This paper investigates a purely qualitative version of Savage's theory for decision mak ing unde... more This paper investigates a purely qualitative version of Savage's theory for decision mak ing under uncertainty. Until now, most rep resentation theorems for preference over acts rely on a numerical representation of util ity and uncertainty where utility and uncer tainty are commensurate. Disrupting the tra dition, we relax this assumption and intro duce a purely ordinal axiom requiring that the Decision Maker (DM) preference between two acts only depends on the relative position of their consequences for each state. Within this qualitative framework, we determine the only possible form of the decision rule and in vestigate some instances compatible with the transitivity of the strict preference.
Proceedings of the AAAI Conference on Artificial Intelligence
This paper proposes incremental preference elicitation methods for multiobjective state space sea... more This paper proposes incremental preference elicitation methods for multiobjective state space search. Our approach consists in integrating weight elicitation and search to determine, in a vector-valued state-space graph, a solution path that best fits the Decision Maker's preferences. We first assume that the objective weights are imprecisely known and propose a state space search procedure to determine the set of possibly optimal solutions. Then, we introduce incremental elicitation strategies during the search that use queries to progressively reduce the set of admissible weights until a nearly-optimal path can be identified. The validity of our algorithms is established and numerical tests are provided to test their efficiency both in terms of number of queries and solution times.
HAL (Le Centre pour la Communication Scientifique Directe), May 5, 2014
This paper studies the likelihood of the existence of a pure Nash equilibrium (PNE) in random pay... more This paper studies the likelihood of the existence of a pure Nash equilibrium (PNE) in random payoff graphical games. Here, players are represented by vertices, they choose a strategy in finite discrete sets of strategies, and the scope of a player's utility function is only local. In this setting, the probability of existence of a PNE has been deeply studied for various graphical structures when the number of players tends to infinity, but only in the two strategies-perplayer case: this paper extends these studies to an arbitrary number of strategies-per-player. We prove theoretically how more strategies-per-player makes the distribution of the number of equilibria get closer to a Poisson distribution. We apply these results to various graph structures and conclude with numerical experiments.
Proceedings of the 9th International Conference on Autonomous Agents and Multiagent Systems Volume 1 Volume 1, 2010
This paper deals with fair assignment problems in decision contexts involving multiple agents. In... more This paper deals with fair assignment problems in decision contexts involving multiple agents. In such problems, each agent has its own evaluation of costs and we want to find a fair compromise solution between individual point of views. Lorenz dominance is a standard decision model used in Economics to refine Pareto dominance while favoring solutions that fairly share happiness among agents. In order to enhance the discrimination possibilities offered by Lorenz dominance, we introduce here a new model called infinite order Lorenz dominance. We establish a representation result for this model using an ordered weighted average with decreasing weights. Hence we exhibit some properties of infinite order Lorenz dominance that explain how fairness is achieved in the aggregation of individual preferences. Then we explain how to solve fair assignment problems of m items to n agents, using infinite order Lorenz dominance and other models used for measuring inequalities. We show that this problem can be reformulated as a 0-1 non-linear optimization problems that can be solved, after a linearization step, by standard LP solvers. We provide numerical results showing the efficiency of the proposed approach on various instances of the paper assignment problem.
Lecture Notes in Economics and Mathematical Systems, 2009
This paper is devoted to the search for Choquet-optimal solutions in multicriteria combinatorial ... more This paper is devoted to the search for Choquet-optimal solutions in multicriteria combinatorial optimization with application to spanning tree problems and knapsack problems. After recalling basic notions concerning the use of Choquet integrals for preference aggregation, we present a condition (named preference for interior points) that characterizes preferences favouring well-balanced solutions, a natural attitude in multicriteria optimization. When using a Choquet integral as preference model, this condition amounts to choosing a submodular (resp. supermodular) capacity when criteria have to be minimized (resp. maximized). Under this assumption, we investigate the determination of Choquet-optimal solutions in the multicriteria spanning tree problem and the multicriteria 0-1 knapsack problem. For both problems, we introduce a linear bound for the Choquet integral, computable in polynomial time, and propose a branch and bound procedure using this bound. We provide numerical experiments that show the actual efficiency of the algorithms on various instances of different sizes.
