CSCI 3210: Computational Game Theory Graphical Games Ref: [AGT] Ch - - PDF document
4/16/18 CSCI 3210: Computational Game Theory Graphical Games Ref: [AGT] Ch 7 https://www.cis.upenn.edu/~mkearns/papers/agt-kearns.pdf Mohammad T . Irfan Email: mirfan@bowdoin.edu Web: www.bowdoin.edu/~mirfan Course Website:
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Mohammad T . Irfan Email: mirfan@bowdoin.edu Web: www.bowdoin.edu/~mirfan Course Website: www.bowdoin.edu/~mirfan/CSCI-3210.html
https://www.cis.upenn.edu/~mkearns/papers/agt-kearns.pdf
u Usual games
u The games we've seen so far u Representation size: O(n mn) [normal form]
u n = # of players, m = # of actions u A player's payoff is specified for each joint action
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u Graphical games
u There's a graph/network u A player's payoff is directly determined by its
neighbors' actions as well as its own action
u Representation size (in normal form):
O(n md+1) + size of the graph
u d is the max degree of a node
u Binary actions: {0, 1} u Player (node) i chooses 1 if at least ti
neighbors choose 1
u Player i chooses 0, if less than ti neighbors
choose 1
4
A pure-strategy NE when all ti’s are 2
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u Another possible NE u Can we turn any of the 0s into 1 and still
maintain NE?
5
Another NE when all ti’s are 2
u Graph/network u Players: n players, each a node u Actions: binary actions {0, 1}
u Player i's action, ai {0, 1}
u Payoffs:
u Player (node) i chooses 1 if at least ti neighbors
choose 1
u Player i chooses 0, if less than ti neighbors choose 1 u Player i's payoff function =
u What is the representation size here? ∈ ai aj
j:neighbor of i
" # $ $ % & ' '−ti +ε " # $ $ % & ' '
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u Games in normal form
u O(n mn)
u Graphical games in normal form
u O(n md+1) + size of the graph u Each node has a payoff matrix u Great when d << n
u Graphical games in parametric form
u O(n) + size of the graph
Increasing size
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u Input: A graphical game in normal form
u A tree graph u Each node/player's payoff matrix is specified w.r.t.
the neighborhood joint actions
u Here, neighborhood contains the player itself
u Want: all pure-strategy NE
u Variant: want one pure-strategy NE
u A two-pass algorithm u Message passing using dynamic programming u First pass: explore "feasible" pure strategies u Second pass: find pure-strategy NE
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u T(v, w): Message that node V sends
down to W, parameterized by all possible pure strategies v and w of V and W, resp.
u T(v, w) = 1 iff there exists an
“upstream NE” from V, consistent with v and w
u The node V and all the nodes upstream
from V are "happy" (W may be unhappy) u T(v, w) = 0 o.w.
U1 Uk U2 L Upstream(V)
v w OK?
1 1 1 1 1 1 1
e.g.
u T(v, w) = 1 iff there exist actions
u1, u2, ... of V's parents U1, U2, ... such that
1.
T(ui, v) = 1, for all parents Ui of V
2.
v is node V's BR when node W plays w and parents play u1, u2, ...
V to W: I'm OK playing v if you play w Every parent Ui to V: We're OK playing ui if you play v Such a (u1, u2, ... ) is a witness vector to T(v, w) = 1
U1 Uk U2 L Upstream(V)
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u W commands to each parent V:
I'm playing w, you play v
u Will lead to a NE, as per the tables
u V gets the command v,
checks witness(es) to T(v,w) = 1, commands parents U1, U2, ... to play as per the witness vector
u Commands propagate upward
U1 Uk U2 L Upstream(V)
u Threshold games of complement
u ti = ½|Ni|
u Want: all pure-strategy NE u Downstream pass
u O(n d 2d) or O(n 2d) with better d.s. u Reason: T(u1, v) = 1 iff there exist
actions x1, x2, ... of U1's parents X1, X2, ... such that (1)... and (2)...
