MinMax Principle in Game Theory – continued…. PARTH

MinMax principle Consider a computation problem P Let A be finite set of deterministic algorithms for solving P Let I be set of all possible inputs of size n for P

MinMax principle Matrix of running times Algorithms |A| Running time of A for input I = C(A,I) Inputs |I| Worst case running times for As (highest of respective column of matrix) Gives the optimal algorithm A Deterministic lower bound = min max C[A,I] A I

MinMax principle Lower bound is actually, Vc in Von Neumann’s principle Vr = Vc -> statement of Von Neumann’s prin. It can be restated as max min E[C(A,I)] = min max E[C(A,I)] I from p A from q A from q I from p Where p and q are payoff matrices

MinMax principle Loomi’s theorem can be restated as max min E[C(A,I)] = min max E[C(A,I)] I from p A € A A from p I € I Yao’s min max principle Take p on left and any distribution q on the right min E[C(A,I)] <= max E[C(A,I)] A € A I € I

MinMax principle

AND-OR tree problem - revisited Replace all AND and OR gates with equivalent combinations of NOR gates For any Las Vegas algorithm, Exponential time >= n0.6… Consider a distribution which sets each leaf independently Leaf = 1 -> probability p Leaf = 0 -> otherwise