We show that the 'erasing-larger-loops-first' (ELLF) method, which was first introduced for erasing loops from the simple random walk on the Sierpiński gasket, does work also for non-Markov random walks, in particular, self-repelling walks to construct a new family of self-avoiding walks on the Sierpiński gasket. The one-parameter family constructed in this method continuously connects the loop-erased random walk and a self-avoiding walk which has the same asymptotic behavior as the 'standard' self-avoiding walk. We prove the existence of the scaling limit and study some path properties: The exponent ν governing the short-time behavior of the scaling limit varies continuously in the parameter. The limit process is almost surely self-avoiding, while it has path Hausdorff dimension 1/ν, which is strictly greater than 1.The self-avoiding walk (SAW) and the loop-erased random walk (LERW) are two typical examples of non-Markov random walks on graphs. The self-avoiding walk is defined by the uniform measure on self-avoiding paths of a given length. In this paper we call this model the 'standard' selfavoiding walk ('standard' SAW), for we shall deal with a family of different walks whose paths are self-avoiding. The loop-erased random walk is a random walk obtained by erasing loops from the simple random walk in chronological order (as soon as each loop is made). Although the LERW has self-avoiding paths, it has a different distribution from that of the 'standard' SAW.
In this paper, we study the egalitarian solution for games with discrete side payment, where the characteristic function is integer-valued and payoffs of players are integral vectors. The egalitarian solution, introduced by Dutta and Ray in 1989, is a solution concept for transferable utility cooperative games in characteristic form, which combines commitment for egalitarianism and promotion of indivisual interests in a consistent manner. We first point out that the nice properties of the egalitarian solution (in the continuous case) do not extend to games with discrete side payment. Then we show that the Lorenz stable set, which may be regarded as a variant of the egalitarian solution, has nice properties such as the Davis-Maschler reduced game property and the converse reduced game property. For the proofs we utilize recent results in discrete convex analysis on decreasing minimization on an M-convex set investigated by Frank and Murota.
In this paper, we study the egalitarian solution for games with discrete side payment, where the characteristic function is integervalued and payoffs of players are integral vectors. The egalitarian solution, introduced by Dutta and Ray in 1989, is a solution concept for transferable utility cooperative games in characteristic form, which combines commitment for egalitarianism and promotion of indivisual interests in a consistent manner. We first point out that the nice properties of the egalitarian solution (in the continuous case) do not extend to games with discrete side payment. Then we show that the Lorenz stable set, which may be regarded a variant of the egalitarian solution, has nice properties such as the Davis and Maschler reduced game property and the converse reduced game property. For the proofs we utilize recent results in discrete convex analysis on decreasing minimization on an M-convex set investigated by Frank and Murota.
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