Le projet Welfare Quality ®(1) vise a produire un standard europeen d’evaluation du bien-etre des... more Le projet Welfare Quality ®(1) vise a produire un standard europeen d’evaluation du bien-etre des bovins, porcs et volailles. Le bien-etre etant multidimensionnel (sante, comportement, absence de stress...), son evaluation globale dans les elevages releve des methodes developpees en evaluation multicritere (Bouyssou, 1990). La construction d’une evaluation globale est cependant soumise a de nombreuses contraintes. En effet, les donnees collectees peuvent etre qualitatives ou ordinales, les mesures varient en precision, pertinence et importance et peuvent etre liees les unes aux autres. Par ailleurs, les dimensions du bien-etre ne peuvent pas totalement se compenser. Par exemple, la bonne sante ne peut pas compenser l’absence d’expression de certains comportements. Ainsi, des situations de compromis, avec des resultats corrects sur toutes les dimensions du bien-etre, seraient a favoriser par rapport a des situations contrastees. Toutes ces specificites compliquent la tâche d’agregation.
This paper is devoted to fair optimization in Multiobjective Markov Decision Processes (MOMDPs). ... more This paper is devoted to fair optimization in Multiobjective Markov Decision Processes (MOMDPs). A MOMDP is an extension of the MDP model for planning under uncertainty while trying to optimize several reward functions simultaneously. This applies to multiagent problems when rewards define individual utility functions, or in multicriteria problems when rewards refer to different features. In this setting, we study the determination of policies leading to Lorenz-nondominated tradeoffs. Lorenz dominance is a refinement of Pareto dominance that was introduced in Social Choice for the measurement of inequalities. In this paper, we introduce methods to efficiently approximate the sets of Lorenz-non-dominated solutions of infinite-horizon, discounted MOMDPs. The approximations are polynomial-sized subsets of those solutions.
This paper is devoted to the the search of robust solutions in state space graphs when costs de p... more This paper is devoted to the the search of robust solutions in state space graphs when costs de pend on scenarios. We first present axiomatic re quirements for preference compatibility with the intuitive idea of robustness. This leads us to pro pose the Lorenz dominance rule as a basis for ro bustness analysis. Then, after presenting com plexity results about the determination of robust solutions, we propose a new sophistication of A* specially designed to determine the set of robust paths in a state space graph. The behavior of the algorithm is illustrated on a small example. Fi nally, an axiomatic justification of the refinement of robustness by an OWA criterion is provided.
International Series in Operations Research & Management Science, 2000
ABSTRACT Decision-making inevitably implies, at some stage, the allocation of rare resources to s... more ABSTRACT Decision-making inevitably implies, at some stage, the allocation of rare resources to some alternatives rather than to others (e.g. deciding how to use one’s income). It is therefore not at all surprising that the question of helping a decision-maker to choose between competing alternatives, projects, courses of action and/or to evaluate them, has attracted the attention of economists. Cost-Benefit Analysis (CBA) is a set of techniques that economists have developed for this purpose. It is based on the following simple and apparently inescapable idea: a project should only be undertaken when its “benefits” outweigh its “costs”.
International Series in Operations Research & Management Science, 2000
In chapter 2, we tried to show that “voting”, although being a familiar activity to almost everyo... more In chapter 2, we tried to show that “voting”, although being a familiar activity to almost everyone, raises many important and difficult questions that are closely connected to the subject of this book. Our main objective in this chapter is similar. We all share the — more or less pleasant — experience of having received “grades” in order to evaluate
International Series in Operations Research & Management Science, 2000
ABSTRACT The increasing development of automatic systems in most sectors of human activities (e.g... more ABSTRACT The increasing development of automatic systems in most sectors of human activities (e.g. manufacturing, management, medicine, etc.) has progressively led to involving computers in many tasks traditionally reserved to humans, even the more “strategic” ones such as control, evaluation and decision-making. The main function of automatic decision systems is to act as a substitute for humans (decision makers, experts) in the execution of repetitive decision tasks. Such systems can be in charge of all or part of the decision process. The main tasks to be performed by automatic decision systems are collecting information (e.g. by sensors), making a diagnosis of the current situation, selecting relevant actions, executing and controlling these actions. Automatisation of these tasks requires the elaboration of computational models able to simulate human reasoning. Such models are, in many respects, comparable to those involved in the scientific preparation of human decisions. Indeed, deciding automatically is also a matter of representation, evaluation and comparison. For this reason, we introduce and discuss some very simple techniques used to design rule-based decision/control systems.
