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Journal articles on the topic 'Partition function form games'

Author: Grafiati

Published: 3 June 2025

Last updated: 19 July 2025

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1

Alonso-Meijide, J. M., M. Álvarez-Mozos, M. G. Fiestras-Janeiro, and A. Jiménez-Losada. "Marginality and convexity in partition function form games." Mathematical Methods of Operations Research 94, no. 1 (2021): 99–121. http://dx.doi.org/10.1007/s00186-021-00748-8.

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AbstractIn this paper an order on the set of embedded coalitions is studied in detail. This allows us to define new notions of superaddivity and convexity of games in partition function form which are compared to other proposals in the literature. The main results are two characterizations of convexity. The first one uses non-decreasing contributions to coalitions of increasing size and can thus be considered parallel to the classic result for cooperative games without externalities. The second one is based on the standard convexity of associated games without externalities that we define using a partition of the player set. Using the later result, we can conclude that some of the generalizations of the Shapley value to games in partition function form lie within the cores of specific classic games when the original game is convex.

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2

ALBIZURI, M. J., J. ARIN, and J. RUBIO. "AN AXIOM SYSTEM FOR A VALUE FOR GAMES IN PARTITION FUNCTION FORM." International Game Theory Review 07, no. 01 (2005): 63–72. http://dx.doi.org/10.1142/s0219198905000405.

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Lucas and Trall (1963) defined the games in partition function form as a generalization of the cooperative games with transferable utility. In our work we propose by means of an axiomatic characterization a solution for such games in partition function form. This solution will be a generalization of the Shapley value (1953).

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3

DO, KIM HANG PHAM, and HENK NORDE. "THE SHAPLEY VALUE FOR PARTITION FUNCTION FORM GAMES." International Game Theory Review 09, no. 02 (2007): 353–60. http://dx.doi.org/10.1142/s021919890700145x.

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Different axiomatizations of the Shapley value for TU games can be found in the literature. The Shapley value has been generalized in several ways to the class of games in partition function form. In this paper we discuss another generalization of the Shapley value and provide a characterization.

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4

Kóczy, László Á. "A recursive core for partition function form games." Theory and Decision 63, no. 1 (2007): 41–51. http://dx.doi.org/10.1007/s11238-007-9030-x.

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5

Hoof, Simon. "Linear-State Differential Games in Partition Function Form." International Game Theory Review 21, no. 04 (2019): 1950006. http://dx.doi.org/10.1142/s0219198919500063.

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We introduce a partition function for [Formula: see text]-player linear-state cooperative differential games. The value of a coalition within a given coalition structure is defined as its noncooperative equilibrium payoff of a game played between the coalitions. We also define two core notions, namely, the cautious and the singleton core. If the game is convex, then the cores are nonempty. In order to illustrate the approach, we consider a symmetric game of pollution accumulation.

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6

FUKUDA, E., S. H. TIJS, R. BRÂNZEI, and S. MUTO. "COMPROMISING IN PARTITION FUNCTION FORM GAMES AND COOPERATION IN PERFECT EXTENSIVE FORM GAMES." International Game Theory Review 08, no. 03 (2006): 329–38. http://dx.doi.org/10.1142/s0219198906000941.

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In this paper reasonable payoff intervals for players in a game in partition function form (p.f.f. game) are introduced and used to define the notion of compromisable p.f.f. game. For a compromisable p.f.f. game a compromise value is defined for which an axiomatic characterization is provided. Also a generic subclass of games in extensive form of perfect information without chance moves is introduced. For this class of perfect extensive form games there is a natural credible way to define a p.f.f. game if the players consider cooperation. It turns out that the p.f.f. games obtained in this way are compromisable.

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7

VAN DEN NOUWELAND, ANNE, and MARCO SLIKKER. "The Position Value for Partition Function Form Network Games." Journal of Public Economic Theory 18, no. 2 (2016): 226–47. http://dx.doi.org/10.1111/jpet.12138.

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8

JU, YUAN. "THE CONSENSUS VALUE FOR GAMES IN PARTITION FUNCTION FORM." International Game Theory Review 09, no. 03 (2007): 437–52. http://dx.doi.org/10.1142/s0219198907001515.

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This paper studies a procedural and axiomatic extension of the consensus value [cf. Ju et al. (2007)] to the class of partition function form games. This value is characterized as the unique function that satisfies efficiency, complete symmetry, the quasi-null player property and additivity. By means of the transfer property, a second characterization is provided. Moreover, it is shown that the consensus value satisfies individual rationality under a superadditivity condition, and well balances the tradeoff between coalitional effects and externality effects. In this respect, explicit differences with other solution concepts are indicated.

