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6 pages, 229 KiB

Open AccessArticle

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 1470

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

Open AccessArticle

Combined Games with Randomly Delayed Beginnings

by F. Thomas Bruss

Mathematics 2021, 9(5), 534; https://doi.org/10.3390/math9050534 - 4 Mar 2021

Viewed by 1530

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

G_{1}

and

G_{2}

. 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)

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15 pages, 391 KiB

Open AccessArticle

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 1933

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)

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26 pages, 799 KiB

Open AccessArticle

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 2268

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)

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21 pages, 449 KiB

Open AccessArticle

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 - 8 Dec 2020

Cited by 5 | Viewed by 1972

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)

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20 pages, 552 KiB

Open AccessFeature PaperArticle

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 - 4 Nov 2020

Cited by 2 | Viewed by 2364

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)

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10 pages, 263 KiB

Open AccessArticle

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 2421

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)

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25 pages, 647 KiB

Open AccessArticle

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 - 6 Jul 2020

Cited by 1 | Viewed by 1914

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)

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