1. Constraint Satisfaction Problems and Games
S Kameshwaran
Oct 22, 2002

2. 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

3. CSP
Given a set of variables, find possible values to the variables which simultaneously satisfy a set of constraints
Applications:
Scheduling
Resource allocation
Computational Molecular Biology
Vision…

4. CSP Given: Find: X: Variables ( X1, X2, …, Xn)
D: Domain Di for variable Xi
R: Set of constraints r (can be logical)
Find:
Solution: Assignment of value to Xi from its domain Di such that all constraints are satisfied

5. 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

6. CSP Example: 8 Queens Problem
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
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

7. CSP Solution Approaches Search algorithms Graph based algorithms
Backtracking
Forward checking
Graph based algorithms
Neural Networks

8. N-Person Non-cooperative Games
N players
Non-cooperative vs. Cooperative: :
Players cannot make binding commitments
Players join and split the gains out of cooperation
Solution concept: Nash Equilibrium

9. N-Person Non-cooperative Games
Normal Form Games
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

10. 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
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

11. Nash Equilibrium: Battle of the Sexes
(Prize Fight, Prize Fight) is a NE: Best responses to each other
(Ballet, Ballet) is a NE: Best responses to each other
Man
Woman
Prize Fight
Ballet
2, 1
0, 0
1, 2
Man
Woman
Prize Fight
Ballet
2, 1
0, 0
1, 2

12. The Welfare Game
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
Government
Pauper
Try to Work
Be Idle
Aid
3, 2
-1, 3
No Aid
-1, 1
0, 0

13. The Welfare Game (Aid, Try to Work) is not NE: Pauper prefers Be Idle
Government
Pauper
Try to Work
Be Idle
Aid
3, 2
-1, 3
No Aid
-1, 1
0, 0

14. The Welfare Game (Aid, Try to Work) is not NE: Pauper prefers Be Idle
(Aid, Be Idle) is not NE: Govt prefers No Aid
Government
Pauper
Try to Work
Be Idle
Aid
3, 2
-1, 3
No Aid
-1, 1
0, 0

15. The Welfare Game (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
Government
Pauper
Try to Work
Be Idle
Aid
3, 2
-1, 3
No Aid
-1, 1
0, 0

16. The Welfare Game (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
No NE
Government
Pauper
Try to Work
Be Idle
Aid
3, 2
-1, 3
No Aid
-1, 1
0, 0

17. 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

18. Mixed Strategies: Advantages
Mathematical point of view:
Convexifies the set: Convex sets are nice to play around as the terrain is well understood

19. Mixed Strategies: Advantages
Mathematical point of view:
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

20. Mixed Strategies: Advantages
Mathematical point of view:
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:
Yes and No (depends on the situation)

21. Mixed Strategies: Interpretation
Expected payoff:
Let payoff with strategy si1 be 1 and si2 be 4
Mixed strategy (½, ½) gives the expected payoff ½+2=2.5

22. Mixed Strategies: Interpretation
Expected payoff:
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

23. Mixed Strategies: Interpretation
Expected payoff:
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

24. Mixed Strategies: Interpretation
Games where multiple strategies can be simultaneously employed
Betting on more than one horse

25. Mixed Strategies: Interpretation
Games where multiple strategies can be simultaneously employed
Betting on more than one horse
Multiple instances of the same game
War Scenario: qij% of pilots use strategy sij

26. Mixed Strategies: Interpretation
Games where multiple strategies can be simultaneously employed
Betting on more than one horse
Multiple instances of the same game
War Scenario: qij% of pilots use strategy sij
Same game repeated infinitely

27. Mixed Strategies: Interpretation
Games where multiple strategies can be simultaneously employed
Betting on more than one horse
Multiple instances of the same game
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

28. CSP as Games CSP C=< X, D, R >
Game induced by C: GC=(S1, …, Sn; U1, …,Un)
n = |X|
Si = Di
Ui(d1, …, dn) = rR[i] k(r) r (d1, d2, …, dn)
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

29. Equilibria and Solutions
Every solution of C is a Nash Equilibrium of GC
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
C may not have a solution but still GC will have a NE

30. Equilibria and SolutionsX

31. Equilibria and Solutions
Not a Solution X

32. Equilibria and Solutions
Not a Solution
Nash Equilibrium: A better solution is not possible by moving a single queen in one move – Non-cooperative X

33. 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

34. Trivia: Non-cooperative Games
Prisoner’s Dilemma in reality: Should IISc water its garden when there is drought in Mysore?
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:
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

35. 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
It is unlikely to be in P
It is also unlikely to be NP-hard as existence of solution is guaranteed

36. Equilibrium Selection
Tracing Procedure (Evolutionary process)
For agent i, there is a probability distribution (mixed strategy) pi, which the other agents expect that i will use
p=(p1, …, pn)
Assumption
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

37. Equilibrium Selection
BR(p)=Best response strategy to the distribution p
Synchronous Process
p: Initial distribution
p0 = p
pk =  BR(pk-1)+(1- )pk-1
0<<=1
If pk converges then the limit point is NE

38. 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
Can be considered as the best possible quasi solution
No general proof that the process will always converge
Susceptible to initial probability distributions

39. Trivia: Solution to 8 Queens Problem
10 distinct solutions X

40. 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

41. Nash Equilibrium: P or NP?
Next..
25/10/02
Nash Equilibrium: P or NP?