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1

On the one-shot two-person zero-sum game in football from a penalty kicker’s perspective

Ekhosuehi V.

Operations Research and Decisions

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2018

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Vol. 28, No. 3

17--27

EN

A penalty kicker’s problem in football has been modelled. The study took into consideration different directions in which the ball can be struck and goalkeepers’ success at defending shots. The strategic form of the game that can be used to predict how the kicker should optimally randomise his strategies has been modelled as a non-linear game-theoretic problem from a professional kicker’s viewpoint. The equilibrium of the game (i.e., the pair of mutually optimal mixed strategies) was obtained from the game-theoretic problem by reducing it to a linear programming problem and the two-phase simplex method was adopted to solve this problem. The optimal solution to the game indicates that the kicker never chooses to kick the ball off target, to the goalpost or to the crossbar, but rather chooses to kick the ball in the opposite direction to the one where the goalkeeper is most likely to successfully defend from past history.

2

Toehold purchase problem: a comparative analysis of two strategies

Banakh I., Banakh T., Vovk M., Trisch P.

ECONTECHMOD : An International Quarterly Journal on Economics of Technology and Modelling Processes

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2015

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Vol. 4, No 1

3--10

EN

Toehold purchase, defined here as purchase of one share in a firm by an investor preparing a tender offer to acquire majority of shares in it, reduces by one the number of shares this investor needs for majority. In the paper we construct mathematical models for the toehold and no-toehold strategies and compare the expected profits of the investor and the probabilities of takeover the firm in both strategies. It turns out that the expected profits of the investor in both strategies coincide. On the other hand, the probability of takeover the firm using the toehold strategy is considerably higher comparing to the no-toehold strategy. In the analysis of the models we apply the apparatus of incomplete Beta functions and some refined bounds for central binomial coefficients.

3

Alokacja zasobu w warunkach niepewności: modele decyzyjne i procedury obliczeniowe

Gaspars H.

Badania Operacyjne i Decyzje

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2007

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nr 1

5-27

PL

W literaturze można znaleźć omówienie deterministycznej i stochastycznej odmiany zagadnienia rozdziału zasobu. Celem niniejszego opracowania jest sformułowanie modeli optymalizacyjnych, mających zastosowanie w zagadnieniu alokacji zasobu w warunkach niepewności, z którą mamy do czynienia wówczas, gdy zyski wynikające ze skierowania danej ilości środka do konkretnej działalności są opisane jako zmienne losowe o nieznanym rozkładzie. Autorka przedstawia cztery modele, uwzględniające różne postawy decydentów wobec stanów natury, które mogą wystąpić. W tym celu odwołuje się do reguł Walda, Hurwicza, Bayes’a i Savage’a. Analizowane są również możliwe procedury obliczeniowe, pozwalające wyznaczyć optymalne rozwiązanie dla każdego przypadku. Oprócz ogólnej metody programowania dynamicznego, wykorzystać można również dwie metody uproszczone, stosowane w deterministycznej wersji zagadnienia, gdy decyzje podejmowane są w warunkach niepewności. Jednakże te dwie procedury wymagają spełnienia dodatkowych założeń, które częściowo różnią się od założeń sformułowanych dla deterministycznej odmiany rozpatrywanego problemu.

EN

The deterministic and stochastic versions of the resource allocation problem have already been discussed in the literature. The goal of this contribution is to formulate optimization models applicable to the problem of resource allocation under uncertainty, which signifies that profits resulting from the assignment of a quantity of resource to a given activity, are defined as random variables with unknown distribution. The author presents four models depending on the attitude of the decision-maker towards states of nature that may occur, and refers to the rules formulated by Wald, Hurwicz, Bayes and Savage to this end. Possible computational procedures, allowing finding the optimal solution for each case, are also analyzed. Apart from the dynamic programming, two simplified methods used for the deterministic version of resource allocation can also be applied when decisions are made under uncertainty. However, these two methods require that the problem fulfill additional assumptions, which are partially different from those formulated for the deterministic approach.