Computing Nash Equilibrium

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Computing Nash Equilibrium

• Presenter Yishay Mansour

2
Outline

• Problem Definition
• Notation
• Last week Zero-Sum game
• This week
• Zero Sum Online algorithm
• General Sum Games
• Multiple players approximate Nash
• 2 players exact Nash

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Model

• Multiple players N1, ... , n
• Strategy set
• Player i has m actions Si si1, ... , sim
• Si are pure actions of player i
• S ?i Si
• Payoff functions
• Player i ui S ? ?

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Strategies

• Pure strategies actions
• Mixed strategy
• Player i pi distribution over Si
• Game P ?i pi
• Product distribution
• Modified distribution
• P-i probability P except for player i
• (q, P-i ) player i plays q other player pj

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Notations

• Average Payoff
• Player i ui(P) EsPui(s) ? P(s)ui(s)
• P(s) ?i pi (si)
• Nash Equilibrium
• P is a Nash Eq. If for every player i
• For any distribution qi
• ui(qi,P-i) ? ui(P)
• Best Response

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Two player games

• Payoff matrices (A,B)
• m rows and n columns
• player 1 has m action, player 2 has n actions
• strategies p and q
• Payoffs u1(pq)pAqt and u2(pq) pBqt
• Zero sum game
• A -B

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Online learning

• Playing with unknown payoff matrix
• Online algorithm
• at each step selects an action.
• can be stochastic or fractional
• Observes all possible payoffs
• Updates its parameters
• Goal Achieve the value of the game
• Payoff matrix of the game define at the end

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Online learning - Algorithm

• Notations
• Opponent distribution Qt
• Our distribution Pt
• Observed cost M(i, Qt)
• Should be MQt, and M(Pt,Qt) Pt M Qt
• cost on 0,1
• Goal minimize cost
• Algorithm Exponential weights
• Action i has weight proportional to bL(i,t)
• L(i,t) loss of action i until time t

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Online algorithm Notations

• Formally
• Number of total steps T is known
• parameter b 0lt b lt 1
• wt1(i) wt(i) bM(i,Qt)
• Zt ? wt(i)
• Pt1(i) wt1(i) / Zt
• Initially, P1(i) gt 0 , for every i

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Online algorithm Theorem

• Theorem
• For any matrix M with entries in 0,1
• Any sequence of dist. Q1 ... QT
• The algorithm generates P1, ... , PT
• RE(AB) ExA ln (A(x) / B(x) )

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Relative Entropy

• For any two distributions A and B
• RE(AB) ExA ln (A(x) / B(x) )
• can be infinite
• B(x) 0 and A(x) ? 0
• Always non-negative
• log is concave
• ? ai log bi ? log ? ai bi
• ? A(x) ln B(x) / A(x) ? ln ? A(x) B(x) / A(x) 0

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Online algorithm Analysis

• Lemma
• For any mixed strategy P
• Corollary

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Online Algorithm Optimization

• b 1/(1 sqrt2 (ln n) / T)
• additional loss
• O(sqrt(ln n )/T)
• Zero sum game
• Average Loss v
• additional loss O(sqrt(ln n )/T)

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Example Zero Sum
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Two players General sum games

• Input matrices (A,B)
• No unique value
• Computational issues
• find some Nash,
• all Nash
• Can be exponentially many
• identity matrix
• Example 2xN

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Computational Complexity

• Complexity of finding a sample equilibrium is
  unknown
• no proof of NP-completeness seems possible
  (Papadimitriou, 94)
• Equilibria with certain properties are NP-Hard
• e.g., max-payoff, max-support
• (Even) for symmetric 2-player games
• ? NE with expected social welfare at least k?
• ? NE with least payoff at least k?
• ? Pareto-optimal NE?
• ? NE with player 1 EU of at least k?
• ? multiple NE?
• ? NE where player 1 plays (or not) a particular
  strategy?

Gilboa Zemel, Conitzer Sandholm
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Two players General sum games

• player 1 best response
• Like for zero sum
• Fix strategy q of player 2
• maximize p (Aqt) such that ?j pj 1 and pj ?0
• dual LP minimize u such that u ? Aqt
• Strong Duality p(Aqt) u p u
• p( u Aq) 0
• complementary system
• Player 2 q(v- pB) 0

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Nash Linear Complementary System

• Find distributions p and q and values u and v
• u ? Aqt
• v ? pB
• p( u Aq) 0
• q(v- pB) 0
• ?j pj 1 and pj ? 0
• ?j qj 1 and qj ? 0

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Two players General sum games

• Assume the support of strategies known.
• p has support Sp and q has support Sq
• Can formulate the Nash as LP

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Approximate Nash

• Assume we are given Nash
• strategies (p,q)
• Show that there exists
• small support
• epsilon-Nash
• Brute force search
• enumerate all small supports!
• Each one requires only poly. time
• Proof!

