computing equilibria christos h. papadimitriou uc berkeley “christos”
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Computing Equilibria
Christos H. Papadimitriou
UC Berkeley
“christos”
Khachiyan lecture, March 2 06
1,-1 -1,1
-1,1 1,-1
4,4 1,5
5,1 0,0
3,3 0,4
4,0 1,1
matching pennies prisoner’s dilemmachicken
Games help us understand rational behavior in competitive situations
Khachiyan lecture, March 2 06
Concepts of rationality
• Nash equilibrium (or double best response)
• Problem: may not exist
• Idea: randomized Nash equilibrium
Theorem [Nash 1951]: Always exists....
Khachiyan lecture, March 2 06
can it be found in polynomial time?
Khachiyan lecture, March 2 06
is it then NP-complete?
No, because a solution always exists
Khachiyan lecture, March 2 06
…and why bother?(a parenthesis)
• Equilibrium concepts provide some of the most intriguing specimens of problems
• They are notions of rationality, aspiring models of behavior
• Efficient computability is an important modeling prerequisite“if your laptop can’t find it, then neither can the market…”
Khachiyan lecture, March 2 06
Complexity of Nash Equilibria?
• Nash’s existence proof relies on Brouwer’s fixpoint theorem
• Finding a Brouwer fixpoint is a hard problem
• Not quite NP-complete, but as hard as any problem that always has an answer can be…
• Technical term: PPAD-complete [P 1991]
Khachiyan lecture, March 2 06
Complexity? (cont.)
• But how about Nash?
• Is it as hard as Brouwer?
• Or are the Brouwer functions constructed in the proof specialized enough so that fixpoints can be computed?
(cf contraction maps)
Khachiyan lecture, March 2 06
An Easier Problem:Correlated equilibrium
4,4 1,5
5,1 0,0
Chicken:
•Two pure equilibria {me, you}
•Mixed (½, ½) (½, ½) payoff 5/2
Khachiyan lecture, March 2 06
Idea (Aumann 1974)
• “Traffic signal”
with payoff 3
• Compare with
Nash equilibrium
• Even better
with payoff 3 1/3
0 ½
½ 0
1/4 1/4
1/4 1/4
1/3 1/3
1/3 0
Probabilities in a lottery drawn by an impartial outsider, and announced to each player separately
Khachiyan lecture, March 2 06
Correlated equilibria
• Always exist (Nash equilibria are examples)
• Can be found (and optimized over) efficiently by linear programming
Khachiyan lecture, March 2 06
Linear programming?
• A variable x(s) for each box s
• Each player does not want to deviate from the signal’s recommendation – assuming that the others will play along
• For every player i and any two rows of boxes s, s':
( ) [ ( ) ( ')] 0 i is
x s u s u s
Khachiyan lecture, March 2 06
Linear programming!
• n players, s strategies each
• ns2 inequalites
• sn variables!
• Nice for 2 or 3 players
• But many players?
Khachiyan lecture, March 2 06
The embarrassing subject of many players
• With games we are supposed to model markets and the Internet
• These have many players
• To describe a game with n players and s strategies per player you need nsn numbers
Khachiyan lecture, March 2 06
The embarrassing subject of many players (cont.)
• These important games cannot require astronomically long descriptions
“if your problem is important, then its input cannot be astronomically long…”
• Conclusion: Many interesting games are
1. multi-player
2. succinctly representable
Khachiyan lecture, March 2 06
e.g., Graphical Games
• [Kearns et al. 2002] Players are vertices of a graph, each player is affected only by his/her neighbors
• If degrees are bounded by d, nsd numbers suffice to describe the game
• Also: multimatrix, congestion, location, anonymous, hypergraphical, …
Khachiyan lecture, March 2 06
Surprise!
Theorem: A correlated equilibrium in a succinct game can be found in polynomial time provided the utility expectation over mixed strategies can be computed in polynomial time.