International Series in Operations Research & Management Science, 2000
ABSTRACT In this chapter, we describe an application that was the theme of a research collaborati... more ABSTRACT In this chapter, we describe an application that was the theme of a research collaboration between an academic institution and a large company in charge of the production and distribution of electricity. We do not give an exhaustive description of the work that was done and of the decision-aiding tool that was developed. A detailed presentation of the first discussions, of the progressive formulation of the problem, of the assumptions chosen, of the hesitations and backtrackings, of the difficulties encountered, of the methodology adopted and of the resulting software would require nearly a whole book. Our purpose is to point out some characteristics of the problem, especially on the modelling of uncertainties. The description was thus voluntarily simplified and some aspects, of minor interest in the framework of this book, were neglected. The main purpose of this presentation is to show how difficult it is to build (or to improvise) a pragmatic decision model that is consistent and sound. It illustrates the interest and the importance of having well-studied formal models at our disposal when we are confronted with a decision problem. Sections 8.2 and 8.3 present the context of the application and the model that was established. Section 8.4 is based on a didactical example: it first illustrates and comments some traditional approaches that could have been used in the application; then it gives a detailed description of the approach that was applied in the concrete case.
International Series in Operations Research & Management Science, 2000
ABSTRACT Our daily life is filled with indicators: I.Q., Dow Jones, GNP, air quality, physicians ... more ABSTRACT Our daily life is filled with indicators: I.Q., Dow Jones, GNP, air quality, physicians per capita, poverty index, social position index, consumer price index, rate of return,… If you read a newspaper, you could feel that these magic numbers rule the world.
This paper investigates a purely qualitative version of Savage's theory for decision mak ing unde... more This paper investigates a purely qualitative version of Savage's theory for decision mak ing under uncertainty. Until now, most rep resentation theorems for preference over acts rely on a numerical representation of util ity and uncertainty where utility and uncer tainty are commensurate. Disrupting the tra dition, we relax this assumption and intro duce a purely ordinal axiom requiring that the Decision Maker (DM) preference between two acts only depends on the relative position of their consequences for each state. Within this qualitative framework, we determine the only possible form of the decision rule and in vestigate some instances compatible with the transitivity of the strict preference.
Proceedings of the AAAI Conference on Artificial Intelligence
This paper proposes incremental preference elicitation methods for multiobjective state space sea... more This paper proposes incremental preference elicitation methods for multiobjective state space search. Our approach consists in integrating weight elicitation and search to determine, in a vector-valued state-space graph, a solution path that best fits the Decision Maker's preferences. We first assume that the objective weights are imprecisely known and propose a state space search procedure to determine the set of possibly optimal solutions. Then, we introduce incremental elicitation strategies during the search that use queries to progressively reduce the set of admissible weights until a nearly-optimal path can be identified. The validity of our algorithms is established and numerical tests are provided to test their efficiency both in terms of number of queries and solution times.
HAL (Le Centre pour la Communication Scientifique Directe), May 5, 2014
This paper studies the likelihood of the existence of a pure Nash equilibrium (PNE) in random pay... more This paper studies the likelihood of the existence of a pure Nash equilibrium (PNE) in random payoff graphical games. Here, players are represented by vertices, they choose a strategy in finite discrete sets of strategies, and the scope of a player's utility function is only local. In this setting, the probability of existence of a PNE has been deeply studied for various graphical structures when the number of players tends to infinity, but only in the two strategies-perplayer case: this paper extends these studies to an arbitrary number of strategies-per-player. We prove theoretically how more strategies-per-player makes the distribution of the number of equilibria get closer to a Poisson distribution. We apply these results to various graph structures and conclude with numerical experiments.