u Existence check: O(2d)
U1 L3
(0,0) { } (1,1) { } (0,0) { } (1,1) { } (0,0) { } (1,1) { } (0,0) { } (1,1) { } (0,0) {(0,0)} (1,0) {(1,1)} (0,1) {(0,0)} (1,1) {(1,1)}
L1 L2 L4
(0,0) {(0,0), (1,0)} (0,1) {(0,0)} (1,0) {(1,1)} (1,1) {(0,1), (1,1)} (0) {(0,0)} (1) {(0,1), (1,0), (1,1)} witness vector
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(0,0) { } (1,1) { }
u Upstream pass
u Nodes send command upward
(e.g. I’m playing 1, you play 0)
u Each pure-strategy NE in O(n d)
U1 L3
(0,0) { } (1,1) { } (0,0) { } (1,1) { } (0,0) { } (1,1) { } (0,0) {(0,0)} (1,0) {(1,1)} (0,1) {(0,0)} (1,1) {(1,1)}
L1 L2 L4
(0,0) {(0,0), (1,0)} (0,1) {(0,0)} (1,0) {(1,1)} (1,1) {(0,1), (1,1)} (0) {(0,0)} (1) {(0,1), (1,0), (1,1)}
(0,0) { } (1,1) { }
u Upstream pass
u Nodes send command upward
(e.g. I’m playing 1, you play 0)
u Each pure-strategy NE in O(n d)
U1 L3
(0,0) { } (1,1) { } (0,0) { } (1,1) { } (0,0) { } (1,1) { } (0,0) {(0,0)} (1,0) {(1,1)} (0,1) {(0,0)} (1,1) {(1,1)}
L1 L2 L4
(0,0) {(0,0), (1,0)} (0,1) {(0,0)} (1,0) {(1,1)} (1,1) {(0,1), (1,1)} (0) {(0,0)} (1) {(0,1), (1,0), (1,1)}
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(0,0) { } (1,1) { }
u Upstream pass
u Nodes send command upward
(e.g. I’m playing 1, you play 0)
u Each pure-strategy NE in O(n d)
U1 L3
(0,0) { } (1,1) { } (0,0) { } (1,1) { } (0,0) { } (1,1) { } (0,0) {(0,0)} (1,0) {(1,1)} (0,1) {(0,0)} (1,1) {(1,1)}
L1 L2 L4
(0,0) {(0,0), (1,0)} (0,1) {(0,0)} (1,0) {(1,1)} (1,1) {(0,1), (1,1)} (0) {(0,0)} (1) {(0,1), (1,0), (1,1)}
u Total running time to compute one pure-
strategy NE
u O(n 2d + n d)
u Each additional pure-strategy NE: O(n d)
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u Binary action u Probabilities: [0,1] u Message from U to V: May exist infinite
number of probabilities (u,v) s.t. T(u,v) = 1
u Good news: These probabilities can be
represented as unions of axis-aligned rectangles within [0,1]x[0,1]
u Bad news: The number of rectangles
exponentially blow up as we go down stream to root!
u Result: exponential time TreeNash algorithm
for MSNE
u Joint mixed strategy such that nobody gains
more than ε by unilateral deviation
u Every player plays ε-best response
u A strategy is ε-BR if it's not more than ε worse than
the actual BR u p* is an ε-NE iff every player i's expected
payoff ui(pi
*, p-i *) satisfies:
u 0-NE is what we call NE
*, p−i * )+ε ≥ max ai
* )
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u Discretize the mixed strategy of every player u How many grid points? Relation with ε? u How does the grid size affect computation? u Different grid size for different players?
1 .2 .4 .6 .8 1 τ 2τ ... ... Grid size = 1/τ
u T(v, w): v is V's probability in
the grid, w is W's prob. in grid
u Size of T(v, w) = (1/τ)2 u Algorithm remains the same u Running time: O(n (1/τ2)d)
u Reason: T(v, w) = 1 iff there exist
grid pt u1, u2, ... of V's parents U1, U2, ... such that (1)... and (2)...
u Existence check (brute force):
O((1/τ2)d)
u Question
u Relation between τ and ε for ε-NE?
U1 Uk U2 L Upstream(V)
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Link: http://www.cis.upenn.edu/~mkearns/ papers/graphgames.pdf
u ...
u Running time (downstream):
O(n (1/τ2)d)
u Is it a polynomial-time algorithm?
u Representation size: O(n md+1) [here m = 2]
k = # of parents
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u For an ε-NE, we only require
τ <= ε/(4d)
u Calculate the running time of TreeNash u Link:
http://www.cis.upenn.edu/~mkearns/ teaching/cgt/revised_approx_bnd.pdf
u Discretize both probability space and payoff
space
u A different algorithm: polynomial in the
input size and 1/ε
u Such algorithms are known as Fully Polynomial
Time Approximation Scheme (FPTAS) u Link:
http://www.bowdoin.edu/~mirfan/papers/ Ortiz_Irfan_AAAI17_Tractable_Algorithms_for _Approximate_Nash_Equilibria.pdf
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u Efficient algorithms for exact NE
computation in trees [polynomial in n and 2d is fine]
u Or prove that it is PPAD-hard u Negative result: No 2-pass dynamic programming
will succeed [Elkind et al., 2006]