Reformulation of Nash Equilibrium with an Application to Interchangeability Yosuke YASUDA Osaka University, Department of Economics [email protected]

August, 2015

1 / 23

Summary of My Talk What is Reformulation? (1st Part) The set of Nash equilibria, if it is nonempty, is identical to the set of minimizers of real-valued function. Connect equilibrium problem to optimization problem. → Similar characterizations are known in the literature. Is it Useful? (2nd Part) Existing results on interchangeability can be derived, in a unified fashion, by lattice structure of optimal solutions. → Completely new results (re-interpretations)! Main Messages Please use/apply/extend my characterization of NE! 2 / 23

Existing Formulations of NE: Equilibrium Approach Strategy profile x∗ ∈ X is called Nash equilibrium if and only if, 1

Inequality (Incentive) Constraints ui (x∗i , x∗−i ) ≥ ui (xi , x∗−i ) for all xi ∈ Xi and for all i ∈ N.

2

Solution to Multivariate Function ui (x∗i , x∗−i ) = max ui (xi , x∗−i ) for all i ∈ N. xi ∈Xi

3

Fixed Point of BR Correspondence x∗ ∈ BR(x∗ ), where BRi (x−i ) = arg max ui (x0i , x−i ). 0 xi ∈Xi

3 / 23

Another Characterization: Optimization Approach Define f : X → R, which aggregates maximum deviation gains (from a fixed action profile x ∈ X) across players.  X 0 f (x) = max ui (xi , x−i ) − ui (xi , x−i ) . (1) 0 i∈N

xi ∈Xi

Theorem 1 A strategy profile x∗ is a Nash equilibrium iff f (x∗ ) = 0. Theorem 2 If there exists an NE, the set of Nash equilibria E ∗ is identical to the set of minimum solutions to f , arg minx∈X f (x).

4 / 23

Slight Modifications Let g(x) = −f (x). The set of Nash equilibria, if it is nonempty, is identical to the set of maximum solutions to g. arg max g(x). x∈X

Let t be a parameter contained in a parameter set T . Then, (1) can be rewritten to incorporate this parameterization.  X 0 f (x, t) = max ui (xi , x−i ; t) − ui (xi , x−i ; t) . (2) 0 i∈N

xi ∈Xi

Could be useful for (monotone) comparative statics: e.g., to analyze conditions under which the set of optimal solutions, or its selection, is increasing in t ∈ T .

5 / 23

Special Case: Best Reply Our formulation can also incorporate best reply, since it is just a (degenerated) Nash equilibrium when there is one strategic player. fi (x, t) = fi (xi ; x−i , t) ui (x0i ; x−i , t) − ui (xi ; x−i , t), = max 0 xi ∈Xi

(Note x−i is fixed and considered as a parameter of the model.) Solving arg minxi ∈Xi fi (xi ; x−i , t) is equivalent to deriving arg maxxi ∈Xi ui (xi ; x−i , t). Comparative statics analysis on best replies can be regarded as a special case of (2) above. Our approach can reproduce all results in the literature, e.g., Milgrom and Roberts (1990); Milgrom and Shannon (1994). 6 / 23

Upper-hemi Continuity The Nash correspondence, a mapping from parameter to Nash equilibria, is upper-hemi continuous under weak assumptions.  X 0 f (x, t) = max ui (xi , x−i ; t) − ui (x; t) . 0 i∈N

xi ∈Xi

Upper-hemi continuity of optimal solutions is given by theorem of the maximum by Berge (1963). Continuity is preserved under addition. Given that ui is continuous (for all i ∈ N ), f is continuous whenever max ui is continuous (for all i ∈ N ). ⇒ Conditions for continuity of max ui guarantees UHC of NE.

