Statistical and Probabilistic Methods in the Game Theory

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Engineering Mathematics".

Deadline for manuscript submissions: closed (30 June 2021) | Viewed by 13846

Special Issue Editor

Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wybrzeze Wyspianskiego 27, PL-50-370 Wrocław, Poland
Interests: mathematics; applied probability; statistics; computer science
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

The purpose of this Special Issue is to highlight the importance of combining probabilistic modeling methods in connection with game theory. When trying to rationally deal with issues in technology, biology, and economics, the states of nature that are difficult to predict, as well as emotions and other determinants of decision-makers, are key. Usually, it is possible to strictly determine the rational behavior under known conditions, but proper consideration of uncertainty is constantly under investigation. We want to present research in this direction in this volume. The subject is included, among others, in monographs such as [1–3]. The calibration of stochastic models should be made by statistical methods (see, e.g., [4])

The theme of this issue is devoted to game-theoretic modeling in a wide range of fields. Different optimality or rationality principles are presented. The problems of stable cooperation and myopic behavior in multi-agent systems will be investigated. Among others, the optimal location and allocation of the resource on the plane and related to its optimal routing in networking are considered.

Game-theoretic models of systems related to the IT market, in particular, mobile operators, cloud, high-performance, and distributed computing, the Internet of things market will be discussed. Competitive situations and various forms of coordination and cooperation will be considered. A comparison of the costs of the system in all these cases will be made, which will make suggestions for changing the design of the organization of the systems.

A new formulation for differential games will be suggested. Studies of game-theoretic models with asymmetric participants and vector payments will be discussed, and concepts of solutions in dynamic games of this type will be proposed. Linear-quadratic dynamic games related to the problem of resource allocation, allowing the construction of potential, will be investigated. In some models, we suppose that the players leave the game at random time instants Ti with known probability distribution Fi, which can be different for different players. The example of differential games with an environment context is represented. The Nash equilibrium is calculated under some circumstances.

Further research on an optimal stopping problem for point processes will be presented. The illustrated examples are extensions of various online auctions and research on the "debugging problem". The typical process of software testing consists of checking subroutines. In the beginning, many kinds of bugs are searching. The consecutive stopping times are moments when the expert stops general testing of modules and starts checking the most important, dangerous types of error. Similarly, in proofreading, it is natural to look at typographic and grammar errors at the same time. Next, we are looking for language mistakes.

Details of other models will be subject to contributed papers.

References

  1. Carmona, R. and Delarue, F. Probabilistic theory of mean field games with applications. II, volume 84 of Probability Theory and Stochastic Modelling. Springer, Cham, 2018.
  2. Bruss, F.T. and Cam, L.L. Game theory, optimal stopping, probability and statistics, volume 35 of Institute of Mathematical Statistics Lecture Notes—Monograph Series. Institute of Mathematical Statistics, Beachwood, OH, 2000. Papers in honor of Thomas S. Ferguson
  3. Aumann, R.J. and Hart, S. Handbook of game theory with economic applications. II, volume 11 of Handbooks in Economics. North-Holland Publishing Co., Amsterdam, 1994.
  4. Ferguson, T.S.; Shapley, L.S.; MacQueen, J.B. Statistics, probability and game theory, volume 30 of Institute of Mathematical Statistics Lecture Notes—Monograph Series. Institute of Mathematical Statistics, Hayward, CA, 1996. Papers in Honor of David Blackwell.

Prof. Krzysztof J. Szajowski
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Published Papers (8 papers)

Order results
Result details
Select all
Export citation of selected articles as:

Research

6 pages, 229 KiB  
Article
Can Producers’ Price War End Up in an Optimal Allocation?
by Marta Kornafel
Mathematics 2022, 10(2), 265; https://doi.org/10.3390/math10020265 - 16 Jan 2022
Viewed by 1091
Abstract
The paper presents a theoretical framework for the phenomenon of the price war in the context of general equilibrium, with special attention to the production system. The natural question that arises is whether Nash-optimal production plans being the reactions to the changing prices [...] Read more.
The paper presents a theoretical framework for the phenomenon of the price war in the context of general equilibrium, with special attention to the production system. The natural question that arises is whether Nash-optimal production plans being the reactions to the changing prices can finally approximate a Nash-optimal production plan at the end of this war. To provide an answer, the production system is described as a parametric-multicriteria game. Referring to some results on the lower semicontinuty of the parametric weak-multicriteria Nash equilibria, we provide a positive answer for the stated problem. Full article
(This article belongs to the Special Issue Statistical and Probabilistic Methods in the Game Theory)
16 pages, 335 KiB  
Article
Combined Games with Randomly Delayed Beginnings
by F. Thomas Bruss
Mathematics 2021, 9(5), 534; https://doi.org/10.3390/math9050534 - 04 Mar 2021
Viewed by 1050
Abstract
This paper presents two-person games involving optimal stopping. As far as we are aware, the type of problems we study are new. We confine our interest to such games in discrete time. Two players are to chose, with randomised choice-priority, between two games [...] Read more.
This paper presents two-person games involving optimal stopping. As far as we are aware, the type of problems we study are new. We confine our interest to such games in discrete time. Two players are to chose, with randomised choice-priority, between two games G1 and G2. Each game consists of two parts with well-defined targets. Each part consists of a sequence of random variables which determines when the decisive part of the game will begin. In each game, the horizon is bounded, and if the two parts are not finished within the horizon, the game is lost by definition. Otherwise the decisive part begins, on which each player is entitled to apply their or her strategy to reach the second target. If only one player achieves the two targets, this player is the winner. If both win or both lose, the outcome is seen as “deuce”. We motivate the interest of such problems in the context of real-world problems. A few representative problems are solved in detail. The main objective of this article is to serve as a preliminary manual to guide through possible approaches and to discuss under which circumstances we can obtain solutions, or approximate solutions. Full article
(This article belongs to the Special Issue Statistical and Probabilistic Methods in the Game Theory)
Show Figures

