Next, we show that Φ satisfies efficiency.
Since ϕ also satisfies efficiency and symmetry, it is the α-Shapley value.
To be suitable in this context, the water treatment process has to satisfy efficiency, technological and cost criteria.
Note that, from Theorem 1, we obtain that the Double Recursive Process satisfies efficiency, non-negativity and claim-boundedness.
A basic goal in mechanism design is to construct mechanisms that simultaneously satisfy efficiency, voluntary participation, and dominant strategy incentive compatibility.
We show the impossibility to simultaneously satisfy efficiency, budget balance and individual rationality at a Bayesian Nash equilibrium, and propose a mechanism in the core of the game.
As noted in [[13], Remark 1.1], if a monoid presentation satisfies efficiency or inefficiency (while it is minimal), then it always contains a minimal number of generators.
Fujinaka (2004) proposes the α-Shapley value Φα through the concept of the α-marginal contribution and uses the above assumptions (1) and (2) to show that Φα satisfies efficiency and symmetry. .
Due to the above Lemma, we will show the following: if a value satisfies efficiency, symmetry and the dummy property, then we can derive players' payoffs allocated by it for all unanimity games.
We firstly recall that, as noted in [[4], Remark 1.1], if a monoid presentation satisfies efficiency or inefficiency (while it is minimal), then it always has a minimal number of generators.
