Mini Quiche

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Mini Quiche
Prepare a double batch of pie crust or purchase a
pie crust mix. Roll out and cut in circles using
a donut cutter or cooking cutter. Place circles
into bottom of muffin pan so that pie crust curls
up the sides by about 1/4 inch. Fill with any
combo of filling. I used bacon, mushrooms,
onions, and swiss for one filling. The other was
broccoli, onions and cheddar cheese. Place just
enough filling so it does not go above lip of
crust. Beat 2 eggs and 3/4 cup sour cream.
Spoon 1 teaspoon of egg mixture over each
quiche. Bake for 20-25 min in 375 oven. Reheat
defrosted quiche for 10 min in a 400
oven. Recipe Archive at Carnegie Mellon's School
of Computer Science (SCS)
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Game Networks Piero La Mura Stanford University
Motivation providing a natural and
computationally efficient framework for
strategic inference in multi-agent decision
problems
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Beer/Quiche Game
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A Nash Equilibrium
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Another Nash Equilibrium
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• Existing frameworks for strategic inference
• include
• normal forms (NF)
• extensive forms (EF)
• sequence forms (SF)
• influence diagrams (ID)
• expected utility networks (EUN).
• Of these,
• EF, SF and ID are dynamic, i.e. capture the
  causal flow of events in the game
• NF, EF and SF are multi-agent representations
• EUN capture strategic independencies.
• G nets are a dynamic, multi-agent generalization
  of EUN.

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Multi-agent Influence Diagrams?
Player1
Player2
This doesnt work. (of course)
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G nets, informally Game networks (G nets) are a
novel class of game representations and
associated inference procedures, which can
exploit strategic separabilities in order to
simplify the inference process. G nets come
with two novel inference procedures one which
selects a single strategic equilibrium as a
function of the game payoffs, and one which
identifies all equilibria .
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• Decision-theoretic background
• Let S be a set of states, A a Boolean algebra on
  S, and
• u S? R a strictly positive (utility) function.
• An expected utility function is a pair (p,u),
  where p is a probability function on A.
• A conditional probability system is a family of
  conditional probability functions p( . F) such
  that, for all E?F?G,
• p(EG)p(EF)p(FG)
• p( . F) is well defined even when p(FG)0,
  i.e. even when a condition F has zero
  probability.
• A conditional expected utility function is a pair
  (p,u) where p is a conditional probability system.

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The expected utility of a possible event E is
defined as u(E)? u(s) p(s E) and the
conditional expected utility of E given F is
defined as u(E F) u(E ? F) / u(F). Finally,
the conditional value of E given F is defined
as v(E F)p(E F) u(E F).
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Ceteris Paribus Comparisons
Let Xkk?N be a set of variables, for M?N, let
XM?k?M Xk , and let x 0(xM0, xN -M 0) be an
arbitrary reference point. Ceteris Paribus
Comparisons for probabilities q(xMxN) p(xM ,
xN-M) / p(xM0 , xN-M) Ceteris Paribus
Comparisons for utilities w(xMxN) u(xM , xN-M)
/ u(xM0 , xN-M)
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Conditional independence of probabilities and
utilities Let A,B,C be a partition of N and
let x0(a0,b0,c0) be an arbitrary reference
point. We say that A is p-independent of B given
C if and only if p(ab,c)p(ab0,c) for all
(a,b,c)?(A,B,C). We say that A is u-independent
of B given C if and only if w(ab,c)w(ab0,c),
where w(ab,c)u(a,b,c)/u(a0,b,c). Finally,
we say that A is strategically independent of B
given C if A is both p- and u- independent of B
given C.
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Health and Wealth
Shohams Additive Utility Independence u(H,W) -
u(-H,-W) u(-H,W) - u(-H,-W)
u(H,-W)-u(-H,-W) u(H,W) u(-H,-W) u(-H,W)
u(H,-W) u-independence u(H,W) /u(-H,-W)
u(-H,W) / u(-H,-W) u(H,-W)/u(-H,-W) u(H,W)
u(-H,-W) u(-H,W) u(H,-W)
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Consequences of strategic independence If
strategic independence holds, decisions regarding
A and B given C can be effectively decentralized
the agent who decides on A does not need to know
about B in order to choose its optimal action
given C, and vice versa. (La Mura and Shoham
1999)
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• A G net is comprised of the following elements
• a finite, ordered set of nodes XN
  (N1,2,...,n) corresponding to a set of
  strategically relevant variables
• a partition I of N which determines the
  identity of the agent responsible for the
  decision at each node (including Nature)
• a set of directed (probability) arcs, with no
  cycles, representing causal dependencies
• a set of undirected (utility) arcs representing
  payoff dependencies
• for each Xk, two functions p(xkxPP(k)) and
  w(xkxUN(k)), where XPP(k) are the probability
  parents of Xk and XUN(k) its utility neighbors.

