Mini QuichePrepare 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)

Game NetworksGame Networks

Piero La MuraStanford University

Motivation:

providing a natural and computationally efficient framework for strategic inference in multi-agent decision problems

Beer/Quiche Game

A Nash Equilibrium

Another Nash Equilibrium

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.

Multi-agent Influence Diagrams?

Player1 Player2This doesn’t work.(of course)

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

.

Decision-theoretic background

Let S be a set of states, A a Boolean algebra on S, and u: S RR 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 EFG,• p(E|G)=p(E|F)p(F|G)• p( . |F) is well defined even when p(F|G)=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.

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).

Ceteris Paribus Comparisons

Let {Xk}kN be a set of variables,

for MN, let XM:=kM Xk ,

and let x 0=(xM0, xN -M

0) be an arbitrary reference

point.

Ceteris Paribus Comparisons for probabilities

q(xM|xN) = p(xM , xN-M) / p(xM0

, xN-M)

Ceteris Paribus Comparisons for utilitiesw(xM|xN) = u(xM , xN-M) / u(xM

0 , xN-M)

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(a|b,c)=p(a|b0,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(a|b,c)=w(a|b0,c), where w(a|b,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.

Health and Wealth

u(...) -W W-H 1 2

H 3 6

u(…) -W W-H 1 2

H 3 4

Shoham’s 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)

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)

A G net is comprised of the following elements:

• a finite, ordered set of nodes XN (N={1,2,...,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(xk|xPP(k)) and w(xk|xUN(k)), where XPP(k) are the probability parents of Xk and XUN(k) its utility neighbors.

S

B F

p(S)=0.9

w1(S|-B,-F)=1

w1(S|B,-F)=1

w1(S|-B,F)=1

w1(S|B,F)=1

w2(S|-F)=1

w1(B|S)=3/2

w1(B|-S)=2/3

w1(F|S)=1/2

w1(F|-S)=1/4

w2(F|S)=1/2

w2(F|-S)=2

reference point: -S,-B,-F(not strong, no beer, no fight)

payoffs = utility relative to reference point, times 12.

Example: the Beer/Quiche game

A1 A2 A3 A4

H1 H2 H3

B1 B2 B3 B4

A1|H0

H0

B1|H0

A2|H1

B2|H1

Another Example: a multi-stage game with observed actions

(e.g., the repeated prisoners’ dilemma)

Bayesian rationality in G nets

A G frame is an incompletely specified G net, where only the probabilities of Nature’s 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 kN and xPP(k) XPP(k), ui(k)(xk|xPP(k))> ui(k)(x’k|xPP(k)) implies p(x’k|xPP(k))=0.

Corollary: For any G frame there exists a corresponding G net which satisfies Bayesian rationality.

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(xk|xPP(k)) ui(k)(p)(xk|xPP(k)) Xk q(xk|xPP(k)) ui(k)(p)(xk|xPP(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)(xk|xPP(k)) := p(xk|xPP(k)) ui(k)(p)(xk|xPP(k)).

Then f has a fixed point by Brouwer’s theorem. A limit point of a sequence of fixed points of f as goes to zero is a Nash (indeed, a sequential) equilibrium.

Global convergence to equilibrium in G nets

Nash equilibria are zeros of the function F(p), defined by

F(p)(xk|xPP(k)):= p(xk|xPP(k)) - v(p)(xk|xPP(k)).

Consider the perturbed problem:

F(p)(xk|xPP(k)):= p(xk|xPP(k)) - f(p)(xk|xPP(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(xk|xPP(k)) - z(xk|xPP(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).

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)(xk|xPP(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.

Computing all equilibria

Let G(p) be defined by

G(p)(xk|xPP(k)):= ui(k)(p)(xPP(k)) F(p)(xk|xPP(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 CCn.

Let G0 be the start system defined by

G0j(p)= j

dj pjdj - j

dj,

where j= (xk|xPP(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 CCn such that, if A and A, then:• the solution set { (p,t) CCn [0,1) : h(p,t)=0 } is a collection of non-overlapping, smooth paths;• the paths move from t=0 to t=1 without backtracking in t;• each geometrically isolated solution of G=0 of multiplicity m has exactly m continuation paths converging to it.

• The Nash equilibria are the real solutions of G = 0 in the region delimited by ui(k)(p)(xk|xPP(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.

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.