Only a third of the way through so far, but confused – you cite “iteration and enforcement mechanisms” as things that make PDs more like Stag Hunts. But, isn’t iteration and enforcement a property that either PD or SH can have? My understanding of the difference between PD and Stag Hunt was about what the actual payoffs are, and that C/C can be a nash equilibrium because you wouldn’t get more payoff by defecting (although D/D is also a nash equilibrium)
In game theory, iterated PD would be a different game than PD. PD as typically defined is a single-shot game. The same is true of stag hunt, battle of the sexes, and many other games: if I say “stag hunt” to a game theorist, they probably don’t ask “single shot or iterated?”. Rather, if I say “stag hunt” and then start talking about iterated strategies, they might be like “oh, you mean iterated stag hunt?”
Similarly with enforcement mechanisms and so on. None of these are assumed by default.
In (single-shot) PD, the “possible strategies” are the moves you can make (or mixtures of moves, if you have randomness available). In iterated PD, however, the strategy space is much more complex: it’s the set of possible iterated strategies, including any possible function of the game history. This gives us a correspondingly much more complex set of equilibria to consider.
It is characteristic of PD that players are incentivised to play away from the Pareto frontier; IE, no Pareto-optimal point is an equilibrium. This is not the case with iterated PD.
It is characteristic of Stag Hunt that there is a Pareto-optimal equilibrium, but there is also another equilibrium which is far from optimal. This is also the case with iterated PD.
Hence my assertion that iterated PD is more like stag hunt.
However, it is furthermore true of iterated PD that there are multiple different Pareto-optimal equilibria, which benefit different players more or less. Also, if players don’t successfully coordinate on one of these equilibria, they can end up in a worse overall state (such as mutual defection forever, due to playing grim-trigger strategies with mutually incompatible demands). This makes iterated PD resemble Battle of the Sexes.
However, it is furthermore true of iterated PD that there are multiple different Pareto-optimal equilibria, which benefit different players more or less. Also, if players don’t successfully coordinate on one of these equilibria, they can end up in a worse overall state (such as mutual defection forever, due to playing grim-trigger strategies with mutually incompatible demands). This makes iterated PD resemble Battle of the Sexes.
I think this paragraph very clearly summarizes your argument. You might consider including it as a TL;DR at the beginning.
Okay. I think I get what you are saying now, but it wasn’t clear on my initial read through.
I did understand on the initial read-through (or, currently think I understand?) that when you say “most games turn out to be Battle of the Sexes in practice”, you mean that there is an emergent property of the iterated game that turns it into Battle of the Sexes.
My current summary of what you are intending to say (correct me if I got it wrong ) is:
1. Most prisoners dilemma games are actually iterated.
2. Iterated prisoners dilemma is actually a different game with a different payoff matrix that has a different set of nash equilibria. Choosing which strategy to play in iterated prisoner’s dilemma is similar to playing Stag Hunt.
3. Then there is a further step where the process of deciding on how to coordinate (meta-strategy?) that you are choosing in a stag hunt is more similar to battle of the sexes.
I think what I was missing the first time through was #2. I was intepreting you to mean “the thing about stag hunts is that they are iterated, and your PD is probably iterated”, where what you actually meant was “your PD is iterated, and iterated PD is actually isomorphic to stag hunt.”
One of the key things I think between the 3 games is whether communication beforehand helps (in single shot games).
In PD communication doesn’t really help much as you there is little reason to trust what the other person.
In SH communication should be able to solve your problem as S-S is optimal for both players.
In BotS communication which results in agreement can at least be trusted as co-ordinating is optimal for both players. Choosing which option to co-ordinate on is another matter.
(assuming you’ve included the pleasure of spiting the other person etc. in the payoff matrix)
Only a third of the way through so far, but confused – you cite “iteration and enforcement mechanisms” as things that make PDs more like Stag Hunts. But, isn’t iteration and enforcement a property that either PD or SH can have? My understanding of the difference between PD and Stag Hunt was about what the actual payoffs are, and that C/C can be a nash equilibrium because you wouldn’t get more payoff by defecting (although D/D is also a nash equilibrium)
In game theory, iterated PD would be a different game than PD. PD as typically defined is a single-shot game. The same is true of stag hunt, battle of the sexes, and many other games: if I say “stag hunt” to a game theorist, they probably don’t ask “single shot or iterated?”. Rather, if I say “stag hunt” and then start talking about iterated strategies, they might be like “oh, you mean iterated stag hunt?”
Similarly with enforcement mechanisms and so on. None of these are assumed by default.
In (single-shot) PD, the “possible strategies” are the moves you can make (or mixtures of moves, if you have randomness available). In iterated PD, however, the strategy space is much more complex: it’s the set of possible iterated strategies, including any possible function of the game history. This gives us a correspondingly much more complex set of equilibria to consider.
It is characteristic of PD that players are incentivised to play away from the Pareto frontier; IE, no Pareto-optimal point is an equilibrium. This is not the case with iterated PD.
It is characteristic of Stag Hunt that there is a Pareto-optimal equilibrium, but there is also another equilibrium which is far from optimal. This is also the case with iterated PD.
Hence my assertion that iterated PD is more like stag hunt.
However, it is furthermore true of iterated PD that there are multiple different Pareto-optimal equilibria, which benefit different players more or less. Also, if players don’t successfully coordinate on one of these equilibria, they can end up in a worse overall state (such as mutual defection forever, due to playing grim-trigger strategies with mutually incompatible demands). This makes iterated PD resemble Battle of the Sexes.
I think this paragraph very clearly summarizes your argument. You might consider including it as a TL;DR at the beginning.
Okay. I think I get what you are saying now, but it wasn’t clear on my initial read through.
I did understand on the initial read-through (or, currently think I understand?) that when you say “most games turn out to be Battle of the Sexes in practice”, you mean that there is an emergent property of the iterated game that turns it into Battle of the Sexes.
My current summary of what you are intending to say (correct me if I got it wrong ) is:
1. Most prisoners dilemma games are actually iterated.
2. Iterated prisoners dilemma is actually a different game with a different payoff matrix that has a different set of nash equilibria. Choosing which strategy to play in iterated prisoner’s dilemma is similar to playing Stag Hunt.
3. Then there is a further step where the process of deciding on how to coordinate (meta-strategy?) that you are choosing in a stag hunt is more similar to battle of the sexes.
I think what I was missing the first time through was #2. I was intepreting you to mean “the thing about stag hunts is that they are iterated, and your PD is probably iterated”, where what you actually meant was “your PD is iterated, and iterated PD is actually isomorphic to stag hunt.”
I had the same confusion.
One of the key things I think between the 3 games is whether communication beforehand helps (in single shot games).
In PD communication doesn’t really help much as you there is little reason to trust what the other person.
In SH communication should be able to solve your problem as S-S is optimal for both players.
In BotS communication which results in agreement can at least be trusted as co-ordinating is optimal for both players. Choosing which option to co-ordinate on is another matter.
(assuming you’ve included the pleasure of spiting the other person etc. in the payoff matrix)
See my reply to Raemon for the aforementioned confusion.