Single-leader multi-follower games model the sequential strategic interactions between one player who chooses first, the leader, and other players, the followers, who respond simultaneously after observing the choice of the leader. In this paper we focus on the concept of subgame perfect equilibrium (SPE for short) in single-leader multi-follower games when the followers are involved in a generalized game and have a non-unique joint best response for each choice of the leader. We first show the connections between the concept of an SPE and those of a pessimistic and optimistic equilibrium, which are usually adopted and extensively studied in this framework. We then prove an existence result for SPEs and a characterization of the set of SPE-outcomes. Specifically, an action profile is shown to be an SPE-outcome if and only if the joint response of the followers to the leader is rational and the leader’s payoff lies between the payoffs achievable in a pessimistic equilibrium and in an optimistic equilibrium. Finally, moving to the broader framework of multi-leader-follower games, we show the existence of SPEs and we identify a subset of SPE-outcomes for a class of multi-leader-follower games where the game played by the leaders involves a generalized weighted potential structure.

We examine a game-theoretic model of vessel sharing agreements in industries endowed with a general class of price functions and classes of convex cost functions. We study the equilibrium structure thereof—in particular, the existence of a unique equilibrium global aggregate as well as the existence of a unique equilibrium—and we provide a comparative statics analysis of consumer welfare with respect to an ordinal measure of concentration of the industry. We show that the a “high degree" of convexity of the cost functions can generate anti-competitive effects. In the presence of linear costs, the model satisfies a weak aggregative form in the sense of aggregative games. By allowing for the nonlinearity of variable cost functions, we further weaken the aggregative nature of the games considered. We provide a refined technique for treating these games in which both the equilibrium structure and the comparative statics analysis are based on the comparison of the equilibrium conditions of the players who strictly decrease their equilibrium strategies within the groups that strictly decrease the group’s equilibrium aggregate from an equilibrium with a smaller global aggregate associated with a less concentrated industry to an equilibrium with a (weakly) larger global aggregate associated with a more concentrated (but possibly identical) industry.

This paper proposes a general definition of an undominated maximal of a relation on a constraint set. No specific requirement is imposed on either the asymmetry of the objective relation or the constraint set (which might, or might not, coincide with the ground set of the objective relation). Several characterizations are formulated that express undominated maximals of an objective relation as maximals of some trace associated with that objective relation. By means of some of these characterizations, the structure of the entire set of undominated maximals is examined in the particular case of relations induced by open and closed convex cones—among them, the weak and strong Pareto dominance—and, in the case of semiorders, that admit certain types of representability. The results of the last part of the examination allow the construction of many examples of relations whose entire sets of maximals and undominated maximals are completely identifiable in an elementary way.

We obtain several characterizations of the existence of greatest elements of a total preorder. The characterizations pertain to the existence of unconstrained greatest elements of a total preorder and to the existence of constrained greatest elements of a total preorder on every nonempty compact subset of its ground set. The necessary and sufficient conditions are purely topological and, in the case of constrained greatest elements, are formulated by making use of a preorder relation on the set of all topologies that can be defined on the ground set of the objective relation. Observing that every function into a totally ordered set can be naturally conceived as a total preorder, we then reformulate the mentioned characterizations in the more restrictive case of an objective function with a totally ordered codomain. The reformulations are expressed in terms of upper semi- and pseudo-continuity by showing a topological connection between the two notions of generalized continuity.

The paper examines the conditions for the existence of maximals of a relation on every nonempty compact subset of its ground set. A preliminary analysis shows that the existence of maximals of a Suzumura-consistent relation is implied by the existence of maximals of the right trace of its transitive closure. Building on this fact, various theorems of the literature are unified by identifying a common topological property of their assumptions that concerns the right trace of the transitive closure of the objective relation. Next, a generalization is provided so as to accommodate some cases of interest to economics. Finally, a necessary and sufficient condition is presented for the existence of maximals on every nonempty compact subset of the ground set of a relation.

This paper examines the conditions for the existence of a maximal element of a relation on every nonempty compact subset of its ground set. A preliminary analysis establishes some connections between the maximals of a relation and those of its right trace. Via this analysis, various results of the literature are unified by identifying a common property of their assumptions that concerns the right trace of the transitive closure of the objective relation. Next, a generalization is provided so as to accommodate some relations of interest to economics. Finally, a necessary and sufficient condition is presented for the existence of a maximal on every nonempty compact subset of the ground set of a relation.