Proceedings of the 9th International Conference on Autonomous Agents and Multiagent Systems Volume 1 Volume 1, 2010
This paper deals with fair assignment problems in decision contexts involving multiple agents. In... more This paper deals with fair assignment problems in decision contexts involving multiple agents. In such problems, each agent has its own evaluation of costs and we want to find a fair compromise solution between individual point of views. Lorenz dominance is a standard decision model used in Economics to refine Pareto dominance while favoring solutions that fairly share happiness among agents. In order to enhance the discrimination possibilities offered by Lorenz dominance, we introduce here a new model called infinite order Lorenz dominance. We establish a representation result for this model using an ordered weighted average with decreasing weights. Hence we exhibit some properties of infinite order Lorenz dominance that explain how fairness is achieved in the aggregation of individual preferences. Then we explain how to solve fair assignment problems of m items to n agents, using infinite order Lorenz dominance and other models used for measuring inequalities. We show that this problem can be reformulated as a 0-1 non-linear optimization problems that can be solved, after a linearization step, by standard LP solvers. We provide numerical results showing the efficiency of the proposed approach on various instances of the paper assignment problem.
Lecture Notes in Economics and Mathematical Systems, 2009
This paper is devoted to the search for Choquet-optimal solutions in multicriteria combinatorial ... more This paper is devoted to the search for Choquet-optimal solutions in multicriteria combinatorial optimization with application to spanning tree problems and knapsack problems. After recalling basic notions concerning the use of Choquet integrals for preference aggregation, we present a condition (named preference for interior points) that characterizes preferences favouring well-balanced solutions, a natural attitude in multicriteria optimization. When using a Choquet integral as preference model, this condition amounts to choosing a submodular (resp. supermodular) capacity when criteria have to be minimized (resp. maximized). Under this assumption, we investigate the determination of Choquet-optimal solutions in the multicriteria spanning tree problem and the multicriteria 0-1 knapsack problem. For both problems, we introduce a linear bound for the Choquet integral, computable in polynomial time, and propose a branch and bound procedure using this bound. We provide numerical experiments that show the actual efficiency of the algorithms on various instances of different sizes.
Le projet Welfare Quality ®(1) vise a produire un standard europeen d’evaluation du bien-etre des... more Le projet Welfare Quality ®(1) vise a produire un standard europeen d’evaluation du bien-etre des bovins, porcs et volailles. Le bien-etre etant multidimensionnel (sante, comportement, absence de stress...), son evaluation globale dans les elevages releve des methodes developpees en evaluation multicritere (Bouyssou, 1990). La construction d’une evaluation globale est cependant soumise a de nombreuses contraintes. En effet, les donnees collectees peuvent etre qualitatives ou ordinales, les mesures varient en precision, pertinence et importance et peuvent etre liees les unes aux autres. Par ailleurs, les dimensions du bien-etre ne peuvent pas totalement se compenser. Par exemple, la bonne sante ne peut pas compenser l’absence d’expression de certains comportements. Ainsi, des situations de compromis, avec des resultats corrects sur toutes les dimensions du bien-etre, seraient a favoriser par rapport a des situations contrastees. Toutes ces specificites compliquent la tâche d’agregation.
This paper is devoted to fair optimization in Multiobjective Markov Decision Processes (MOMDPs). ... more This paper is devoted to fair optimization in Multiobjective Markov Decision Processes (MOMDPs). A MOMDP is an extension of the MDP model for planning under uncertainty while trying to optimize several reward functions simultaneously. This applies to multiagent problems when rewards define individual utility functions, or in multicriteria problems when rewards refer to different features. In this setting, we study the determination of policies leading to Lorenz-nondominated tradeoffs. Lorenz dominance is a refinement of Pareto dominance that was introduced in Social Choice for the measurement of inequalities. In this paper, we introduce methods to efficiently approximate the sets of Lorenz-non-dominated solutions of infinite-horizon, discounted MOMDPs. The approximations are polynomial-sized subsets of those solutions.
