This document presents an optimization approach to characterizing Nash equilibria in games. It shows that the set of Nash equilibria is identical to the set of solutions that minimize an objective function defined over strategy profiles. This allows the equilibrium problem to be framed as an optimization problem. The approach provides a unified way to derive existing results on interchangeability of equilibria in zero-sum and supermodular games, by relating the properties of the objective function to the structure of the optimal solution set.

1. 580 Optimization Approach to Nash Equilibria
with Applications to Interchangeability
Yosuke YASUDA
Osaka University, Department of Economics
(visiting ISEG, University of Lisbon)
yasuda@econ.osaka-u.ac.jp
July, 2021
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2. Summary
What is Optimization Approach? (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? Any application? (2nd Part)
Existing results on interchangeability can be derived, in a
unified fashion, by lattice structure of optimal solutions.
→ NEW contribution! (looking for ompletely new results...)
Main Messages
Please use/apply/extend our characterization of NE!
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3. Traditional Formulations of NE
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
xi∈Xi
ui(xi, x∗
−i) for all i ∈ N.
3 Fixed Point of BR Correspondence
x∗
∈ BR(x∗
),
where BRi(x−i) = arg max
x0
i∈Xi
ui(x0
i, x−i).
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4. Alternative: Optimization Approach
Define f : X → R, which aggregates maximum deviation gains
(from a fixed action profile x ∈ X) across players.
f(x) =
X
i∈N
max
x0
i∈Xi
ui(x0
i, x−i) − ui(xi, x−i)
. (1)
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, i.e., arg minx∈X f(x).
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5. Example: Prisoner’s Dilemma
Table: Payoff Matrix
C D
C 3, 3 0, 4
D 4, 0 1, 1
Table: Values of f
C D
C 2 (= (4 − 3) + (4 − 3)) 1 (= (1 − 0) + 0)
D 1 (= 0 + (1 − 0)) 0 (= 0 + 0)
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6. Extensions
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
x∈X
g(x).
f can be replaced by ˜
f : X → R, using a sign-preserving
function ψi : R+ → R+ such that ψi(x) = 0 iff x = 0.
˜
f(x) =
X
i∈N
ψi
max
x0
i∈Xi
ui(x0
i, x−i) − ui(xi, x−i)
.
Let t be a parameter contained in a parameter set T. Then,
(1) can be rewritten to incorporate this parameterization.
f(x, t) =
X
i∈N
max
x0
i∈Xi
ui(x0
i, x−i; t) − ui(xi, x−i; t)
.
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7. From “Sum” to “Product”
The parallel characterization can be available when the objective
function is replaced with the product of deviation gains.
f(x) = Πi∈N
1 + max
x0
i∈Xi
ui(x0
i, x−i) − ui(xi, x−i)
. (2)
Note by construction that f(x) ≥ 1 holds for any x ∈ X.
Lemma 3
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, (2) converges to the product of players’ payoff
differences, which may look similar to the Nash product.
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8. Interchangeability for Two-Person Games
Let x = (x1, x2) and x0 = (x0
1, x0
2) be two distinct NE.
Definition 4
A pair of Nash equilibia x and x0 is called interchangeable if
(x1, x0
2) and (x0
1, 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).
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9. Introduction to Lattice (1)
Let = be a binary relation on a non-empty set S.
Definition 5
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∗.
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10. Introduction to Lattice (2)
Definition 6
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. (3)
(3) trivially holds with equality whenever x0 and x00 are
ordered, i.e., x0 = x00 or x0 5 x00.
If ≤ in (3) 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. (4)
If in (4) is replaced with , h is strictly supermodular.
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11. Results by Topkis (1978)
Fact 7 (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 8 (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 9
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.
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12. Supermodular Game
Let us define u as the sum of the payoff functions.
u(x) = u1(x) + u2(x) for all x ∈ X.
Lemma 10
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.
Lemma 11
u (= u1 + u2) is supermodular (resp. strictly supermodular) for
any two-person supermodular (resp. strictly supermodular) games.
Supermodularity is preserved under addition.
The converse is not true, e.g., zero-sum game.
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13. Proof of Lemma 10 (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. (5)
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
x1∈X1
u1(x1, x2) − u1(x1, x2)
+ max
x2∈X2
u2(x1, x2) − u2(x1, x2).
Now consider a pair of unordered strategy profiles, x0 = (x0
1, x0
2)
and x00 = (x00
1, x00
2). W.o.l.g, assume x0
1 =1 x00
1 and x0
2 52 x00
2.
Join: x0 ∧ x00 = (x00
1, x0
2)
Meet: x0 ∨ x00 = (x0
1, x00
2)
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14. Proof of Lemma 10 (2)
The corresponding values of f are expressed by
f(x0
∧ x00
) = max
x1∈X1
u1(x1, x0
2) − u1(x00
1, x0
2)
+ max
x2∈X2
u2(x00
1, x2) − u2(x00
1, x0
2).
f(x0
∨ x00
) = max
x1∈X1
u1(x1, x00
2) − u1(x0
1, x00
2)
+ max
x2∈X2
u2(x00
1, x2) − u2(x0
1, x00
2).
Substituting them into (5), 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.
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15. Proof of Lemma 10 (3)
f(x0
∧ x00
) + f(x0
∨ x00
) − {f(x0
) + f(x00
)}
= − { u1(x00
1, x0
2) + u2(x00
1, x0
2) + u1(x0
1, x00
2) + u2(x0
1, x00
2)}
+
u1(x0
1, x0
2) + u2(x0
1, x0
2) + u1(x00
1, x00
2) + u2(x00
1, x00
2)
= − { 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
)}.
Since u = u1 + u2 is constant for a zero-sum game, we obtain:
Lemma 12
For any two-person zero-sum game, f is submodular over mixed
strategy profiles with any order of strategy space for each player.
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16. Main Result: Zero-Sum Game
Theorem 13
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 = (x0
1, x0
2) are two distinct NE.
We can always construct an order that satisfies
x1 =1 x0
1 and x2 52 x0
2.
By Lemma 9 (i), E∗ is sublattice.
⇒ x ∧ x0 = (x0
1, x2) and x ∨ x0 = (x1, x0
2) are both NE.
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17. Main Result: Supermodular Game
Since the existence of (pure strategy) NE for supermodular games
is guaranteed by Topkis (1979), we obtain the next theorem.
Theorem 14
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.
The above results also hold for log-supermodular games.
If ui is log-supermodular, then ˆ
f must be submodular.
(Note f = 0 iff ˆ
f = 0)
ˆ
f(x) =
X
i∈N
max
x0
i∈Xi
ln ui(x0
i, x−i) − ln ui(xi, x−i)
.
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18. Conclusion
Summary: Provide a functional formulation of NE.
Enables us to analyze Nash equilibrium as a solution to an
optimization problem without any constraints.
May connect non-cooperative game theory with other related
fields such as OR and CS.
Natural Reaction: So what? Is it really useful?
→ New insight on interchangeability of NE.
Existing results on two person (i) zero-sum games and (ii)
supermodular games can be derived, in a unified fashion:
The set of minimizers of submodular function is sublattice.
Applications to symmetric, potential, cooperative games (via
Nash implementation), etc? → Future research!
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