The utility functions of both players sum up to 0
- If the utility function of each player has concavity, then there exist a unique equilibrium.
- VCG can maximize the social welfare given the individuals are all “selfish”
- There are game players,
- The actions of the players are denoted as
- Real demand
- for its action
Following Nisan’s work, the terms “mechanisms” and “incentive compatible” are defined as
- For every player , every , , , , and every , where and , then
- , then the mechanism is incentive compatible.
Specifically, among those incentive compatible mechanisms, the Vickrey-Clarke-Groves (VCG) mechanism is the mostly used one.
The VCG generally seeks to maximize the social welfare of all players in one game, where the social welfare is calculated as
. So the goal function of VCG is
1. Bargaining problems Scenarios
Bargaining problems represent situations in which
- There is a conflict of interest about agreements.
- Individual have the possibility of concluding a mutually beneficial agreements.
- No agreement may be imposed on any individual without his approval
2. Bargaining problem Definition
Example: Suppose 2 players must split one unit of good. If no agreement is reached, then players do not receive anything. We define the following notations.
- : the set of possible agreements
- : the disagreement outcome
- : each player i has preferences, represented by a utility function over
Definition: a bargaining problem is then defined as a pair of where and . We assume that
- is a convex and compact set
- There exists some such that (i.e., for some i)
- Pareto Efficiency
- A bargaining solution is Pareto efficient if there does not exist a such that and for some .
- An inefficient outcome is unlikely, since it leaves space for renegotiation.
- Let be such that if and only if and . Then .
- If the players are indistinguishable, the agreement should not discriminate between them.
- Invariance to Equivalent Payoff Representations
- Given a bargaining problem , consider a different bargaining problem , for some .
- Utility functions are only representation of preferences over outcomes. A transformation of the utility function that maintaining the same ordering over preferences (such as linear transformation) should not alter the outcome of bargaining process.
- Independence of Irrelevant Alternatives
- Let and be two bargaining problems such that , if , then .
4. Nash Bargaining Solution
Definition: We say that a pair of payoffs is a Nash bargaining solution if it solves the following optimization problem
- subject to
We use to denote the Nash Bargaining Solution
- Existence of an optimal solution: since the set is compact and the objective function of the problem is continuous, there exists an optimal solution for the problem
- Uniqueness of the optimal solution: the objective function of the problem is strictly quasi-concave. Therefore, the problem has a unique solution.
Proposition: Nash bargaining solution is the unique bargaining solution that satisfies the 4 axioms.
 Game Theory with Engineering Applications: Nash Bargaining Solution, by Asu Ozdaglar, MIT 2010