1. Nash Equilibrium: P or NP?
S Kameshwaran
Oct 25, 2002

2. Complexity prelims P NP
Set of problems that have algorithms which have polynomial running time (worst case) in the input length NP
Given a solution, checking its correctness is possible in polynomial time
But finding a solution is not assured in polynomial time

3. Complexity prelims Polynomial reductions NP-Complete:
Two problems X and Y
X p Y: If there is a polynomial algorithm that takes an instance X` of X and creates an instance Y` of Y, such that they are equivalent in terms of the solutions
NP-Complete:
Most difficult problems in NP
If any of these problems can be solved in polynomial time then all problems in NP can be solved in polynomial time

4. Complexity prelims Cook’s Theorem: If Y p X, where Y  NP-Complete
SAT is NP-Complete
SAT  NP
For all X  NP, X p SAT
If Y p X, where Y  NP-Complete
X is NP-Hard
X  NP, then X  NP-Complete

5. Nash Equilibrium
A strategy combination is in NE if each player chooses a strategy that is the best response to the strategies chosen by others
Issues:
Existence of NE for a game
Uniqueness of NE
Pareto-optimality

6. Nash Equilibrium: Dining in Bangalore
(TGIF, TGIF) is good for both the players
(Pizza Hut, Pizza Hut) is not a admissible NE
Man
Woman
Pizza Hut
TGIF
10, 10
0, 0
20, 20

7. Nash Equilibrium: Battle of the Sexes
Both NE are Pareto-optimal and both are admissible
Man
Woman
Prize Fight
Ballet
2, 1
0, 0
1, 2

8. Nash Equilibrium N players Action set of player i: Si={si1, …, sim(i)}
Utility to player i: ui: S  R
Strategy of player i: pi={pi1, …, pim(i)}
pij >= 0, jpij = 1
Utility of using strategy p: ui(p) = s ui(s) j pj(s)

9. ui(pi, p*-i) <= ui(p*i, p*-i)
Nash Equilibrium
N players
Action set of player i: Si={si1, …, sim(i)}
S = i Si
Utility to player i: ui: S  R
Strategy of player i: pi={pi1, …, pim(i)}
pij >= 0, jpij = 1
Utility of using strategy p: ui(p) = s ui(s) j pj(s)
Nash Equilibrium: p* is NE if for all i, and for all pi
ui(pi, p*-i) <= ui(p*i, p*-i)

10. Some more notations xij(p) = ui(sij,p-i) zij(p) = xij(p) – ui(p)
Other players use mixed strategy profile p-i and i use pure strategy sij
zij(p) = xij(p) – ui(p)
p* is NE iff z(p*) <= 0
gij(p) = max[zij(p), 0]
p* is NE iff gij(p*) = 0

11. NE as a fixed point of a function
Define y: P  P
p* is NE iff it is a fixed point of y
y is a continuous function of a compact set P to itself, so a fixed point exists (Brouwer’s FPT)
Nash’s proof of existence of NE for N-person games

12. NE as a fixed point of a correspondence
Best response correspondence: BR: P  P
Point to set function
BR(p) = arg maxq[i ui(qi, p-i)]
BR(p) = Set of best response strategies to p
p* is a NE iff it is a fixed point of best response correspondence, p*BR(p*)
BR is non-empty, closed and convex valued, hence by Kakutani’s FPT, fixed point exists

13. NE as a solution to complementarity problem
Find a pair of vectors x, y which are complementary (orthogonal) to each other
Used to prove complementary slackness and optimality in mathematical programming
p* is NE iff p* and z(p*) are orthogonal
Two-person games: Linear complementary problems

14. NE as a minimum of a function on a polytope
Define v: P  R
v(p) = i j [gij(p)]2
v is a continuous, differentiable function on polytope P
p* is NE iff it is global minimum of v,
v(p*) = 0
Other formulations: Stationary point, semi-algebraic set

15. Two person zero sum games
2 Players
For every strategy x of player 1 and strategy y of player 2, payoff to 1+ payoff to 2=0
What player one loses is gained by the other
Player 1 wants to minimize his losses
Determine the optimal mixed strategy that minimizes his loss
Player 2 wants to maximize his profits
Determine the optimal mixed strategy that maximizes his profit