Corollaries: All succinct games in the literature
Khachiyan lecture, March 2 06
U 0x max ix
0x TU 1y
0y
need to show dual is infeasible
show it is unbounded
Khachiyan lecture, March 2 06
Lemma [Hart and Schmeidler, 89]:For every y there is an x such that
xUTy = 0
•and in fact, x is the product of thesteady-state distributions of the Markovchains implied by y
•Idea: run “ellipsoid against hope”
Khachiyan lecture, March 2 06
Leonid Khachiyan [1953-2005]
Khachiyan lecture, March 2 06
1 0TxU y
2 0Tx U y
... 0Tkx U y
These k inequalities are themselves infeasible!
Khachiyan lecture, March 2 06
TU 1y
0y infeasible
TXU 1y
0y also infeasible
UXT 0x
0x just need to solve
Khachiyan lecture, March 2 06
as long as we can solve…
given a succinct representation of a game,
and a product distribution x,
find the expected utility of a player,
in polynomial time.
Khachiyan lecture, March 2 06
And it so happens that…
…in all known cases,
this problem can be solved
by applying one, two, or all three
of the following tricks:
• Explicit enumeration
• Dynamic programming
• Linearity of expectation
Khachiyan lecture, March 2 06
Corollaries:
• Graphical games (on any graph!)
• Polymatrix games
• Hypergraphical games
• Congestion games and local effect games
• Facility location games
• Anonymous games
• Etc…
Khachiyan lecture, March 2 06
Back to Nash complexity: summary
2-Nash 3-Nash 4-Nash … k-Nash …
1-GrNash 2-GrNash 3-GrNash … d-GrNash …
|||
Theorem (with Paul Goldberg, 2005):All these problems are equivalent
Khachiyan lecture, March 2 06
From d-graphical games to d2-normal-form games
• Color the graph with d2 colors• No two vertices affecting the same vertex
have the same color• Each color class is represented by a single
player who randomizes among vertices, strategies
• So that vertices are not “neglected:” generalized matching pennies
Khachiyan lecture, March 2 06
From k-normal-form games to graphical games
• Idea: construct special, very expressive graphical games
• Our vertices will have 2 strategies each
• Mixed strategy = a number in [0,1]
(= probability vertex plays strategy 1)
• Basic trick: Games that do arithmetic!
Khachiyan lecture, March 2 06
“Multiplicationis the name of the gameand each generationplays the same…”
Khachiyan lecture, March 2 06
The multiplication game
x
y
z = x · y
“affects”
w0 0
0 1
if w plays 0,then it gets xy.if it plays 1,then it gets z,but z gets punished
z wins whenit plays 1 and w plays 0
Khachiyan lecture, March 2 06
From k-normal-form games to 3-graphical games (cont.)
• At any Nash equilibrium, z = x y
• Similarly for +, -, “brittle comparison”
• Construct graphical game that checks the equilibrium conditions of the normal form game
• Nash equilibria in the two games coincide
Khachiyan lecture, March 2 06
Finally, 4 players
• Previous reduction creates a bipartite graph of degree 3
• Carefully simulate each side by two players, refining the previous reduction
Khachiyan lecture, March 2 06
Nash complexity, summary
2-Nash 3-Nash 4-Nash … k-Nash …
1-GrNash 2-GrNash 3-GrNash … d-GrNash …
|||
Theorem (with Paul Goldberg, 2005):All these problems are equivalentTheorem (with Costas Daskalakis andPaul Goldberg, 2005): …and PPAD-complete
Khachiyan lecture, March 2 06
Nash is PPAD-complete
• Proof idea: Start from a PPAD-complete stylized version of Brouwer on the 3D cube
• Use arithmetic games to compute Brouwer functions
• Brittle comparator problem solved by averaging
Khachiyan lecture, March 2 06
Open problems
• Conjecture 1: 3-player Nash is also PPAD-complete
• Conjecture 2: 2-player Nash can be found in polynomial time
• Approximate equilibria? [cf. Lipton Markakis and Mehta 2003]
Khachiyan lecture, March 2 06
In November…
• Conjecture 1: 3-player Nash is also PPAD-complete
• Proved!! [Chen&Deng05, DP05]
Khachiyan lecture, March 2 06
In December…
• Conjecture 2: 2-player Nash is in P• PPAD-complete [Chen&Deng05b]
Khachiyan lecture, March 2 06
game over!
Khachiyan lecture, March 2 06
Thank You!
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