Fall 2016 Yang Cai Lecture 05 Overview so far Recap Games rationality solution concepts Existence Theorems for Nash equilibrium Nashs theorem for general games via Brouwer ID: 651514
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Slide1
Comp/Math 553: Algorithmic Game Theory
Fall 2016
Yang Cai
Lecture
05Slide2
Overview so far
Recap:
Games, rationality, solution concepts
Existence Theorems for Nash equilibrium:
Nash’s theorem for general games (via
Brouwer
)
von Neumann’s theorem for 2-player ZS games (via LP duality)
This lecture:
algorithms for Nash equilibrium beyond two player zero-sum games (which can be solved using linear programming)
Overarching goal:
How complex computationally is it to find the Nash equilibrium of the game, for the game theorist and the players themselves? Slide3
Algorithms for Nash
Equilibria
Support Enumeration Algorithms
Algorithms for Symmetric Games
The Lemke-
Howson
AlgorithmSlide4
Algorithms for Nash
Equilibria
Algorithms for Symmetric Games
The Lemke-
Howson
Algorithm
Support Enumeration AlgorithmsSlide5
Support Enumeration Algorithms
How better would my life be if I knew the support of the Nash equilibrium?
… and the game is 2-player?
Claim:
Can find Nash equilibrium (
x
,
y
) using linear programming.
Setting:
Let (
R
,
C
) be an
m
by
n
game, and suppose a friend revealed to us the supports and respectively of the Row and Column players’ mixed strategies at some equilibrium of the game.
s.t
.
andSlide6
Support Enumeration Algorithms
How better would my life be if I knew the support of the Nash equilibrium?
… and the game is 2-player?
for guessing the support of each player’s strategy at a Nash
eq
for solving the LP
Runtime:
Corollary:
Existence of rational
equilibria
in 2-player games.
Proof:
Follows from the correctness of the support enumeration algorithm. If there is a Nash equilibrium with supports
S
R
and
S
C
for the two players, then the
polytope
of the corresponding LP is non-empty. Any vertex of that
polytope
is a Nash equilibrium, whose coordinates are rational and whose bit complexity is polynomial in the description of the game.Slide7
Support Enumeration for
n-player games
How better would my life be if I knew the support of the Nash equilibrium?
… and the game is
n
-player?
Challenge:
Even if the support
Σ
p
of each player at a Nash equilibrium is given to us, the problem we need to solve is still not an LP.
Polynomial of degree (
n
-1) in variables
x
11
, x
12
,…,
x
nk
(suppose
k
strategies per player)
So need to solve a system of polynomial equations and inequalities on
n
k
variables, where the involved polynomials have degree
n
-1.
Can be done to accuracy
ε
in time polynomial in
n
nk
log (1/ε). (using tools from the existential theory of the
reals
)
Overall time: polynomial in
nnk
log (1/ε) (c.f. input size: n
kn
= n
2n log k )Slide8
Algorithms for Nash
Equilibria
The Lemke-
Howson
Algorithm
Support Enumeration Algorithms
Algorithms for Symmetric GamesSlide9
Symmetries in Games
Def:
An n-player game is
symmetric
iff
:
- all players have the same strategy set:
S
= {1
,…,
k
}
- there exists a function
f
such that every player’s utility can be written
as:
u
p
(s
) =
f
(
s
p
;
n
1
(
s
-p
)
,…,
n
k
(
s
-p
) )
number of the other players choosing each strategy in
S
E.g. :
- congestion games, with same source destination pairs for each player
Thm
[Nash ’51]:
Always exists a Nash equilibrium in which every player uses the same mixed strategy.
- Rock-Paper-Scissors, guess 2/3 of the average
Description Size:
O(min
{
k
n
k-1
,
k
n
})Slide10
Existence of a Symmetric Equilibrium
Recall Nash’s function:
restrict Nash’s function on the set:
Thought
Experiment:
crucial observation:
Nash’s function maps points of the above set to itself as every player will perform the same updateSlide11
Algorithms for
n-player symmetric games
A symmetric equilibrium
x
= (
x
1
,
x
2
, …,
x
k
)
can be found as follows [Papadimitriou-
Roughgarden
’04]:
- guess the support of
x
: 2
k
possibilities
- write down a set of polynomial equations and inequalities corresponding to the equilibrium conditions, for the guessed support
- polynomial equations and inequalities of degree
n
in
k
variables (
cf
kn
for general games)
can be solved approximately in time polynomial in
n
k
log(1/ε)
using tools from the existential theory of the
reals
polynomial in the size of the input for
k
up to about
n
(
n
1-
δ
for all
δ
)Slide12
Symmetrization
R
,
C
C
T
,
R
T
R
,
C
x
/ |x|
1
y
/ |y|
1
x
y
x
y
Symmetric Equilibrium
Equilibrium
0, 0
0, 0
Any Equilibrium
Equilibrium
In fact we show that
[
Gale-Kuhn-Tucker 1950
]
w.l.o.g
. suppose that R, C have positive entries
Proof: On the board.Slide13
Symmetrization
R
,
C
R
T
,C
T
C, R
x
y
x
y
x
y
Symmetric Equilibrium
Equilibrium
0,0
0,0
Any Equilibrium
Equilibrium
In fact
[…]
Hence, essentially as hard to solve symmetric 2-player games as it is to solve general 2-player games
Open:
- Reduction from 3-player games to symmetric 3-player games?Slide14
Algorithms for Nash
Equilibria
The Lemke-
Howson
Algorithm
Support Enumeration Algorithms
Parenthesis: Symmetric GamesSlide15
Polytopes
101
Convex Polytope in
R
n
:
The intersection of
n
-dimensional
hyperplanes
(or half-spaces) of the form
a
T
x ≥ c, where a is a vector in
R
n
and
c
a scalar.
