1. A Technique for Reducing Normal Form Games to Compute a Nash Equilibrium Vincent Conitzer and Tuomas Sandholm Carnegie Mellon University, Computer Science Department

2. Nash equilibrium b1 b2 b3 a1 4, 0 1, 2 0, 4 a2 0, 3 2, 2 2, 3 a3 1, 4
0.25
0.5
0.25 b1 b2 b3 a1
4, 0
1, 2
0, 4 a2
0, 3
2, 2
2, 3 a3
1, 4
0.25
0.5
0.25

3. Computing Nash equilibria (normal-form games)
The complexity of computing a (single) Nash equilibrium is a well-known open problem
Search-based methods [Porter et al. AAAI04, Sandholm et al. AAAI05] clearly (worst-case) exponential
Famous Lemke-Howson [Lemke & Howson 64] algorithm also (worst-case) exponential [Savani & von Stengel FOCS04]
Finding equilibria with certain properties (e.g. social welfare maximizing) is NP-complete (even inapproximable) [Gilboa & Zemel 89, Conitzer & Sandholm IJCAI03/extended draft]

4. Recursive techniques for computing Nash equilibria
It would be nice to have a recursive technique
reduce game to one or more smaller games
such that solution to original game can easily be recovered from solutions to the smaller game(s)
If technique is always applicable, we get an algorithm for computing Nash equilibria
If not, could still be useful as preprocessing step
One such technique: (iterated) dominance
[Conitzer & Sandholm AAAI05] generalized eliminability criterion is not recursive
Here we go for something completely different…

5. Required structure on original game Ov1 v2 … vl t1 t2 tn u1
c11, b1
c12, b1
c1n, b1 u2
c21, b2
c22, b2
c2n, b2 uk
ck1, bk
ck2, bk
ckn, bk s1
a1, d11
a2, d12
al, d1l s2
a1, d21
a2, d22
al, d2l sm
a1, dm1
a2, dm2
al, dml H G
That is: against any fixed vj, all the si give the row player the same utility aj
against any fixed ui, all the tj give the column player the same utility bi

6. Solve for equilibrium of G (recursively)t1 t2 … tn s1 s2 sm G
Obtain
Equilibrium distributions pG(si), pG(tj)
Player’s expected payoffs in equilibrium πr, πc

7. Reduced game R v1 v2 … vl t u1
ΣjpG(tj)c1j, b1 u2
ΣjpG(tj)c2j, b2 uk
ΣjpG(tj)ckj, bk s
a1,ΣipG(si)di1
a2, ΣipG(si)di2
al, ΣipG(si)d1l
πr, πc H
Expected payoffs when row player plays the equilibrium of G, column player plays vi
Expected payoffs when both players play the equilibrium of G
Theorem. pR(ui), pR(s)pG(si); pR(vj), pR(t)pG(tj) constitutes a Nash equilibrium of original game.

8. Example
0.5
0.25
0.25
0.5
0.5 v1 t1 t2 u1
2, 2
0, 3
2, 3 s1
1, 2
4, 0
0, 4 s2
1, 4 t1 t2 s1
4, 0
0, 4 s2
0.5
0.5
0.25
0.5
0.25
0.5
0.5 v1 t u1
2, 2
1, 3 s
0.5
0.5

9. A more difficult exampleb1 b2 b3 a1
4, 0
1, 2
0, 4 a2
0, 3
2, 2
2, 3 a3
1, 4
v1 = b2
t1 = b1
t2 = b3
u1 = a2
2, 2
0, 3
2, 3
s1 = a1
1, 2
4, 0
0, 4
s2 = a3
1, 4
= the game that we solved before!
But how (in general) do we find the correct labeling of the strategies as ui, si , vj , tj? Can it be done in polynomial time?

