't have the mathematical knowledge to take this much further, so I'd be happy to hand it over to T Worthington if he wishes to find %'s for mission 2-4.


He probably has experimental/statistical data on this. He's the publisher.


I want to know mathematically optimal spy strategy
 
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Christopher M
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david707 wrote:
1awesomeguy wrote:
david707 wrote:
Basically I don't have the mathematical knowledge to take this much further, so I'd be happy to hand it over to T Worthington if he wishes to find %'s for mission 2-4.


He probably has experimental/statistical data on this. He's the publisher.


I want to know mathematically optimal spy strategy


Fail three missions.
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david707 wrote:
Basically I don't have the mathematical knowledge to take this much further, so I'd be happy to hand it over to T Worthington if he wishes to find %'s for mission 2-4.


I wouldn't know how to do the math except by brute forcing it.

But I suspect the mathematically optimal solution would lie between always failing the first mission and never failing, just not sure where.

The model would have to be recursive - giving the observer the % of times that a spy would fail the first mission as a parameter in evaluating the likelihood that each player was a spy.
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My favorite times is being a spy on a two man team in the first round and convincing everyone it was the other guy who sabotaged the mission...

...and then sticking it to them again! angry
So many enemies. Good times.
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Hockey Mask wrote:
My favorite times is being a spy on a two man team in the first round and convincing everyone it was the other guy who sabotaged the mission...

...and then sticking it to them again! angry
So many enemies. Good times.
Strong disagree with your components issues, BUT TOTALLY AGREE WITH THIS. Going up against someone one-on-one (and convincing everyone else you're good) is one of the more fun aspects of this game.

But I also love passing the first two missions as a spy. Both are so enjoyable.
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david707 wrote:
Conclusion
Spies always playing blue into M1 is better for the spies then spies always playing red into M1, however I suspect the optimal strategy is for spies to play red a certain % of the time, which is what I plan to investigate next.


Given that the only reason to play blue as a spy, by your own argument, is that "always play red" is exploitable, this is certainly the case. Here's a handwaving attempt at quantifying how often this frequency is: let's make some simplifying assumptions:

-There is only one spy on the team
-The spy's goal is to avoid giving the Resistance more information than they already have (will bias things toward blue carding more often because failing the mission has nonzero utility for spies)

Let S be the proportion of players that are spies, and Q be P(P|S), the probability that a spy hides if on a team.

Redcarding is always going to give information in this artificial case unless S = 1/K, where K is the number of players on the team. Note that this also suggests that if 1/K < S, hiding is almost certainly a bad play because everyone on the given team is already below average likelihood to be a spy even if it fails (note that in 5-player 1/K for mission 1 is 1/2, while 1/K for mission 3 is 1/3 < 2/5)

So let's look at blue-carding. Bayes' Theorem says that if we don't want a third party to gain information based on a passing mission, we need to choose Q such that:

S = QS/(QS + 1-S)

The problem is that this means Q = 1.* Which implies always bluecarding, which really just demonstrates that this simplified model is oversimplified and not that useful. You would have to somehow quantify the utility of a failed mission to compare to this. Which means that, while not impossible, I don't think it's likely that we'll find a reasonable answer to this question without some sort of simulation.

*(this is true even if you recalculate everything so that a resistance member not on the team is taking his own status as a resistance member into account)
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david707 wrote:
...
Assumptions:

The resistance know the spies Strategy:

In the case of "Always play blue" the resistance know that's what the spies will do and in the case of "Always sabotage" the resistance know that's what the spies will do.
...

I think this is a bad assumption. The purpose of a spy sometimes not sabotaging the first mission is so that resistance will not know the spy strategy. What you may have shown is that given only the choice between always sabotage or never sabotage the first mission, never sabotage is better. However I strongly suspect that a spy should sabotage some percentage (other than 100% or 0%) of the time.

btw: Using red and blue to mean sabotage and not sabotage is very confusing as the mission vote cards are not color coded (at least not in my set). I assume red is sabotage, since the spys have red cards.
 
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The mission cards are color coded, red and blue, for pass and fail, in my set. What edition do you have?
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