Figure 1

15 pages, 391 KiB  
Article
Cooperative Stochastic Games with Mean-Variance Preferences
by Elena Parilina and Stepan Akimochkin
Mathematics 2021, 9(3), 230; https://doi.org/10.3390/math9030230 - 25 Jan 2021
Viewed by 1538
Abstract
In stochastic games, the player’s payoff is a stochastic variable. In most papers, expected payoff is considered as a payoff, which means the risk neutrality of the players. However, there may exist risk-sensitive players who would take into account “risk” measuring their stochastic [...] Read more.
In stochastic games, the player’s payoff is a stochastic variable. In most papers, expected payoff is considered as a payoff, which means the risk neutrality of the players. However, there may exist risk-sensitive players who would take into account “risk” measuring their stochastic payoffs. In the paper, we propose a model of stochastic games with mean-variance payoff functions, which is the sum of expectation and standard deviation multiplied by a coefficient characterizing a player’s attention to risk. We construct a cooperative version of a stochastic game with mean-variance preferences by defining characteristic function using a maxmin approach. The imputation in a cooperative stochastic game with mean-variance preferences is supposed to be a random vector. We construct the core of a cooperative stochastic game with mean-variance preferences. The paper extends existing models of discrete-time stochastic games and approaches to find cooperative solutions in these games. Full article
(This article belongs to the Special Issue Statistical and Probabilistic Methods in the Game Theory)
Show Figures

Figure 1

26 pages, 799 KiB  
Article
Mixed Mechanisms for Auctioning Ranked Items
by Estrella Alonso, Joaquín Sánchez-Soriano and Juan Tejada
Mathematics 2020, 8(12), 2227; https://doi.org/10.3390/math8122227 - 15 Dec 2020
Viewed by 1766
Abstract
This paper deals with the problem of designing and choosing auctioning mechanisms for multiple commonly ranked objects as, for instance, keyword auctions in search engines on Internet. We shall adopt the point of view of the auctioneer who has to select the auction [...] Read more.
This paper deals with the problem of designing and choosing auctioning mechanisms for multiple commonly ranked objects as, for instance, keyword auctions in search engines on Internet. We shall adopt the point of view of the auctioneer who has to select the auction mechanism to be implemented not only considering its expected revenue, but also its associated risk. In order to do this, we consider a wide parametric family of auction mechanisms which contains the generalizations of discriminatory-price auction, uniform-price auction and Vickrey auction. For completeness, we also analyze the Generalized Second Price (GSP) auction which is not in the family. The main results are: (1) all members of the family satisfy the four basic properties of fairness, no over-payment, optimality and efficiency, (2) the Bayesian Nash equilibrium and the corresponding value at risk for the auctioneer are obtained for the considered auctions, (3) the GSP and all auctions in the family provide the same expected revenue, (4) there are new interesting auction mechanisms in the family which have a lower value at risk than the GSP and the classical auctions. Therefore, a window opens to apply new auction mechanisms that can reduce the risk to be assumed by auctioneers. Full article
(This article belongs to the Special Issue Statistical and Probabilistic Methods in the Game Theory)
Show Figures

Figure 1

21 pages, 449 KiB  
Article
A Differential Game with Random Time Horizon and Discontinuous Distribution
by Anastasiia Zaremba, Ekaterina Gromova and Anna Tur
Mathematics 2020, 8(12), 2185; https://doi.org/10.3390/math8122185 - 08 Dec 2020
Cited by 5 | Viewed by 1543
Abstract
One class of differential games with random duration is considered. It is assumed that the duration of the game is a random variable with values from a given finite interval. The cumulative distribution function (CDF) of this random variable is assumed to be [...] Read more.
One class of differential games with random duration is considered. It is assumed that the duration of the game is a random variable with values from a given finite interval. The cumulative distribution function (CDF) of this random variable is assumed to be discontinuous with two jumps on the interval. It follows that the player’s payoff takes the form of the sum of integrals with different but adjoint time intervals. In addition, the first interval corresponds to the zero probability of the game to be finished, which results in terminal payoff on this interval. The method of construction optimal solution for the cooperative scenario of such games is proposed. The results are illustrated by the example of differential game of investment in the public stock of knowledge. Full article
(This article belongs to the Special Issue Statistical and Probabilistic Methods in the Game Theory)
Show Figures

