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The Collaborator's Dilemma

Started by droqen, February 22, 2023, 02:03:38 PM

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The Prisoner's Dilemma is a well-known game theory 'game':

Two prisoners (players) are each given the option to stay silent, or snitch on the other.
If they both stay silent, they both get 1 year in prison.
If they both snitch, they both get 5 years.
In one snitches and the other stays silent, the snitch gets off scot-free and the loyal silent one gets all 10 years.

I've been thinking of simple alternative 'payoff square'; it started while observing cars at an intersection:

Two cars (players) are each given the option to accelerate through the intersection, or slow down and wait for the other.
If they both wait, they both lose a few seconds of time. (and then they have to play another round.)
If they both accelerate, they crash and both lose hours of time to insurance, repairs, costs, and whatnot.
If one accelerates and the other slows, the fast car loses nothing, and the slower car loses a few seconds of time.

What does this look like in an environment where two (or more) collaborators have to come to an agreement together?

Two collaborators (players) are each given the option to agree with the other and support their idea, or commit to proposing their own idea.
If they both agree, they both lose some time because no idea was proposed for them to agree with. (And play again.)
If they both propose their own idea, they both lose some time because there was no consensus. (And play again.)
If one proposes an idea and the other agrees, they have come to a decision.

Obviously this is a really stripped down and silly version of how agreements are formed, but so are the prisoner's dilemma and the cars-at-an-intersection dilemma stripped down and silly versions of those real-world examples. The interesting thing here is that it describes a dynamic which gives rise to hierarchies: I've understood on an intuitive level for some time that it's useful for someone to take responsibility for a thing, to take charge, but this game makes it more clear to me exactly why that is.


Aside: It's easier for smaller groups with no explicit leader (or, let us say, shot-caller) because the probability shakes out better: playing this game with two players, there's a 50% chance in every randomly played round that we will come to a consensus. With three players, a 33% chance. (Or: 2 players have a 1/2 chance, 3 players have a 1/3 chance, etc.)
Also, with smaller groups, the rounds go faster, but that's not an interesting new observation: everybody knows that already.

There's no winning strategy for this game at the level of individual players. If everyone does the same thing as everyone else, the winning chance cannot get any better than those probabilities given above. The only way to develop a better winning strategy (and one is clearly available) is through communication -- and asymmetry.


P.S. Any kind of hierarchy/leadership/whatever solution to this game only works if the group maintains a good strategic consensus - and I'm not advocating for any kind of dictatorship here, you can have a 100% success rate in a group where the 'shot-caller' is constantly changing as long as everyone agrees who it is for any given decision!

It's as important for that one person to propose an idea every time as it is for everyone else to agree every time; this is something that I didn't quite get, I found it easier to assume that agreeing was a net good. It is only a net good to be 100% agreeable if there is someone else who is proposing an idea (i.e. "taking responsibility/initiative").

P.P.S. Some factors noticeably not covered in this simple game:
 * In a group of 10 people you definitely prefer the best of 10 ideas rather than 1 random idea from 1 random person. For simplicity's sake I am not considering idea quality, only group consensus.
 * I am only considering group utility - I think this is a good way to think about things, but there is a Prisoner's Dilemma-like factor to take into account when individuals value having their idea chosen rather than someone else's.


Also I need to test my math at some point, is the best success rate for 10 disorganized players really 10%?
Is there a better individual strategy for such a group than 'choose to lead 10% of the time, choose to agree 90% of the time'?


P.P.P.S. A great low-key example that I didn't fit into the above description is the iconic dilemma between a bunch of people deciding what to get together for lunch. It's best when exactly one person takes initiative and proposes an idea, and everyone else agrees to it.