Date of Submission


Date of Award


Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name

Computer Science


Theoretical Statistics and Mathematics Unit (TSMU-Bangalore)


Parthasarathy, Thiruvenkatachari (TSMU-Bangalore; ISI)

Abstract (Summary of the Work)

This monograph deals with the area from game theory known as co operativo games. Except the last chapter on NTU games, it deals with transferable utility games. llere we will introiduce and diseus the involved game theuretie notions and set a mathematical hase for the chapters to come.In 1944 von Neamann and Morgenstern(15) introduced a theory of solutions for n-person games in characteristic function form in which cooperation and coalition formation is a crucial aspect. The primary mathematical concern regarding this model is the existence of solutions or stable set. In 1968 Lucas|19 described a tem person game which has no solution. However, researchers have gone on to identify properties of such solutions when they exint and their relationship with other known concepts, in particular, the core. Sharkey13| defined and studied the concept of largenesn of the cure which arose while he was studying an econonie problenm invulvingcust allneation. He showed that largenes of the tore isa sufficient condition fe the the core to he a stahle set.We present in this this some results concerning the coincidene of the stable set anl the cor. This las been particulary done through the concept of large core. We alo present in conditions for the core to be large and quite a few examples giving insight into the resulte proved.Slharkey 43] proved that largeness of core is sufficient for the cure to be a stable set. We identify a xubclass namely the syametrie games, where largeness of the une turns ont to he also unnecessary and leads to other interesting and ensy to check randitius lor stability of the core in symmetrie gamem. Subsequently ne answer the art ion if erry x actgae has a large cure We prove that foe games with 5 players or moery extent game need not posses a large core, haoever, in the wulaelaan of symmetric games largenesoi of core and the stability of the core turn out to be mguivalett to the concept of the gae being exact. For general TU games with a or 4 plasenery esact game has a large core. Su for atutally balanced symrtrie ganes Large core, stable core and exactuess turn out to br equivalent.However, for general TU gan, largeness of the core always impiks rore stability and there are examples where the core is stable hut noe 1.L. INTRODUCTIONlarge I is known that under the estendability condition introduced by Kikuta and Shapiey 17 the core is a stable set but the core may not be lange. In this thersis, we show that the Kikuta-Shapley condition is sufficient for the core to le stable as well as lange for TU games with foe oe les number of plays. We promide a coenter exanper uhen the number af players is si. We then introduastrongre extendability condition and show that this condition in neccesary aud saffiejent for the ore to be large.The core of a TU RAe is perhaps the most intutive and easiest e lutiun concrpt in Couperative (lame Thecny (13). The uther approach to solution concepla is the stahle seta intmduced by von Neumana and Morgeatem(45. Thas far the relation hetwees the twu mant crucially important salutica couopts for caoperative game han been investi Rated in the contest of symnetric tranvderable utility games and this has heen related to the aotion of large core We have further investi gated the relation between the vun- Nrumana-Morgeetern stability of the oe and the largeness of it in the case of aon-trancferable utility (NTU) Rames. The main findings am basically estensiuns of existing results obrtained in (4. 14. 13) in the case NTU gan, which are very similar to the reutia in TU cases.


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