7 / 23

Further Characterization For each i ∈ N , consider a function ψi : R+ → R+ such that ψi (0) = 0, and define f˜ : X → R, using {ψi }i∈N as follows.  X  0 ˜ f (x) = ψi max ui (xi , x−i ) − ui (xi , x−i ) . 0 i∈N

xi ∈Xi

(3)

Let Zi be a set of nonnegative real numbers such that ψi takes 0. Zi = {z ∈ R+ |ψi (z) = 0}. Theorem 3 Fix {ψi }i∈N in (3). Then, the following holds. (i) For any game, every NE x∗ must satisfy f˜(x∗ ) = 0. (ii) Strategy profile x that satisfies f˜(x) = 0 must be an NE for any game, iff Zi = {0} for all i ∈ N . (iii) If Zi = {0} for all i ∈ N and there is an NE, E ∗ is identical to the set of minimum solutions to f˜, arg minx∈X f˜(x). 8 / 23

From “Sum” to “Product” The parallel characterization can be available when the objective function is replaced with the product of deviation gains.   0 f (x) = Πi∈N 1 + max ui (xi , x−i ) − ui (xi , x−i ) . (4) 0 xi ∈Xi

Note by construction that f (x) ≥ 1 holds for any x ∈ X. Theorem 4 A strategy profile x∗ is an NE iff f (x∗ ) = 1. If there is an NE, E ∗ is identical to the set of minimum solutions to f , arg minx∈X f (x). 1 can be replaced with any ε > 0. (Then, f (x∗ ) = εn ) As ε goes to 0, (4) converges to the product of players’ payoff differences, which may look similar to the Nash product. 9 / 23

Relation with Nash Demand Game Each player i = 1, 2 chooses her demand, di (> ui ). (d1 , d2 ) realizes iff (d1 , d2 ) ∈ B, the set of feasible payoffs. If (d1 , d2 ) ∈ / B, players receive (u1 , u2 ). ⇒ Multiplicity of equilibria due to payoff discontinuity. Introducing h(d1 , d2 ), Nash (1953) considers the smoothed game. Payoff becomes di h(d1 , d2 ), where h(d1 , d2 ) = 1 for d ∈ B. h rapidly shrinks to 0 as d move away from B. Fact 5 (Nash, 1953) NE of the smoothed game is unique and characterized by arg max (d1 − u1 )(d2 − u2 )h(d1 , d2 ) ← Potential Function d>u

which converges to the Nash barging solution. arg max (u1 − u1 )(u2 − u2 ). u(>u)∈B

10 / 23

Interchangeability for Two-Person Games Let x = (x1 , x2 ) and x0 = (x01 , x02 ) be two distinct NE. Definition 6 A pair of Nash equilibia x and x0 is called interchangeable if (x1 , x02 ) and (x01 , x2 ) constitute NE of the same game. Our approach can explain the following existing results on interchangeability, independently derived in the literature: 1

for a zero-sum game any pairs of its mixed strategy Nash equilibria are interchangeable (Luce and Raiffa, 1957).

2

for a supermodular game where each player’s strategy space is totally ordered, any unordered pairs of the pure strategy Nash equilibria are interchangeable (Echenique, 2003).

3

equilibrium set of a strictly supermodular game with totally ordered strategy spaces, is totally ordered (Vives, 1985). 11 / 23

Introduction to Lattice Let = be a binary relation on a non-empty set S. Definition 7 The pair (S, =) is a partially ordered set if, for x, y, z in S, = is reflexive: x = x. transitive: x = y and y = z implies x = z. antisymmetric: x = y and y = x implies x = y. A partially ordered set (S, =) is totally ordered if for x and y in S either x = y or y = x is satisfied, and is called chain. called lattice if any two elements have a least upper bound (join, ∨) and a greatest lower bound (meet, ∧) in the set. A subset S ∗ (⊂ S) is called sublattice if x0 ∧ x00 ∈ S ∗ and x0 ∨ x00 ∈ S ∗ hold for any x0 , x00 ∈ S ∗ . 12 / 23

