Optimality Conditions for Semivectorial Bilevel Convex Optimal Control Problems

We present optimality conditions for bilevel optimal control problems where the upper level, to be solved by a leader, is a scalar optimal control problem and the lower level, to be solved by several followers, is a multiobjective convex optimal control problem. Multiobjective optimal control problems arise in many application areas where several conflicting objectives need to be considered. Minimize several objective functionals leads to solutions such that none of the objective functional values can be improved further without deteriorating another. The set of all such solutions is referred to as efficient (also called Pareto optimal, noninferior, or nondominated) set of solutions. The lower level of the semivectorial bilevel optimal control problems can be considered to be associated to a ”grande coalition” of a p-player cooperative differential game, every player having its own objective and control function. We consider situations in which these p-?players react as ”followers” to every decision imposed by a ”leader” (who acts at the so-called upper level). The best reply correspondence of the followers being in general non uniquely determined, the leader cannot predict the followers choice simply on the basis of his rational behavior. So, the choice of the best strategy from the leader point of view depends of how the followers choose a strategy among his best responses. In this paper, we will consider two (extreme) possibilities: (i) the optimistic situation, when for every decison of the leader, the followers will choose a strategy amongst the efficient controls which minimizes the (scalar) objective of the leader; in this case the leader will choose a strategy which minimizes the best he can obtain amongst all the best responses of the followers: (ii) the pessimistic situation, when the followers can choose amongst the efficient controls one which maximizes the (scalar) objective of the leader; in this case the leader will choose a strategy which minimizes the worst he could obtain amongst all the best responses of the followers. This paper continues the research initiated in [17] where existence results for these problems have been obtained.

Global emission ceiling versus international cap and trade: What is the most efficient system when countries act non-cooperatively?

We model climate negotiations as a two-stage game. In the first stage of the game, players have to agree on a global emission cap (GEC). In the second stage, they non-cooperatively choose either their emission level or their emission quota, depending on whether emission trading is allowed, under the cap that potentially binds them together. A three heterogenous player quadratic game serves as a base for the analysis. In this framework, when the cap is non-binding, there exists a unique Nash equilibrium. When the emission cap is binding, among all the coupled constraints Nash equilibria, we select a normalized equilibrium by solving a variational inequality, which has a unique solution. In both scenarios – with and without emission trading – we show that there exists a non-empty range of values for which setting a binding cap improves all players’ payoff. It also appears that for some values of the cap, all players get a higher payoff under the GEC system alone than under the international cap and trade (ITC) system alone. Thus, the introduction of a GEC outperforms the ITC system both in terms of emission reduction and of payoff gains.environmental game, climate change, international cap and trade system, national emission quotas, global emission cap, normalized equilibria, variational and quasi-variational inequalities.

Altruistic Behavior and Correlated Equilibrium Selection

This paper studies new refinement concepts for correlated equilibria based on altruistic behavior of the players and which generalize some refinement concepts introduced by the authors in previous papers for Nash equilibria. Effectiveness of the concepts, relations with the corresponding notions for Nash equilibria and with other correlated equilibrium refinements are investigated. The analysis of the topological properties of the set of solutions concludes the paper.correlated equilibrium, altruistic behavior, refinement

Slightly Altruistic Equilibria in Normal Form Games

We introduce a refinement concept for Nash equilibria (slightly altruistic equilibrium) defined by a limit process and which captures the idea of reciprocal altruism as presented in Binmore (2003). Existence is guaranteed for every finite game and for a large class of games with a continuum of strategies. Results and examples emphasize the (lack of) connections with classical refinement concepts. Finally, it is shown that under a pseudo-monotonicity assumption on a particular operator associated to the game it is possible, by selecting slightly altruistic equilibria, to eliminate those equilibria in which a player can switch to a strategy that is better for the others without leaving the set of equilibria.

