1. christos alatzidis constantina galbogini

2.  The Complexity of Computing a Nash Equilibrium  Constantinos Daskalakis  Paul W. Goldberg  Christos H. Papadimitriou  Non-Cooperative Games (11/1951)  John Nash

3.  Any continuous map from a compact (closed and bounded) and convex (without holes) subset of the Euclidean space into itself always has a fixed point  Suggests an interesting computational total search problem:  Given a continuous function from some compact and convex set to itself, find a fixed point

4.  For a meaningful definition of Brouwer we need to address two questions:  How do we specify a continuous map from some compact and convex set to itself?  How do we deal with irrational fixed points?

5.  Fix the compact and convex set to be the unit cube [0, 1] m  Assume that the function F is given by an efficient algorithm Π F which, for each point x of the cube written in binary, computes F(x)  F obeys a Lipschitz condition:  x 1, x 2 є [0, 1] m : d(F(x 1 ), F(x 2 )) ≤ K * d(x 1, x 2 )  d(.,.) is the Euclidean distance  K is the Lipschitz constant of F  ensures that approximate fixed points can be localized by examining the value F(x) when x ranges over a discretized grid over the domain

6.  We can deal with irrational solutions in a similar manoeuvre as with Nash  Only seeking approximate fixed points  Strong guarantee :  for any e, there is an e -approximate fixed point x, such that d(F(x), x) ≤ e

7.  Suppose that the players in a game have chosen some (mixed) strategies  Unless these already constitute a Nash equilibrium, some of the players will be unsatisfied, and will wish to change to some other strategies  Can construct a preference function from the set of players‘ strategies to itself, that indicates the movement that will be made by any unsatisfied players

8.  A fixed point of such a function is a point that is mapped to itself, a Nash equilibrium  Brouwer's theorem guarantees that such a fixed point exists  An approximate fixed point corresponds to an approximate Nash equilibrium  Nash reduces to Brouwer

9.  Is a state where no one person can improve, given what others are doing!  Thus given the choices of the other players, I choose to do what is the best for me (maximize my payoff). Nash’s Idea : we cannot predict the result of the choices of multiple decision makers if we analyze those decisions in isolation. Instead, we must ask what each player would do, taking into account the decision-making of the others.

10.  Finite game:  n persons each associated with a finite set of pure strategies  Corresponding to each player i there is a payoff function p i  Mixed strategy:  Collection of n o n negative numbers ( c iα ) which have:  Unit sum  One to one correspondence with a players pure strategies

11. A mixed strategy of a player i : Where π ia will indicate the i th players a th pure strategy and c iα ≥ 0 Use suffixes i,j,k for players. α, β,γ to indicate various pure strategies of a player. Symbols s i, t i, r i etc. will indicate mixed strafegies of player i.

12. Also we shall use s* or t* to denote an n tuple of mixed strategies, where s * = (s 1...s n ). We also introduce ( s*; t i ) to stand for (s 1...s i-1,t i,s i+1..s n ). Formally an n-tuple s* will be an equilibrium point if for every player i: (1)

13.  If and c iα > 0 then we say that s i uses pure strategy π i α And because p i (s i... s n ) is linear we can see that from (1): (2)  Thus for p i ( s* ) to be an equilibrium point:  (3), i ntuitively this means that the maximization will have to be done for every player  Now if s* = (s 1...s n ) and then, and for (3) to hold:  c i α = 0 whenever.  Explanation: If pure strategy π i α is not optimal for i, then it will not be used.  Thus if π iα is used in s* then

14.  Theorem 1: Every finite game has an equilibrium point  Based in Brouwer's fixed point theorem  Every continous function from a closed unit ball from D n to D n, has at least one fixed point x in D n (s.t f(x) = x).

15.  Proof of Theorem 1 :  Let s* be an n-tuple of mixed strategies  p i (s*) : payoff for player i  p iα (s*) : payoff for i if he changes and chooses to use his α th pure strategy  Define φ iα (s*) = max[0,p iα (s*) - p i (s*)], φ will be a continuous function of s*  Also define

16. From (4), we have a set of n-tupples (s’ 1, s’ 2, …, s’ n ). We suppose that these are fixed points of the mapping T: S* -> S*’. We must show that these points are the equilibrium points Consider any n-tupple s*. In s* the i th player’s mixed strategy s i will use certain of his pure strategies. Some one of these strategies, say π ia, must be “least profitable” so that p ia (S*) ≤ p i (S*) This will make φ ia (S*) = 0

17. If S* is fixed under T, for any i and β, φ i β ( S*) = 0. This means no player can improve his pay-off by moving to a pure strategy π i β. This is just a criterion for an equilibrium point ! Conversely, if S* is an equilibrium point, all φ ’s vanish, making S* a fixed point under T Since the space of n-tuples is a cell the Brouwer fixed point theorem requires that T must have at least on fixed point S*, which must be an equilibrium point.

18. A beautiful mind... Even if he played perfectly, game theory does not state that he should have won!! The bar and the blonde beautiful woman... With your friends already going for brunettes, you have no competition to go for the blonde!!!!

19. The end..