TheN-armed bandit with unimodal structure

In this paper we study a special class of bandit problems, which are characterized by a unimodal structure of the expected rewards of the arms. In Section 1, the motivation for studying this problem is explained. In the next two sections, two different decision procedures are analyzed, which are based on a stochastic approximation of the best arm of the bandit. Finally, in Section 4, a special procedure is discussed and some numerical data are presented, which were obtained by applying it to a concreteN-armed bandit with unimodal structure.     

On two properties of an unequal probability sampling scheme

Summary For an inclusion probability proportional to size (IPPS) sampling scheme recently proposed by Saxena, Singh and Srivastava (1986), it is shown that under certain simple verifiable conditions (1) the Horvitz-Thompson (1952) estimator based on it has a smaller variance than the variance of the Hansen-Hurwitz (1943) estimator based on probability proportional to size (PPS) sampling with replacement (WR) both involving the same size-measures and the expected sample size in the former being equal to the number of draws in the latter and (2) the Yates-Grundy (1953) estimator for the variance of the Horvitz-Thompson estimator based on this IPPS scheme is uniformly non-negative.     

On monotone optimal decision rules and the stay-on-a-winner rule for the two-armed bandit

Summary Consider the following optimization problem: Find a decision rule such thatw(x, (x))=max _a w(x, a) for allx under the constraint (x)D (x). We give conditions for the existence of monotone optimal decision rules . The term monotone is used in a general sense. The well-known stay-on-a-winner rules for the two-armed bandit can be characterized as monotone decision rules by including the stage number intox and using a special ordering onx. This enables us to give simple conditions for the existence of optimal rules that are stay-on-a-winner rules. We extend results ofBerry andKalin/Theodorescu to the case of dependent arms.     

On the estimation of location parameters in the multivariate one sample and two sample problems

Summary In this paper we consider the problem of estimating the vectors of location parameters in the multivariate one sample and two sample problems. These estimators are obtained through the use of the multivariate rank order statistics such as theWilcoxon or the normal scores statistic considered by the authors inPuri, Sen [1966] andSen, Puri [1967] for the corresponding testing problems. The distribution of these estimators is shown to be symmetric with respect to the parameters being estimated. These estimators are translation invariant, robust and asymptotically normal. Their asymptotic relative efficiencies with respect to the estimators based on the vector of means and medians are discussed by applying the criterion ofWilks generalized variance [Anderson, p. 166]. In particular, it is shown that the estimators based on the multivariate normal scores statistics are asymptotically as efficient as the ones based on the method of least squares when the parent distributions are normal. Research sponsored by National Science Foundation Grant No. GP-12462, and by Research Grant, GM-12868 from the N.I.H., Public Health Service.     

A note on strategies for bandit problems with infinitely many arms

A bandit problem consisting of a sequence of n choices (n) from a number of infinitely many Bernoulli arms is considered. The parameters of Bernoulli arms are independent and identically distributed random variables from a common distribution F on the interval [0,1] and F is continuous with F(0)=0 and F(1)=1. The goal is to investigate the asymptotic expected failure rates of k-failure strategies, and obtain a lower bound for the expected failure proportion over all strategies presented in Berry et al. (1997). We show that the asymptotic expected failure rates of k-failure strategies when 0<b1 and a lower bound can be evaluated if the limit of the ratio F(1)–F(t) versus (1–t)^b exists as t1^– for some b>0.     

On probability models in voting theory

This paper illustrates the use of probability models to study properties of voting rules. In particular, a simple occupancy distribution and its limiting Dlrichlet form are introduced, corresponding to simplifying assumptions about voters' preferences. We use as illustrations the occurrence probability of the Condorcet Paradox, a vintage problem in social choice theory, along with the related concept of Condorcet efficiency, a measure of goodness for voting rules. Further examples include properties of a lottery rule and the vulnerability of certain voting rules to strategic manipulation. Prospects for future work are indicated.     

On the probability of entry in potential competition cases

This paper explores, both theoretically and empirically, the extent to which having ‘other potential entrants’ to a market injected into potential-competition cases has been a worthwhile enterprise. This two-pronged attack leads to the conclusion that (he attempt to specifically identify potential entrants probably results in our deluding ourselves as to the likelihood that entry will occur, and that unless a convincing argument can be made to the effect that any given potential entrant from a stockpile of unnamed potential entrants is ‘reasonably likely’ to elect the entry option, expanding lhat stockpile will be similarly misleading.     

