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Comparing the notions of optimality in strategic games and soft constraints (and CP nets). K. R. Apt*, F. Rossi**, K. B. Venable** * CWI and Univ. of Amsterdam ** University of Padova, Italy. Main aim. To compare the notion of optimality used in many formalisms

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Comparing the notions of optimality in strategic games and soft constraints and cp nets

Comparing the notions of optimality in strategic games and soft constraints(and CP nets)

K. R. Apt*, F. Rossi**, K. B. Venable**

* CWI and Univ. of Amsterdam

** University of Padova, Italy


Main aim
Main aim soft constraints

  • To compare the notion of optimality used in many formalisms

  • To exploit results in one field and reuse them in the other field (ex. computational results)

  • Strategic games

    • Agent interaction while pursuing their own interest (payoff function)

  • CP nets

    • Agent’s qualitative and conditional preferences

  • Soft constraints

    • Agent’s quantitative preferences


Outline
Outline soft constraints

  • CP nets

  • Strategic games

  • Relation between CP nets and games

  • Soft constraints

  • Relation between soft constraints and games


Cp statements
CP statements soft constraints

  • Conditional ceteris paribus preference statements

  • Example:

    • If there is fish, I prefer white wine to red all else being equal

    • Written asfish: white wine > red wine

    • Ceteris paribus interpretation

      • {fish, white wine, peaches} > {fish, red wine, peaches}

      • {fish, white wine, peaches} >? {fish, red wine, strawberries}


Cp nets
CP-nets soft constraints

  • Graphical representation of a set of CP statements

  • A directed graph over features X1, … , Xn

    • each feature may depend on some others (its parents): arc from parents to child

  • A set of conditional preference tables

    • Each row is a cp statement for Xi given values for its parents

    • Complete tables: one row for each instatiation of the parents

    • Total order on domain of Xi

Ref.: Boutilier, Brafman, Hoos, Poole,1999.


Cp net an example

Main soft constraints

course

Independent

feature

Dependent

feature

Wine

Independent

feature

Fruit

CP net: an example

fish>meat

Conditional Preference Table

peaches > strawberries


Cp net semantics

(fish, white wine, peaches) soft constraints

worsening flip

(fish, red wine, peaches)

CP-net semantics

  • Worsening flip: changing the value of an attribute in a way that is less preferred in some statement. Example:

  • An outcome O1 is preferred to O2 iff there is a sequence of worsening flips from O1 to O2

  • Ordering induced: preorder

  • Optimal outcomes: no flip brings to a better outcome


Example

Fish, white, peaches soft constraints

Main

course

Fish, red, peaches

Fish, white, berries

Fish, red, berries

Wine

meat, red, peaches

meat, red, berries

meat, white, peaches

Fruit

meat, white, berries

Example

fish>meat

peaches > strawberries


Finding optimal outcomes in cp nets
Finding optimal outcomes in CP nets soft constraints

  • From a CP-net to a set of hard constraints:

    • For each cp statement A: b1>b2, we get the constraint a → b1 (a implies b)

    • An outcome is optimal for the CP-net iff it is a solution of these constraints

  • Acyclic CP-net

    • No cycles in the graph

    • Linear time: sweep forward in the topological order


Strategic games
Strategic games soft constraints

  • A set of players 1,.., n

  • For each player i:

    • A set of strategies Si

    • A total order iover Si depending on s-i (a joint strategy of all players but player i): payoff function

  • Example (prisoner’s dilemma): 2 players, 2 strategies (ci, ni) for each player i


Nash equilibria
Nash equilibria soft constraints

  • A strategy si is a best response for i to s-i if sii s’i for all s’i in Si

  • A joint strategy s is a Nash equilibrium if each si is a best response to s-i

    • Also: for all i, for all s’i in Si, si i s’i

    • No player has regrets on the strategy he chose

    • But there could be better joint strategies if more than one player changed its strategy

  • In the example, one Nash equilibria (NE): (N1,N2)


Pareto efficient joint strategies
Pareto efficient joint strategies soft constraints

  • No other joint strategy is better or equal for all agents, and better for at least one

  • Example:

    • (N1,N2): unique Nash equilibrium

    • All other joint strategies are Pareto efficient (PE)


From cp nets to games
From CP-nets to games soft constraints

  • Given a CP-net N, we build the game g(N)

    • Players: features

    • Strategies of player i: domain of feature xi

    • Payoff function of player i: CP table for xi

      • Given s-i, s’i >i si iff s-i|par(xi) : s’i >i si in the cp table for variable i

  • Thm: opt(N) = NE(g(N))


