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. Find sources: "Parthasarathy's theorem" – news · newspapers · books · scholar · JSTOR (February 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 other words, one of the players is forbidden to use a pure strategy).

The theorem is attributed to the Indian mathematician Thiruvenkatachari Parthasarathy.

Theorem Let {displaystyle X} and {displaystyle Y} stand for the unit interval {displaystyle [0,1]} ; {displaystyle {mathcal {M}}_{X}} denote the set of probability distributions on {displaystyle X} (with {displaystyle {mathcal {M}}_{Y}} defined similarly); and {displaystyle A_{X}} denote the set of absolutely continuous distributions on {displaystyle X} (with {displaystyle A_{Y}} defined similarly).

Suppose that {displaystyle k(x,y)} is bounded on the unit square {displaystyle Xtimes Y={(x,y):0leq x,yleq 1}} and that {displaystyle k(x,y)} is continuous except possibly on a finite number of curves of the form {displaystyle y=phi _{k}(x)} (with {displaystyle k=1,2,ldots ,n} ) where the {displaystyle phi _{k}(x)} are continuous functions. For {displaystyle mu in M_{X},lambda in M_{Y}} , define {displaystyle k(mu ,lambda )=int _{y=0}^{1}int _{x=0}^{1}k(x,y),dmu (x),dlambda (y)=int _{x=0}^{1}int _{y=0}^{1}k(x,y),dlambda (y),dmu (x).} Then {displaystyle max _{mu in {mathcal {M}}_{X}},inf _{lambda in A_{Y}}k(mu ,lambda )=inf _{lambda in A_{Y}},max _{mu in {mathcal {M}}_{X}}k(mu ,lambda ).} This is equivalent to the statement that the game induced by {displaystyle k(cdot ,cdot )} has a value. Note that one player (WLOG {displaystyle Y} ) is forbidden from using a pure strategy.

Parthasarathy goes on to exhibit a game in which {displaystyle max _{mu in {mathcal {M}}_{X}},inf _{lambda in {mathcal {M}}_{Y}}k(mu ,lambda )neq inf _{lambda in {mathcal {M}}_{Y}},max _{mu in {mathcal {M}}_{X}}k(mu ,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, number 2.

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