Preference-based

x2=c2: n1 > c1 x2=n2: n1 > c1 X1=c1: n2 > c2 X1=n1: n2 > c2

x2 depends on x1



Reduced CP-nets


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)

C2 C1 N1 3, 3 4, 0
N2 0, 4 1, 1

(N1,N2): unique Nash equilibrium All other joint strategies are Pareto efficient (PE)
Dominance between strategies


A strategy si is never a best response for i if it is not a best response to any joint strategy s-i In the example: for each player i, Ci is never a best response

In the example, one Nash equilibria (NE): (N1,N2)
Pareto efficient joint strategies


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

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

From game G to n(G) From n(G) to r(n(G)) Hard constraints for r(n(G)) Optimals of r(n(G)) = Nash equilibria of G
G n n(G) r Nash equilibria of G = optimals of r(n(G)) r(n(G))
Main course
Fish, red, peaches
Fish, white, berries
Wine
Fish, red, berries
white > red
red > white
Wine
meat, red, peaches
peaches > strawberries
Fruit
meat, white, peaches
N2
3, 3 0, 4 4, 0 1, 1
Pure Nash equilibria
C1 N1

C2 3, 3
4, 0
N2 0, 4
1, 1

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 pure Nash equilibrium if each si is a best response to s-i
Two features: x1, x2 D(x1)={c1, n1} D(x2)={c2, n2} x1 and x2 independent For x1: n1 > c1 For x2: n2 > c2

x2 depends on x1

CP-net techniques in games

Hard constraints: fish, peaches, fish white, meat red Meat is nbr for main course Strawberries is nbr for fruit Once meat is eliminated, red is nbr for wine Nash equilibrium: fish, white, peaches
C2
C1 3, 3
N2
0, 4
N1
4, 0
1, 1
Elimination of dominated strategies

G NBR G’

G’ subgame of G For all i, each si in Si-S’i is never a best response for i in G
Example: reduced CP-net

Two features: x1, x2 D(x1)={c1, n1} D(x2)={c2, n2} x1 depends on x2


x2=c2: n1 > c1 x2=n2: n1 > c1 X1=c1: n2 > c2 X1=n1: n2 > c2

For the game:

From games to CP-nets

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
Main course fish meat
Wine
white > red
red > white
Wine
peaches > strawberries
Fruit
Example: optimals and Nash equilibria

Unique optimal for CP-net: (fish, white, peaches)
Outline
1. 2.
3.
Strategic games Relation between CP nets and games Relation between soft constraints and games
Parametrized Strategic games

A set of players 1,.., n For each player i:
Eliminate from the strategies of each players those that are never a best response
Nash equilibria and strategy elimination


If G NBR G’, then s Nash equilibrium of G iff Nash equilibrium of G’ In the example: Ci is nbr, thus G’ has one row and one column, which is the unique Nash equilibrium 1,1

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
fish>meat
Main course fish meat
Optimality in strategic games, CP nets and soft constraints
Main aim





To compare the notion of optimality used in many formalisms To throw the basis for exploiting results in one field and reuse them in the other field 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
meat, red, berries
meat, white, berries
CP net Param. Strategic Game
fish>meat Main course
Three players: 1 = main course, 2 = wine, 3 = fruit Two strategies for each player: S1= {meat, fish} S2={red, white} S3={peaches, strawberries} Payoff functions For 1 main course: fish > meat, always For 2 wine: fish, -- white > red meat, -- red > white For 3 fruit: peaches > strawberries, always