Improvement Dynamics in Games with Strategic
Complementarities
∗
Nikolai S. Kukushkin
†
Satoru Takahashi
‡
Tetsuo Yamamori
§
Abstract
In a finite game with strategic complementarities, every strategy profile is con-
nected to a Nash equilibrium with a finite individual improvement path. If, addi-
tionally, the strategies are scalar, then every strategy profile is connected to a Nash
equilibrium with a finite individual best response improvement path.
Key words: Nash equilibrium, Better response dynamics, Best response dy-
namics, Game with strategic complementarities
∗
The first author acknowledges financial support from the Russian Foundation for Basic Research
(grant 02-01-00854) and from a presidential grant for the state support of the leading scientific schools
(NSh-1843.2003.01); he also thanks Universidad Carlos III de Madrid, Departamento de Econom´ıa, and
personally Francisco Marhuenda for their hospitality. The other authors would like to thank Michihiro
Kandori, Akihiko Matsui, Jim Friedman, Claudio Mezzetti, and Federico Echenique for helpful comments.
All the three thank an anonymous referee and the Editor of this journal for bringing us together and
suggesting the merger of two independent works.
†
Russian Academy of Sciences, Dorodnicyn Computing Center, 40, Vavilova, Moscow 119991 Russian
Federation. E-mail: ququ@ccas.ru
‡
Department of Economics, Harvard University, Cambridge, MA 02138, USA. E-mail:
stakahas@fas.harvard.edu
§
Graduate School of Economics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033,
JAPAN. E-mail: ee27004@mail.ecc.u-tokyo.ac.jp
1
1 Introduction
Learning and adaptation in strategic games is an important and interesting topic, attract-
ing much attention (Milgrom and Roberts, 1990, 1991; Young, 1993; Kandori and Rob,
1995; Monderer and Shapley, 1996; Milchtaich, 1996; Fudenberg and Levine, 1998; Fried-
man and Mezzetti, 2001; Kukushkin, 2004). Quite often, the analysis of sophisticated
scenarios involving random moves crucially depends on simple properties of the game,
expressed in terms of improvement paths. It is essential, therefore, to understand when
we can expect which property to hold.
For instance, when Young (1993) found the importance of the condition that every
strategy profile should be connected to a Nash equilibrium with a best response path,
he was unable to point out a single natural class of games where the property had been
established. Since then, much work has been done; in particular, that property holds in
every finite potential game as defined by Monderer and Shapley (1996), who provide quite
a list of such games. Still, we cannot say that the subject is thoroughly investigated.
Convergence properties of improvement dynamics in games with strategic complemen-
tarities have been considered since the very beginning: Topkis (1979) and Vives (1990)
showed that every best response improvement path starting from a certain region in
the outcomes space eventually reaches a Nash equilibrium. Milgrom and Roberts (1990)
studied rather general adaptive scenarios without explicitly considering their convergence.
Kandori and Rob (1995) established the convergence to a Nash equilibrium of every best
response path in every finite, symmetric, and strictly supermodular game with scalar
strategies; Kukushkin (2004) established the same property of best response improve-
ment paths in games with additive aggregation. Friedman and Mezzetti (2001) showed
that every strategy profile in a finite game with strategic complementarities and one-
dimensional strategy sets can be connected to a Nash equilibrium with an improvement
path.
This paper strengthens the last result considerably: it turns out that Friedman and
Mezzetti’s statement holds in a multidimensional context as well, whereas their conditions
ensure the possibility to reach a Nash equilibrium with a best response improvement path.
Both extensions are more than purely technical hair-splitting. Although most games
with strategic complementarities in the literature have scalar strategies, multidimensional
models are also important: e.g., in the context of industrial organization, it is natural for
a firm to choose several parameters (Vives, 2003, Section 4.4). The difference between
better response and best response dynamics is crucial, e.g., for the scenario of Young
(1993) mentioned above.
2
Strategic complementarities are interpreted in the broadest sense available in the litera-
ture: we assume single crossing (Milgrom and Shannon, 1994) and pseudosupermodularity
(Agliardi, 2000) conditions. The proofs use the technique of binary relations suggested in
Kukushkin (1999, 2004).
The attention here is restricted to finite games. Similar results can be obtained for
the infinite (topological) case too; however, the proofs need rather heavy tools such as
the notion of improvement paths parameterized with transfinite numbers, outlined in
Kukushkin (2000).
Section 2 contains basic formal definitions. Section 3 reproduces conditions usually
associated with the term “strategic complementarities.” Section 4 contains the main
theorems and examples showing the impossibility of easy generalizations. The proofs are
deferred to Section 5.
2 Basic Notions
A strategic game is defined by a finite set of players N, and strategy sets X
i
and ordinal
utility functions u
i
on X =
Q
i∈N
X
i
for all i ∈ N. In this pap er, we assume that each X
i
is a finite set.
For every player i ∈ N, the best response correspondence R
i
(·) is defined in the usual
way:
R
i
(x
−i
) = {y
i
∈ X
i
| ∀z
i
∈ X
i
[u
i
(y
i
, x
−i
) ≥ u
i
(z
i
, x
−i
)]},
where x
−i
∈ X
−i
=
Q
j∈N \{i}
X
j
.