10. Let’s try to use satisfiability
4, 0
1, 2
0, 4 a2
0, 3
2, 2
2, 3 a3
1, 4
Say that v(σ) = true if we label σ as one of the si or tj (that is, we put it “in” G)
If a1, a2 are both in G, then b1 must also be in G because a1, a2 get different payoffs against b1
Equivalently, v(a1) and v(a2)  v(b1)
or (-v(a1) or -v(a2) or v(b1))
Theorem: satisfaction of all such clauses  the condition is satisfied

11. Clauses for the exampleb1 b2 b3 a1
4, 0
1, 2
0, 4 a2
0, 3
2, 2
2, 3 a3
1, 4
v(a1) and v(a2)  v(b1) and v(b2) and v(b3)
v(a1) and v(a3)  v(b1) and v(b3)
v(a2) and v(a3)  v(b2) and v(b3)
v(b1) and v(b2)  v(a1) and v(a2)
v(b1) and v(b3)  v(a1) and v(a3)
v(b2) and v(b3)  v(a1) and v(a2) and v(a3)
Complete characterization of solutions:
Set at most one variable to true for each player (does not reduce game)
Set all variables to true (G = whole game!)
Only nontrivial solution: set v(a1), v(a3), v(b1), v(b3) to true

12. Algorithm to find nontrivial solution:
Simple algorithm
Algorithm to find nontrivial solution:
Start with any two variables for the same agent set to true
Follow the implications
If all variables set to true, start with next pair of variables

13. Solving the example with the algorithm (pass 1)b1 b2 b3 a1
4, 0
1, 2
0, 4 a2
0, 3
2, 2
2, 3 a3
1, 4
v(a1) and v(a2)  v(b1) and v(b2) and v(b3)
v(a1) and v(a3)  v(b1) and v(b3)
v(a2) and v(a3)  v(b2) and v(b3)
v(b1) and v(b2)  v(a1) and v(a2)
v(b1) and v(b3)  v(a1) and v(a3)
v(b2) and v(b3)  v(a1) and v(a2) and v(a3)
Variables set to true:
v(a1)
v(a2)
v(b1)
v(b2)
v(b3)
v(a3)

14. Solving the example with the algorithm (pass 2)b1 b2 b3 a1
4, 0
1, 2
0, 4 a2
0, 3
2, 2
2, 3 a3
1, 4
v(a1) and v(a2)  v(b1) and v(b2) and v(b3)
v(a1) and v(a3)  v(b1) and v(b3)
v(a2) and v(a3)  v(b2) and v(b3)
v(b1) and v(b2)  v(a1) and v(a2)
v(b1) and v(b3)  v(a1) and v(a3)
v(b2) and v(b3)  v(a1) and v(a2) and v(a3)
Variables set to true:
v(a1)
v(a3)
v(b1)
v(b3)

15. Theorem. Requires at most O((#rows+#columns)4) clause applications
Algorithm complexity
Theorem. Requires at most O((#rows+#columns)4) clause applications
That is, quadratic if the game is square
Can improve in practice by caching previous results

16. Does this help in finding e. g. social welfare maximizing equilibrium
Does this help in finding e.g. social welfare maximizing equilibrium? (Unfortunately, no) 1 1 v1 t1 t2 u1
1, 1
4, 0
0, 0 s1
0, 4
3, 3 s2
2, 2 t1 t2 s1
3, 3
0, 0 s2
2, 2 1 1
Choose best equilibrium
But this one is better!  1 v1 t u1
1, 1
4, 0 s
0, 4
3, 3 1

17. ALAGIU games
Definition. A game is an ALAGIU (Any Lower Action Gives Identical Utility) game if
the strategy spaces are subsets of the real numbers, and
given that you choose a lower number than your opponent, it does not matter which one you choose
E.g. any 1-item auction format where the lower bidder never wins and never pays anything
Vendor game:

18. Finite ALAGIU games can be solved completely, in linear time, using the technique
Proof. One of three cases must apply: t1 t2 … tn u1
c11, b1
c12, b1
c1n, b1 s1 s2 sm v1 t1 t2 … tn s1
a1, d11 s2
a1, d21 sm
a1, dm1 G G
Row has the largest strategy
Column has the largest strategy v1 t1 t2 … tn u1
hr, hc
c11, b1
c12, b1
c1n, b1 s1
a1, d11 s2
a1, d21 sm
a1, dm1 G
Both have the largest strategy

19. Conclusions
To compute a Nash equilibrium, we presented a way to reduce game O to smaller game R based on solution to subgame G satisfying a certain condition
A subgame G satisfying the condition can be found in polynomial time (if it exists)
Technique does not extend to e.g. welfare maximizing equilibria
ALAGIU games (in particular, “most” 1-item auctions) can be solved completely using the technique, in polynomial time
Thank you for your attention!