## LESSWRONGLW

3 min read15th Oct 20191 comment

# 15

## 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.