Décision, Risque, Interactions Sociales

In most decisions individuals must to choose between options that involve some uncertainty about their outcomes and their effect on their well-being. Experimental studies suggest that, in making these decisions, individuals often deviate from the paradigm of classical decision theory, even in relatively simple situations. In carefully controlled studies psychologists show, in more complex situations, that individual choices are sensitive to the description of the options, their contextualization and elicitation method. In social context individuals care not just about their outcomes but also about the outcomes and the intentions of those around them.

This thesis is divided in three independent chapters. The first one deals with additively separable preferences on the set of lotteries. This study leads to a non-linear expected utility representation and a weak form of event-separability of preferences. Also, I deduce a simple axiomatic foundation of an entropy modified expected utility. In the second chapter, I provide a general choice model under risk with social interactions. The third chapter of the thesis has to do with the potential of quantum probability theory in the von Neumann and Morgenstern framework. Each chapter focus on a particular generalization of the expected utility model and I am going to present in the next paragraphs the general ideas and theories that are common to most of them.

Decision theory has a long history since the emergence  of probabilities. The first natural criterion of decision making under risk is the standard expectation value. This naive criterion has been challenged by the St. Petersburg paradox in the 18th century and solved by Bernoulli (1738-1954) that postulates the expected utility criterion. This criterion has been axiomatised by von Neumann and Morgenstern (1944) (modern presentations refer usually to Marschak (1950), Herstein and Milnor (1953), Luce and Raiffa (1957), Jensen (1967) or Fishburn (1970) for exogenous probabilities). This formulation is tractable, it defines the attitude toward risk and it is applicable to many academic fields, especially those related to the theory of non cooperative games. Expected utility is based on the independence axiom and has a normative appeal. However, many experimental results have shown that this decision criterion was questionable. The most popular is definitely the Allais paradox (1953) which leads to the definition of two more general phenomena, common ration effect and common consequence effect. Both phenomena have been reproduced by Kahneman and Tversky (1979) (problems 1,2 and 3,4 respectively). MacCrimmon and Larsson (1979) provide a detailed study of the paradoxes of the independence axiom in the context of risk and radical uncertainty. The descriptive accuracy of expected utility has led to generalizations that we can classify into three non exhaustive and non-mutually exclusive categories.

The first class of generalizations weakens the independence axiom. The contribution of the first chapter is in this class. The sophistication of the expected utility theories with a clear axiomatic framework, identifying the weakening of the independence axiom, allows to better understand the normative and descriptive aspects of these theories. For example, weighted expected utility  proposed by Chew and MacCrimmon (1979) where the independence axiom holds for lotteries in the same equivalence class and rank dependent expected utility axiomatised by Quiggin (1982) in the risk where the independence axiom holds for co-monotonic lotteries.

The second class of generalizations rejects the independence axiom. One example is the local expected utility initiated by Machina (1982) and developed by Chew and Nishimura (1992), Chew and Hui (1995) and for a recent contribution Chaterjee and Krishna (2011). This approach preserves the weak order condition and requires a notion of differentiability of the representation of the preferences. The expected utility becomes a local notion (in the topological sense) because differentiable functional can be considered as locally linear. This approach allows great flexibility and a generalization of criteria attitudes towards risk . Another example is the theory of Luce (see Luce (2000) for a compilation of all of his work and Wakker (2000) for a short summary). Luce built an alternative theory from psychological concepts and found the majority of standard models from behavioural axioms with a concatenation operation between lotteries.

The third class consists of generalizations based on experimental approaches focusing on the descriptive aspect of the individual decision-making, identifying utility functionals that reproduce the experimental results. The best known example is the prospect theory of Kahneman and Tversky (1979) and refinement, the cumulative prospect theory (i.e., the rank-dependent prospect theory (1992)). Another example is the TAX model (Transfer of Attention Exchange) developed by Brinbaum and Chavez (1997).

Rank-dependent expected utility is certainly the contribution that has had the most success in decision theory (e.g., see Weber and Kirsner (1997), Diecidue and Wakker (2001) and Mongin (2009) for arguments highlighting the rank dependent expected utility). This theory has, both in the context of risk or uncertainty  , and in many models  to accommodate violations of expected utility theory. In addition, the functional associated with this theory preserves interesting properties such as stochastic dominance. Empirically, the rank-dependent expected utility leads to better results than expected utility or weighted expected utility for choice situations referring to the Allais paradox. In more general situations, rank-dependent expexted utility does not perform better than expected utility or other theories.