// zero-sum properties // minimum expected time user 1 can guarantee to eventually send a packet <<usr1>>R{"time"}min=?[F s1=3] // minimum expected time users 2 and 3 can guarantee to eventually send their packets <<usr2,usr3>>R{"time"}min=?[F s2=3 & s3=3] // maximum probability user 1 can guarantee to send a packet <<usr1>>Pmax=?[F s1=3] // maximum probability users 2 and 3 can guarantee to send their packets <<usr2,usr3>>Pmax=?[F (s2=3 & s3=3)] // maximum probability user 1 can guarantee to send a packet within a deadline <<usr1>>Pmax=?[F (s1=3 & t<=D)] // maximum probability users 2 and 3 can guarantee to send their packets within a deadline <<usr2,usr3>>Pmax=?[F (s2=3 & s3=3 & t<=D)] // nonzero-sum properties // probability user 1 eventually sends a packet // and users 2 and 3 form a coalition to send two packets <<usr1:usr2,usr3>>max=? (P[F s1=3] + P[F s2=3 & s3=3]) // should be 1.0+1.0=2.0 as can collaborate // i.e. first one sends and then the other does afterwards as no deadline // expected time user 1 eventually sends a packet // and users 2 and 3 form a coalition to send two packets <<usr1:usr2,usr3>>min=? (R{"time"}[F s1=3] + R{"time"}[F s2=3 & s3=3]) // probability user 1 eventually sends a packet by a deadline // and users 2 and 3 form a coalition to send two packets by a deadline <<usr1:usr2,usr3>>max=? (P[F (s1=3 & t<=D)] + P[F (s2=3 & s3=3 & t<=D)]) // probability user 1 eventually sends a packet by a deadline // and users 2 and 3 form a coalition to send two packets <<usr1:usr2,usr3>>max=? (P[F (s1=3 & t<=D)] + P[F (s2=3 & s3=3)]) // probability user 1 eventually sends a packet // and users 2 and 3 form a coalition to send two packets by a deadline <<usr1:usr2,usr3>>max=? (P[F (s1=3)] + P[F (s2=3 & s3=3 & t<=D)]) // Nonzero-sum properties (N-player) // expected time each user eventually sends a packet <<usr1:usr2:usr3>>min=? (R{"time"}[F s1=3] + R{"time"}[F s2=3] + R{"time"}[F s3=3]) // probability each user eventually sents a packet <<usr1:usr2:usr3>>max=? (P[F s1=3] + P[F s2=3] + P[F s3=3]) // probability each user eventually sents a packet within a deadline <<usr1:usr2:usr3>>max=? (P[F (s1=3 & t<=D)] + P[F (s2=3 & t<=D)] + P[F (s3=3 & t<=D)]) // expected time user 1 eventually sends a packet // and users 2 and 3 form a coalition to send two packets (adding the time for 2 and 3 to send (to allow a comparision with the n-player case) <<usr1:usr2,usr3>>min=? (R{"time"}[F s1=3] + R{"time23"}[F s2=3 & s3=3])