Constraint Satisfaction Problems
and Games
S Kameshwaran
Oct 22, 2002
Outline
 Introduction to CSP
 Introduction to N-Person Non-cooperative
Games
 Nash Equilibrium revisited: Mixed Strategies
 Mapping CSP to a Game
 Tracing Procedure (Evolutionary Process) to
find NE
CSP
 Given a set of variables, find possible values
to the variables which simultaneously satisfy
a set of constraints
 Applications:
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Scheduling
Resource allocation
Computational Molecular Biology
Vision…
CSP
 Given:
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X: Variables ( X1, X2, …, Xn)
D: Domain Di for variable Xi
R: Set of constraints r (can be logical)
 Find:
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Solution: Assignment of value to Xi from its
domain Di such that all constraints are satisfied
CSP
 A constraint is called k-ary constraint if it
connects k variables
 k(r): arity of constraint r
 (d1, d2, …, dn) r: The assignment di to Xi
satisfies constraint r
 Characteristic Function: r (d1, d2, …, dn)
1 if (d1, d2, …, dn) r
 0 otherwise

CSP
 Example: 8 Queens Problem
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Place the 8 queens on the chess board such that no queen
attacks the others
No two queens should be placed on the same row, or on
the same column or on the same diagonal
8 variables
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Xi=j: Queen on ith row is placed on jth column
Constraint r: No two queens should be placed in the same column
 Binary Constraint: k(r)=2
 Xi is not equal to Xj
CSP
 Solution Approaches
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Search algorithms
Backtracking
 Forward checking
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Graph based algorithms
Neural Networks
N-Person Non-cooperative
Games
 N players
 Non-cooperative vs. Cooperative: :
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Players cannot make binding commitments
Players join and split the gains out of
cooperation
 Solution concept: Nash Equilibrium
N-Person Non-cooperative
Games
 Normal Form Games
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N players
Si=Strategy set of player i (Pure Strategy)
Single simultaneous move: each player i chooses a
strategy siSi
Nobody observes others’ move
The strategy combination (s1, s2, …, sN) gives payoff (u1,
u2, …, uN) to the N players
All the above information is known to all the players and
it is common knowledge
Nash Equilibrium
 Nash Equilibrium is a strategy combination s*= (s1*,
s2*, …, sN*), such that si* is a best response to (s1*,
…,si-1*,si+1*,…, sN*), for each i

