Nash Equilibria In Graphical Games On Trees Revisited

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Nash Equilibria In Graphical Games On Trees
Revisited
(To appear in ACM EC06)

• Edith Elkind
• Leslie Ann Goldberg
• Paul Goldberg
• (University of Warwick)

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Normal Form Games(with 2 actions per player)

• finite set of players 1, , n
• each player has 2 actions
• (pure strategies) 0 and 1
• payoffs of the ith player Pi 0, 1n ? R

2 0
0 1
1 0
0 3
Row player
Column player
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Mixed Strategies

• pure strategy action
• mixed strategy probability distribution over
  actions
• pi Prob i plays 1
• expected payoff of the ith player
• for a strategy profile p (p1, , pn)
• EP(i) EPi(a) Probai1 pi

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Nash Equilibrium

• Nash equilibrium a strategy profile such that
• noone wants to deviate given other players
  strategies, i.e., each players strategy is a
  best response to others strategies
• (0, 0) and (1, 1) are both NE.
• any other NE?

2 0
0 1
1 0
0 3
Row player
Column player
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Finding NE in 2-player 2-action Games
2 0
0 1
1 0
0 3
Row player
Column player
BR(C)
Suppose R plays 1 w.p. r EP(C) from playing 0
(1-r)1 EP(C) from playing 1 r3 1-r gt 3r
iff r lt ¼
Suppose C plays 1 w.p. c EP(R) from playing 0
(1-c)2 EP(R) from playing 1 c1 (1-c)2 gt c
iff c lt 2/3
c
r
1
NE r1/4, c2/3
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NE for n-player 2-action games

• (poly-time) algorithm for NE in n-player games?
• representation payoffs to each player for every
  action profile (vector in 0, 1n) n2n numbers
• graphical games
• players are associated with the vertices of a
  graph
• each players payoff depends on his own action
  and actions of his neighbors
• n players, max degree d gt n2d1 numbers

W
t0, u0, v0, w0 12 t1, u0, v0, w0 31
. t1, u1, v1, w1 -6
Ws payoffs (16 cases)
T
V
U
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Related Work

• Bounded-degree trees
• Exp-time algorithm/poly-time approximation
  algorithm to find all NE (Kearns, Littmann,
  Singh, UAI 2001)
• ??? poly-time algorithm to find a single NE
  (Kearns, Littmann, Singh, NIPS2001)
• General graphs
• can it be NP-hard? no NE always exists
• hardness notion for total functions
  PPAD-hardness
• NE in graphical games with d 3 is PPAD-complete
  (GP, DGP, STOC06)

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Our Results

• Algorithm in NIPS01 paper is incorrect (does not
  always output a NE)
• We fix the NIPS01 algorithm, but
• our algorithm runs in poly-time on paths
• with a trick, also on cycles
• can be used to find all NE (rather than a single
  one)
• there is a graph of pathwidth 2 on which our
  algorithm (and all algorithms that use the basic
  approach of the UAI01 paper) runs in exp time
• The problem remains PPAD-complete for bounded
  pathwidth graphs
• Open question what if pathwidth 1?

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Algorithm for Trees

• Recall from 2-player case best response function
• Potential best response v is a PBR to w iff
  when W plays w,
  there is a NE for T in
  which V plays v.
• Bottom-up approach
  information propagates
  from the leaves to the root

c
c BR(r)
r
W
vPBRV(w)
V
T
U3
U1
U2
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Computing PBR Example

• Payoffs to U 0 if UV, 1 if U?V
• EP(U) from playing 0 v EP(U) from playing 1
  1-v
• Payoffs to V
• P0001, P001-9, P1009, P101-1, PU1W0 for all
  U, W
• EP(V) from playing 0
  (1-u)(1-w)1(1-u)w
  (-9)u(1-w)9uw(-1)
• V is indifferent btw 0 and 1 iff w (8u1)/10
  f(u)

v
u
1
1
(v, u) ? (f(u), v)
.5
.5
1
w
1
.1
.9
v
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Computing PBR General Case

• EP(V) from playing 0 a1uwa2ua3wa4
• EP(V) from playing 1 b1uwb2ub3wb4
• V is indifferent between 0 and 1 iff
  w f(u) AuB/CuD
• PBRV(W)L0 U f(PBRU(V)) U L1
• For paths, we can show that for any V, PBRV(W)
  consists of polynomially many segments
  (rectangles if degenerate)

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Computing PBR on Trees

• Trees similar algorithm
• Indifference function w L1(u1, , un)/L2(u1,
  , un)
• Potential best response can be exponential in
  size!