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9

Huang, Chen-Ying, and Tomas Sjöström. "Implementation of the recursive core for partition function form games." Journal of Mathematical Economics 42, no. 6 (2006): 771–93. http://dx.doi.org/10.1016/j.jmateco.2005.08.009.

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10

Grabisch, Michel, and Yukihiko Funaki. "A coalition formation value for games in partition function form." European Journal of Operational Research 221, no. 1 (2012): 175–85. http://dx.doi.org/10.1016/j.ejor.2012.02.036.

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11

Bolger, Edward M. "A Class of Efficient Values for Games in Partition Function Form." SIAM Journal on Algebraic Discrete Methods 8, no. 3 (1987): 460–66. http://dx.doi.org/10.1137/0608038.

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12

Skibski, Oskar. "Complexity of Computing the Shapley Value in Partition Function Form Games." Journal of Artificial Intelligence Research 77 (August 2, 2023): 1237–74. http://dx.doi.org/10.1613/jair.1.14648.

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We study the complexity of computing the Shapley value in partition function form games. We focus on two representations based on marginal contribution nets (embedded MC-nets and weighted MC-nets) and five extensions of the Shapley value. Our results show that while weighted MC-nets are more concise than embedded MC-nets, they have slightly worse computational properties when it comes to computing the Shapley value: two out of five extensions can be computed in polynomial time for embedded MC-nets and only one for weighted MC-nets.

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13

HERNÁNDEZ-LAMONEDA, L., J. SÁNCHEZ-PÉREZ, and F. SÁNCHEZ-SÁNCHEZ. "THE CLASS OF EFFICIENT LINEAR SYMMETRIC VALUES FOR GAMES IN PARTITION FUNCTION FORM." International Game Theory Review 11, no. 03 (2009): 369–82. http://dx.doi.org/10.1142/s0219198909002364.

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In this paper we study linear symmetric solutions for the space of games in partition function form with n players. In particular, we provide an expression for all linear, symmetric and efficient solutions. Furthermore, adding an additional axiom, we identify a unique value satisfying these properties.

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14

Hu, Cheng-Cheng, and Yi-You Yang. "An axiomatic characterization of a value for games in partition function form." SERIEs 1, no. 4 (2010): 475–87. http://dx.doi.org/10.1007/s13209-009-0004-9.

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15

Alonso-Meijide, J. M., M. Álvarez-Mozos, and M. G. Fiestras-Janeiro. "Power Indices and Minimal Winning Coalitions for Simple Games in Partition Function Form." Group Decision and Negotiation 26, no. 6 (2017): 1231–45. http://dx.doi.org/10.1007/s10726-017-9542-x.

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16

Sánchez-Pérez, Joss. "A Note on a Class of Solutions for Games with Externalities Generalizing the Shapley Value." International Game Theory Review 17, no. 03 (2015): 1550003. http://dx.doi.org/10.1142/s0219198915500036.

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In this paper we study a family of extensions of the Shapley value for games in partition function form with n players. In particular, we provide a complete characterization for all linear, symmetric, efficient and null solutions in these environments. Finally, we relate our characterization result with other ways to extend the Shapley value in the literature.

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17

Nessah, Rabia, Moussa Larbani, and Tarik Tazdaït. "Coalitional ZP-Equilibrium in Games and its Existence." International Game Theory Review 17, no. 03 (2015): 1550002. http://dx.doi.org/10.1142/s0219198915500024.

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We introduce a solution concept for games in normal form with undetermined parameters, coalitional ZP-equilibrium, based on the notions of Z-equilibrium of [Zhukovskii and Chikrii [1994] Linear quadratic differential games, Kiev, Naoukova Doumka] and ZS-equilibrium of [Larbani and Lebbah [1999] A concept of equilibrium for a game under uncertainity. Europ. J. Oper. Res.117, 145–156]. For each coalition structure, ZP-equilibrium ensures both the stability of the partition and equilibrium of coalitional strategies (in Pareto sense). We show that under some quasiconcavity conditions on payoff functions, the coalitional ZP-equilibrium exists in compact, convex and continuous normal form games involving undetermined parameters.

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18

Naumova, Natalia I. "Computation Problems for Envy Stable Solutions of Allocation Problems with Public Resources." Contributions to Game Theory and Management 14 (2021): 302–11. http://dx.doi.org/10.21638/11701/spbu31.2021.22.

