1. MinMax Principle in Game Theory – continued….
PARTH

2. 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

3. 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

4. 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

5. 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

6. MinMax principle

7. 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