On Strong Centerpoints

Let $P$ be a set of $n$ points in $\mathbb{R}^d$ and $\mathcal{F}$ be a family of geometric objects. We call a point $x \in P$ a strong centerpoint of $P$ w.r.t $\mathcal{F}$ if $x$ is contained in all $F \in \mathcal{F}$ that contains more than $cn$ points from $P$, where $c$ is a fixed constant. A strong centerpoint does not exist even when $\mathcal{F}$ is the family of halfspaces in the plane. We prove the existence of strong centerpoints with exact constants for convex polytopes defined by a fixed set of orientations. We also prove the existence of strong centerpoints for abstract set systems with bounded intersection

On Representative Social Capital

This paper analyzes data for a random sample drawn from the Dutch population who reveal their propensity to invest and reward investments in building up social capital by means of an economic experiment.We find substantial heterogeneity and asymmetries in the propensity to invest and in the propensity to reward investments.In particular, we find strong evidence that the young, elderly, and low educated individuals invest relatively less, but are relatively more likely to reward investments in social capital.On the other hand, labor market participation, income, and religion do not have any significant impact on behavior in the experiment. Series: CentER Discussion PaperSocial Capital Investments;Experimental Economics;Representative samples

Actions and Beliefs: Estimating Distribution-Based Preferences Using a Large Scale Experiment with Probability Questions on Expectations

We combine the choice data of proposers and responders in the ultimatum game, their expectations elicited in the form of subjective probability questions, and the choice data of proposers ("dictator") in a dictator game to estimate a structural model of decision making under uncertainty.We use a large and representative sample of subjects drawn from the Dutch population.Our results indicate that there is considerable heterogeneity in preferences for equity in the population.Changes in preferences have an important impact on decisions of dictators in the dictator game and responders in the ultimatum game, but a smaller impact on decisions of proposer's in the ultimatum game, a result due to proposers subjective expectations about responders' decisions.The model which uses subjective data on expectations has better predictive power and lower noise level than a model which assumes that players have rational expectations.ultimatum game;inquity aversion;subjective expectations

On Representative Social Capital

This paper analyzes data for a random sample drawn from the Dutch population who reveal their capacity to provide and sustain social capital by their propensity to invest and reward investments by means of an economic experiment. We have three main results. First, we find that heterogeneity in behavior is characterized by several asymmetries -- men, the young and elderly, and low educated individuals invest relatively less, but reward significantly more investments. Second, higher expected levels of investments have a positive and significant effect on the level of investments themselves, corroborating the presence of social norms. Third, we compare our results with a laboratory experiment conducted with a student sample. We find that the student sample provides a lower bound of the population level of social capital.Social Capital Investments, Social Norms, Experiment, Representative Sample

Actions and Beliefs: Estimating Distribution-Based Preferences Using a Large Scale Experiment with Probability Questions on Expectations

We combine the choice data of proposers and responders in the ultimatum game, their expectations elicited in the form of subjective probability questions, and the choice data of proposers (“dictators”) in a dictator game to estimate a structural model of decision making under uncertainty. We use a large and representative sample of subjects drawn from the Dutch population. Our results indicate that there is considerable heterogeneity in preferences for equity in the population. Changes in preferences have an important impact on decisions of dictators in the dictator game and responders in the ultimatum game, but a smaller impact on decisions of proposers in the ultimatum game, a result due to proposer’s subjective expectations about responders’ decisions. The model which uses subjective data on expectations has better predictive power and lower noise level than a model which assumes that players have rational expectations. Afin de déceler un modèle structurel de prise de décision dans l'incertitude, nous combinons les données recueillies par i) les choix des deux types de joueurs dans le jeu de l'ultimatum (ceux qui proposent et ceux qui répondent), ii) leurs attentes obtenues sous la forme de questions à probabilité subjective, et iii) les choix des dictateurs (premiers joueurs) dans le jeu du dictateur. Nous utilisons un large échantillon représentatif des sujets dans la population néerlandaise. Nos résultats indiquent une hétérogénéité considérable dans la population sur les préférences en matière d'équité. La diversité de ces préférences a un impact important sur les décisions des dictateurs dans le jeu du dictateur, et sur les répondeurs dans le jeu de l'ultimatum, mais un impact plutôt moindre sur les décisions des joueurs qui proposent dans le jeu de l'ultimatum. Cette dernière observation est due aux attentes subjectives de ces joueurs face aux décisions des répondeurs. Le modèle qui utilise des données subjectives sur les attentes des joueurs prévoit mieux les choix des sujets et réduit plus le bruit qu'un modèle qui suppose que les joueurs ont des attentes rationnelles., attentes subjectives, aversion à l'injustice, jeu de l'ultimatum

Small Strong Epsilon Nets

Let P be a set of n points in $\mathbb{R}^d$. A point x is said to be a centerpoint of P if x is contained in every convex object that contains more than $dn\over d+1$ points of P. We call a point x a strong centerpoint for a family of objects $\mathcal{C}$ if $x \in P$ is contained in every object $C \in \mathcal{C}$ that contains more than a constant fraction of points of P. A strong centerpoint does not exist even for halfspaces in $\mathbb{R}^2$. We prove that a strong centerpoint exists for axis-parallel boxes in $\mathbb{R}^d$ and give exact bounds. We then extend this to small strong $\epsilon$-nets in the plane and prove upper and lower bounds for $\epsilon_i^\mathcal{S}$ where $\mathcal{S}$ is the family of axis-parallel rectangles, halfspaces and disks. Here $\epsilon_i^\mathcal{S}$ represents the smallest real number in $[0,1]$ such that there exists an $\epsilon_i^\mathcal{S}$-net of size i with respect to $\mathcal{S}$.Comment: 19 pages, 12 figure

On Representative Social Capital

This paper analyzes data for a random sample drawn from the Dutch population who reveal their capacity to provide and sustain social capital by their propensity to invest and reward investments by means of an economic experiment. We have three main results. First, we find that heterogeneity in behavior is characterized by several asymmetries -- men, the young and elderly, and low educated individuals invest relatively less, but reward significantly more investments. Second, higher expected levels of investments have a positive and significant effect on the level of investments themselves, corroborating the presence of social norms. Third, we compare our results with a laboratory experiment conducted with a student sample. We find that the student sample provides a lower bound of the population level of social capital