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!!!!
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