The quantum monty hall problem

2002 ◽  
Vol 2 (5) ◽  
pp. 355-366
Author(s):  
G.M. D'Ariano ◽  
R.D. Gill ◽  
M. Keyl ◽  
B. Kummerer ◽  
H. Maassen ◽  
...  
Keyword(s):  
Optimal Strategy ◽  
Decision Problem ◽  
Prior Information ◽  
Classical Case ◽  
Statistical Decision ◽  
Statistical Decision Problem ◽  
Finite Set ◽  
Quantum Version ◽  
Person Game ◽  
Classical Game

We consider a quantum version of a well-known statistical decision problem, whose solution is, at first sight, counter-intuitive to many. In the quantum version a continuum of possible choices (rather than a finite set) has to be considered. It can be phrased as a two person game between a player P and a quiz master Q. Then P always has a strategy at least as good as in the classical case, while Q's best strategy results in a game having the same value as the classical game. We investigate the consequences of Q storing his information in classical or quantum ways. It turns out that Q's optimal strategy is to use a completely entangled quantum notepad, on which to encode his prior information.

More Data or Better Data? A Statistical Decision Problem

2017 ◽  
Vol 84 (4) ◽  
pp. 1583-1605 ◽  
Author(s):  
Jeff Dominitz ◽  
Charles F. Manski
Keyword(s):  
Data Collection ◽  
Decision Problem ◽  
Low Cost ◽  
Statistical Decision Theory ◽  
Sample Design ◽  
Statistical Decision ◽  
Finite Samples ◽  
Statistical Decision Problem ◽  
Sample Member

AbstractWhen designing data collection, crucial questions arise regarding how much data to collect and how much effort to expend to enhance the quality of the collected data. To make choice of sample design a coherent subject of study, it is desirable to specify an explicit decision problem. We use the Wald framework of statistical decision theory to study allocation of a budget between two or more sampling processes. These processes all draw random samples from a population of interest and aim to collect data that are informative about the sample realizations of an outcome. They differ in the cost of data collection and the quality of the data obtained. One may incur lower cost per sample member but yield lower data quality than another. Increasing the allocation of budget to a low-cost process yields more data, while increasing the allocation to a high-cost process yields better data. We initially view the concept of “better data” abstractly and then fix attention on two important cases. In both cases, a high-cost sampling process accurately measures the outcome of each sample member. The cases differ in the data yielded by a low-cost process. In one, the low-cost process has non-response and in the other it provides a low-resolution interval measure of each sample member’s outcome. In these settings, we study minimax-regret sample design for prediction of a real-valued outcome under square loss; that is, design which minimizes maximum mean square error. The analysis imposes no assumptions that restrict the unobserved outcomes. Hence, the decision maker must cope with both the statistical imprecision of finite samples and the partial identification of the true state of nature.

A continuous time inventory model

1966 ◽  
Vol 3 (2) ◽  
pp. 538-549 ◽  
Author(s):  
J. A. Bather
Keyword(s):  
Inventory Control ◽  
Discrete Time ◽  
Optimal Policy ◽  
Continuous Time ◽  
Decision Problem ◽  
Optimization Problem ◽  
Continuous Treatment ◽  
Inventory Level ◽  
Statistical Decision ◽  
Statistical Decision Problem

This paper discusses an optimization problem arising in the theory of inventory control. Much of the previous work in this field has been focused on the Arrow-Harris-Marschak model, [1], [2], in which the inventory level can be modified only at the instants of discrete time. Here, we shall be concerned with a continuous time analogue of the model, in an attempt to avoid the difficulties experienced in solving the basic integral equations. The approach was suggested by recent investigations of a statistical decision problem, [3], [5], which exploited the advantages of a continuous treatment. Although the ideas discussed here are relatively straightforward and involve strong assumptions as to the behavior of the inventory, the explicit character of the optimal policy is encouraging and particular solutions might nevertheless provide useful restocking procedures.

A continuous time inventory model

1966 ◽  
Vol 3 (02) ◽  
pp. 538-549 ◽  
Author(s):  
J. A. Bather
Keyword(s):  
Inventory Control ◽  
Discrete Time ◽  
Optimal Policy ◽  
Continuous Time ◽  
Decision Problem ◽  
Optimization Problem ◽  
Continuous Treatment ◽  
Inventory Level ◽  
Statistical Decision ◽  
Statistical Decision Problem

This paper discusses an optimization problem arising in the theory of inventory control. Much of the previous work in this field has been focused on the Arrow-Harris-Marschak model, [1], [2], in which the inventory level can be modified only at the instants of discrete time. Here, we shall be concerned with a continuous time analogue of the model, in an attempt to avoid the difficulties experienced in solving the basic integral equations. The approach was suggested by recent investigations of a statistical decision problem, [3], [5], which exploited the advantages of a continuous treatment. Although the ideas discussed here are relatively straightforward and involve strong assumptions as to the behavior of the inventory, the explicit character of the optimal policy is encouraging and particular solutions might nevertheless provide useful restocking procedures.

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