The main contribution of the paper is the proof that any element in the convex hull of a decomposably antichain-convex set is Pareto dominated by at least one element of that set. Building on this result, the paper demonstrates the disjointness of the convex hulls of two disjoint decomposably antichain-convex sets, under the assumption that one of the two sets is upward. These findings are used to obtain a number of consequences on: the structure of the set of Pareto optima of a decomposably antichain-convex set; the separation of two decomposably antichain-convex sets; the convexity of the set of maximals of an antichain-convex relation; the convexity of the set of maximizers of an antichain-quasiconcave function. Emphasis is placed on the invariance of the solution set of a problem under its “convexification.” Some entailments in the field of mathematical economics of the results of the paper are briefly discussed.

Considered are imperfectly discriminating contests in which players may possess private information about the primitives of the game, such as the contest technology, valuations of the prize, cost functions, and budget constraints. We find general conditions under which a given contest of incomplete information admits a unique pure-strategy Nash equilibrium. In particular, provided that all players have positive budgets in all states of the world, existence requires only the usual concavity and convexity assumptions. Information structures that satisfy our conditions for uniqueness include independent private valuations, correlated private values, pure common values, and examples of interdependent valuations. The results allow dealing with inactive types, asymmetric equilibria, population uncertainty, and the possibility of resale. It is also shown that any player that is active with positive probability ends up with a positive net rent.

A preference relation is a binary endorelation B over an arbitrary ground set X of alternatives. The condition that B is a strict partial order (or, equivalently that B is transitive and asymmetric) is often viewed as an expression of the coherence of the preference relation; the stronger condition that B is a strict weak order (or, equivalently that B is negatively transitive and asymmetric) is often viewed as an expression of the rankability of the alternatives in X. As is known, the requisite that a relation B on a ground set X is a strict partial order is equivalent to the assumption that for every pair (y,x) in X×X the implication y∈B(x)⇒B(y)⊂B(x) is satisfied (with B(t) denoting {s∈X:(s,t)∈B} for all t∈X and with ⊂ denoting strict inclusion). The paper introduces the new notions of a strongly coherent relation and that of a completely coherent relation: a strongly coherent relation is defined as one that satisfies the previous one-way implication as a double implication; a completely coherent relation is defined as one whose restriction to any subset of its ground set is strongly coherent. Three are the principal results of the paper that justify its title: (1) any relation on a finite ground set is a strict weak order if and only if it is strongly coherent; (2) any Ferrers relation is a strict weak order if and only if it is strongly coherent; (3) any relation is a strict weak order if and only if it is completely coherent.

We introduce the notion of an antichain-convex set to extend Debreu (1954)’s version of the second welfare theorem to economies where either the aggregate production set or preference relations are not convex. We show that – possibly after some redistribution of individuals’ wealth – the Pareto optima of some economies which are marked by certain types of non-convexities can be spontaneously obtained as valuation quasiequilibria and equilibria: both equilibrium notions are to be understood in Debreu (1954)’s sense. From a purely structural point of view, the mathematical contribution of this work is the study of the conditions that guarantee the convexity of the Minkowski sum of finitely many possibly non-convex sets. Such a study allows us to obtain a version of the Minkowski\Hahn–Banach separation theorem which dispenses with the convexity of the sets to be separated and which can be naturally applied in standard proofs of the second welfare theorem; in addition – and equally importantly – the study allows to get a deeper understanding of the conditions on the single production sets of an economy that guarantee the convexity of their aggregate.

A preference relation is a binary endorelation B over an arbitrary ground set X of alternatives. The requisite that B is a strict partial order relation is equivalent to positing that the implication y is an element of B(x) double right arrow B(y) subset of B(x) - with B(t) denoting {s is an element of X : (s, t) is an element of B} for all t is an element of X and with subset of denoting strict inclusion - holds true for every pair (y, x) in X x X. The paper introduces the notion of a coherent order relation, defined as one that satisfies the previous one-way implication as a double implication. Examined are some properties of coherent order relations and how they connect with strict weak order relations and with transitive relations. In particular, the paper shows various conditions for a transitive relation to be a coherent order relation and for a coherent order relation to be a strict weak order relation. (C) 2019 Elsevier Inc. All rights reserved.

