Unilateral precommitment lets people win at “Almost Free Lunches”.
One way to model precommitment is as a sequential game: first player 1 chooses a number, then player 1 has the option of either showing that number to player 2 or keeping it hidden, then player 2 chooses a number. Optimal play is for player 1 to pick £1,000,000 and show that number, and then for player 2 to choose £999,999.99.
An interesting feature of this is that player 1′s precommitment helped player 2 even more than it helped player 1. Player 1 is “taking one for the team”, but still winning big. This distinguishes it from games like chicken, where precommitment is a threat that allows the precommitter to win the larger share. Though this means that if either player can precommit (rather than one being pre-assigned to go first as player 1) then they’d both prefer to have the other one be the precommitter.
This benefit of precommitment does not extend to the two option version (n2 vs. n1). In that version, player 2 is incentivized to say “n1” regardless of what player 1 commits to, so unilateral precommitment doesn’t help them avoid the Nash Equilibrium. As in the prisoner’s dilemma.
Is there a name for this type of equilibrium, where a player can pre-commit in a way where the best response leaves the first player very well-off, but not quite optimally well-off? What about if it is a mixed strategy (e.g. consider the version of this game where the player who gave the larger number gets paid nothing).
I think another key difference between PD and traveller/AFL is that in the PD variant, (n2, n1) is a Pareto outcome—you can’t improve the first player’s outcome without making the second one worse off. However, in the other problem, (0,0) is very very far from being Pareto.
Unilateral precommitment lets people win at “Almost Free Lunches”.
One way to model precommitment is as a sequential game: first player 1 chooses a number, then player 1 has the option of either showing that number to player 2 or keeping it hidden, then player 2 chooses a number. Optimal play is for player 1 to pick £1,000,000 and show that number, and then for player 2 to choose £999,999.99.
An interesting feature of this is that player 1′s precommitment helped player 2 even more than it helped player 1. Player 1 is “taking one for the team”, but still winning big. This distinguishes it from games like chicken, where precommitment is a threat that allows the precommitter to win the larger share. Though this means that if either player can precommit (rather than one being pre-assigned to go first as player 1) then they’d both prefer to have the other one be the precommitter.
This benefit of precommitment does not extend to the two option version (n2 vs. n1). In that version, player 2 is incentivized to say “n1” regardless of what player 1 commits to, so unilateral precommitment doesn’t help them avoid the Nash Equilibrium. As in the prisoner’s dilemma.
Is there a name for this type of equilibrium, where a player can pre-commit in a way where the best response leaves the first player very well-off, but not quite optimally well-off? What about if it is a mixed strategy (e.g. consider the version of this game where the player who gave the larger number gets paid nothing).
I think another key difference between PD and traveller/AFL is that in the PD variant, (n2, n1) is a Pareto outcome—you can’t improve the first player’s outcome without making the second one worse off. However, in the other problem, (0,0) is very very far from being Pareto.