16. Two person zero sum games
Minimax theorem:
Expected minimum of the maximum loss = Expected maximum of the minimum profit
The above can be converted into two linear programs which are dual to each other
LP  P
Simplex algorithm can solve it efficiently though it is not bounded in polynomial time

17. Two person general sum games
Two person non-zero sum games can be formulated as linear complementary problems (LCP)
LCP:
Given a matrix M and a vector q
Find w and z, such that
w – Mz = q
w >= 0, z >= 0, and wizi = 0 for all i
Solution technique: Lemke-Howson Algorithm

18. Two person general sum games
LCP
Can be characterized as LPs (Mangasarian, 1978)
The solution to LCP can be obtained by optimizing a suitable LP
Finding the suitable objective function is not easy (depends on matrix M)
Possible Approach:
Characterize M in terms of two person games (M consists of payoff matrices of both players)
Infer the complexity of games using M

19. Two person general sum games
Lemke-Howson Algorithm
Finds a sample equilibrium
Incomplete for finding all equilibria
Computational complexity: still unknown
Exponential lower bound is shown (Murty, 1978)
Can be exponential even on a zero-sum game
Approximation theory cannot be used as there is no objective function is used and one cannot know how close is the solution to the optimal

20. Two person general sum games
Reduction of a SAT to symmetric two-person game (Conitzer, Sandholm, 2002)
Following are NP-hard (even for symmetric case)
Deciding whether more than one NE exists
Deciding whether a given strategy is played in NE
Deciding whether a given strategy is never played in NE
Deciding if a Pareto-optimal NE exists

21. N person games NE as a min of a function v(p) = i j [gij(p)]2
Constraints: pij >= 0, jpij = 1
Global minima correspond to NE
Local minima may exist
Algorithms:
Continuous differentiable function, so any global optimization technique can be used
Penalty function methods
Convergence of these algorithms are very slow

22. N person games Alternate formulation:
Non-linear complementarity problem
Sequence of approximations by LCP
Similar to Newton’s method
Scarf’s algorithms
Worst case complexity is exponential in N

23. Bayesian-Nash Equilibrium
Games of incomplete information: A player may not the utility of other players
Most real world applications induce such games: Auctions, markets, bargaining, etc
Modifications:
Ti: Set of types a player can belong to
Eg: In bargaining, the seller assumes that buyer belongs to type [10,25], which means that buyer may value the good anywhere between 10 and 25
Let q be the joint probability distribution on T = i Ti
q and T are known to all players
Each player knows his own type ti  Ti

24. Bayesian Nash Equilibrium
N players
Action set of player i: Ai={ai1, …, aim(i)}
A = i Ai
Utility to player i: ui: A  T  R (utility depends on the type of the player)
Strategy of player i, i : Ti  Ai
If either Ti or Ai is infinite then the strategy space is infinite
Strategy is now a function
Utility of using strategy  : ui( , ti) = Expected utility over all the possible types for other players given the type of i is ti

25. Bayesian Nash Equilibrium
Complexity for 2 player case
Reduction from Set cover (Conitzer, Sandholm, 2002)
Deciding whether a BNE exists is NP-complete
If randomization is allowed, then BNE always exists for finite games
Infinite case:
Strategy space is infinite
Finding the utility of a player amounts to optimizing over a function space
Function spaces are generally infinite dimensional
Assuring the existence of a solution is not possible

26. Bayesian Nash Equilibrium
Games arising out of market design
Infinite Games
Given the action combination from players, utility to each player is calculated by some optimization problem
some resource allocation algorithm which moves the goods from sellers to buyers to maximize some criteria
Finding the expected utility:
What is the complexity of finding the utility of a player who uses a strategy  ?
What is the complexity of deciding the existence of BNE in infinite games?

27. References
Complementarity and Fixed Point Problems, Ed: M. L. Balinski and R. W. Cottle, North-Holland Publishing Company, 1978
Computational complexity of complementary pivot methods, K. G. Murty
Characterization of LCPs as LPs, O. L. Mangasarian
Complexity results about NE, V. Conitzer and T. Sandholm, 2002
Computation of equilibria in finite games, R. D. McKelvey and A. McLennan, 1996

28. Combinatorial Markets
Next Week..
Combinatorial Markets
Prof. Y. Narahari