1. Computation of Nash Equilibrium Jugal Garg Georgios Piliouras

2. Recap – Two player game 2

3. Nash Equilibrium 3

4. Best Response Polyhedron 4 Complementarity Conditions ( CC )

5. Characterization 5

6. Quadratic Programming Formulation 6

7. Zero-sum Game 7

8. Zero-sum Game 8

9. Support Enumeration Algorithm 9

10. Support Enumeration Algorithm How many possible support? 10

11. Support Enumeration Algorithm 11

12. Support Enumeration Algorithm 12

13. Corollaries 13

14. Lemke-Howson Algorithm (1964) 14

15. Preliminaries 15

16. Best Response Polyhedron (Recall) 16

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

18. LCP Characterization 18

19. Preliminaries 19

20. Fully-labeled Points 20

21. Fully-labeled Characterization 21

22. Lemke-Howson Algorithm 22 (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 ) 23

24. Structural Results 24

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

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 ) 26

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. 27

28. Approximate Nash Equilibrium 28

29. Approximate Nash Equilibrium 29

30. FPTAS for Constant Rank Games 30

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

32. Complexity Considerations 32 Nash You Are here

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 33

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? 34

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 problems for which the validity of an (input, output) pair can be verified in polynomial time by some algorithm. 35

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. 36

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 problems for 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! 37

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. 38

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. 39

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

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