(s1*, s2*, s3*) is a Nash Equilibrium (3 player game) iff
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s1* is the best response of 1, if 2 chooses s2* and 3 chooses s3*
s2* is the best response of 2, if 1 chooses s1* and 3 chooses s3*
s3* is the best response of 3, if 1 chooses s1* and 2 chooses s2*
Note: It is a simultaneous game and nobody knows what
exactly the choice of other agents
Nash Equilibrium assumes correct and consistent beliefs
Nash Equilibrium: Battle of the
Sexes
Woman
Man
Prize Fight
Ballet
Prize Fight
2, 1
0, 0
Ballet
0, 0
1, 2
 (Prize Fight, Prize Fight) is a NE: Best responses to
each other
 (Ballet, Ballet) is a NE: Best responses to each
other
The Welfare Game
Pauper
Government
Try to Work
Be Idle
Aid
3, 2
-1, 3
No Aid
-1, 1
0, 0
 Government wishes to aid a pauper if he searches for work
but not otherwise
 Pauper searches for work only if he cannot depend on
government aid
The Welfare Game
Pauper
Government
Try to Work
Be Idle
Aid
3, 2
-1, 3
No Aid
-1, 1
0, 0
 (Aid, Try to Work) is not NE: Pauper prefers Be Idle
The Welfare Game
Pauper
Government
Try to Work
Be Idle
Aid
3, 2
-1, 3
No Aid
-1, 1
0, 0
 (Aid, Try to Work) is not NE: Pauper prefers Be Idle
 (Aid, Be Idle) is not NE: Govt prefers No Aid
The Welfare Game
Pauper
Government
Try to Work
Be Idle
Aid
3, 2
-1, 3
No Aid
-1, 1
0, 0
 (Aid, Try to Work) is not NE: Pauper prefers Be Idle
 (Aid, Be Idle) is not NE: Govt prefers No Aid
 (No Aid, Be Idle) is not NE: Pauper prefers Try to Work
The Welfare Game
Pauper
No NE
Government
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
Try to Work
Be Idle
Aid
3, 2
-1, 3
No Aid
-1, 1
0, 0
(Aid, Try to Work) is not NE: Pauper prefers Be Idle
(Aid, Be Idle) is not NE: Govt prefers No Aid
(No Aid, Be Idle) is not NE: Pauper prefers Try to Work
(No Aid, Try to Work) is not NE: Govt prefers Aid
Mixed Strategies
 Pure Strategy: Player i chooses strategy sij
from set Si
 Mixed Strategy: Player i chooses strategy sij
with probability qij (qij>=0, j qij=1)
 Every pure strategy is also a mixed strategy
 Payoff in mixed strategies is the expected
payoff
Mixed Strategies: Advantages
 Mathematical point of view:
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Convexifies the set: Convex sets are nice to play
around as the terrain is well understood
Mixed Strategies: Advantages
 Mathematical point of view:
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Convexifies the set: Convex sets are nice to play
around as the terrain is well understood
Existence of Nash Equilibrium for finite games:
Kakutani fixed point theorem
Mixed Strategies: Advantages
 Mathematical point of view:
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Convexifies the set: Convex sets are nice to play
around as the terrain is well understood
Existence of Nash Equilibrium for finite games:
Kakutani fixed point theorem
 Practical point of view:
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Yes and No (depends on the situation)
Mixed Strategies: Interpretation
 Expected payoff:
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Let payoff with strategy si1 be 1 and si2 be 4
Mixed strategy (½, ½) gives the expected payoff
½+2=2.5
Mixed Strategies: Interpretation
 Expected payoff:
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Let payoff with strategy si1 be 1 and si2 be 4
Mixed strategy (½, ½) gives the expected payoff
½+2=2.5
It means a sure payoff of 2.5 is equivalent to a gamble
where the payoffs are 1 and 4, each with probability ½
Mixed Strategies: Interpretation
 Expected payoff:
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Let payoff with strategy si1 be 1 and si2 be 4
Mixed strategy (½, ½) gives the expected payoff
½+2=2.5
It means a sure payoff of 2.5 is equivalent to a gamble
where the payoffs are 1 and 4, each with probability ½
The above interpretation will not make sense if the
payoff is money
It is true only for utilities
Mixed Strategies: Interpretation
 Games where multiple strategies can be
simultaneously employed
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Betting on more than one horse
Mixed Strategies: Interpretation
 Games where multiple strategies can be
simultaneously employed
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Betting on more than one horse
 Multiple instances of the same game
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War Scenario: qij% of pilots use strategy sij
Mixed Strategies: Interpretation
 Games where multiple strategies can be
simultaneously employed
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Betting on more than one horse
 Multiple instances of the same game
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War Scenario: qij% of pilots use strategy sij
 Same game repeated infinitely
Mixed Strategies: Interpretation
 Games where multiple strategies can be
simultaneously employed
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Betting on more than one horse
 Multiple instances of the same game
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War Scenario: qij% of pilots use strategy sij
 Same game repeated infinitely
 For a single game: The probability distribution is
the opponents’ estimation of player i’s decision
CSP as Games
 CSP C=< X, D, R >
 Game induced by C: GC=(S1, …, Sn; U1, …,Un)
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n = |X|
Si = Di
Ui(d1, …, dn) = rR[i] k(r) r (d1, d2, …, dn)
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R[i] = Constraint set that includes variable i
The payoff function counts the number of satisfied constraints
connecting that variable, taking every constraint along with its
arity
Instead of arity one can use different weights
Equilibria and Solutions
 Every solution of C is a Nash Equilibrium of
GC
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For a solution, all constraints are satisfied, so no
agent can improve its payoff by assuming a
different value
 All Nash Equilibriums are not solutions
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C may not have a solution but still GC will have
a NE
Equilibria and Solutions
X
X
X
X
X
X
X
X
Equilibria and Solutions
X
X
X
X
X
X
X
X
 Not a Solution
Equilibria and Solutions
X
X
X
X
X
X
X
X
 Not a Solution
 Nash Equilibrium: A better solution is not possible
by moving a single queen in one move – Noncooperative
Trivia: Non-cooperative Games
 A solution better to some agents may be
available, but cannot be reached by a
decision of single agent alone
 Cooperation or non-cooperation depends on
the game
 Non-cooperation need not be due to conflict
in goal, but may be due to communication
costs
Trivia: Non-cooperative Games
 Prisoner’s Dilemma in reality: Should IISc water its
garden when there is drought in Mysore?
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Consider the following situation: Drought in Mysore but
not in Bangalore
Saved water from Bangalore can be transported to
Mysore
Decision making of an agent in Bangalore:
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If all saves water, his saving will not contribute much
If nobody saves water, his saving will not contribute much
So, better not to save
Equilibria and Solutions
 The complexity of the CSP depends on its
structure
 Finding solution to C  Finding NE to GC
 Complexity of finding NE is not known
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It is unlikely to be in P
It is also unlikely to be NP-hard as existence of
solution is guaranteed
Equilibrium Selection
 Tracing Procedure (Evolutionary process)
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For agent i, there is a probability distribution (mixed
strategy) pi, which the other agents expect that i will use
p=(p1, …, pn)
Assumption
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Limited computational power
Agents are Bayesian decision makers
Each agent estimates its best strategy depending on p
Value of p is updated based on the previous outcome
Equilibrium Selection
 BR(p)=Best response strategy to the
distribution p
 Synchronous Process
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p: Initial distribution
p0 = p
pk =  BR(pk-1)+(1- )pk-1

0<<=1
 If pk converges then the limit point is NE
Equilibrium Selection
 Computation of BR(p) for an agent is
computationally taxing if it is connected with large
number of variables
 The process may converge to a NE that may not be
a solution
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Can be considered as the best possible quasi solution
 No general proof that the process will always
converge
 Susceptible to initial probability distributions
Trivia: Solution to 8 Queens
Problem
X
X
X
X
X
X
X
X
 10 distinct solutions
 http://www.math.utah.edu/~alfeld/queens/queens.html
References
 Equilibrium Theory and Constraint Networks,
Francesco Ricci, 1992
 Games and Decisions, Luce and Raiffa, Dover
Publications, 1957
 Games and Information: An Introduction to Game
Theory, Eric Rasmusen, Basil Blackwell Publishers,
1989
Next..
 25/10/02
Nash Equilibrium: P or NP?