Payoff-dependant Balancedness and Cores.

We prove the non-emptiness of the core of an NTU game satisfying a condition of payoff-dependent balancedness, based on transfer rate mappings. We also define a new equilibrium condition on transfer rates and we prove the existence of core payoff vectors satisfying this condition. The additional requirement of transfer rate equilibrium refines the core concept and allows the selection of specific core payoff vectors. Lastly, the class of parameterized cooperative games is introduced. This new setting and its associated equilibrium–core solution extend the usual cooperative game framework and core solution to situations depending on an exogenous environment. A non-emptiness result for the equilibrium–core is also provided in the context of a parametrized cooperative game. Our proofs borrow mathematical tools and geometric constructions from general equilibrium theory with non-convexities. Applications to extant results taken from game theory and economic theory are given.Balancedness; Cooperative game; Core; Parametrized game;

Payoff-dependent balancedness and cores (revised version)

We prove the non-emptiness of the core of an NTU game satisfying a condition of payoff-dependent balancedness, based on transfer rate mappings. We also define a new equilibrium condition on transfer rates and we prove the existence of core payoff vectors satisfying this condition. The additional requirement of transfer rate equilibrium refines the core concept and allows the selection of specific core payoff vectors. Lastly, the class of parametrized cooperative games is introduced. This new setting and its associated equilibrium-core solution extend the usual cooperative game framework and core solution to situations depending on an exogenous environment. A non-emptiness result for the equilibrium-core is also provided in the context of a parametrized cooperative game. Our proofs borrow mathematical tools and geometric constructions from general equilibrium theory with non convexities. Applications to extant results taken from game theory and economic theory are given.balancedness, cooperative game, core, parametrized game

Dominating Set Games

In this paper we study cooperative cost games arising from domination problems on graphs.We introduce three games to model the cost allocation problem and we derive a necessary and su cient condition for the balancedness of all three games.Furthermore we study concavity of these games.game theory;cost allocation;cooperative games

Payoff-dependent balancedness and cores (revised version)

We prove the non-emptiness of the core of an NTU game satisfying a condition of payoff-dependent balancedness, based on transfer rate mappings. We also define a new equilibrium condition on transfer rates and we prove the existence of core payoff vectors satisfying this condition. The additional requirement of transfer rate equilibrium refines the core concept and allows the selection of specific core payoff vectors. Lastly, the class of parametrized cooperative games is introduced. This new setting and its associated equilibrium-core solution extend the usual cooperative game framework and core solution to situations depending on an exogenous environment. A non-emptiness result for the equilibrium-core is also provided in the context of a parametrized cooperative game. Our proofs borrow mathematical tools and geometric constructions from general equilibrium theory with non convexities. Applications to extant results taken from game theory and economic theory are given

Online vertex-coloring games in random graphs

Consider the following one-player game. The vertices of a random graph on n vertices are revealed to the player one by one. In each step, also all edges connecting the newly revealed vertex to preceding vertices are revealed. The player has a fixed number of colors at her disposal, and has to assign one of these to each vertex immediately. However, she is not allowed to create any monochromatic copy of some fixed graph F in the process. For n → ∞, we study how the limiting probability that the player can color all n vertices in this online fashion depends on the edge density of the underlying random graph. For a large family of graphs F, including cliques and cycles of arbitrary size, and any fixed number of colors, we establish explicit threshold functions for this edge density. In particular, we show that the order of magnitude of these threshold functions depends on the number of colors, which is in contrast to the corresponding offline coloring proble

The Investment Management Game: Extending the Scope of the Notion of Core

The core is a dominant solution concept in economics and cooperative game theory; it is predominantly used for profit, equivalently cost or utility, sharing. This paper demonstrates the versatility of this notion by proposing a completely different use: in a so-called investment management game, which is a game against nature rather than a cooperative game. This game has only one agent whose strategy set is all possible ways of distributing her money among investment firms. The agent wants to pick a strategy such that in each of exponentially many future scenarios, sufficient money is available in the right firms so she can buy an optimal investment for that scenario. Such a strategy constitutes a core imputation under a broad interpretation, though traditional formal framework, of the core. Our game is defined on perfect graphs, since the maximum stable set problem can be solved in polynomial time for such graphs. We completely characterize the core of this game, analogous to Shapley and Shubik characterization of the core of the assignment game. A key difference is the following technical novelty: whereas their characterization follows from total unimodularity, ours follows from total dual integralityComment: 16 pages. arXiv admin note: text overlap with arXiv:2209.0490

Bounds on Ramsey Games via Alterations

This note contains a refined alteration approach for constructing H-free graphs: we show that removing all edges in H-copies of the binomial random graph does not significantly change the independence number (for suitable edge-probabilities); previous alteration approaches of Erdos and Krivelevich remove only a subset of these edges. We present two applications to online graph Ramsey games of recent interest, deriving new bounds for Ramsey, Paper, Scissors games and online Ramsey numbers.Comment: 9 page

Randomised Load Balancing

Due to the increased use of parallel processing in networks and multi-core architectures, it is important to have load balancing strategies that are highly efficient and adaptable to specific requirements. Randomised protocols in particular are useful in situations in which it is costly to gather and update information about the load distribution (e.g. in networks). For the mathematical analysis randomised load balancing schemes are modelled by balls-into-bins games, where balls represent tasks and bins computers. If m balls are allocated to n bins and every ball chooses one bin at random, the gap between maximum and average load is known to grow with the number of balls m. Surprisingly, this is not the case in the multiple-choice process in which each ball chooses d > 1 bins and allocates itself to the least loaded. Berenbrink et al. proved that then the gap remains ln ln(n) / ln(d). This thesis analyses generalisations and variations of the multiple-choice process. For a scenario in which batches of balls are allocated in parallel, it is shown that the gap between maximum and average load is still independent of m. Furthermore, we look into a process in which only predetermined subsets of bins can be chosen by a ball. Assuming that the number and composition of the subsets can change with every ball, we examine under which circumstances the maximum load is one. Finally, we consider a generalisation of the basic process allowing the bins to have different capacities. Adapting the probabilities of the bins, it is shown how the load can be balanced over the bins according to their capacities