On solving a linear program with one quadratic constraint

A solution procedure for linear programs with one convex quadratic constraint is suggested. The method finds an optimal solution in finitely many iterations.Fractional programming, quadratic programming.

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Sunto Si propone una procedura di soluzione per un programma lineare con un vincolo quadratico convesso.Il metodo perviene a una soluzione ottima in un numero finito di iterazioni.

     

In-work policies in Europe: Killing two birds with one stone?

Generous social assistance has been held responsible for inactivity traps and social exclusion in several European countries, hence the recent trend of promoting employment through in-work transfers. Yet, the relative consensus on the need for ‘making work pay’ policies is muddied by a number of concerns relative to the design of the reforms and the treatment of the family dimension. Relying on EUROMOD, a EU-15 integrated tax-benefit microsimulation software, we simulate two types of in-work benefits. The first one is means-tested on family income, in the fashion of the British Working Family Tax Credit, while the second is a purely individualized policy. Both reforms are built on the same cost basis (after behavioral responses) and simulated in three European countries suspected to experience large poverty traps, namely Finland, France and Germany. The potential labor supply responses to the reforms and the subsequent redistributive impacts are assessed for each country using a structural discrete-choice model. We compare how both reforms achieve poverty reduction and social inclusion (measured as the number of transitions into activity). All three countries present different initial conditions, including existing tax-benefit systems and distribution of incomes and wages. These sources of heterogeneity are exploited together with different labor supply elasticities to explain the cross-country differences in the impact of the reforms.     

On determining probability forecasts from betting odds

We show that the probabilities determined from betting odds using Shin’s model are more accurate forecasts than those determined using basic normalization or regression models. We also provide empirical evidence that some bookmakers are significantly different sources of probabilities in terms of forecasting accuracy, and that betting exchange odds are not always the best source, especially in smaller markets. The advantage of using Shin probabilities and the differences between bookmakers decrease with an increasing market size.     

On consensus through communication without a commonly known protocol

The present paper extends the standard model of pairwise communication among Bayesian agents to cases where the structure of the communication protocol is not commonly known. We show that, even under standard strict conditions on the structure of the protocols and the nature of the transmitted signals, a consensus may never be reached if very little asymmetric information about the protocol is introduced.     

On the existence of a probability measure compatible with a total preorder on a Boolean algebra

If an economic agent's beliefs about the relative likelihood of events are characterized by a total preorder ? on the algebra $\text{A}$ of events, the problem arises to know under which conditions, ? is representable by a probability measure. Here we show that there exists a probability measure compatible with a total preorder on a Boolean algebra, if and only if, the Boolean algebra is well bounded, weakly Archimedean, and perfectly separable, this last condition substituting for Villegas' monotone condition used in Chateauneuf and Jaffray (1984); if σ-additivity is required. Villegas' monotone condition, must merely be added.     

On the normality of probability distributions of inflation and GNP forecasts

This study analyzes mean probability distributions reported by ASA-NBER forecasters on two macroeconomic variables, GNP and the GNP implicit price deflator. In the derivation of expectations, a critical assertion has been that the aggregate average expectation can be regarded as coming from a normal distribution. We find that, in fact, this assumption should be rejected in favor of distributions which are more peaked and skewed. For IPD, they are mostly positively skewed, and for nominal GNP the reverse is true. We then show that a non-central scaled t-distribution fit the empirical distributions remarkably well. The practice of using the degree of consensus across a group of predictions as a measure of a typical forecasters' uncertainty about the prediction is called to question.     

On characterizing the probability of survival in a large competitive economy

We consider a Walrasian exchange economy in which an agent is characterized by a utility function, a random endowment vector, and a function that specifies the minimum expenditure necessary for survival at a given price system. If at any equilibrium price system, the income of the agent is no more than the minimum expenditure for survival, it is ruined. The main results characterize the probability of ruin when the number of agents is large. The implications of stochastic dependence among agents are explored.