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Nash Linear Complementary System

• Find distributions p and q and values u and v
• u ? Aqt
• v ? pB
• p( u Aq) 0
• q(v- pB) 0
• ?j pj 1 and pj ? 0
• ?j qj 1 and qj ? 0

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Lemke Howson

• Define labeling
• For strategy p (player 1)
• Label i if (pi0) where i action of player 1
• Label j if action j (payer 2) is best response
  to p
• bj p ? bkp
• Similar for player 2
• Label j if (qj0) where j action of player 2
• Label i if action i (payer 1) is best response
  to q
• ai q ? ajq

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LM algo

• strategy (p,q) is Nash if and only if
• Each label k is either a label of p or q (or
  both)
• Proof!
• Example

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Lemke-Howson Example
G1
G2
a3
a5
(0,0,1)
(0,1)
1
2
4
(0,1/3,2/3)
4
2
(1/3,2/3)
1
a1
3
(2/3,1/3)
5
(1,0,0)
a4
(2/3,1/3,0)
(1,0)
5
3
(0,1,0)
a2
U2
U1
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Lemke-Howson Example
G1
G2
a3
a5
(0,0,1)
(0,1)
1
2
4
(0,1/3,2/3)
4
2
(1/3,2/3)
1
a1
3
(2/3,1/3)
5
(1,0,0)
a4
(2/3,1/3,0)
(1,0)
5
3
(0,1,0)
a2
U2
U1
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LM non-degenerate

• Two player game is non-degenerate if
• given a strategy (p or q)
• with support k
• At most k pure best responses
• Many equivalent definitions
• Theorem For a non-degenerate game
• finite number of p with m labels
• finite number of q with n labels

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LM Graphs

• Consider distributions where
• player 1 has m labels
• player 2 has n labels
• Graph (per player)
• join nodes that share all but 1 label
• Product graph
• nodes are pair of nodes (p,q)
• edges if (p,p) an edge then (p,q)-(p,q) edge

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LM

• completely labeled node
• node that has mn labels
• Nash!
• node k-almost completely labeled
• all labeling but label k.
• edge k-almost completely labeled
• all labels on both sides except label k
• artificial node (0,0)

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LM Paths

• Any Nash Eq.
• connected to exactly one vertex which is
• k-almost completely labeled
• Any k-almost completely labeled node
• has two neighbors in the graph
• Follows from the non-degeneracy!

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LM algo

• start at (0,0)
• drop label k
• follow a path
• end of the path is a Nash

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Lemke-Howson Algorithm
a3
a5
(0,0,1)
G1
G2
(0,1)
1
2
4
(0,1/3,2/3)
4
2
(1/3,2/3)
1
a1
3
(2/3,1/3)
5
(1,0,0)
a4
(2/3,1/3,0)
(1,0)
5
3
(0,1,0)
a2
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Lemke-Howson Algorithm
a3
a5
G2
(0,0,1)
G1
(0,1)
1
2
4
(0,1/3,2/3)
4
2
(1/3,2/3)
1
a1
3
(2/3,1/3)
5
(1,0,0)
a4
(2/3,1/3,0)
(1,0)
5
3
(0,1,0)
a2
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Lemke-Howson Algorithm
a3
a5
(0,0,1)
G1
G2
(0,1)
1
2
4
(0,1/3,2/3)
4
2
1
(1/3,2/3)
a1
3
(2/3,1/3)
5
(1,0,0)
a4
(2/3,1/3,0)
(1,0)
5
3
(0,1,0)
a2
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Lemke-Howson Other Equilibria
a3
a5
G1
(0,0,1)
G2
(0,1)
1
2
4
(0,1/3,2/3)
4
2
1
(1/3,2/3)
a1
3
(2/3,1/3)
5
(1,0,0)
a4
(2/3,1/3,0)
(1,0)
5
3
(0,1,0)
a2
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LM Theorem

• Consider a non-degenerate game
• Graph consists of disjoint paths and cycles
• End points of paths are Nash
• or (0,0)
• Number of Nash is odd.

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LM Sketch of Proof

• Deleting a label k
• making support larger
• making BR smaller
• Smaller BR
• solve for the smaller BR
• subtract from dist. until one component is zero
• Larger support
• unique solution (since non-degenerate)