A Composite Run-to-the-Bank Rule for Multi-Issue Allocation Situations

In this paper, we propose a new extension of the run-to-the-bank rule for bankruptcy situations to the class of multi-issue allocation situations.We show that this rule always yields a core element and that it satisfies self-duality. We characterise our rule by means of a new consistency property, issue-consistency.allocation;bankruptcy;cooperative games;consistency

Nash Equilibria in 2 × 2 × 2 Trimatrix Games with Identical Anonymous Best-Replies

This paper introduces the class of 2 × 2 × 2 trimatrix games with identical anonymous best-replies. For this class a complete classification on the basis of the Nash equilibrium set is provided.trimatrix games;Nash equilibrium;best-reply correspondences;symmetric games

A Taxonomy of Best-Reply Multifunctions in 2x2x2 Trimatrix Games

This paper provides an overview of the various shapes the best- reply multifunctions can take in 2×2×2 trimatrix games. It is shown that, unlike in 2×2 bimatrix games, the best replies to the opponents’ pure strategies do not completely determine the structure of the Nash equilibrium set.non-cooperative games;best-reply multifunction;Nash equilibrium

Cooperation in Capital Deposits

The rate of return earned on a deposit can depend on its term, the amount of money invested in it, or both. Most banks, for example, offer a higher interest rate for longer term deposits. This implies that if one individual has capital available for investment now, but needs it in the next period, whereas the opposite holds for another individual, then they can both benefit from cooperation since it allows them to invest in a longer term deposit. A similar situation arises when the rate of return on a deposit depends on the amount of capital invested in it. Although the benefits of such cooperative behavior may seem obvious to all individuals, the actual participation of an individual depends on what part of the revenues he eventually receives. The allocation of the jointly earned benefits to the investors thus plays an important part in the stability of the cooperation. This paper provides a game theoretical analysis of this allocation problem. Several classes of corresponding deposit games are introduced. For each class, necessary conditions for a nonempty core are provided, and allocation rules that yield core-allocations are examined.Cooperative game theory;capital deposits.

Effects of turbulence and rotation on protostar formation as a precursor to seed black holes

Context. The seeds of the first supermassive black holes may have resulted from the direct collapse of hot primordial gas in $\gtrsim 10^4$ K haloes, forming a supermassive or quasistar as an intermediate stage. Aims. We explore the formation of a protostar resulting from the collapse of primordial gas in the presence of a strong Lyman-Werner radiation background. Particularly, we investigate the impact of turbulence and rotation on the fragmentation behaviour of the gas cloud. We accomplish this goal by varying the initial turbulent and rotational velocities. Methods. We performed 3D adaptive mesh refinement simulations with a resolution of 64 cells per Jeans length using the ENZO code, simulating the formation of a protostar up to unprecedentedly high central densities of $10^{21}$ cm$^{-3}$, and spatial scales of a few solar radii. To achieve this goal, we employed the KROME package to improve modelling of the chemical and thermal processes. Results. We find that the physical properties of the simulated gas clouds become similar on small scales, irrespective of the initial amount of turbulence and rotation. After the highest level of refinement was reached, the simulations have been evolved for an additional ~5 freefall times. A single bound clump with a radius of $2 \times 10^{-2}$ AU and a mass of ~$7 \times 10^{-2}$ M$_{\odot}$ is formed at the end of each simulation, marking the onset of protostar formation. No strong fragmentation is observed by the end of the simulations, regardless of the initial amount of turbulence or rotation, and high accretion rates of a few solar masses per year are found. Conclusions. Given such high accretion rates, a quasistar of $10^5$ M$_{\odot}$ is expected to form within $10^5$ years.Comment: 18 pages, 7 figures, fixed typos, added references and clarified some details; accepted for publication in A&