83 research outputs found
Lexikografikus allokációk a hozzárendelési játékokban
Két új lexikografikus allokációs eljárást vizsgálunk: a leximin és a leximax eljárásokat.
Ezek abban hasonlítanak a jól ismert marginális allokációs eljáráshoz, hogy (i) a kifizetések
meghatározása itt is a játékosok egy eleve adott prioritási sorrendjében történik; (ii) ha az
eredmény egy mag-elosztás, akkor a kapott allokáció a magnak egy extremális eleme. A
két új eljárás viszont nem a koalíciós értékekből állapítja meg az egyes kifizetéseket, hanem
a mag-elosztásokra vonatkozó alsó, illetve felső korlátokat igyekszik, amennyire csak
lehetséges, kielégíteni. Két f˝o kérdésre keressük a választ, néhány általános észrevételtől eltekintve
főként a mindig nem üres maggal rendelkező hozzárendelési játékokra fókuszálva:
(1) Mag-elosztást kapunk-e bármelyik játékos-sorrend esetén? (2) Megkapjuk-e mindegyik
extremális mag-elosztást valamilyen játékos-sorrenddel
The kernel is in the least core for permutation games
Permutation games are totally balanced transferable utility cooperative
games arising from certain sequencing and re-assignment optimization problems.
It is known that for permutation games the bargaining set and the core coincide, consequently, the kernel is a subset of the core. We prove that for permutation
games the kernel is contained in the least core, even if the latter is a lower dimensional subset of the core. By means of a 5-player permutation game
we demonstrate that, in sense of the lexicographic center procedure leading to the nucleolus, this inclusion result can not be strengthened. Our 5-player permutation
game is also an example (of minimum size) for a game with a non-convex kernel
Weighted nucleoli and dually essential coalitions
We consider linearly weighted versions of the least core and the (pre)nuceolus and
investigate the reduction possibilities in their computation. We slightly extend some
well-known related results and establish their counterparts by using the dual game.
Our main results imply, for example, that if the core of the game is not empty, all
dually inessential coalitions (which can be weakly minorized by a partition in the dual
game) can be ignored when we compute the per-capita least core and the per-capita
(pre)nucleolus from the dual game. This could lead to the design of polynomial time
algorithms for the per-capita (and other monotone nondecreasingly weighted versions
of the) least core and the (pre)nucleolus in specific classes of balanced games with
polynomial many dually essential coalitions
Lexicographic allocations and extreme core payoffs: the case of assignment games
We consider various lexicographic allocation procedures for coalitional games with transferable utility where the payoffs are computed in an externally given order of the players. The common feature of the methods is that if the allocation is in the core, it is an extreme point of the core. We first investigate the general relationship between these allocations and obtain two hierarchies on the class of balanced games.
Secondly, we focus on assignment games and sharpen some of these general relationship. Our main result is the coincidence of the sets of lemarals (vectors of lexicographic maxima over the set of dual coalitionally rational payoff vectors), lemacols (vectors of lexicographic maxima over the core) and extreme core points. As byproducts, we show that, similarly to the core and the coalitionally rational payoff set, also the dual coalitionally rational payoff set of an assignment game is determined by the individual and mixed-pair coalitions, and present an efficient and elementary way to compute these basic dual coalitional values. This provides a way to compute the Alexia value (the average of all lemacols) with no need to obtain the whole coalitional function of the dual assignment game
Universal characterization sets for the nucleolus in balanced games
We provide a new mo dus op erandi for the computation of the nucleolus in co op-
erative games with transferable utility. Using the concept of dual game we extend
the theory of characterization sets. Dually essential and dually saturated coalitions
determine b oth the core and the nucleolus in monotonic games whenever the core
is non-empty. We show how these two sets are related with the existing charac-
terization sets. In particular we prove that if the grand coalition is vital then the
intersection of essential and dually essential coalitions forms a characterization set
itself. We conclude with a sample computation of the nucleolus of bankruptcy games
- the shortest of its kind
On bargaining sets of supplier-firm-buyer games
We study a special three-sided matching game, the so-called supplier-firm-buyer game, in which buyers and sellers (suppliers) trade indirectly through middlemen (firms). Stuart (1997) showed that all supplier-firm-buyer games have non-empty core. We show that for these games the core coincides with the classical bargaining set (Davis and Maschler, 1967), and also with the Mas-Colell bargaining set (Mas-Colell, 1989)
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