This paper is devoted to the the search of robust solutions in state space graphs when costs de p... more This paper is devoted to the the search of robust solutions in state space graphs when costs de pend on scenarios. We first present axiomatic re quirements for preference compatibility with the intuitive idea of robustness. This leads us to pro pose the Lorenz dominance rule as a basis for ro bustness analysis. Then, after presenting com plexity results about the determination of robust solutions, we propose a new sophistication of A* specially designed to determine the set of robust paths in a state space graph. The behavior of the algorithm is illustrated on a small example. Fi nally, an axiomatic justification of the refinement of robustness by an OWA criterion is provided.
International Series in Operations Research & Management Science, 2000
ABSTRACT Decision-making inevitably implies, at some stage, the allocation of rare resources to s... more ABSTRACT Decision-making inevitably implies, at some stage, the allocation of rare resources to some alternatives rather than to others (e.g. deciding how to use one’s income). It is therefore not at all surprising that the question of helping a decision-maker to choose between competing alternatives, projects, courses of action and/or to evaluate them, has attracted the attention of economists. Cost-Benefit Analysis (CBA) is a set of techniques that economists have developed for this purpose. It is based on the following simple and apparently inescapable idea: a project should only be undertaken when its “benefits” outweigh its “costs”.
International Series in Operations Research & Management Science, 2000
In chapter 2, we tried to show that “voting”, although being a familiar activity to almost everyo... more In chapter 2, we tried to show that “voting”, although being a familiar activity to almost everyone, raises many important and difficult questions that are closely connected to the subject of this book. Our main objective in this chapter is similar. We all share the — more or less pleasant — experience of having received “grades” in order to evaluate
International Series in Operations Research & Management Science, 2000
ABSTRACT The increasing development of automatic systems in most sectors of human activities (e.g... more ABSTRACT The increasing development of automatic systems in most sectors of human activities (e.g. manufacturing, management, medicine, etc.) has progressively led to involving computers in many tasks traditionally reserved to humans, even the more “strategic” ones such as control, evaluation and decision-making. The main function of automatic decision systems is to act as a substitute for humans (decision makers, experts) in the execution of repetitive decision tasks. Such systems can be in charge of all or part of the decision process. The main tasks to be performed by automatic decision systems are collecting information (e.g. by sensors), making a diagnosis of the current situation, selecting relevant actions, executing and controlling these actions. Automatisation of these tasks requires the elaboration of computational models able to simulate human reasoning. Such models are, in many respects, comparable to those involved in the scientific preparation of human decisions. Indeed, deciding automatically is also a matter of representation, evaluation and comparison. For this reason, we introduce and discuss some very simple techniques used to design rule-based decision/control systems.
International Series in Operations Research & Management Science, 2000
ABSTRACT In this chapter, we describe an application that was the theme of a research collaborati... more ABSTRACT In this chapter, we describe an application that was the theme of a research collaboration between an academic institution and a large company in charge of the production and distribution of electricity. We do not give an exhaustive description of the work that was done and of the decision-aiding tool that was developed. A detailed presentation of the first discussions, of the progressive formulation of the problem, of the assumptions chosen, of the hesitations and backtrackings, of the difficulties encountered, of the methodology adopted and of the resulting software would require nearly a whole book. Our purpose is to point out some characteristics of the problem, especially on the modelling of uncertainties. The description was thus voluntarily simplified and some aspects, of minor interest in the framework of this book, were neglected. The main purpose of this presentation is to show how difficult it is to build (or to improvise) a pragmatic decision model that is consistent and sound. It illustrates the interest and the importance of having well-studied formal models at our disposal when we are confronted with a decision problem. Sections 8.2 and 8.3 present the context of the application and the model that was established. Section 8.4 is based on a didactical example: it first illustrates and comments some traditional approaches that could have been used in the application; then it gives a detailed description of the approach that was applied in the concrete case.
International Series in Operations Research & Management Science, 2000
ABSTRACT Our daily life is filled with indicators: I.Q., Dow Jones, GNP, air quality, physicians ... more ABSTRACT Our daily life is filled with indicators: I.Q., Dow Jones, GNP, air quality, physicians per capita, poverty index, social position index, consumer price index, rate of return,… If you read a newspaper, you could feel that these magic numbers rule the world.
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