A family of oligopolies that possess a unique equilibrium was identified in the second authors doctoral dissertation. For such a family, it is therein specified a class of functions-economically related to the price function of a Cournot oligopoly - that satisfy a particular type of quasi-concavity. The first part of the present article (i) conceptualizes that type of quasi-concavity by introducing the notion of demi-concavity, (ii) considers two possible variants and (iii) provides some calculus properties. The second part, by relying on the results on demi-concavity, proves a Cournot equilibrium uniqueness theorem which is new for the journal literature and subsumes various results thereof. A third part shows an example that illustrates the novelty of the result.

Attention is focussed on a type of strategic alliance of the container shipping industry: vessel sharing agreements. In such consortia carriers jointly provide—but independently sell—a liner service. The strategic alliances studied in this work have not been extensively analyzed in the theoretical literature; a new model is proposed that embodies their main distinguishing features. By it, an examination is provided of the effects on equilibrium prices, equilibrium aggregate quantities and consumer welfare of the formation and enlargement of vessel sharing agreements. A positive answer is developed to the question raised in the title of the present work that supports a laissez-faire policy for these consortia.

We prove the existence of a unique pure-strategy Nash equilibrium in nice games with isotone chain-concave best reply functions and compact strategy sets. We show a preliminary fixpoint uniqueness argument which provides sufficient assumptions on the best replies of a nice game for the existence of exactly one Nash equilibrium. Then we examine the necessity and sufficiency of the conditions on the utility functions for such assumptions to be satisfied; in particular, we find necessary and sufficient conditions for the isotonicity and concavity∖chain-concavity of best reply functions. We extend the results on Nash equilibrium uniqueness to nice games with upper unbounded strategy sets and we present “dual” results for games with isotone convex∖chain-convex best reply functions. A final extension to Bayesian games is exhibited.

This state-of-the-art collection of papers on the theory of Cournotian competition focuses on two main subjects: oligopolistic Cournot competition and contests. The contributors present various applic

We consider a two-group contest game with weakest-link social composition functions and convex cost functions and prove the existence of a unique group-proof Nash equilibrium. Such a refinement of the Nash equilibrium prescribes the same communication possibilities as those required by a coalition-proof Nash equilibrium-in the precise sense of Bernheim et al. (J Econ Theory 42: 1-12, 1987)-only among the contenders of the same group and between the two groups. We show how a fictitious game with "most inefficient fictitious contenders" can be constructed to prove the existence of a unique group-proof Nash equilibrium of the original two-group contest game. An example evidences that cautious arguments on the (twice) differentiability of cost functions must be used in such a construction.

In this Introduction to the Festschrift we identify a game structure that is commonly analyzed in many-albeit not all-works of Prof. Okuguchi. We define the games characterized by this structure as Cournotian games and we show that the archetypical Cournot model of oligopolistic competition and other models of economic interest are special instances of these games. We then pass to a brief summary of the contributions collected in this volume that are related to Cournotian games.

It is well known that a symmetric game has only symmetric (pure strategy) Nash equilibria if its best-reply correspondences admit only increasing selections and its strategy sets are totally ordered. Several nonexamples of the literature show that this result is generally false when the totality condition of the relation that orders the strategy sets is simply dropped. Making use of the structure of interaction functions, this note provides sufficient conditions for the symmetry of all (pure strategy) Nash equilibria in symmetric games where best-reply correspondences admit only increasing selections, but strategy sets are not necessarily totally ordered.

We examine the coalition-proofness and Pareto properties of Nash equilibria in pure strategy \(\sigma \)-interactive games with strategic substitutes and increasing/decreasing externalities. For this class of games: (i) we prove the equivalence among the set of Nash equilibria, the set of coalition-proof Nash equilibria under strong Pareto dominance and the set of Nash equilibria that are not strongly Pareto dominated by other Nash equilibria; (ii) we prove that the fixpoints of some “ extremal” selections from the joint best reply correspondence are both coalition-proof Nash equilibria under weak Pareto dominance and not weakly Pareto dominated by other Nash equilibria. We also provide an order-theoretic characterization of the set of Nash equilibria and show various applications of our results.