Kalai-Smorodinsky Bargaining Solution Equilibria

Multicriteria games describe strategic interactions in which players, having more than one criterion to take into account, don't have an a-priori opinion on the rel- ative importance of all these criteria. Roemer (2005) introduces an organizational interpretation of the concept of equilibrium: each player can be viewed as running a bargaining game among criteria. In this paper, we analyze the bargaining problem within each player by considering the Kalai-Smorodinsky bargaining solution. We provide existence results for the so called Kalai-Smorodinsky bargaining solution equilibria for a general class of disagreement points which properly includes the one considered in Roemer (2005). Moreover we look at the refinement power of this equilibrium concept and show that it is an effective selection device even when combined with classical refinement concepts based on stability with respect to perturbations such as the the extension to multicriteria games of the Selten's (1975) trembling hand perfect equilibrium concept.

Including Social Nash Equilibria in Abstract Economies

We consider quasi-variational problems (variational problems having constraint sets depending on their own solutions) which appear in concrete economic models such as social and economic networks, financial derivative models, transportation network congestion and traffic equilibrium. First, using an extension of the classical Minty lemma, we show that new upper stability results can be obtained for parametric quasi-variational and linearized quasi-variational problems, while lower stability, which plays a fundamental role in the investigation of hierarchical problems, cannot be achieved in general, even on very restrictive conditions. Then, regularized problems are considered allowing to introduce approximate solutions for the above problems and to investigate their lower and upper stability properties. We stress that the class of quasi-variational problems include social Nash equilibrium problems in abstract economies, so results about approximate Nash equilibria can be easily deduced.quasi-variational, social Nash equilibria, approximate solution, closed map, lower semicontinuous map, upper stability, lower stability

On Multicriteria Games with Uncountable Sets of Equilibria

The famous Harsanyi's (1973) Theorem states that generically a finite game has an odd number of Nash equilibria in mixed strategies. In this paper, we show that for finite multicriteria games (games with vector-valued payoffs) this kind of result does not hold. In particular, we show, by examples, that it is possible to find balls in the space of games such that every game in this set has uncountably many equilibria so that uncountable sets of equilibria are not nongeneric in multicriteria games. Moreover, we point out that, surprisingly, all the equilibria of the games cor- responding to the center of these balls are essential, that is, they are stable with respect to every possible perturbation on the data of the game. However, if we consider the scalarization stable equilibrium concept (introduced in De Marco and Morgan (2007) and which is based on the scalarization technique for multicriteria games), then we show that it provides an effective selection device for the equilibria of the games corresponding to the centers of the balls. This means that the scalarization stable equilibrium concept can provide a sharper selection device with respect to the other classical refinement concepts in multicriteria games.

Social Networks: Equilibrium Selection and Friendliness

Given their importance in determining the outcome of many economic interactions, different models have been proposed to determine how social networks form and which structures are stable. In Bala and Goyal (2000), the one-sided link formation model has been considered, which is based on a noncooperative game of network formation. They found out that the empty networks, the wheel in the one-way flow of benefits case and the center sponsored star in the two-way flow case play a fundamental role, since they are strict Nash equilibria of the corresponding games for certain classes of payoff functions. In this paper, firstly we prove that all these network structures are in weakly dominated strategies whenever there are no strict Nash equilibria. Then, we exhibit a more accurate selection device between these network architectures by considering ‘altruistic behavior’ refinements. Such refinements, that we investigate here in the framework of finite strategy sets games, have been introduced by the authors in previous papers.