Example cp net

Fish, white, peaches soft constraints

Main

course

Fish, red, peaches

Fish, white, berries

Fish, red, berries

Wine

meat, red, peaches

meat, red, berries

meat, white, peaches

Fruit

meat, white, berries

Example – CP net

fish>meat

peaches > strawberries


Example players and strategies
Example – players and strategies soft constraints

  • Three players: 1 = main course, 2 = wine, 3 = fruit

  • Two strategies for each player:

    • 1: meat, fish

    • 2: red, white

    • 3: peaches, strawberries


Example payoff functions
Example: payoff functions soft constraints

  • For wine:

    • fish, --  white > red

    • meat, --  red > white

  • For main course:

    • --, --  fish > meat

  • For fruit:

    • --, --  peaches > strawberries


Example optimals and nash equilibria
Example: optimals and Nash equilibria soft constraints

  • Unique optimal for CP-net: (fish, white, peaches)

  • For the game:

    • Nash equilibrium: (fish, white, peaches)


From games to cp nets
From games to CP-nets soft constraints

  • Given a game G, we build a CP-net n(G):

    • Feature xi: player i

    • Domain of xi: strategies for player i

    • Parents of xi: all the other features

    • CP table of xi: s-i: si > s’i if si >i s’i given s-i

  • Thm.: NE(G) = opt(n(G))


Example1
Example soft constraints

  • Two features: x1, x2

  • D(x1)={c1, n1}

  • D(x2)={c2,n2}

  • x1 depends on x2

    • x2=c2: n1 > c1

    • x2=n2: n1 > c1

  • x2 depends on x1

    • X1=c1: n2 > c2

    • X1=n1: n2 > c2

  • Hard constraints:

    • x2=c2 → x1=n1

    • x2=n2 → x1=n1

    • x1=c1 → x2=n2

    • x1=n1 → x2=n2

  • Unique solution: x1=n1, x2=n2


Reduced cp nets
Reduced CP-nets soft constraints

  • If y is a parent of x, but the preference over the domain of x does not depend on y, then we can remove y from the parents of x  eliminate rows

  • From a CP net N to its reduced version r(N)


Example reduced cp net

Two features: x1, x2 soft constraints

D(x1)={c1, n1}

D(x2)={c2, n2}

x1 depends on x2

x2=c2: n1 > c1

x2=n2: n1 > c1

x2 depends on x1

X1=c1: n2 > c2

X1=n1: n2 > c2

Two features: x1, x2

D(x1)={c1, n1}

D(x2)={c2, n2}

x1 and x2 independent

For x1: n1 > c1

For x2: n2 > c2

Example: reduced CP-net


Cp net techniques in games

n soft constraints

G

n(G)

r

r(n(G))

Nash equilibria of G

= optimals of r(n(G))

CP-net techniques in games

  • From game G to n(G)

  • Hard constraints for r(n(G))

  • Optimals of r(n(G)) = Nash equilibria of G


Games and acyclic cp nets

n soft constraints

G

n(G)

r

Nash equilibrium of G

= optimal of r(n(G))

r(n(G)) acyclic

linear

time

Games and acyclic CP-nets

  • From game G to r(n(G))

  • If r(n(G)) is acyclic, then G has one Nash equilibrium, and linear time to find it


Soft constraints the c semiring framework
Soft Constraints: soft constraintsthe c-semiring framework

  • Variables{X1,…,Xn}=X

  • Domains{D(X1),…,D(Xn)}=D

  • Soft constraints

    • each constraint involves some of the variables

    • a preference is associated with each assignment of the variables

  • Set of preferences A

    • Totally or partially ordered (induced by +)

    • Combination operator (x)

    • Top and bottom element (1, 0)

    • Formally defined by a c-semiring <A,+,x,0,1>


Soft constraints
Soft constraints soft constraints

  • Soft constraint: a pair c=<f,con> where:

    • Scope: con={Xc1,…, Xck} subset of X

    • Preference function :

      f: D(Xc1)x…xD(Xck) → A

      tuple (v1,…, vk) → p preference

  • Hard constraint: a soft constraint where for each tuple (v1,…, vk)

    f (v1,…, vk)=1 the tuple is allowed

    f (v1,…, vk)=0 the tuple is forbidden


Solutions
Solutions soft constraints

  • Solution: assignment to all variables

  • Preference of a solution:

    • Combination (via x) of the preferences of the partial assignments given by the constraints

  • Optimal solutions: those with the highest preference


Some instances of soft constraints
Some instances of soft constraints soft constraints

