christos alatzidisconstantina galbogini

The Complexity of Computing a Nash Equilibrium Constantinos Daskalakis Paul W. Goldberg Christos H. Papadimitriou

Non-Cooperative Games (11/1951) John Nash

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

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?

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 ΠF which, for each point x of the cube written in binary, computes F(x)

F obeys a Lipschitz condition: x1, x2 є [0, 1]m : d(F(x1), F(x2)) ≤ K * d(x1, x2)

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

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

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

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

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.

Finite game: n persons each associated with a finite set of pure

strategies Corresponding to each player i there is a payoff function pi

Mixed strategy: Collection of non negative numbers (ciα) which have:

Unit sum One to one correspondence with a players pure

strategies

1ic

a

iii cs

• A mixed strategy of a player i :

Where πia will indicate the ith players ath pure strategy and ciα ≥ 0

Use suffixes i,j,k for players. α, β ,γ to indicate various pure strategies of a player. Symbols si, ti, ri etc. will indicate mixed strafegies of player i.

Also we shall use s* or t* to denote an n tuple of mixed strategies, where s* = (s1...sn).

We also introduce (s*;ti) to stand for (s1...si-1,ti,si+1..sn).

Formally an n-tuple s* will be an equilibrium point if for every player i: (1)

)]*,([max*)( iir

i rspspi

If and ciα > 0 then we say that si uses pure strategy πiα And because pi(si ... sn) is linear we can see that from (1):

(2) Thus for pi(s*) to be an equilibrium point:

(3), intuitively this means that the maximization will have to be done for every player

Now if s* = (s1...sn) and then ,and for (3) to hold: ciα = 0 whenever . Explanation: If pure strategy πiα is not optimal for i, then it

will not be used. Thus if πiα is used in s* then

iii cs

)]*;([max)]*;([max iiii

rsprsp

i

*)]([max*)( spsp ii

a

iαiai cs a

iaiai spcsp *)(*)(

*)(max*)( spsp iia

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 Dn

to Dn, has at least one fixed point x in Dn (s.t f(x) = x).

Proof of Theorem 1: Let s* be an n-tuple of mixed strategies pi(s*) : payoff for player i

piα(s*) : payoff for i if he changes and chooses to use his αth pure strategy

Define φiα(s*) = max[0,piα(s*) - pi(s*)], φ will be a continuous function of s*

Also define

a

iii

i si

sss

*)(1

*)('

• 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 ith

player’s mixed strategy si will use certain of his pure strategies. Some one of these strategies, say πia, must be “least profitable” so that pia(S*) ≤ pi(S*)This will make φia(S*) = 0

• If S* is fixed under T, for any i and β, φiβ(S*) = 0. This means no player can improve his pay-off by moving to a pure strategy πiβ. 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.

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!!!!

The end..