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We consider generalizations of TU games with restricted cooperation in partition function form and propose their interpretation as allocation problems with several public resources. Either all resources are goods or all resources are bads. Each resource is distributed between points of its set and permissible coalitions are subsets of the union of these sets. Each permissible coalition estimates each allocation of resources by its gain/loss function, that depends only on the restriction of the allocation on that coalition. A solution concept of "fair" allocation (envy stable solution) was proposed by the author in (Naumova, 2019). This solution is a simplification of the generalized kernel of cooperative games and it generalizes the equal sacrifice solution for claim problems. An allocation belongs to this solution if there do not exist special objections at this allocation between permissible coalitions. For several classes of such problems we describe methods for computation selectors of envy stable solutions.

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19

Toumasatos, Evangelos, and Stein Ivar Steinshamn. "Coalition Formation with Externalities: The Case of the Northeast Atlantic Mackerel Fishery in a Pre- and Post-Brexit Context." International Game Theory Review 20, no. 02 (2018): 1850001. http://dx.doi.org/10.1142/s0219198918500019.

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The partition function approach is applied to study coalition formation in the Northeast Atlantic mackerel fishery in the presence of externalities. Atlantic mackerel is mainly exploited by the European Union (EU), the United Kingdom (UK), Norway, the Faroe Islands and Iceland. Two games are considered. First, a four-player game where the UK is still a member of the EU. Second, a five-player game where the UK is no longer a member of the union. Each game is modeled in two stages. In the first stage, players form coalitions following a predefined set of rules. In the second stage, given the coalition structure that has been formed, each coalition chooses the economic strategy that maximizes its own net present value of the fishery, given the behavior of the other coalitions. The game is solved using backward induction to obtain the set of Nash equilibria coalition structures in pure strategies, if any. We find that the current management regime is among the stable coalition structures in all eight scenarios of the four-player game but in only one case of the five-player game. In addition, stability in the five-player game is sensitive to the growth function applied and the magnitude of the stock elasticity parameter.

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20

Ruan, Lang, Jin Chen, Qiuju Guo, Xiaobo Zhang, Yuli Zhang, and Dianxiong Liu. "Group Buying-Based Data Transmission in Flying Ad-Hoc Networks: A Coalition Game Approach." Information 9, no. 10 (2018): 253. http://dx.doi.org/10.3390/info9100253.

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In scenarios such as natural disasters and military strikes, it is common for unmanned aerial vehicles (UAVs) to form groups to execute reconnaissance and surveillance. To ensure the effectiveness of UAV communications, repeated resource acquisition issues and transmission mechanism designs need to be addressed urgently. Since large-scale UAVs will generate high transmission overhead due to the overlapping resource requirements, in this paper, we propose a resource allocation optimization method based on distributed data content in a Flying Ad-hoc network (FANET). The resource allocation problem with the goal of throughput maximization is constructed as a coalition game framework. Firstly, a data transmission mechanism is designed for UAVs to execute information interaction within the coalitions. Secondly, a novel mechanism of coalition selection based on group-buying is investigated for UAV coalitions to acquire data from the central UAV. The data transmission and coalition selection problem are modeled as coalition graph game and coalition formation game, respectively. Through the design of the utility function, we prove that both games have stable solutions. We also prove the convergence of the proposed approach with coalition order and Pareto order. Based on simulation results, coalition order based coalition selection algorithm (CO-CSA) and Pareto order based coalition selection algorithm (PO-CSA) are proposed to explore the stable coalition partition of system model. CO-CSA and PO-CSA can achieve higher data throughput than the contrast onetime coalition selection algorithm (Onetime-CSA) (at least increased by 34.5% and 16.9%, respectively). Besides, although PO-CSA has relatively lower throughput gain, its convergence times is on average 50.9% less than that of CO-CSA, which means that the algorithm choice is scenario-dependent.

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21

Гусев, Василий, Vasily Gusev, Владимир Мазалов, and Vladimir Mazalov. "Owen-stable coalition partitions in games with vector payoffs." Mathematical Game Theory and Applications 10, no. 3 (2019): 3–23. http://dx.doi.org/10.17076/mgta3_6.

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The paper is devoted to the study of multicriteria cooperative games with vector payoffs and coalition partition. The imputation which is based on the concept of the Owen value is proposed. We use it for the definition of stable coalition partition for bicriteria games. In three person cooperative game with 0-1 characteristic function the conditions under which the coalition partition is stable are found.

