One interesting strategy that does not achieve the Pareto boundary:
Defect with a higher probability if the opponent gives you a worse deal. This way, you at least have some probability of cooperation if both agents have ideas of fairness skewed away from each other, but you limit (and can completely remove) the incentive to be unfair.
For example, if you think (12, 12) is fair, and they think (11, 13) is fair, then you can offer to accept their (11, 13) with %80 probability. Their expected utility is 0.8x13 + 0.2x8 = 12. This is the same for them as if they agree with you, so there’s no incentive for them to skew their idea of fairness. The expected payoff ends up being (10.4, 12). It’s not as good as (12, 12) or (11, 13), but at least it’s better than (8, 8).
Furthermore, if they also use this strategy, you will end up deciding on something somewhere between (12, 12) and (11, 13) with a higher probability. I think the expected payoff matrix will end up being (11, 12).
Edit:
I came up with a modification to put it in the Pareto boundary.
Introduce a third agent. Let’s call the agents Alice, Bob, and Charlie.
If Alice and Bob disagree on what’s fair, Bob gets what Alice thinks is fair for him to have, Alice gets what she thinks it’s fair for Bob to have, and Charlie gets as much as possible while Alice and Bob get that much. Similarly for when Bob and Charlie or Charlie and Alice disagree. Since joining together like this means that they’ll get value that would otherwise be wasted if it was just the other two, there’s incentive to join.
If it’s possible, but difficult, for one to bribe another without being detected by the third, this can be fixed by making it so they get just enough less to make up for it.
If it’s not difficult, you could increase the number of agents so that bribery would be unfeasible.
If there’s ever a deal that Alice, Bob, and Charlie are involved in, then you’d have to introduce someone else to get it to work. Ultimately, the idea fails if everyone has to make a deal together.
One interesting strategy that does not achieve the Pareto boundary:
Defect with a higher probability if the opponent gives you a worse deal. This way, you at least have some probability of cooperation if both agents have ideas of fairness skewed away from each other, but you limit (and can completely remove) the incentive to be unfair.
For example, if you think (12, 12) is fair, and they think (11, 13) is fair, then you can offer to accept their (11, 13) with %80 probability. Their expected utility is 0.8x13 + 0.2x8 = 12. This is the same for them as if they agree with you, so there’s no incentive for them to skew their idea of fairness. The expected payoff ends up being (10.4, 12). It’s not as good as (12, 12) or (11, 13), but at least it’s better than (8, 8).
Furthermore, if they also use this strategy, you will end up deciding on something somewhere between (12, 12) and (11, 13) with a higher probability. I think the expected payoff matrix will end up being (11, 12).
Edit:
I came up with a modification to put it in the Pareto boundary.
Introduce a third agent. Let’s call the agents Alice, Bob, and Charlie.
If Alice and Bob disagree on what’s fair, Bob gets what Alice thinks is fair for him to have, Alice gets what she thinks it’s fair for Bob to have, and Charlie gets as much as possible while Alice and Bob get that much. Similarly for when Bob and Charlie or Charlie and Alice disagree. Since joining together like this means that they’ll get value that would otherwise be wasted if it was just the other two, there’s incentive to join.
If it’s possible, but difficult, for one to bribe another without being detected by the third, this can be fixed by making it so they get just enough less to make up for it.
If it’s not difficult, you could increase the number of agents so that bribery would be unfeasible.
If there’s ever a deal that Alice, Bob, and Charlie are involved in, then you’d have to introduce someone else to get it to work. Ultimately, the idea fails if everyone has to make a deal together.
“Exploitable” because your opponent gets the ‘fair’ Pareto outcome, you do worse, and they don’t do worse.
They have no advantage doing so.
You can also make it so that they get a little less than what you consider fair.