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Example the Beer/Quiche game
reference point -S,-B,-F (not strong, no beer,
no fight) payoffs utility relative to
reference point, times 12.
p(S)0.9 w1(S-B,-F)1 w1(SB,-F)1 w1(S-B,F)1 w
1(SB,F)1 w2(S-F)1
S
w1(FS)1/2 w1(F-S)1/4 w2(FS)1/2 w2(F-S)2
w1(BS)3/2 w1(B-S)2/3
B
F
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Another Example a multi-stage game with
observed actions (e.g., the repeated prisoners
dilemma)
A1H0
B1H0
A2H1
B2H1
A1
A2
A3
A4
H1
H2
H3
H0
B1
B2
B3
B4
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Bayesian rationality in G nets A G frame is an
incompletely specified G net, where only the
probabilities of Natures actions (but not those
of the other agents) are specified. Theorem Any
finite game in extensive form has a G frame
representation. A G net satisfies Bayesian
rationality if, for all k?N and xPP(k) ? XPP(k),
ui(k)(xkxPP(k))gt ui(k)(xkxPP(k)) implies
p(xkxPP(k))0. Corollary For any G frame
there exists a corresponding G net which
satisfies Bayesian rationality.
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• Strategic equilibrium in G nets
• A Nash equilibrium (in the agent-strategic form)
  is a conditional probability system p?? such
  that, for all q ??,
• ?Xk p(xkxPP(k)) ui(k)(p)(xkxPP(k)) ? ?Xk
  q(xkxPP(k)) ui(k)(p)(xkxPP(k)).
• Let f? ?? ? be defined by f? ?z (1- ?)v,
  where
• z is the conditional probability system which
  gives uniform probability to all the possible
  actions, and
• v(p)(xkxPP(k)) p(xkxPP(k))
  ui(k)(p)(xkxPP(k)).
• Then f? has a fixed point by Brouwers theorem. A
  limit point of a sequence of fixed points of f?
  as ? goes to zero is a Nash (indeed, a
  sequential) equilibrium.

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Global convergence to equilibrium in G nets Nash
equilibria are zeros of the function F(p),
defined by F(p)(xkxPP(k)) p(xkxPP(k)) -
v(p)(xkxPP(k)). Consider the perturbed
problem F?(p)(xkxPP(k)) p(xkxPP(k)) -
f?(p)(xkxPP(k)). This can be rewritten as
?F0(p) (1- ?)F(p), where F is the target system
whose zeros we want to find, and F0 is the
trivial system p(xkxPP(k)) - z(xkxPP(k)), whose
unique solution is p z. Then F? defines a
convex-linear homotopy h(p,t) F1-t. Its
homotopy paths are very well behaved, and can be
tracked using standard computational methods
(Morgan 1987).
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• First equilibrium
• For G nets with generic payoffs, the end point
  of the homotopy path generated by h(p,t)
  identifies a unique sequential equilibrium (first
  equilibrium).
• The first equilibrium is uniquely determined by
  the G net structure, and can be computed using
  standard path-tracking techniques.
• Most importantly, strategically independent
  variables do not affect the values of
  F(p)(xkxPP(k)). It follows that, in the presence
  of strategic independencies, convergence to the
  zeros of a large system can be reduced to
  convergence to the zeros of smaller,
  strategically independent subsystems.

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Computing all equilibria Let G(p) be defined by
G(p)(xkxPP(k)) ui(k)(p)(xPP(k))
F(p)(xkxPP(k)). Since u is strictly positive,
the zeros of G coincide with the zeros of F.
Note that G(p) is a vector of polynomial
functions, whose zeros include all the Nash
equilibria. A Nash equilibrium may not have any
homotopy path converging to it in Rn yet, as it
turns out, one can always get at all of them in
the complex space Cn.
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• Let G0 be the start system defined by
• G0j(p) ?jdj pjdj - ?jdj,
• where j (xkxPP(k)), dj is the degree of Gj(p),
  and ?j and ?j are generic complex constants. Then
  G0(p) 0 has ?j dj solutions.
• Let h(p,t) be the homotopy defined by h(p,t)(1 -
  t) G0(p) t G(p) then the following result
  holds.
• Theorem (Morgan 1987) Given G, there are sets of
  measure zero, A? and A? in Cn such that, if ? ?
  A? and ? ? A?, then
• the solution set (p,t)? Cn ? 0,1) h(p,t)0
  is a collection of non-overlapping, smooth
  paths
• the paths move from t0 to t1 without
  backtracking in t
• each geometrically isolated solution of G0 of
  multiplicity m has exactly m continuation paths
  converging to it.

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• The Nash equilibria are the real solutions of G
  0 in the region delimited by ui(k)(p)(xkxPP(k))
  ? 1
• The mixed-strategy equilibria are the solutions
  with multiplicity higher than one
• The homotopy paths are very well-behaved, and
  can be tracked using standard computational
  methods
• The procedure applies to general games yet, as
  in the single-equilibrium case, one can exploit
  the strategic separabilities in the G net
  structure in order to simplify the computations.

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• Conclusions
• G nets are a novel multi-agent framework for
  strategic inference
• G nets are more modular and compact than
  standard game-theoretic representations, thanks
  to a structured representation of probabilities,
  utilities and expected utilities
• moreover, they come with two novel computational
  procedures which can take advantage of the extra
  structure in order to simplify the inference
  process
• the first procedure selects a unique sequential
  equilibrium as a function of the G net structure
• while the second identifies all Nash equilibria.