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22

DERKS, JEAN, and STEF TIJS. "ON MERGE PROPERTIES OF THE SHAPLEY VALUE." International Game Theory Review 02, no. 04 (2000): 249–57. http://dx.doi.org/10.1142/s0219198900000214.

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Given a transferable utility game, where the players merge into subgroups described by a partition, we address the following question: under which conditions on the characteristic function and partition, merging is beneficial if the Shapley value is applied. Our results can be positioned among the search for well-defined classes of games where merging of players is possible without utility loss in case the Shapley value is chosen as the outcome of the game, and we will report on two of these classes of games arising from telecommunication problems.

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23

Bloch, Francis, and Anne van den Nouweland. "Expectation formation rules and the core of partition function games." Games and Economic Behavior 88 (November 2014): 339–53. http://dx.doi.org/10.1016/j.geb.2014.10.012.

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24

Albizuri, M. Josune. "Games with externalities: games in coalition configuration function form." Mathematical Methods of Operations Research 72, no. 1 (2010): 171–86. http://dx.doi.org/10.1007/s00186-010-0314-5.

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25

Pinto, Tiago, Michael Wooldridge, and Zita Vale. "Consumer Flexibility Aggregation Using Partition Function Games With Non-Transferable Utility." IEEE Access 9 (2021): 51519–35. http://dx.doi.org/10.1109/access.2021.3069416.

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26

Bolger, E. M. "A set of axioms for a value for partition function games." International Journal of Game Theory 18, no. 1 (1989): 37–44. http://dx.doi.org/10.1007/bf01248494.

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27

Tiago, Pinto, Wooldridge Michael, and Vale Zita. "Consumer flexibility aggregation using partition function games with non-transferable utility." IEEE Access 9 (March 29, 2021): 51519–35. https://doi.org/10.1109/ACCESS.2021.3069416.

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This paper explores the aggregation of electricity consumers flexibility. A novel coalitional game theory model for partition function games with non-transferable utility is proposed. This model is used to formalize a game in which electricity consumers find coalitions among themselves in order to trade their consumption flexibility in the electricity market. Utility functions are defined to enable measuring the players preferences. Two case studies are presented, including a simple illustrative case, which assesses and explains the model in detail; and a large-scale scenario based on real data, comprising more than 20,000 consumers. Results show that the proposed model is able to reach solutions that are more suitable for the consumers when compared to the solutions achieved by traditional aggregation techniques in power and energy systems, such as clustering-based methodologies. The solutions found by the proposed model consider the perspectives from all players involved in the game and thus are able to reflect the rational behaviour of the involved players, rather than imposing an aggregation solution that is only beneficial from the perspective of the aggregator.

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28

HATZINIKITAS, AGAPITOS, and IOANNIS SMYRNAKIS. "CLOSED BOSONIC STRING PARTITION FUNCTION IN TIME INDEPENDENT EXACT pp-WAVE BACKGROUND." International Journal of Modern Physics A 21, no. 05 (2006): 995–1013. http://dx.doi.org/10.1142/s0217751x06025493.

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The modular invariance of the one-loop partition function of the closed bosonic string in four dimensions in the presence of certain homogeneous exact pp -wave backgrounds is studied. In the absence of an axion field, the partition function is found to be modular invariant and equal to the free field partition function. The partition function remains unchanged also in the presence of a fixed axion field. However, in this case, the covariant form of the action suggests summation over all possible twists generated by the axion field. This is shown to modify the partition function. In the light-cone gauge, the axion field generates twists only in the worldsheet σ-direction, so the resulting partition function is not modular invariant, hence wrong. To obtain the correct partition function one needs to sum over twists in the t-direction as well, as suggested by the covariant form of the action away from the light-cone gauge.

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29

Krishnamurthy, Dvijotham, Soumen Chakrabarti, and Subhasis Chaudhuri. "New closed-form bounds on the partition function." Machine Learning 72, no. 3 (2008): 205–29. http://dx.doi.org/10.1007/s10994-008-5072-8.

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30

Ahlgren, Scott, Olivia Beckwith, and Martin Raum. "Scarcity of congruences for the partition function." American Journal of Mathematics 145, no. 5 (2023): 1509–48. http://dx.doi.org/10.1353/ajm.2023.a907704.