Sublattice and Interchangeability Assume each player’s strategy is totally ordered, e.g., Xi ⊂ R. → For any pair of strategy profiles, its meet/join constitutes strategy profile where every player takes smaller/larger strategy. 0. If E ∗ is (non-empty) lattice, then The existence of maximum and minimum NE is guaranteed. 1. If E ∗ is sublattice, then Any unordered pairs of the NE must be interchangeable. ← Let x = (x1 , x2 ) and x0 = (x01 , x02 ) be two unordered NE. Sublattice property implies that both (x01 , x2 ) and (x1 , x02 ) must be elements in E ∗ and hence NE. → This does not imply interchangeability of ordered NE. 2. If E ∗ is chain, then All NE must be totally ordered. 13 / 23

Submodular and Supermodular Functions Definition 8 A real-valued function h defined over a lattice S is called a submodular function if, for any x0 , x00 ∈ S, h satisfies h(x0 ∧ x00 ) + h(x0 ∨ x00 ) − {h(x0 ) + h(x00 )} ≤ 0.

(5)

(5) trivially holds with equality whenever x0 and x00 are ordered, i.e., x0 = x00 or x0 5 x00 . If ≤ in (5) is replaced with ≥, h is called supermodular. h : S → R becomes a strictly submodular function if, for any unordered pair x0 , x00 ∈ S, i.e., x0  x00 and x0  x00 , h satisfies h(x0 ∧ x00 ) + h(x0 ∨ x00 ) − {h(x0 ) + h(x00 )} < 0.

(6)

If < in (6) is replaced with >, h is strictly supermodular. 14 / 23

Lattice Properties Fact 9 (Topkis, 1978) If h is a submodular function on a lattice S, then the set S ∗ of points at which h attains its minimum on S is a sublattice of S. Fact 10 (Topkis, 1978) If h is st. submodular on a lattice S, then the set S ∗ of points at which h attains its minimum on S is a chain. Assume X is a lattice. The above facts imply the following. Lemma 11 Suppose that f is defined by (1) and E ∗ is nonempty. Then, (i) If f is submodular on X, then E ∗ is a sublattice of X. (ii) If f is st. submodular on X, then E ∗ is a chain. 15 / 23

Supermodularity and Complementarity If, for all i ∈ N , ui is a supermodular function ⇐⇒ Supermodular game. st. supermodular function ⇐⇒ St. supermodular game. h : X → R is called a quasi-supermodular function if, for all x and x0 in X, h(x) ≥ h(x ∧ x0 ) implies h(x ∨ x0 ) ≥ h(x0 ), and h(x) > h(x ∧ x0 ) implies h(x ∨ x0 ) > h(x0 ). A game with strategic complementarities (GSC) is a weaker notion than supermodular game, which requires that each player’s best reply is (weakly) increasing in other players’ strategies. Fact 12 (Milgrom and Shannon, 1994) A quasi-supermodular game is identical to a GSC. 16 / 23

Supermodular Game Let us define u as the sum of the payoff functions. u(x) = u1 (x) + u2 (x) for all x ∈ X. Lemma 13 u (= u1 + u2 ) is supermodular for any supermodular games, and u is st. supermodular for any st. supermodular games. Supermodularity is preserved under addition. The converse is not true, e.g., zero-sum game. Theorem 14 For any two-person game with totally ordered strategy space for each player, (i) f is submodular iff u is supermodular. (ii) f is st. submodular iff u is st. supermodular. 17 / 23

Proof (1) Recall that f is a submodular function if, for any x0 , x00 ∈ X, f (x0 ∧ x00 ) + f (x0 ∨ x00 ) − {f (x0 ) + f (x00 )} ≤ 0.

