Contracting for innovation under knightian uncertainty

At any given point in time, the collection of assets existing in the economy is observable. Each asset is a function of a set of contingencies. The union taken over all assets of these contingencies is what we call the set of publicly known states. An innovation is a set of states that are not publicly known along with an asset (in a broad sense) that pays contingent on those states. The creator of an innovation is an entrepreneur. He is represented by a probability measure on the set of new states. All other agents perceive the innovation as ambiguous: each of them is represented by a set of probabilities on the new states. The agents in the economy are classified with respect to their attitude towards this Ambiguity: the financiers are (locally) Ambiguity-seeking while the consumers are Ambiguity-averse. An entrepreneur and a financier come together when the former seeks funds to implement his project and the latter seeks new profit opportunities. The resulting contracting problem does not fall within the standard theory due to the presence of Ambiguity (on the financier’s side) and to the heterogeneity in the parties’ beliefs. We prove existence and monotonicity (i.e., truthful revelation) of an optimal contract. We characterize such a contract under the additional assumption that the financiers are globally Ambiguity-seeking. Finally, we re-formulate our results in an insurance framework and extend the classical result of Arrow [4] and the more recent one of Ghossoub. In the case of an Ambiguity-averse insurer, we also show that an optimal contract has the form of a generalized deductible.

Oligopolistic competition under knightian uncertainty

This artic/e applies a theorem of Nash equilibrium under uncertainty (Dow & Werlang, 1994) to the classic Coumot model of oligopolistic competition. It shows, in particular, how one can map all Coumot equilibrium (which includes the monopoly and the null solutions) with only a function of uncertainty aversion coefficients of producers. The effect of variations in these parameters over the equilibrium quantities are studied, also assuming exogenous increases in the number of matching firms in the game. The Cournot solutions under uncertainty are compared with the monopolistic one. It shows principally that there is an uncertainty aversion level in the industry such that every aversion coefficient beyond it induces firms to produce an aggregate output smaller than the monopoly output. At the end of the artic/e equilibrium solutions are specialized for Linear Demand and for Coumot duopoly. Equilibrium analysis in the symmetric case allows to identify the uncertainty aversion coefficient for the whole industry as a proportional lack of information cost which would be conveyed by market price in the perfect competition case (Lerner Index).

Nash equilibrium under knightian uncertainty: breaking down backward induction (extensively revised version)

We define Nash equilibrium for two-person normal form games in the presence of uncertainty, in the sense of Knight(1921). We use the fonna1iution of uncertainty due to Schmeidler and Gilboa. We show tbat there exist Nash equilibria for any degree of uncertainty, as measured by the uncertainty aversion (Dow anel Wer1ang(l992a». We show by example tbat prudent behaviour (maxmin) can be obtained as an outcome even when it is not rationaliuble in the usual sense. Next, we break down backward industion in the twice repeated prisoner's dilemma. We link these results with those on cooperation in the finitely repeated prisoner's dilemma obtained by Kreps-Milgrom-Roberts-Wdson(1982), and withthe 1iterature on epistemological conditions underlying Nash equilibrium. The knowledge notion implicit in this mode1 of equilibrium does not display logical omniscience.

Nash equilibrium under knightian uncertainty: breaking-down backward induction

We present two alternative definitions of Nash equilibrium for two person games in the presence af uncertainty, in the sense of Knight. We use the formalization of uncertainty due to Schmeidler and Gilboa. We show that, with one of the definitions, prudent behaviour (maxmin) can be obtained as an outcome even when it is not rationalizable in the usual sense. Most striking is that with the Same definition we break down backward induction in the twice repeated prisoner's dilemma. We also link these results with the Kreps-Milgrom-Roberts-Wilson explanation of cooperation in the finitely repeated prisoner's dilemma.

A notion of subgame perfect Nash equilibrium under knightian uncertainty

We define a subgame perfect Nash equilibrium under Knightian uncertainty for two players, by means of a recursive backward induction procedure. We prove an extension of the Zermelo-von Neumann-Kuhn Theorem for games of perfect information, i. e., that the recursive procedure generates a Nash equilibrium under uncertainty (Dow and Werlang(1994)) of the whole game. We apply the notion for two well known games: the chain store and the centipede. On the one hand, we show that subgame perfection under Knightian uncertainty explains the chain store paradox in a one shot version. On the other hand, we show that subgame perfection under uncertainty does not account for the leaving behavior observed in the centipede game. This is in contrast to Dow, Orioli and Werlang(1996) where we explain by means of Nash equilibria under uncertainty (but not subgame perfect) the experiments of McKelvey and Palfrey(1992). Finally, we show that there may be nontrivial subgame perfect equilibria under uncertainty in more complex extensive form games, as in the case of the finitely repeated prisoner's dilemma, which accounts for cooperation in early stages of the game.

