Luck and Skill in Unequal Choice Games

Imagine you and I are playing Bruno Faidutti’s Citadels.


I am the assassin and can murder a character of my choice. You, knowing I have the assassin and will try to murder you, must choose between 2 characters: one that will award a symbolic value of ‘1’ if not murdered (say, the thief and I have no money), and one that will award the much stronger symbolic value of ‘3’ (say, the warlord and you have several red districts). I will not know which one you chose when I announce my target, and your symbolic value will be ‘0’ if I murder you. Which character should you choose?

According to game theory, an equilibrium strategy would involve randomly choose one using a probability model of 75% for 3 and 25% for 1, assuming no prior data about a specific opponent. Equilibrium for the assassin would involve randomly choose murder using a probability model of 75% for 3 and 25% for 1, assuming no prior data about a specific opponent.

I believe it is fair to say that if two computers, using these strategies, played a game out like this, the end result would essentially be decided by chance (after compensating for first player advantage).

There are those that have argued the counter-intuitive point that when people make these sorts of choices it is also decided by chance. The argument goes something like this: while both players make their choice for particular reasons, they have no substantive grounds for believing that their opponent will choose one character over the other. This is justified since actual gamers don’t generally decide their moves based on probability rolls or equilibrium strategies. Rather, both players are making decisions based on phantom data (the presumed move of their opponent), and because of this, their own decisions are effectively random.


I am going to attempt to build a case against this argument.

My first piece of evidence is that some players are measurably better and worse at these sorts of challenges. The worse assassin and non assassin players tend to choose each of the options roughly half of the time, while the better assassin and non assassin players will favour choosing the ‘3’ much more often than the ‘1′  (closer to the equilibrium model). The weaker players likely do this because they are primed to consider all options equally, even when results are wildly unequal, merely by the availability of the choices. Bomber is a simple game that quickly reveals the folly of this sort of thinking.

In Bomber, each turn players alternate being ‘bomber’ or ‘diffuser’ and reveal their moves simultaneously. The bomber can choose to immediately bomb by revealing 10 fingers, or reduce their timer bomb by 1 finger (timer starts at 9 fingers). If the bomber sets off a bomb, they win. The diffuser can either diffuse an immediate bomb by revealing 10 fingers, or stall their opponent’s timer by revealing 0 fingers.


An immediate bomb pretty much always wins the game because of errors on behalf of the diffuser, who should diffuse 8 times out of 9 but usually doesn’t. Experienced players learn to nearly always diffuse, while new players tend to try and stall their opponent’s timers far too often and lose because of it.


Once two players have established a reputation for equilibrium proportions in their choice frequencies, metagaming becomes a crucial and meaningful factor. A player well known for assassinating the stronger character, for instance, may occasionally deviate from the optimal strategy and murder the weaker character more often than would be ideal from a computing standpoint if the other player has become accustomed to selecting the weaker character. Meta-gaming works well against humans when it doesn’t against computers precisely because humans are unable to make random choices, and instead use patterns of behaviour to determine their best course of action. Establishing a pattern in order to break it is a meaningful and effective human vs human strategy.

There are only two instances in which I do not doubt that luck plays a deciding role. One is with two new players that continually choose both characters with equal frequency. The other is with two experienced players that fail to deviate from the equilibrium strategy described in game theory in order to gain small advantages. However, the number of games that fall into either of these camps is relatively small.

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