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abstract: The arithmetic properties of the ordinary partition function $p(n)$ have been the topic of intensive study for the past century. Ramanujan proved that there are linear congruences of the form $p(\ell n+\beta)\equiv 0$ $({\rm mod}\;\ell)$ for the primes $\ell=5,7,11$, and it is known that there are no others of this form. On the other hand, for every prime $\ell\geq 5$ there are infinitely many examples of congruences of the form $p(\ell Q^m n+\beta)\equiv 0$ $({\rm mod}\;\ell)$ where $Q\geq 5$ is prime and $m\geq 3$. This leaves open the question of the existence of such congruences when $m=1$ or $m=2$ (no examples in these cases are known). We prove in a precise sense that such congruences, if they exist, are exceedingly scarce. Our methods involve a careful study of modular forms of half integral weight on the full modular group which are related to the partition function. Among many other tools, we use work of Radu which describes expansions of such modular forms along square classes at cusps of the modular curve $X(\ell Q)$, Galois representations and the arithmetic large sieve.

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31

AMBJØRN, J., D. BOULATOV та V. A. KAZAKOV. "THE BOSONIC STRING REPRESENTED AS Φ3 GRAPHS: NEW MONTE-CARLO SIMULATIONS". Modern Physics Letters A 05, № 10 (1990): 771–85. http://dx.doi.org/10.1142/s0217732390000871.

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We discuss a new method for measuring the critical exponent γ for the partition function of the bosonic string. The statistics seems very good and the fit to γconsistent with the assumed asymptotic form for the partition function for dimensions d=1–6. The results are in agreement with analytical results when the target space dimension is d=0, but disagree when d=1. We conjecture that this is due to the appearance of logarithmic corrections to the asymptotic form of the partition function. These corrections might persist for d>1 and might render the determination of γquite difficult.

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32

GAMBARELLI, GIANFRANCO. "TRANSFORMING GAMES FROM CHARACTERISTIC INTO NORMAL FORM." International Game Theory Review 09, no. 01 (2007): 87–104. http://dx.doi.org/10.1142/s0219198907001291.

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A technique is proposed to represent games in characteristic function form as games in normal form, enabling the former to exploit the concepts of solution of the latter. A new solution for games in characteristic function form is then introduced and some properties are found. A generalization of a result of von Neumann and Morgenstern is thus obtained.

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33

Meng, Fanyong, and Xiaohong Chen. "Cooperative Fuzzy Games with Convex Combination Form." Asia-Pacific Journal of Operational Research 33, no. 01 (2016): 1650007. http://dx.doi.org/10.1142/s021759591650007x.

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In this paper, a new class of cooperative fuzzy games named fuzzy games with convex combination form is introduced. This kind of fuzzy games considers two aspects of information. One is the contribution of the players to the associated crisp coalitions; the other is their participation levels. The explicit expression of the Shapley function is given, which is equal to the production of the Shapley function on crisp games and the player participation levels. Meanwhile, the relationship between the fuzzy core and the Shapley function is studied. Surprisingly, the relationship between them does coincide as in crisp case. Furthermore, some desirable properties are researched. Finally, an example is provided to illustrate the difference in fuzzy coalition values and the player Shapley values for four types of fuzzy games.

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34

Chen, Ji Wen, Jin Sheng Zhang, Zhi Wang, and Jing Kun Wang. "Function Module Dynamic Partition for Product Innovation Design." Applied Mechanics and Materials 58-60 (June 2011): 2095–100. http://dx.doi.org/10.4028/www.scientific.net/amm.58-60.2095.

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The reasonable functional modules partition is crucial to technical solution of function in product innovation design. Technical evolution factors are not considered in current product module partition method. The correlation of customer demand and product function unit, function unit flow correlation and function technology correlation are synthesized for function module partition in product innovation design. Based on function base expression, function chain and function structure is established to provide basis of function correlation analysis. Function correlation matrix is established by combining the correlation matrix of customer demand and product function unit, the function unit flow correlation matrix and function technology correlation matrix. The dynamic cluster analysis of fuzzy equivalence matrix is used to form function module. The function module partitions are evaluated by polymerization degree and coupling degree. The presented dynamic module partition method has strong distinguishing ability.

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35

Liu, Chenwei, Shuwen Xiang, Yanlong Yang, and Enquan Luo. "Cooperative Games Based on Coalition Functions in Biform Games." Axioms 12, no. 3 (2023): 296. http://dx.doi.org/10.3390/axioms12030296.