A notion of subgame perfect nash equilibrium under knightian uncertainty

We define a subgame perfect Nash equilibrium under Knightian uncertainty for two players, by means of a recursive backward induction procedure. We prove an extension of the Zermelo-von Neumann-Kuhn Theorem for games of perfect information, i. e., that the recursive procedure generates a Nash equilibrium under uncertainty (Dow and Werlang(1994)) of the whole game. We apply the notion for two well known games: the chain store and the centipede. On the one hand, we show that subgame perfection under Knightian uncertainty explains the chain store paradox in a one shot version. On the other hand, we show that subgame perfection under uncertainty does not account for the leaving behavior observed in the centipede game. This is in contrast to Dow, Orioli and Werlang(1996) where we explain by means of Nash equilibria under uncertainty (but not subgame perfect) the experiments of McKelvey and Palfrey(1992). Finally, we show that there may be nontrivial subgame perfect equilibria under uncertainty in more complex extensive form games, as in the case of the finitely repeated prisoner's dilemma, which accounts for cooperation in early stages of the game .

Toward a rational-choice foundation of non-additive theories

A classical argument of de Finetti holds that Rationality implies Subjective Expected Utility (SEU). In contrast, the Knightian distinction between Risk and Ambiguity suggests that a rational decision maker would obey the SEU paradigm when the information available is in some sense good, and would depart from it when the information available is not good. Unlike de Finetti's, however, this view does not rely on a formal argument. In this paper, we study the set of all information structures that might be availabe to a decision maker, and show that they are of two types: those compatible with SEU theory and those for which SEU theory must fail. We also show that the former correspond to "good" information, while the latter correspond to information that is not good. Thus, our results provide a formalization of the distinction between Risk and Ambiguity. As a consequence of our main theorem (Theorem 2, Section 8), behavior not-conforming to SEU theory is bound to emerge in the presence of Ambiguity. We give two examples of situations of Ambiguity. One concerns the uncertainty on the class of measure zero events, the other is a variation on Ellberg's three-color urn experiment. We also briefly link our results to two other strands of literature: the study of ambiguous events and the problem of unforeseen contingencies. We conclude the paper by re-considering de Finetti's argument in light of our findings.

The management of innovation under ambiguity

De récents développements en théorie de la decision ont largement enrichi notre connaissance de la notion d'incertitude knightienne, usuellement appelée ambiguïté. Néanmoins ces dévelopement tardent à être intégrés au coeur de la théorie économique. Nous suggérons que l'analyse de phénonèmes économiques tel que l'innovation et la Recherche et Développement gagnerait à intégrer les modèles de décision en situation d'ambiguïté. Nous étayons notre propos en analysant l'allocation des droits de propriété d'une découverte. Les deux premières parties de la présentation s'inspire d'un modèle d'Aghion et de Tirole, The Management of Innovation, portant sur l'allocation des droits de propriété entre une unité de recherche et un investisseur. Il est démontré qu'un désaccord entre les agents sur la technologie de recherche affecte leur niveau d'effort, l'allocation des droits de propriété et l'allocation des revenus subséquents. Finalement, nous examinons une situation où plusieurs chercheurs sont en compétition en s'inspirant du traitement de l'incertitude de Savage. La présence d'ambuïgité affecte le comportement des agents et l'allocation des droits de propriétés de manière qui n'est pas captée en assumant l'hypothèse de risque.

Does uncertainty cause inertia in decision making? An experimental study of the role of regret aversion and indecisiveness

Previous research has shown that often there is clear inertia in individual decision making---that is, a tendency for decision makers to choose a status quo option. I conduct a laboratory experiment to investigate two potential determinants of inertia in uncertain environments: (i) regret aversion and (ii) ambiguity-driven indecisiveness. I use a between-subjects design with varying conditions to identify the effects of these two mechanisms on choice behavior. In each condition, participants choose between two simple real gambles, one of which is the status quo option. I find that inertia is quite large and that both mechanisms are equally important.

Risk, ambiguity, and diversification

Attitudes toward risk influence the decision to diversify among uncertain options. Yet, because in most situations the options are ambiguous, attitudes toward ambiguity may also play an important role. I conduct a laboratory experiment to investigate the effect of ambiguity on the decision to diversify. I find that diversification is more prevalent and more persistent under ambiguity than under risk. Moreover, excess diversification under ambiguity is driven by participants who stick with a status quo gamble when diversification among gambles is not feasible. This behavioral pattern cannot be accommodated by major theories of choice under ambiguity.

Non-emptiness of the alpha-core

We prove non-emptiness of the alpha-core for balanced games with non-ordered preferences, extending and generalizing in several aspects the results of Scarf (1971), Border (1984), Florenzano (1989), Yannelis (1991) and Kajii (1992). In particular we answer an open question in Kajii (1992) regarding the applicability of the non-emptiness results to models with infinite dimensional strategy spaces. We provide two models with Knightian and voting preferences for which the results of Scarf (1971) and Kajii (1992) cannot be applied while our non-emptiness result applies.

Integral representation with convex capacities that are squeeze of (additive) probability measures

In this paper I will investigate the conditions under which a convex capacity (or a non-additive probability which exhibts uncertainty aversion) can be represented as a squeeze of a(n) (additive) probability measure associate to an uncertainty aversion function. Then I will present two alternatives forrnulations of the Choquet integral (and I will extend these forrnulations to the Choquet expected utility) in a parametric approach that will enable me to do comparative static exercises over the uncertainty aversion function in an easy way.