Socio-legal systems and implementation of the Nash solution in Debreu–Hurwicz equilibrium

In this article we combine Debreu’s (Proc Natl Acad Sci 38(10):886–893, 1952) social system with Hurwicz’s (Econ Design 1(1):1–14, 1994; Am Econ Rev 98(3):577–585, 2008) ideas of embedding a “desired” game form into a “natural” game form that includes all feasible behavior, even if it is “illegal” according to the desired form. For the resulting socio-legal system we extend Debreu’s concepts of a social system and its social equilibria to a socio-legal system with its Debreu–Hurwicz equilibria. We build on a more general version of social equilibrium due to Shafer and Sonnenschein (J Math Econ 2(3):345–348, 1975) that also generalizes the dc-mechanism of Koray and Yildiz (J Econ Theory 176:479–502, 2018) which relates implementation via mechanisms with implementation via rights structures as introduced by Sertel (Designing rights: invisible hand theorems, covering and membership. Tech. rep. Mimeo, Bogazici University, 2001). In the second part we apply and illustrate these new concepts via an application in the narrow welfarist framework of two person cooperative bargaining. There we provide in a socio-legal system based on Nash’s demand game an implementation of the Nash bargaining solution in Debreu–Hurwicz equilibrium.

Existence of Walrasian equilibria with discontinuous, non-ordered, interdependent and price-dependent preferences, without free disposal, and without compact consumption sets

We extend a result on existence of Walrasian equilibria in He and Yannelis (Econ Theory 61:497–513, 2016) by replacing the compactness assumption on consumption sets made there by the standard assumption that these sets are closed and bounded from below. This provides a positive answer to a question explicitly raised in He and Yannelis (Econ Theory 61:497–513, 2016). Our new equilibrium existence theorem generalizes many results in the literature as we do not require any transitivity or completeness or continuity assumption on preferences, initial endowments need not be in the interior of the consumption sets, preferences may be interdependent and price-dependent, and no monotonicity or local non satiation is needed for any of the agents.

Reduced-form mechanism design and ex post fairness constraints

This paper incorporates fairness constraints into the classic single-unit reduced-form implementation problem (Border in Economet J Econ Soc, 59(4):1175–1187, 1991, Econ Theory 31(1):167–181, 2007; Che et al. in Econometrica 81(6): 2487–2520, 2013; Manelli and Vincent in Econometrica, 78(6):1905–1938, 2010) with two agents. To do so, I use a new approach that utilizes the results from Kellerer (Math Ann 144(4):323–344, 1961) and Gutmann et al. (Ann Prob 19:1781–1797, 1991). Under realistic assumptions on the constraints, the conditions are transparent and can be verified in polynomial time.

Strategy-proof mechanism design with non-quasi-linear preferences: ex-post revenue maximization for an arbitrary number of objects

We consider the multi-object allocation problem with monetary transfers where each agent obtains at most one object (unit-demand). We focus on allocation mechanisms satisfying individual rationality, non-wastefulness, equal treatment of equals, and strategy-proofness. Extending the result of Kazumura et al. (J Econ Theory 188:105036, 2020b), we show that for an arbitrary number of agents and objects, the minimum price Walrasian is the unique ex-post revenue maximizing mechanism among the mechanisms satisfying no subsidy in addition to the four properties, and that no subsidy in this result can be replaced by no bankruptcy on the positive income effect domain.

Serial dictatorship mechanisms with reservation prices: heterogeneous objects

We adapt a set of mechanisms introduced by Klaus and Nichifor (Econ Theory 70:665–684, 2020), serial dictatorship mechanisms with (individual) reservation prices, to the allocation of heterogeneous indivisible objects, e.g., specialist clinic appointments. We show how the characterization of serial dictatorship mechanisms with reservation prices for homogeneous indivisible objects (Klaus and Nichifor 2020, Theorem 1) can be adapted to the allocation of heterogeneous indivisible objects by adding neutrality: mechanism $$\varphi $$ φ satisfies minimal tradability, individual rationality, strategy-proofness, consistency, independence of unallocated objects, neutrality, and non wasteful tie-breaking if and only if there exists a reservation price vector r and a priority ordering $$\succ $$ ≻ such that $$\varphi $$ φ is a serial dictatorship mechanism with reservation prices based on r and $$\succ $$ ≻ .