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In this paper, we try to study a class of biform games with the coalition function from the cooperation of players. For this purpose, we interpret the biform games as cooperative games by defining a characteristic function of minimax representation based on the coalition function and giving the core and Shapley value as cooperative solutions. The relations between the coalition function and the characteristic function are investigated in terms of additivity and convexity, and the properties associated with the characteristic function, such as individual rationalities and cores, are compared with the corresponding results. The relations among the solutions of the normal-form game, biform game, and cooperative game are discussed with several examples.

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36

Tetzlaff, Ulrich A. W. "On normalization constants for closed queueing networks with finite local buffers." Journal of Applied Probability 35, no. 3 (1998): 600–607. http://dx.doi.org/10.1239/jap/1032265208.

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We present new closed form solutions for partition functions used to normalize the steady-state flow balance equations of certain Markovian type queueing networks. The results focus on single class closed product form networks with state space constraints at the queueing stations. They are achieved by combining the partition function of the open network, having finite local buffers with a delta function in order to fix the number of customers in the system.

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Tetzlaff, Ulrich A. W. "On normalization constants for closed queueing networks with finite local buffers." Journal of Applied Probability 35, no. 03 (1998): 600–607. http://dx.doi.org/10.1017/s0021900200016259.

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We present new closed form solutions for partition functions used to normalize the steady-state flow balance equations of certain Markovian type queueing networks. The results focus on single class closed product form networks with state space constraints at the queueing stations. They are achieved by combining the partition function of the open network, having finite local buffers with a delta function in order to fix the number of customers in the system.

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38

Leinartene, A. B., and A. P. Lyapin. "Applying computer algebra systems to study Chaundy-Bullard identities for the vector partition function with weight." Программирование, no. 2 (April 15, 2024): 79–83. http://dx.doi.org/10.31857/s0132347424020105.

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An algorithm for obtaining the Chaundy-Bullard identity for a vector partition function with weight that uses computer algebra methods is proposed. To automate this process in Maple, an algorithm was developed and implemented that calculates the values of the vector partition function with weight by finding non-negative solutions of systems of linear Diophantine equations that are used to form the identities involved. The algorithm’s input data is represented by the set of integer vectors that form a pointed lattice cone and by some point from this cone, and the Chaundy-Bullard identity for the vector partition function with weight is its output. The code involved is stored in the depository and is ready-to-use. An example demonstrating the algorithm’s operation is given.

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39

Chander, Parkash. "Stability of the merger-to-monopoly and a core concept for partition function games." International Journal of Game Theory 49, no. 4 (2020): 953–73. http://dx.doi.org/10.1007/s00182-020-00721-5.

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40

Ahlgren, Scott, Patrick Allen, and Shiang Tang. "Congruences like Atkin’s for the partition function." Transactions of the American Mathematical Society, Series B 9, no. 33 (2022): 1044–64. http://dx.doi.org/10.1090/btran/128.

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Let p ( n ) p(n) be the ordinary partition function. In the 1960s Atkin found a number of examples of congruences of the form p ( Q 3 ℓ n + β ) ≡ 0 ( mod ℓ ) p( Q^3 \ell n+\beta )\equiv 0\pmod \ell where ℓ \ell and Q Q are prime and 5 ≤ ℓ ≤ 31 5\leq \ell \leq 31 ; these lie in two natural families distinguished by the square class of 1 − 24 β ( mod ℓ ) 1-24\beta \pmod \ell . In recent decades much work has been done to understand congruences of the form p ( Q m ℓ n + β ) ≡ 0 ( mod ℓ ) p(Q^m\ell n+\beta )\equiv 0\pmod \ell . It is now known that there are many such congruences when m ≥ 4 m\geq 4 , that such congruences are scarce (if they exist at all) when m = 1 , 2 m=1, 2 , and that for m = 0 m=0 such congruences exist only when ℓ = 5 , 7 , 11 \ell =5, 7, 11 . For congruences like Atkin’s (when m = 3 m=3 ), more examples have been found for 5 ≤ ℓ ≤ 31 5\leq \ell \leq 31 but little else seems to be known. Here we use the theory of modular Galois representations to prove that for every prime ℓ ≥ 5 \ell \geq 5 , there are infinitely many congruences like Atkin’s in the first natural family which he discovered and that for at least 17 / 24 17/24 of the primes ℓ \ell there are infinitely many congruences in the second family.

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41

GUSTAVSSON, ANDREAS. "ON THE HOLOMORPHICALLY FACTORIZED PARTITION FUNCTION FOR ABELIAN GAUGE THEORY IN SIX DIMENSIONS." International Journal of Modern Physics A 17, no. 03 (2002): 383–93. http://dx.doi.org/10.1142/s0217751x02006018.