Figure 1

20 pages, 552 KiB  
Article
Prioritised Learning in Snowdrift-Type Games
by Maria Kleshnina, Sabrina S. Streipert, Jerzy A. Filar and Krishnendu Chatterjee
Mathematics 2020, 8(11), 1945; https://doi.org/10.3390/math8111945 - 04 Nov 2020
Cited by 2 | Viewed by 1752
Abstract
Cooperation is a ubiquitous and beneficial behavioural trait despite being prone to exploitation by free-riders. Hence, cooperative populations are prone to invasions by selfish individuals. However, a population consisting of only free-riders typically does not survive. Thus, cooperators and free-riders often coexist in [...] Read more.
Cooperation is a ubiquitous and beneficial behavioural trait despite being prone to exploitation by free-riders. Hence, cooperative populations are prone to invasions by selfish individuals. However, a population consisting of only free-riders typically does not survive. Thus, cooperators and free-riders often coexist in some proportion. An evolutionary version of a Snowdrift Game proved its efficiency in analysing this phenomenon. However, what if the system has already reached its stable state but was perturbed due to a change in environmental conditions? Then, individuals may have to re-learn their effective strategies. To address this, we consider behavioural mistakes in strategic choice execution, which we refer to as incompetence. Parametrising the propensity to make such mistakes allows for a mathematical description of learning. We compare strategies based on their relative strategic advantage relying on both fitness and learning factors. When strategies are learned at distinct rates, allowing learning according to a prescribed order is optimal. Interestingly, the strategy with the lowest strategic advantage should be learnt first if we are to optimise fitness over the learning path. Then, the differences between strategies are balanced out in order to minimise the effect of behavioural uncertainty. Full article
(This article belongs to the Special Issue Statistical and Probabilistic Methods in the Game Theory)
Show Figures

Figure 1

10 pages, 263 KiB  
Article
Secretary Problem with Possible Errors in Observation
by Marek Skarupski
Mathematics 2020, 8(10), 1639; https://doi.org/10.3390/math8101639 - 23 Sep 2020
Cited by 1 | Viewed by 1918
Abstract
The classical secretary problem models a situation in which the decision maker can select or reject in the sequential observation objects numbered by the relative ranks. In theoretical studies, it is known that the strategy is to reject the first 37% of objects [...] Read more.
The classical secretary problem models a situation in which the decision maker can select or reject in the sequential observation objects numbered by the relative ranks. In theoretical studies, it is known that the strategy is to reject the first 37% of objects and select the next relative best one. However, an empirical result for the problem is that people do not apply the optimal rule. In this article, we propose modeling doubts of decision maker by considering a modification of the secretary problem. We assume that the decision maker can not observe the relative ranks in a proper way. We calculate the optimal strategy in such a problem and the value of the problem. In special cases, we also combine this problem with the no-information best choice problem and a no-information second-best choice problem. Full article
(This article belongs to the Special Issue Statistical and Probabilistic Methods in the Game Theory)
Show Figures

Figure 1

25 pages, 647 KiB  
Article
When Inaccuracies in Value Functions Do Not Propagate on Optima and Equilibria
by Agnieszka Wiszniewska-Matyszkiel and Rajani Singh
Mathematics 2020, 8(7), 1109; https://doi.org/10.3390/math8071109 - 06 Jul 2020
Cited by 1 | Viewed by 1579
Abstract
We study general classes of discrete time dynamic optimization problems and dynamic games with feedback controls. In such problems, the solution is usually found by using the Bellman or Hamilton–Jacobi–Bellman equation for the value function in the case of dynamic optimization and a [...] Read more.
We study general classes of discrete time dynamic optimization problems and dynamic games with feedback controls. In such problems, the solution is usually found by using the Bellman or Hamilton–Jacobi–Bellman equation for the value function in the case of dynamic optimization and a set of such coupled equations for dynamic games, which is not always possible accurately. We derive general rules stating what kind of errors in the calculation or computation of the value function do not result in errors in calculation or computation of an optimal control or a Nash equilibrium along the corresponding trajectory. This general result concerns not only errors resulting from using numerical methods but also errors resulting from some preliminary assumptions related to replacing the actual value functions by some a priori assumed constraints for them on certain subsets. We illustrate the results by a motivating example of the Fish Wars, with singularities in payoffs. Full article
(This article belongs to the Special Issue Statistical and Probabilistic Methods in the Game Theory)
Show Figures

Figure 1

Back to TopTop