Computing Equilibria

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Computing Equilibria

• Christos H. Papadimitriou
• UC Berkeley
• christos

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Games help us understand rational behavior in
competitive situations
matching pennies
prisoners dilemma
chicken
1,-1 -1,1
-1,1 1,-1
4,4 1,5
5,1 0,0
3,3 0,4
4,0 1,1
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Concepts of rationality

• Nash equilibrium (or double best response)
• Problem may not exist
• Idea randomized Nash equilibrium
• Theorem Nash 1951 Always exists.

. . .
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can it be found in polynomial time?
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is it then NP-complete?
No, because a solution always exists
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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 cant find it, then neither can
  the market

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Complexity of Nash Equilibria?

• Nashs existence proof relies on Brouwers
  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

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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)

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An Easier ProblemCorrelated equilibrium
Chicken

• Two pure equilibria me, you
• Mixed (½, ½) (½, ½) payoff 5/2

4,4 1,5
5,1 0,0
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Idea (Aumann 1974)

• Traffic signal
• with payoff 3
• Compare with
• Nash equilibrium
• Even better
• with payoff 3 1/3

0 ½
½ 0
Probabilities in a lottery drawn by an impartial
outsider, and announced to each player separately
1/4 1/4
1/4 1/4
1/3 1/3
1/3 0
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Correlated equilibria

• Always exist (Nash equilibria are examples)
• Can be found (and optimized over) efficiently by
  linear programming

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Linear programming?

• A variable x(s) for each box s
• Each player does not want to deviate from the
  signals recommendation assuming that the
  others will play along
• For every player i and any two rows of boxes s,
  s'

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Linear programming!

• n players, s strategies each
• ns2 inequalites
• sn variables!
• Nice for 2 or 3 players
• But many players?

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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

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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
• multi-player
• succinctly representable

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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,

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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

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U
show it is unbounded
need to show dual is infeasible
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Lemma Hart and Schmeidler, 89 For every y
there is an x such that xUTy 0

• and in fact, x is the product of the
• steady-state distributions of the Markov
• chains implied by y
• Idea run ellipsoid against hope

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Leonid Khachiyan 1953-2005
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These k inequalities are themselves infeasible!
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also infeasible
infeasible
UXT
just need to solve
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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.

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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

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Corollaries

• Graphical games (on any graph!)
• Polymatrix games
• Hypergraphical games
• Congestion games and local effect games
• Facility location games
• Anonymous games
• Etc

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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
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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

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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!

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Multiplicationis the name of the gameand each
generationplays the same
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The multiplication game
x
z wins when it plays 1 and w plays 0
affects
z x y
w
if w plays 0, then it gets x?y. if it plays
1, then it gets z, but z gets punished
0 0
0 1
y
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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

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Finally, 4 players

• Previous reduction creates a bipartite graph of
  degree 3
• Carefully simulate each side by two players,
  refining the previous reduction

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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
Theorem (with Costas Daskalakis and Paul
Goldberg, 2005) and PPAD-complete
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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

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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

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In November

• Conjecture 1 3-player Nash is also
  PPAD-complete
• Proved!! ChenDeng05, DP05

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In December

• Conjecture 2 2-player Nash is in P
• PPAD-complete ChenDeng05b

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game over!
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Thank You!