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We use holomorphic factorization to find the partition functions of an Abelian two-form chiral gauge-field on a flat six-torus. We prove that exactly one of these partition functions is modular invariant. It turns out to be the one that previously has been found in a Hamiltonian formulation.

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42

Bachrach, Yoram, Reshef Meir, Kyomin Jung, and Pushmeet Kohli. "Coalitional Structure Generation in Skill Games." Proceedings of the AAAI Conference on Artificial Intelligence 24, no. 1 (2010): 703–8. http://dx.doi.org/10.1609/aaai.v24i1.7620.

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We consider optimizing the coalition structure in Coalitional Skill Games (CSGs), a succinct representation of coalitional games. In CSGs, the value of a coalition depends on the tasks its members can achieve. The tasks require various skills to complete them, and agents may have different skill sets. The optimal coalition structure is a partition of the agents to coalitions, that maximizes the sum of utilities obtained by the coalitions. We show that CSGs can represent any characteristic function, and consider optimal coalition structure generation in this representation. We provide hardness results, showing that in general CSGs, as well as in very restricted versions of them, computing the optimal coalition structure is hard. On the positive side, we show that the problem can be reformulated as constraint satisfaction on a hyper graph, and present an algorithm that finds the optimal coalition structure in polynomial time for instances with bounded tree-width and number of tasks.

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43

Singhal, Shiksha, and Veeraruna Kavitha. "Coalition Formation Resource Sharing Games in Networks." ACM SIGMETRICS Performance Evaluation Review 49, no. 3 (2022): 57–58. http://dx.doi.org/10.1145/3529113.3529132.

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Cooperative game theory deals with systems where players want to cooperate to improve their payoffs. But players may choose coalitions in a non-cooperative manner, leading to a coalition-formation game. We consider such a game with several players (willing to cooperate) and a possible adamant player (unwilling to cooperate) involved in resource-sharing. Here, the strategy of a player is the set of players with whom it wants to form a coalition. Given a strategy profile, an appropriate partition of coalitions is formed; players in each coalition maximize their collective utilities leading to a non-cooperative resource-sharing game among the coalitions, the (unique) utilities at the resulting equilibrium are shared via Shapley-value; these shares define the utilities of players for the given strategy profile in the coalition-formation game. We also consider the utilitarian solution to derive the price of anarchy.

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44

Hoyle, David C. "Replica analysis of the lattice-gas restricted Boltzmann machine partition function." Journal of Statistical Mechanics: Theory and Experiment 2023, no. 1 (2023): 013301. http://dx.doi.org/10.1088/1742-5468/acaf83.

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Abstract We study the expectation value of the logarithm of the partition function of large binary-to-binary lattice-gas restricted Boltzmann machines (RBMs) within a replica-symmetric ansatz, averaging over the disorder represented by the parameters of the RBM Hamiltonian. Averaging over the Hamiltonian parameters is done with a diagonal covariance matrix. Due to the diagonal form of the parameter covariance matrix not being preserved under the isomorphism between the Ising and lattice-gas forms of the RBM, we find differences in the behaviour of the quenched log partition function of the lattice-gas RBM compared to that of the Ising RBM form usually studied. We obtain explicit expressions for the expectation and variance of the lattice-gas RBM log partition function per node in the thermodynamic limit. We also obtain explicit expressions for the leading order finite size correction to the expected log partition function per node, and the threshold for the stability of the replica-symmetric approximation. We show that the stability threshold of the replica-symmetric approximation is equivalent, in the thermodynamic limit, to the stability threshold of a recent message-passing algorithm used to construct a mean-field Bethe approximation to the RBM free energy. Given the replica-symmetry assumption breaks down as the level of disorder in the spin-spin couplings increases, we obtain asymptotic expansions, in terms of the variance controlling this disorder, for the replica-symmetric log partition function and the replica-symmetric stability threshold. We confirm the various results derived using simulation.

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45

Чернов, Андрей Владимирович, and Andrey Chernov. "Differential games in a Banach space on a fixed chain." Mathematical Game Theory and Applications 12, no. 3 (2020): 89–118. http://dx.doi.org/10.17076/mgta_2020_3_22.

