# Always Discard Fascist Policies

## Burden of proof

I enjoy the game Secret Hitler. If you're unfamilliar with the game, the rules
can be found here: https://secrethitler.com/assets/Secret_Hitler_Rules.pdf
In this game, like in many social deception games, players love to make claims
about what you should _always_ or _never_ do when playing. In general, I reject
such claims since most pure strategies in hidden information games are
exploitable. The true mixed-strategy Nash equilibrium for Secret Hitler
inevitably will involve non-zero and non-100% probabilities for most actions.

Some examples of bad 100% policies that I've heard:

* As president, if you pass two fascist policies to your chancellor, always
  claim you drew three fascist policies rather than going head-to-head against
  your chancellor.
* Always assume presidents follow the above bullet point (and thus suspect only
  the chancellor when two players go head-to-head).
* As Hitler, always play policies identically to a liberal unless it would cause
  the liberals to win the game _immediately_.
* As a Fascist, if given the chance to view Hitler's party affiliation, always
  claim Hitler is liberal.

So with this in mind, I recognize that if I make the claim "as a liberal, Always
Discard Fascist Policies", I'm placing a heavy burden of proof on myself.
Nonetheless, I intend to prove that this is indeed _always_ the optimal
strategy.

## Assumption

However I need to make one assumption. I've never played with a group who
rejected this assumption. But I'd be curious to hear if readers feel
differently. The assumption is that *liberal players always immediately and
truthfully reveal all information they have available to them*.

## The Setup:

Imagine Alice and Bob are playing Secret Hitler. The group has approved
Alice as president and Bob as chancellor. Alice is a liberal and does not
concretely know Bob's party affiliation. Alice draws three tiles. Two are
liberal policies and one is fascist. Alice has the option to either:

* Discard the fascist tile and pass off the two liberal tiles to Bob, denying
him any choice in what he does.

OR

* Discard one of the liberal tiles and allow Bob to select whether to play a
liberal or fascist tile.

## Proof

Since liberals always honestly reveal all information available to them, we know
that Bob knows everything Alice claims to know. Therefore Bob's information is a
superset of Alice's (or any liberal's) information.

Now we'll sample a random real number `p` in the range `[0,1]` which acts as
Bob's source of randomness in case his optimal strategy is probabilistic.

We have two cases:
1. Based on Bob's information and `p`; the optimal Fascist strategy if Bob is
a Fascist is to play a Liberal policy and avoid being found out.

2. Based on Bob's information and `p`; the optimal Fascist strategy if Bob is
a Fascist is to (if possible) play a Fascist policy and go head-to-head against
Alice.

In the first case, it doesn't matter what Alice does. Whether she hands Bob
two liberal policies or a fascist and a liberal policy, Bob will play a liberal
policy and the group gains no information about Bob (since _anyone_ fascist or
liberal would play a liberal policy in this case).

In the _second_ case, Bob will play a fascist policy. However, by our assumption
that we are in case 2, this is strictly worse for liberals (from Bob's
perspective as a fascist and therefore from the liberals' perspective as well
since their information is a subset of his) than if he had instead been forced
to play a liberal policy.

## Technical Flaws
After writing this, I realized there are two technical flaws in the above proof
that I don't believe make a difference in practice. I'm curious how relevant
readers think these are:

1. Alice _does_ have one piece of information which Bob doesn't have. She knows
that she drew two liberal policies and one fascist policy as opposed to two
fascist and one liberal policy. If Bob's optimal strategy for some reason hinges
on what Alice discarded and Alice knows this, this might be a reason for her to
try to briefly fool him into thinking she discarded a fascist policy (the rules
expressly forbid talking while a policy is being enacted). I don't see how this
might be the case but I also can't prove that it never is.

2. (Only applies to games with at least 7 players; in <=6-player games, fascists strictly have at least as much information as liberals)
Knowing what someone claims is not the same as knowing what they know. Alice
knows that she is a liberal whereas Bob only knows that Alice claims to be a
liberal.
To see how this concretely could make a difference, imagine this is a
game with >=7 players and Alice suspects Bob is Hitler and has managed to subtly
convince Bob that she's a fellow fascist; Bob may play the fascist policy in
hopes that Alice will back him up and claim to have drawn three fascist
policies. I have not seen anything like this happen in practice and it certainly
is not the case most of the time I get into arguments with people about what to
discard in the 2L1F case. IME This argument comes up in <=6-player games as
often as it comes up in larger games.

## In Practice

Now I recognize that in practice, most Secret Hitler players are not perfectly
able to determine optimal strategy and therefore one might claim Alice is
giving Bob the option with the hope that he'll "make a mistake" and reveal
himself as Fascist when that was an inferior move. This could be the case if,
for instance, Alice is a more experienced player than Bob. However, IME newer
players almost universally tend to err the other way. That is, they hide and
act as liberals even when the proper strategy calls for them to play a Fascist
policy and risk outing themselves.

Furthermore, I've found that most players attach too much weight to the
decisions the chancellor makes when given a choice. In other words they see
a liberal policy come out and decide to trust the chancellor as a liberal even
when that clearly would have been an optimal fascist strategy as well. This
seems to be an instance of over-reliance on the representativeness heuristic.
One way to avoid falling prey to this mistake is simply not to give them the
choice in the first place.