Complexity of Strong Implementability
📝 Abstract
We consider the question of implementability of a social choice function in a classical setting where the preferences of finitely many selfish individuals with private information have to be aggregated towards a social choice. This is one of the central questions in mechanism design. If the concept of weak implementation is considered, the Revelation Principle states that one can restrict attention to truthful implementations and direct revelation mechanisms, which implies that implementability of a social choice function is easy to check. For the concept of strong implementation, however, the Revelation Principle becomes invalid, and the complexity of deciding whether a given social choice function is strongly implementable has been open so far. In this paper, we show by using methods from polyhedral theory that strong implementability of a social choice function can be decided in polynomial space and that each of the payments needed for strong implementation can always be chosen to be of polynomial encoding length. Moreover, we show that strong implementability of a social choice function involving only a single selfish individual can be decided in polynomial time via linear programming.
💡 Analysis
We consider the question of implementability of a social choice function in a classical setting where the preferences of finitely many selfish individuals with private information have to be aggregated towards a social choice. This is one of the central questions in mechanism design. If the concept of weak implementation is considered, the Revelation Principle states that one can restrict attention to truthful implementations and direct revelation mechanisms, which implies that implementability of a social choice function is easy to check. For the concept of strong implementation, however, the Revelation Principle becomes invalid, and the complexity of deciding whether a given social choice function is strongly implementable has been open so far. In this paper, we show by using methods from polyhedral theory that strong implementability of a social choice function can be decided in polynomial space and that each of the payments needed for strong implementation can always be chosen to be of polynomial encoding length. Moreover, we show that strong implementability of a social choice function involving only a single selfish individual can be decided in polynomial time via linear programming.
📄 Content
Mechanism design is a classical area of noncooperative game theory [7] and microeconomics [2] which studies how privately known preferences of several people can be aggregated towards a social choice. Applications include the design of voting procedures, the writing of contracts among parties, and the construction of procedures for deciding upon public projects. Recently, the study of the Internet has fostered the interest in algorithmic aspects of mechanism design [6].
In the classical social choice setting considered in this paper, there are n selfish agents, which must make a collective decision from some finite set X of possible social choices. Each agent i has a private value θ i ∈ Θ i (called the agent’s type), which influences the preferences of all agents over the alternatives in X . Formally, this is modeled by a valuation function V i : X × Θ → Q for each agent i, where Θ = Θ 1 × • • • × Θ n . Every agent i reports some information s i from a set S i of possible bids of i to the mechanism designer who must then choose an alternative from X based on these bids. The goal of the mechanism designer is to implement a given social choice function f : Θ → X , that is, to make sure that the alternative f (θ ) is always chosen in equilibrium when the vector of true types is θ = (θ 1 , . . . , θ n ). To achieve this, the mechanism designer hands out a payment P i (θ ) to each agent i, which depends on the bids. Each agent then tries to maximize the sum of her valuation and payment by choosing an appropriate bid depending on her type. A mechanism Γ = (S 1 , . . . , S n , g, P) is defined by the sets S 1 , . . . , S n of possible bids of the agents, an outcome function g : S 1 × • • • × S n → X , and the payment scheme P = (P 1 , . . . , P n ).
In the most common concept called weak implementation, the mechanism Γ is said to implement the social choice function f if some (Bayesian) equilibrium of the noncooperative game defined by the mechanism yields the outcome specified by f . An important result known as the Revelation Principle (cf.
[2, p. 884]) states that a social choice function is weakly implementable if and only if it can be truthfully implemented by a direct revelation mechanism, which means that f can be implemented by a mechanism with S i = Θ i for all i and truthful reporting as an equilibrium that yields the outcome specified by f . As a result, the question whether there exists a mechanism that weakly implements a given social choice function f can be easily answered in time polynomial in |Θ| by checking for negative cycles in complete directed graphs on the agents’ type spaces with changes of valuations as edge weights (cf. [4]).
However, there is an obvious drawback in considering weak implementation: Although a mechanism Γ may have some equilibrium that yields the outcome specified by f , there may be other equilibria that yield different outcomes. Thus, the concept of weak implementation heavily relies on the implicit assumption that the agents always play the equilibrium that the mechanism designer wants if there is more than one.
The standard way to avoid this problem is to consider the more robust concept of implementation called strong implementation. A mechanism Γ is said to strongly implement the social choice function f if every equilibrium of Γ yields the outcome specified by f . For strong implementation, the Revelation Principle does not hold, so one cannot, in general, restrict attention to direct revelation mechanisms and truthful implementations when trying to decide whether a social choice function is strongly implementable. In tackling the question whether a given social choice function can be strongly implemented, it is not even a priori clear that one can restrict attention to finite sets S i or polynomially sized payments. Thus, to the best of our knowledge, the complexity has been open so far. The main result of this paper is that strong implementability of a social choice function can be decided in polynomial space. In particular, if a social choice function can be strongly implemented, our results show that each of the payments in a mechanism that strongly implements it can be chosen to be of polynomial encoding length. It seems unlikely that the problem is contained in NP, at least the characterizations of strong implementability developed so far require an exponential number of (polynomially sized) certificates. We conjecture that deciding strong implementability is in fact PSPACE-complete. However, we show that the problem can be solved in polynomial time in case of a single agent.
We are given n agents identified with the set N = {1, . . . , n} and a finite set X of possible social choices. For each agent i, there is a finite set Θ i of possible types and we write
The true type θ i of agent i is known only to the agent herself. However, there is a commonly known probability distribution p : Θ → Q on Θ satisfying p(θ ) ≥ 0 for all θ ∈ Θ and ∑ θ ∈Θ p(θ ) = 1.
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