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The paper is devoted to obtaining the sufficient conditions for existence of epsilon-equilibrium in the sense of piecewise program strategies in antagonistic games associated with nonlinear non-autonomous controlled differential equation in a Banach space and cost functional of a general enough form. The concept of piecewise program strategies in such a game is defined on the base of a concept of Volterra set chain for a right-hand operator of the corresponding integral equation controlled by the opponent players and according to a given partition of the time segment. As example we consider the game associated with a nonlinear pseudoparabolic partial differential equation governing the evolution of electric field in a semiconductor.

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BRIEC, WALTER. "BENEFIT FUNCTION AND DUALITY IN FINITE NORMAL FORM GAMES." International Game Theory Review 09, no. 03 (2007): 495–513. http://dx.doi.org/10.1142/s0219198907001564.

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Luenberger (1992, 1994) introduced a function he terms the benefit function, that converts preferences into a numerical function and has some cardinal meaning. In this paper, we show that the benefit function enjoys many interesting properties in a game theory context. We point out that the benefit function can be adapted to compare the mixed profiles of a game. Along this line, inspired from the Luenberger's approach, we propose a dual framework and establish a characterization of Nash equilibriums in terms of the benefit function. Moreover, some criterions are provided to identify the efficient mixed strategies of a game (which differ from the Pareto efficient strategies). Finally, we go a bit further proposing some issue in comparing profiles and equilibriums of a game. This we do using the so-called Σ-subdifferential of the benefit function.

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Saad, Walid, Zhu Han, Rong Zheng, Are Hjorungnes, Tamer Basar, and H. Vincent Poor. "Coalitional Games in Partition Form for Joint Spectrum Sensing and Access in Cognitive Radio Networks." IEEE Journal of Selected Topics in Signal Processing 6, no. 2 (2012): 195–209. http://dx.doi.org/10.1109/jstsp.2011.2175699.

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48

DOLAN, LOUISE, and CHIARA R. NAPPI. "THE RAMOND–RAMOND SELF-DUAL FIVE-FORM'S PARTITION FUNCTION ON T10." Modern Physics Letters A 15, no. 19 (2000): 1261–73. http://dx.doi.org/10.1142/s0217732300001547.

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In view of the recent interest in formulating a quantum theory of Ramond–Ramond p-forms, we exhibit an [Formula: see text] invariant partition function for the chiral four-form of Type IIB string theory on the ten-torus. We follow the strategy used to derive a modular invariant partition function for the chiral two-form of the M-theory five-brane. We also generalize the calculation to self-dual quantum fields in space–time dimension 2p = 2 + 4k, and display the [Formula: see text] automorphic forms for odd p > 1. We relate our explicit calculation to a computation of the B-cycle periods, which are discussed in the work of Witten.

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49

Qian, Wei-Liang, Kai Lin, Rui-Hong Yue, Yogiro Hama, and Takeshi Kodama. "On the Partition Temperature of Massless Particles in High-Energy Collisions." Symmetry 15, no. 11 (2023): 2035. http://dx.doi.org/10.3390/sym15112035.

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Although partition temperature derived using the Darwin–Fowler method is exact for simple scenarios, the derivation for complex systems might reside in specific approximations whose viability is not ensured if the thermodynamic limit is not attained. This work elaborates on a related problem relevant to relativistic high-energy collisions. On the one hand, it is simple enough that closed-form expressions can be obtained precisely for the one-particle distribution function. On the other hand, the resulting expression is not an exponential form, and therefore, it is not straightforward that the notion of partition function could be implied. Specifically, we derive the one-particle distribution function for massless particles where the phase-space integration is performed exactly for the underlying canonical ensemble consisting of a given number of particles. We discuss the viability of the partition temperature in this case. Possible implications of the obtained results regarding the observed Tsallis distribution in transverse momentum spectra in high-energy collisions are also addressed.

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50

ZHANG, SUHONG, ZIJING LI, YUANXING GUI, and WEI WANG. "PARTITION FUNCTION OF THE GAS-LIQUID COEXISTENT SYSTEM." International Journal of Modern Physics B 21, no. 21 (2007): 3755–64. http://dx.doi.org/10.1142/s0217979207037648.

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In this article the equilibrious gas-liquid coexistent system is studied, and a new expression of partition function (PF) corresponding to the two-phase region is derived. Based on this expression, the horizontal line in the isotherm of pressure versus volume is obtained naturally for a finite particle system (i.e., without the necessity of taking the thermodynamic limit). Extending this PF, we can gain a unitive form of the one-component fluid in any system (i.e., one-phase or multi-phase). Then the whole isotherm will have reasonable statistical foundation. The VDW fluid system is discussed as a concrete example.

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