Technical Note Alternative Theorem: Equilibrium in Abstract Economies

We prove a new theorem for the existence of equilibrium in discontinuous games in which the players’ preferences are neither complete nor transitive. Our result is an alternative version of Shafer and Sonnenschein ([1975] J. Math. Econ. 2, 345–348), He and Yannelis ([2016] Econ. Theory 61, 497–513), Reny ([2016] Econ. Theory Bull.) and Carmona and Prodczeck ([2016] Econ. Theory 61, 457–478).

Monte Carlo sampling processes and incentive compatible allocations in large economies

Monte Carlo simulation is used in Hammond and Sun (Econ Theory 36:303–325, 2008. 10.1007/s00199-007-0279-7) to characterize a standard stochastic framework involving a continuum of random variables that are conditionally independent given macro shocks. This paper presents some general properties of such Monte Carlo sampling processes, including their one-way Fubini extension and regular conditional independence. In addition to the almost sure convergence of Monte Carlo simulation considered in Hammond and Sun (2008), here we also consider norm convergence when the random variables are square integrable. This leads to a necessary and sufficient condition for the classical law of large numbers to hold in a general Hilbert space. Applying this analysis to large economies with asymmetric information shows that the conflict between incentive compatibility and Pareto efficiency is resolved asymptotically for almost all sampling economies, following some similar results in McLean and Postlewaite (Econometrica 70:2421–2453, 2002) and Sun and Yannelis (J Econ Theory 134:175–194, 2007. 10.1016/j.jet.2006.03.001).

Approximation Guarantee of OSP Mechanisms: The Case of Machine Scheduling and Facility Location

Obvious strategyproofness (OSP) is an appealing concept as it allows to maintain incentive compatibility even in the presence of agents that are not fully rational, i.e., those who struggle with contingent reasoning (Li in Am Econ Rev 107(11):3257–3287, 2017). However, it has been shown to impose some limitations, e.g., no OSP mechanism can return a stable matching (Ashlagi and Gonczarowski in J Econ Theory 177:405–425, 2018). We here deepen the study of the limitations of OSP mechanisms by looking at their approximation guarantees for basic optimization problems paradigmatic of the area, i.e., machine scheduling and facility location. We prove a number of bounds on the approximation guarantee of OSP mechanisms, which show that OSP can come at a significant cost. However, rather surprisingly, we prove that OSP mechanisms can return optimal solutions when they use monitoring—a novel mechanism design paradigm that introduces a mild level of scrutiny on agents’ declarations (Kovács et al. in WINE 9470:398–412, 2015).

Refinements and higher-order beliefs: a unified survey

This paper presents a simple framework that allows us to survey and relate some different strands of the game theory literature. We describe a “canonical” way of adding incomplete information to a complete information game. This framework allows us to give a simple “complete theory” interpretation (Kreps in Game theory and economic modelling. Clarendon Press, Oxford, 1990) of standard normal form refinements such as perfection, and to relate refinements both to the “higher-order beliefs literature” (Rubinstein in Am Econ Rev 79:385–391, 1989; Monderer and Samet in Games Econ Behav 1:170–190, 1989; Morris et al. in Econ J Econ Soc 63:145–157, 1995; Kajii and Morris in Econ J Econ Soc 65:1283–1309, 1997a) and the “payoff uncertainty approach” (Fudenberg et al. in J Econ Theory 44:354–380, 1988; Dekel and Fudenberg in J Econ Theory 52:243–267, 1990).