(7)

If the above inequality is strict for any unordered pairs, f is a st. submodular function. Since we consider two-person games, f (x) = max u1 (x1 , x2 ) − u1 (x1 , x2 ) x1 ∈X1

+ max u2 (x1 , x2 ) − u2 (x1 , x2 ). x2 ∈X2

Now consider a pair of unordered strategy profiles, x0 = (x01 , x02 ) and x00 = (x001 , x002 ). W.o.l.g, assume x01 =1 x001 and x02 52 x002 . Join: x0 ∧ x00 = (x001 , x02 ) Meet: x0 ∨ x00 = (x01 , x002 ) 18 / 23

Proof (2) The corresponding values of f are expressed by f (x0 ∧ x00 ) = max u1 (x1 , x02 ) − u1 (x001 , x02 ) x1 ∈X1

+ max u2 (x001 , x2 ) − u2 (x001 , x02 ). x2 ∈X2

0

00

f (x ∨ x ) = max u1 (x1 , x002 ) − u1 (x01 , x002 ) x1 ∈X1

+ max u2 (x001 , x2 ) − u2 (x01 , x002 ). x2 ∈X2

Substituting them into (7), max ui parts will be canceled out. The next equality illustrates that the (st.) submodularity of f is completely characterized by the (st.) supermodularity of u.

19 / 23

Proof (3) f (x0 ∧ x00 ) + f (x0 ∨ x00 ) − {f (x0 ) + f (x00 )} = − { u1 (x001 , x02 ) + u2 (x001 , x02 ) + u1 (x01 , x002 ) + u2 (x01 , x002 )}  + u1 (x01 , x02 ) + u2 (x01 , x02 ) + u1 (x001 , x002 ) + u2 (x001 , x002 ) = − { u1 (x0 ∧ x00 ) + u2 (x0 ∧ x00 ) + u1 (x0 ∨ x00 ) + u2 (x0 ∨ x00 )}  + u1 (x0 ) + u2 (x0 ) + u1 (x00 ) + u2 (x00 ) = u(x0 ) + u(x00 ) − {u(x0 ∧ x00 ) + u(x0 ∨ x00 )}. Lemma 13, combined with Theorem 14, implies the following. Corollary 15 For any two-person (i) supermodular games, f is submodular. (ii) st. supermodular games, f is st. submoduler. 20 / 23

Interchangeability for Zero-Sum Game Theorem 16 For a zero-sum game, any pairs of its (mixed strategy) Nash equilibria are interchangeable. Proof. E ∗ is nonempty. Existence of NE by Nash (1950) Existence of minimax solution by Neumann (1928). Suppose that x = (x1 , x2 ) and x0 = (x01 , x02 ) are two distinct NE. We can always construct an order that satisfies x1 =1 x01 and x2 52 x02 . By Lemma 11 (i), E ∗ is sublattice. ⇒ x ∧ x0 = (x01 , x2 ) and x ∨ x0 = (x1 , x02 ) are both NE. 21 / 23

Interchangeability for Supermodular Game Since the existence of (pure strategy) NE for supermodular games is guaranteed by Topkis (1979), we obtain the next theorem. Theorem 17 Assume strategy space for each player is totally ordered. Then, (i) For a two-person supermodular game, any unordered pairs of its pure strategy NE are interchangeable. (ii) For a two-person st. supermodular game, all pure strategy NE are totally ordered. Theorem 18 Assume that strategy space for each player is totally ordered. Then, for a two-person GSC, any unordered pairs of its pure strategy NE are interchangeable. 22 / 23

Conclusion Summary: I provide a reformulation/reinterpretation of NE. Enables us to analyze Nash equilibrium as a solution to a simple optimization problem without any constraints. May bridge a gap (if it exists) between non-cooperative game theory and other related fields such as OR and CS. Natural Reaction: So what? Is it really useful? → As an application, I revisit interchangeability of NE. Existing results on two person (i) zero-sum games and (ii) supermodular games can be derived, in a unified fashion, by lattice structure of optimal solutions: The set of minimizers of submodular function is sublattice.

23 / 23