Reactions of metal complexes with Lewis bases and their utilisation in the selective filtration of smoke

This thesis is concerned with internal redox reactions of metal complexes and the application of such reactions in the selective filtration of tobacco smoke. The reactions of Fe(II), Fe(III) and Cu(II) quinoneoximic complexes and of some other complexes of these metals with Lewis bases (e.g. Ph3P and py) have been investigated. From the results obtained and from a critical consideration of analogous reactions reported in the literature the behaviour of metal chelates towards Lewis bases has been assessed. It has been established that the reaction between a metal chelate and a Lewis base may lead to: (i) Adduct formation, (ii) condensation and (iii) to an internal redox reaction, M(chel)n LB[over arrow] M(chel)n-1(LB)x + chel* The latter is favoured when the metal in M(chel)n has a stable lower oxidation state, by Lewis bases capable of E[eta]—bonding and by chelating ligands exerting a strong ligand field. The type of behaviour shown by chel* has been considered. The dissociated radical may: (i) Abstract a hydrogen atom to give the protonated ligand, (ii) dimerise, (iii) undergo coupling or degradation. A detailed investigation of the reactions of Fe(l-nqo)3 and Fe(2-nqo)3 with Ph3P (including kinetic studies) showed that these complexes undergo internal redox reactions to give iron(II) complexes and products arising from the deoxygenation of the nqo* radical. Mechanisms for these resctions have been proposed. The behaviour of quinoneoximic complexes of iron towards the Lewis base CO has been examined both directly and indirectly by studying the behaviour of nqoH towards Fe(CO)5 or Fe(CO)5/PhNH2. Fe(CO)5 and 1-nqoH or 2-nqoH afforded complexes of the type Fe(nqo)2, without formation of CO adducts or deoxygenation products. In the presence of PhNH2 complexes of type Fe(nqo)2 and Fe(nqo-A)2 resulted, where nqo-A is a species arising from the coupling of the nqo ligand with PhNH2. Both systems also afforded various organic products. On the basis of Mossbauer and other techniques oligomeric structures have been suggested for the complexes Fe(nqo)2 and Fe(nqo-A)2. Whereas CO did not react with Fe(nqo)2 or Fe(nqo)2.2py, it reacted with Fe(nqo)3, possibly via an internal redox process. For Fe(2-nqo)3 products arising from a deoxygenated ligand and a complex tentatively formulated as the nitrene species, Fe(2-nq)2, were isolated. The reactivity of the Fe(nqo), complexes towards CO has been successfully utilised in the selective filtration of tobacco smoke. Both Fe(1-nqo)3, and Fe(2-nqo)3 were shown to be capable of significantly reducing the level of CO in tobacco smoke under smoking conditions when supported on suitable filters. Reduction of the levels of other Lewis bases in tobacco smoke, e.g. nicotine and furfural, was observed. The complexes Fe(nqo)2, Fe(nqo)2.2py and Fe(nqo)3 reacted readily with NO to give complicated mixtures, but their effectiveness in removing NO from tobacco smoke has not been unambiguously established

Illuminating Massive Black Holes With White Dwarfs: Orbital Dynamics and High Energy Transients from Tidal Interactions

White dwarfs (WDs) can be tidally disrupted only by massive black holes (MBHs) with masses less than $\sim10^5 M_\odot$. These tidal interactions feed material to the MBH well above its Eddington limit, with the potential to launch a relativistic jet. The corresponding beamed emission is a promising signpost to an otherwise quiescent MBH of relatively low mass. We show that the mass transfer history, and thus the lightcurve, are quite different when the disruptive orbit is parabolic, eccentric, or circular. The mass lost each orbit exponentiates in the eccentric-orbit case leading to the destruction of the WD after several tens of orbits. We examine the stellar dynamics of clusters surrounding MBHs to show that single-passage WD disruptions are substantially more common than repeating encounters. The $10^{49}$ erg s$^{-1}$ peak luminosity of these events makes them visible to cosmological distances. They may be detectible at rates of as many as tens per year by instruments like Swift. In fact, WD-disruption transients significantly outshine their main-sequence star counterparts, and are the most likely tidal interaction to be detected arising from MBHs with masses less than $10^5 M_\odot$. The detection or non-detection of such WD-disruption transients by Swift is, therefore, a powerful tool to constrain lower end of the MBH mass function. The emerging class of ultra-long gamma ray bursts all have peak luminosities and durations reminiscent of WD disruptions, offering a hint that WD-disruption transients may already be present in existing datasets.Comment: Revised following response from refere