# Is there a commitment mechanism which allows players to achieve pareto optimal solutions?

For any non-cooperative game, is there some commitment mechanism which will can guarantee that players chose a pareto-optimal solution?

Or, are there known conditions for this to obtain?

EDIT: I elaborate what I mean by "commitment mechanism" below: Informally, I mean something like a "contract" that players can sign and are guaranteed to follow in the future.
So there is an additional round at the start of the game where each player, independently, decides to sign this contract (or not).

For instance, coordination in the prisoner's dilemma can be achieved by offering each player the option of signing a contract which states: "If both players sign this contract, then I will cooperate".

• Please clarify what you mean by "commitment mechanism". If player can commit to a PO solution they can choose it but it is not clear within the context of your question what it means to be able to commit. – Giskard Sep 20 '16 at 6:25
• Now I really do not know what your question is. As long as you allow for such automatically enforced contract I see no reason why the contract "If both players sign this contract, then I will cooperate" is not applicable in all games. (Where "cooperate" means playing the strategy needed to reach the selected PO outcome.) – Giskard Sep 20 '16 at 19:08
• I'm looking for a proof, though ;). At a high level, it might not always work because any given PO might not be sufficiently good by all players' standards. Battle of the sexes seems like it would require the players to agree to act based on the output of a commonly observed coin flip, which is not something I've granted, so far, in my description. – capybaralet Sep 20 '16 at 19:28
• If you want a "proof" you need to clarify how the "commitment mechanism" interacts with the game. If players are forced to sign it then obviously you can achieve what you want. If players are not, then it will not happen if one player has a strictly dominant strategy. If the mechanism adds repeated games then that gets more interesting. – VCG Sep 21 '16 at 2:12
• If that's how you frame it, then it literally is a different game. You could simply add a new move for each player with the payoffs equivalent to what happens under the contract. (Like if 1 signs and the other doesn't). Or if they sign before playing the game, then you do an SPNE. – VCG Sep 22 '16 at 19:04

Expanding on @VCG's comment, which pretty much answered the question:

You can model this by adding a simultanous first move (to sign or not to sign) for both players. If they both choose to sign the contract, a predesignated outcome $A$ is realized and they get the corresponding payoffs. If they do not both sign, the original game is played.

Whether they will sign the contract or not will depend on their beliefs about what payoff they could expect in equilibrium. If a player's expected payoff from the original game is larger than what they would get in $A$ they will not sign.

(There are similar considerations in Aumann's correlated equilibrium, though there is no enforcable contract there.)

An example for when you would not sign the enforcable contract:
Consider a game of chicken with the payoffs

.
(Source: Wikipedia)

Suppose players could sign a contract forcing them to chose the "Chicken" strategy. A player will not do so if for some reason she expects that the other player will play chicken anyway, allowing her to collect a payoff of 7. In fact, using forward induction not signing may serve as a signal: The only reason he would not sign the contract is because she expects to win a larger than 6 payoff. The only way she can do this is by playing "Dare". Thus if the other player was willing to sign the contract, she she effectively signalled that she intends to play "Dare" by not signing the contract. Unfortunately according to the logic of forward induction the other player will also not sign.