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

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

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]

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…

Required structure on original game O v1 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

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

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.

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

A more difficult example b1 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?

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

Clauses for the example 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) 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

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

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)

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)

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

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

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:

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

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!