Vladimir Ivanovich Danilov

Cores and Stable Blockings

This chapter is devoted to the issue of stable outcomes, that is those outcomes which are rejected by no coalition of agents. The existence of such outcomes depends only on the stability of the blocking generated by a given mechanism. We investigate here stable blocking relations. We begin with a few examples and give some useful instruments (Section 4.1). In Sections 4.2–4.4 we discuss three classes of stable blockings: additive blockings, almost additive blockings, and convex blockings. The main finding is that for almost additive blockings a family of coalitions which reject alternatives out of the core, can be equipped with a laminar structure (Theorem (4.4.7)). Section 4.5 reviews a series of necessary conditions to warrant the stability of a given blocking. In particular, convexity and almost-additivity turn out to be necessary for the stability of maximal blockings. In Section 4.6, we develop a veto-procedure in order to find elements in the core. The procedure yields single-element outcomes for any maximal convex blocking.

Nash-consistent Mechanisms

This chapter is devoted to Nash-consistent mechanisms, that is mechanisms possessing Nash equilibria at every preference profile. In Section 2.1, we examine a few examples, then proceed to investigate blockings generated by Nash-consistent mechanisms (Section 2.2). In Section 2.3, we show that the correspondence of equilibrium outcomes exhibit a somewhat stronger property than monotonicity, which is called strong monotonicity. In Section 2.4, we describe Nash-implementable SCCs. In the more-than-two-agents case, the class of Nash-implementable SCCs coincides with the class of strongly monotone SCCs. The case of two agents is considered in Section 2.5. In Section 2.6, we discuss acceptable mechanisms, that is consistent mechanisms whose outcomes are Pareto optimal.

Strongly Consistent Mechanisms

In this chapter, we examine mechanisms that have strong Nash equilibria for any preference profile. We start with some examples (Section 5.1), then investigate in more detail a strongly consistent mechanism with tokens (Section 5.2). Section 5.3 addresses the issue somewhat more theoretically. In particular, we show that for any maximal blocking B, there exists a mechanism whose set of equilibrium outcomes coincides with the core of the blocking B. Further (Section 5.4) we consider direct core mechanisms, that is SCFs whose outcomes are in the core. The existence of a strongly consistent selector from the core depends on a property of the underlying blocking, namely “lam-inability”. In Section 5, we introduce several equivalent characterizations of laminable blockings, in particular an elimination procedure for finding strong Nash equilibria. Then (Section 5.6) we provide examples of laminable blockings and in Section 7, formulate a necessary and sufficient condition for lam-inability in terms of the blocking relation itself. In Section 8, we tackle the case of neutral laminable blockings. The Appendix provides insights on the strong implementation issue.

Strategy-proof Mechanisms

In this chapter we examine strategy-proof mechanisms, i.e., mechanisms that endow every agent with the best (called dominant) strategy for each permissible preference profile. Based on the revelation principle, we construct, for every strategy-proof mechanism, an equivalent direct non-manipulable mechanism. The key characteristic of such a mechanism is the agent’s effective region in the set of outcomes. From this point of view, we study the structure of non-manipulable mechanisms in both the universal and the single-peaked environments (Sections 3.1 and 3.2). The convex structure of the outcome set yields an affine environment and allows us to mix strategy-proof mechanisms. In Section 3.3 we conjecture that any non-manipulable mechanism (within an affine environment) is a probability mixture of duplet and unilateral non-manipulable mechanisms. In the following two sections, we study the properties of Groves mechanisms in transferable environments, in particular, the issue of efficiency. We present some efficiency evaluations and efficiency criteria for Groves mechanisms.