Analysis of a Secret Hitler Scenario

Se­cret Hitler is a so­cial de­cep­tion game in the tra­di­tion of mafia, the re­sis­tance and Avalon [1]. You can read the rules here if you aren’t fa­mil­iar. I haven’t played so­cial de­cep­tion games reg­u­larly since 2016 but in my mind it’s a re­ally good game that rep­re­sented the state of the art in the genre at that time. I’m go­ing to dis­cuss an in­ter­est­ing situ­a­tion in which I rea­soned poorly.

I was a liberal in a ten player game. The ini­tial table set up, dis­play­ing rele­vant play­ers was,

We passed a fas­cist ar­ti­cle in the first round. The next gov­ern­ment was Marek as pres­i­dent and Chan­cel­lor a player 3 to the right of Marek [2]. Marek passed a fas­cist ar­ti­cle and in­spected Sam and de­clared Sam was fas­cist to Sam’s coun­ter­claim that Marek was fas­cist. Sam was to the left of Marek so we had no data about him. The table gen­er­ally sup­ported Sam but I leaned to­wards be­liev­ing Marek.

At the be­gin­ning of the game my view of pos­si­bil­ities looked roughly like:

But of course I’m already ap­prox­i­mat­ing. From my per­spec­tive an in­di­vi­d­ual is only 49 to be Fas­cist and the events that two in­di­vi­d­u­als are fas­cist is not in­de­pen­dent. A more care­ful calcu­la­tion would have been:

  • There are pos­si­ble dis­tri­bu­tions of Fas­cists in which of them both Marek and Sam are good for a prob­a­bil­ity of

  • There are ways for Marek to be Fas­cist and Sam to be Liberal. Of course so we get again.

  • The re­main­ing event that both are evil must then be .

So a bet­ter per­spec­tive would have been:

I don’t think I could do this math con­sis­tently in game though so I’ll do the rest of the anal­y­sis with my origi­nal pri­ors. I’ve in­cluded it here for refer­ence in your fu­ture 10 player games of Se­cret Hitler.

When Marek de­clared Sam was Fas­cist the only sce­nario which is con­fi­dently elimi­nated is that both are liberal. Any rea­son­able liberal player is truth pro­mot­ing and has no rea­son to lie. At first glance it seems that the pos­si­bil­ity that both are fas­cist is also elimi­nated as the fas­cists should have no rea­son to fight each other. This doesn’t strictly hold though. The fas­cists only need one fas­cist in a gov­ern­ment to likely sink it and if they rea­son that the liber­als will rea­son that one of them must be good then they have good rea­son to pick fights with each other. But in prac­tice in an ac­cu­sa­tion situ­a­tion the table of­ten opts to pick nei­ther so its a risky move.

Now we get into the ques­tion­able de­duc­tion I made dur­ing the game. If Marek is a fas­cist and he in­spects a liberal he has a choice to make. He can ac­cuse the liberal of be­ing fas­cist to sow dis­trust among the liber­als. Or he can tell the truth to gar­ner trust with the liber­als. If Marek is liberal and he in­spects a fas­cist then he has no choice to make. He will de­clare that the fas­cist is a fas­cist.

In the mo­ment I figured there was about a 50-50 chance that a fas­cist would choose to call a liberal a fas­cist or a liberal. Let’s call fas­cists who lie about liberal’s iden­tities bold and fas­cists who tell the truth timid. I dis­counted the pos­si­bil­ity that both were fas­cist and I rea­soned with prob­a­bil­ities from the first square. So given that Marek had ac­cused Sam that meant that Marek was a liberal with prob­a­bil­ity 75%.

The prob­lem is I didn’t ad­e­quately up­date on Marek’s fas­cist pres­i­dency. In my men­tal model of the game it’s not the im­prob­a­ble to draw 3 fas­cist ar­ti­cles. This men­tal model is de­rived from the fact that it gen­er­ally hap­pens once or twice a game. But its still an un­likely event in the sense that I should con­sider it ev­i­dence that the pres­i­dent was fas­cist. From the re­ports of the first group 3 fas­cist ar­ti­cles had already been buried. Even if I dis­trust them I can still guess at least 2 fas­cist ar­ti­cles were buried. So the prob­a­bil­ity that 3 fas­cist ar­ti­cles were drawn again is at most . Go­ing into the in­ves­ti­ga­tion I had a be­lief that Marek was Fas­cist with prob­a­bil­ity about 50% but I should have already up­dated to ~88% that he was a fas­cist as op­posed to an un­lucky liberal (100% of the time he didn’t draw 3 fas­cist ar­ti­cles he buried a liberal and is a fas­cist and half the rest of the time he’s a fas­cist by my prior). Given that, even with my con­jec­ture that Fas­cists make false ac­cu­sa­tions only half the time I should have guessed Marek was more likely fas­cist than Sam. Marek’s ac­cu­sa­tion demon­strated Marek is not a timid fas­cist which I con­jec­tured to be half the fas­cist prob­a­bil­ity mass. By Bayes’ The­o­rem I should have up­dated to Marek be­ing fas­cist with prob­a­bil­ity:

I definitely com­puted badly in the mo­ment. I think my model build­ing was also bad in a num­ber of other ways which are harder for me to put num­bers on:

  • I thought Marek was un­likely to pick a fight since he seemed rel­a­tively new and he was very quiet af­ter his ac­cu­sa­tion. In my mind peo­ple lie about other peo­ple’s iden­tities pre­pared to fight and Marek seemed sort of timid. The coun­ter­point to this is ly­ing is the ob­vi­ous level one strat­egy for fas­cists. A rea­son­able per­son might think its just what fas­cists are sup­posed to do.

  • I didn’t fac­tor in the prob­a­bil­ity that both were fas­cists at all.

  • All the play­ers seemed to jump on Marek. I think part of the rea­son I defended him was a con­trar­ian bias. But be­ing con­trar­ian against too big a con­sen­sus in Se­cret Hitler is im­por­tant. If ev­ery­one agrees about some­thing then some fas­cists are agree­ing.

  • On the meta level I was very sleepy and new my­self to not be rea­son­ing that well or re­mem­ber­ing ba­sic facts that ac­cu­rately. I prob­a­bly should have deferred to the group.

Thanks for read­ing and let me know if you have any other thoughts about the po­si­tion.

  1. The de­vel­op­ment mafia → re­sis­tance → avalon → Se­cret Hitler rep­re­sents sub­stan­tial progress in board gam­ing tech­nol­ogy. There’s also a lot of amaz­ing ad­ja­cent games like Two Rooms and a Boom and One Night Ul­ti­mate Were­wolf. I’m thank­ful to live in a time of ex­traor­di­nary board game tech­nolog­i­cal progress. ↩︎

  2. Our meta was such that the chan­cel­lor­ship ro­tated coun­ter­clock­wise as the pres­i­dency ro­tated clock­wise to see the max­i­mum num­ber of play­ers. In later games our meta up­dated to make the chan­cel­lor 3 to the left of the pres­i­dent and nein­ing to them if we got a fa­vor­able re­sult which seems to be a very pow­er­ful strat­egy for the liber­als. ↩︎