Parthasarathy's theorem

Parthasarathy's theorem This article relies largely or entirely on a single source. Relevant discussion may be found on the talk page. Please help improve this article by introducing citations to additional sources. Trova fonti: "Parthasarathy's theorem" – news · newspapers · books · scholar · JSTOR (febbraio 2016) In mathematics – and in particular the study of games on the unit square – Parthasarathy's theorem is a generalization of Von Neumann's minimax theorem. It states that a particular class of games has a mixed value, provided that at least one of the players has a strategy that is restricted to absolutely continuous distributions with respect to the Lebesgue measure (in altre parole, one of the players is forbidden to use a pure strategy).

The theorem is attributed to the Indian mathematician Thiruvenkatachari Parthasarathy.

Theorem Let {stile di visualizzazione X} e {stile di visualizzazione Y} stand for the unit interval {stile di visualizzazione [0,1]} ; {stile di visualizzazione {matematico {M}}_{X}} denote the set of probability distributions on {stile di visualizzazione X} (insieme a {stile di visualizzazione {matematico {M}}_{Y}} defined similarly); e {stile di visualizzazione A_{X}} denote the set of absolutely continuous distributions on {stile di visualizzazione X} (insieme a {stile di visualizzazione A_{Y}} defined similarly).

Supporre che {stile di visualizzazione k(X,y)} is bounded on the unit square {displaystyle Xtimes Y={(X,y):0leq x,yleq 1}} e quello {stile di visualizzazione k(X,y)} is continuous except possibly on a finite number of curves of the form {displaystyle y=phi _{K}(X)} (insieme a {displaystyle k=1,2,ldots ,n} ) dove il {stile di visualizzazione phi _{K}(X)} are continuous functions. Per {displaystyle mu in M_{X},lambda in M_{Y}} , definire {stile di visualizzazione k(in ,lambda )=int _{y=0}^{1}int _{x=0}^{1}K(X,y),dm (X),dlambda (y)=int _{x=0}^{1}int _{y=0}^{1}K(X,y),dlambda (y),dm (X).} Quindi {stile di visualizzazione massimo _{mu in {matematico {M}}_{X}},inf_{lambda in A_{Y}}K(in ,lambda )=inf_{lambda in A_{Y}},massimo _{mu in {matematico {M}}_{X}}K(in ,lambda ).} This is equivalent to the statement that the game induced by {stile di visualizzazione k(cdot ,cdot )} has a value. Note that one player (WLOG {stile di visualizzazione Y} ) is forbidden from using a pure strategy.

Parthasarathy goes on to exhibit a game in which {stile di visualizzazione massimo _{mu in {matematico {M}}_{X}},inf_{lambda in {matematico {M}}_{Y}}K(in ,lambda )neq inf _{lambda in {matematico {M}}_{Y}},massimo _{mu in {matematico {M}}_{X}}K(in ,lambda )} which thus has no value. There is no contradiction because in this case neither player is restricted to absolutely continuous distributions (and the demonstration that the game has no value requires both players to use pure strategies).

References T. Parthasarathy 1970. On Games over the unit square, SIAM, volume 19, numero 2.

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