  • Each instance is characterized by a c-semiring

    <A, +, x, 0, 1>

  • Classical constraints: <{0,1}, logical or, logical and, 0, 1>

    • Satisfy all constraints

  • Fuzzy constraints: <[0,1], max, min, 0, 1>

    • Maximize the minimum preference

  • Weighted constraints (N):<N+, min, +, +, 0>

    • Minimize the sum of the costs


Example2

X soft constraints

Y

Z

(a,a)  0.4

(a,b)  0.1

(b,a)  0.3

(b,b)  0.5

(a,a)  0.4

(a,b)  0.3

(b,a)  0.1

(b,b)  0.5

Example

  • Fuzzy CSPs

    • maximize the minimum preference

  • Optimal solution: x=y=z=b, with preference 0.5


Combination operator
Combination operator soft constraints

  • Extensive (always): for all a,b in A, a x b  a,b

  • Idempotent: for all a in A, a x a = a

    • Ex.: max, min, and

    • Ex. of instances: fuzzy, classical

    • It is possible to apply soft constraint propagation

  • Strictly monotonic: for all a,b in A, a x b

    • Ex.: sum, product

    • Ex. of instances: weighted

  • It cannot be idempotent and strictly monotonic at the same time


From soft csps to games a local approach
From soft CSPs to games: a local approach soft constraints

  • From a soft CSP P to a game L(P)

    • Players: one for each variable

    • Strategies for a player i: all values in domain of xi

    • Payoff of player i for joint strategy s: preference for assignment s in constraints involving xi


Example3
Example soft constraints

X

Y

Z

  • Three players x,y,z

  • Two strategies a,b

  • Payoff functions

    • For x: p(aa-)=0.4, ... (same for z)

    • For y:

      • p(aaa) = min(0.4,0.4) = 0.4

      • p(aab) = min(0.4,0.3)=0.3

      • ...

  • Two Nash equilibria: aaa and bbb

  • Optimal solutions: only bbb

(a,a)  0.4

(a,b)  0.1

(b,a)  0.3

(b,b)  0.5

(a,a)  0.4

(a,b)  0.3

(b,a)  0.1

(b,b)  0.5


Strictly monotonic combination
Strictly monotonic combination soft constraints

  • In general, no relationship between optimal solutions of P and Nash equilibria of L(P)

  • However, some relationship exist if combination is strictly monotonic

  • Thm.:Soft CSP P with strictly monotonic combination  Opt(P)  NE(L(P))


Classical csps
Classical CSPs soft constraints

  • Classical constraints are combined via logical and (which is not strictly monotonic)

  • However, if we consider consistent CSPs, the result holds

  • Thm.: consistent CSP  Sol(P)  NE(L(P))


A global mapping
A global mapping soft constraints

  • Given an SCSP P, build a game GL(P):

    • Players = variables

    • Strategies = domain values

    • Payoff for player x for strategy s: preference value for that assignment (by looking at all constraints)

  • Note: same payoff for all players

  • Theorem: Opt(P)  NE(GL(P))

  • Subset relation for all classes of SCSPs


From strategic games to scsps
From strategic games to SCSPs soft constraints

  • From a game G to an SCSP L’(G):

    • Variables = players (n)

    • Domains = strategies

    • Semiring = Cartesian product of n semirings

    • For each variable xi, one constraint involving xi and its neighbourhood

      • pref(t) = (d1,...,dn), where dj = 1j for j  i, and di = pi(t)

  • Thm.: Game G  opt(L’(G)) = PE(G)


Example4

(c1,c2) soft constraints (0,7)

(c1,n2)  (0,6)

(n1,c2)  (0,10)

(n1,n2)  (0,9)

X1

x2

(c1,c2)  (7,0)

(c1,n2)  (10,0)

(n1,c2)  (6,0)

(n1,n2)  (9,0)

Example

Semiring: weighted x weighted

  • Pareto efficient joint strategies: all but (1,1)

  • Optimal solutions:

    • (c,c) with pref. (7,7)

    • (n,c) with pref. (10,6)

    • (c,n) with pref. (6, 10)


Computational issues
Computational issues soft constraints

  • If opt(P) = NE(G) or opt(P)=PE(G)

    • Exploit the SCSPs/CP net machinery and tractability results to compute exactly NEs and PEs

  • If opt(P)  NE(G)

    • finding Opt(P) gives a lower approximation of NE(G)


Summary of main results
Summary of main results soft constraints

  • Optimal outcomes in CP nets are Nash equilibria in corresponding strategic games

  • Same mapping for soft constraints 

    • In general, no relation between optimals and NE

    • If strictly monotonic SCSPs  optimals  NE

  • Global mapping  optimals  NE

  • From games to SCSPs:

    • Optimals = Pareto efficient joint strategies