1. Computation of Nash Equilibrium JugalGargGeorgiosPiliouras

2. Recap – Two player game

3. Nash Equilibrium

4. Best Response Polyhedron Complementarity Conditions (CC)

5. Characterization

6. Quadratic Programming Formulation

7. Zero-sum Game

8. Zero-sum Game

9. Support Enumeration Algorithm

10. Support Enumeration Algorithm • How many possible support?

11. Support Enumeration Algorithm

12. Support Enumeration Algorithm

13. Corollaries

14. Lemke-Howson Algorithm (1964)

15. Preliminaries

16. Best Response Polyhedron (Recall)

17. Linear Complementarity Problem (LCP) Formulation Complementarity Conditions (CC)

18. LCP Characterization

19. Preliminaries

20. Fully-labeled Points

21. Fully-labeled Characterization

22. Lemke-Howson Algorithm (0,0) This path is called k-almost fully-labeled – only label k is missing

23. Convergence • No cycling • Cannot comes back to starting vertex (0,0) • Has to terminate in finite time • Exponential in worst case (Savani, von Stengel’04)

24. Structural Results

25. Break Q. How we get existence and oddness of the number of equilibria from Lemke-Howson algorithm?

26. PTIME for Special Cases • Zero-sum games - LP formulation • Both rank(A) and rank(B) are constant (Lipton, Markakis, Mehta’04) • Either rank(A) or rank(B) is constant (G., Jiang, Mehta’11) • Proof on board • Rank-based hierarchy (rank(A+B)) • Zero-sum games = rank-0 games • Rank-1 games (Adsul, G., Mehta, Sohoni’ 11)

27. Rank-1 Games • Number of Nash equilibrium can be exponential (von Stengel’12) • For any c, rank-1 games can have c number of connected components (Kannan-Theobold’07) • Rank-1 games are strictly general than zero-sum games and LP (Dantzig’47) • In general rank-1 QP is NP-hard to solve • surprisingly those arising from rank-1 games are polynomial time solvable.

28. Approximate Nash Equilibrium

29. Approximate Nash Equilibrium

30. FPTAS for Constant Rank Games

31. Quasi-polynomial Time Algorithm (Lipton, Markakis, Mehta’03)

32. Complexity Considerations You Are here Nash 32

33. Standard Complexity Classes • P: The set of decision problems for which some algorithm can provide an answer in polynomial time • NP: set of all decision problems for which for the instances where the answer is "yes“, we can verify in polynomial time that the answer is indeed yes. • coNP: Same as above with yes->no

34. Equilibrium Computation • The goal is to find a function that maps objects (games) to the mixed strategy profiles. • This is NOT a decision problem (yes/no). • Examples: • Add two numbers and find the outcome • Is the sum of two numbers odd?

35. Function Complexity Classes • FP: The set of function problems for which some algorithm can provide an answer in polynomial time • FNP: set of all function problemsfor which the validity of an (input, output) pair can be verified in polynomial time by some algorithm.

36. Function Complexity Classes • FP: A binary relation P(x,y) is in FP if and only if there is a deterministic polynomial time algorithm that,given x,can find some y such that P(x,y) holds. • FNP: A binary P(x,y), where y is at most polynomially longer than x, is in FNP if and only if there is a deterministic polynomial time algorithm that can determine whether P(x,y) holds given both x and y.

37. Function Complexity Classes • FP: The set of function problems for which some algorithm can provide an answer in polynomial time • FNP: set of all function problemsfor which the validity of an (input, output) pair can be verified in polynomial time by some algorithm. • TFNP: The subclass of FNP for which existence of solution is guaranteed for every input!

38. Non-constructive arguments • Local Search: Every directed acyclic graph must have a sink. • Pigeonhole Principle:If a function maps n elements to n-1 elements, then there is a collision. • Handshaking lemma: If a graph has a node of odd degree, then it must have another. • End of line: If a directed path has an unbalanced node, then it must have another.

39. From non-constructive arguments to complexity classes • PLS: All problems in TFNP whose existence proof is implied by Local Search arg. • PPP:All problems in TFNP whose existence proof is implied by the Pigeonhole Principle. • PPA: All problems in TFNP whose existence proof is implied by the Handshaking lemma. • PPAD: All problems in TFNP whose existence proof is implied by the End-of-line argument.

40. From non-constructive arguments to complexity classes FNP PLS PPP PPA PPAD P

41. PPAD [Papadimitriou 1994] Suppose that an exponentially large graph with vertex set {0,1}nis defined by two circuits: possible previous P(v)=u and N(u)=v Finding Nash equilibria, even in 2-person games is PPAD –complete. node id node id v u [Daskalakis, Goldberg, Papadimitriou 05], [Chen, Deng 06’] node id node id N P possible next GivenP and N: If0n is an unbalanced node, find another unbalanced node. Otherwise say “yes”. END OF THE LINE: { Search problems in FNP reducible to END OF THE LINE} PPAD =