Often described compactly in matrix form:
A
x
≥
b
, where
A
is an
m
n
matrix and
b
is a vector
in
R
n
.
This inequality specifies the
polytope
that is the intersection of half-spaces:
A
iT
x ≥ b
i ,
i=1,…,m
(*)
Extreme-point of a Convex
Polytope
:
A point
x
such that
(*) is satisfied with at least
n
(linearly-independent
1
) inequalities tight.
Edge of a Convex
Polytope
:
Subset of the
polytope
where
n
-1 (linearly independent) inequalities are tight.
1
a collection of inequalities
B
i
T
x
≥
d
i
,
i
= 1, …,
k
are called
linearly independent
if the vectors
B
1
,…,
B
k
are linearly independent vectors of
R
nSlide16
The Lemke-
Howson Algorithm (1964)
Problem: Find an exact equilibrium of a 2-player game.
Since there exists a rational equilibrium this task is feasible.
Idea of LH:
Perform pivoting steps between the corners of a
polytope
related to the game until a Nash equilibrium is found.
Assumption (
w.l.o.g
.):
The game given in the input is a
symmetric game
, i.e.
Polytope
of Interest:
Assumption 2 (
w.l.o.g
):
At every corner of the
polytope
exactly
n
out of the 2
n
inequalities are tight.
(perturb original payoff entries with exponentially small noise to achieve this;
equilibria
of the new game are approximate eq. of original game of very high accuracy, and these can be converted to exact
equilibria
(exercise) )Slide17
The Lemke-
Howson Algorithm
At corner (0,0,…,0) all pure strategies are present. Call any corner of the
polytope
where this happens a
democracy.
Lemma:
If a vertex
z
≠0
of the
polytope
is a democracy, then is a Nash eq.
Proof:
Hence:
Def: Pure strategy
i
is
represented
at a corner
z
of the
polytope
if at least one of the following is tight:
At a democracy we have the following implication:Slide18
The Lemke-
Howson Algorithm
Start at the corner (0,0,…,0).
By non-degeneracy there are exactly
n
edges of the
polytope
adjacent to the (0,0,…,0) corner. Each of these edges corresponds to un-tightening one of the inequalities.
Select an arbitrary pure strategy, say pure strategy
n
, and un-tighten . This corresponds to an edge of the
polytope
adjacent to 0. Jump to the other endpoint of this edge.
If the obtained vertex
z
is a democracy, then a Nash equilibrium has been found because
z
≠0.
Otherwise, one of the strategies 1,…,
n
-1, say strategy
j
, is represented twice, by both
was already tight
just became tight
Question:
I will
untighten
one of the above. What happens if I require ?
A: I am going to return to (0,0,…,0), since I would be walking on the edge of the
polytope
that brought me here.
So let me
untighten
the other one, requiring .Slide19
The Lemke-
Howson Algorithm
If the obtained vertex is a democracy, then a Nash equilibrium has been found.
Otherwise, one of the strategies 1,…,
n
-1, is represented twice. This strategy is doubly represented because one of its inequalities was tight before the step, and the other one became tight after the step was taken. To proceed, un-tighten the former.
This defines a directed walk on the vertices of the
polytope
, starting at the democracy (0,0,…,0), and with every intermediate vertex having all of 1,…,
n
-1 represented, and exactly one of them represented twice.
The walk proceeds by un-tightening one of the two inequalities of the doubly represented strategy, namely the one that does not bring it back to where it came from.
The walk can keep going unless a democracy is encountered.
Claim:
The walk will settle on a democracy ≠ 0.
Proof:
Next slide. Slide20
Proof that Lemke-
Howson Terminates
Claim: The walk will settle on a democracy ≠ 0.
Proof:
Consider the vertices of the
polytope
that are either democracies or have only 1,…,
n
-1 represented (and exactly one of them represented twice).
Define a graph whose nodes are the aforementioned vertices, and whose edges are defined as follows:
a vertex where
n
is not represented and where
j
is represented twice has two neighbors corresponding to the vertices reached when un-tightening either
zj
≥ 0 or
R
j
z
≤1
a vertex that is a democracy has one neighbor corresponding to un-tightening whichever of
z
n
≥ 0 or
R
n
z
≤1 is tight
Clearly every node in this graph has degree ≤ 2, hence it comprises paths and cycles. In fact, nodes with degree 1 are democracies and only democracies have degree 1.
(0,0,…,0) has degree 1 so it is sitting on a path of this graph, whose other endpoint is another democracy ≠0
Lemke-
Howson
visits the vertices on the path where (0,0,…,0) sits, by the definition of the graph and the definition of the algorithm.Slide21
Lemke-
Howson Example
3 0 0
2 2 2
0 3 0Slide22
Post Mortem
The Lemke-
Howson algorithm:
- provides an alternative proof that a Nash equilibrium exists in 2-player games;
- moreover, it shows that there always exists a rational equilibrium in 2-player games;
it works by virtue of a
parity argument
: it identifies a directed path on the vertices of the
polytope
; since that path has a source, it must also have a sink.
worst-case running time: exponential in the number of strategies
[
Savani
-von Stengel’04]
.
there are analogs of the Lemke-
Howson
algorithm for multi-player games working with manifolds instead of
polytopes
( [
Rosenmuller
’71] and [Wilson ’71])