We introduce a number of binary relations on X (y, x ∈ X and i ∈ N):
y
i
x ⇐⇒ [y
−i
= x
−i
& u
i
(y) > u
i
(x)];
y x ⇐⇒ ∃i ∈ N [y
i
x];
y
∗
i
x ⇐⇒ [y
−i
= x
−i
& u
i
(y) > u
i
(x) & y
i
∈ R
i
(x
−i
)];
y
∗
x ⇐⇒ ∃i ∈ N [y
∗
i
x].
A strategy profile x ∈ X is a Nash equilibrium if and only if x is a maximizer for ,
i.e., if y x is impossible for any y ∈ X; equivalently, x ∈ X is a Nash equilibrium if and
only if x is a maximizer for
∗
.
An (individual) improvement path is a finite or infinite sequence {x
k
}
k=0,1,...,
such that
x
k+1
x
k
whenever k ≥ 0 and x
k+1
is defined. A best response improvement path is a
3
finite or infinite sequence {x
k
}
k=0,1,...,
such that x
k+1
∗
x
k
whenever k ≥ 0 and x
k+1
is
defined.
Putting together the terminology of Monderer and Shapley (1996), Milchtaich (1996),
and Friedman and Mezzetti (2001), we introduce the following definitions.
A game has the finite improvement path (FIP) property if there exists no infinite im-
provement path. A game has the finite best response improvement path (FBRP) property
if there exists no infinite best response improvement path. A game has the weak FIP
property if, for every x ∈ X, there exists a finite improvement path {x
0
, . . . , x
m
} such
that x
0
= x and x
m
is a Nash equilibrium, i.e., if every strategy profile is connected to a
Nash equilibrium with a finite improvement path. A game has the weak FBRP property
if, for every x ∈ X, there exists a finite best response improvement path {x
0
, . . . , x
m
}
such that x
0
= x and x
m
is a Nash equilibrium.
It is easy to see that the following chain of implications holds:
FIP ⇒ FBRP ⇒ weak FBRP ⇒ weak FIP.
Each of these properties characterizes the global convergence property in a correspond-
ing class of learning and adaptive dynamics. A stationary Markov process on X is said to
be a better-reply (respectively best-reply) dynamic if (1) for each strategy profile x that
is not a Nash equilibrium, there exists x
0
such that x
0
x (respectively x
0
∗
x) and the
one step transition probability from x to x
0
is positive, and (2) every Nash equilibrium
is absorbing. It is easy to show that a game has the FIP (respectively FBRP) property
if and only if, under any better-reply (respectively best-reply) dynamic, the sequence of
strategy profiles from any initial strategy profile almost surely converges to a Nash equi-
librium. Alternatively, if we assume those dynamics to have full support, i.e., for any x
0
such that x
0
x (respectively x
0
∗
x), the one step transition probability from x to x
0
is
positive, then we obtain the weak versions of the above statements. See Friedman and
Mezzetti (2001), Kandori and Rob (1995), and Milchtaich (1996) for examples of better-
and best-reply dynamics.
3 Games with Strategic Complementarities
For the term to be applicable to a game, the latter must satisfy several requirements,
which can be formalized in somewhat different ways.
First, we assume that each X
i
is a lattice; therefore, X and every X
−i
are lattices too.
4
Second, there must be a complementarity condition on strategies of different players.
The basic choice is between Topkis’s (1979) cardinal increasing differences condition:
[y
i
≥ x
i
& y
−i
≥ x
−i
] ⇒ [u
i
(y) − u
i
(x
i
, y
−i
) ≥ u
i
(y
i
, x
−i
) − u
i
(x)] (3.1)
and Milgrom and Shannon’s (1994) ordinal single crossing condition:
[y
i
≥ x
i
& y
−i
≥ x
−i
] ⇒ [sign(u
i
(y) − u
i
(x
i
, y
−i
)) ≥ sign(u
i
(y
i
, x
−i
) − u
i
(x))], (3.2)
where i ∈ N, x
i
, y
i
∈ X
i
, x
−i
, y
−i
∈ X
−i
, and sign(t) is −1 if t < 0, 0 if t = 0, and 1 if
t > 0 (although subtraction is used in the second definition, the property itself is purely
ordinal).
Finally, if strategies of a given player i are multidimensional, there must be a com-
plementarity condition on different dimensions. Here we have even three versions: the
cardinal supermodularity condition
u
i
(x
i
∨ y
i
, z
−i
) − u
i
(x
i
, z
−i
) ≥ u
i
(y
i
, z
−i
) − u
i
(x
i
∧ y
i
, z
−i
); (3.3)
the ordinal quasisupermodularity condition (Milgrom and Shannon, 1994)
sign(u
i
(x
i
∨ y
i
, z
−i
) − u
i
(x
i
, z
−i
)) ≥ sign(u
i
(y
i
, z
−i
) − u
i
(x
i
∧ y
i
, z
−i
)); (3.4)
the, also ordinal, pseudosupermodularity condition (Agliardi, 2000)
sign(max{u
i
(x
i
∨ y
i
, z
−i
) − u
i
(x
i
, z
−i
), u
i
(x
i
∨ y
i
, z
−i
) − u
i
(y
i
, z
−i
)}) ≥
sign(max{u
i
(x
i
, z
−i
) − u
i
(x
i
∧ y
i
, z
−i
), u
i
(y
i
, z
−i
) − u
i
(x
i
∧ y
i
, z
−i
)}). (3.5)
Again, i ∈ N, x
i
, y
i
∈ X
i
, and z
−i
∈ X
−i
; sign(t) has the same meaning as above.
The following implications are easy to check: (3.1) ⇒ (3.2); (3.3) ⇒ (3.4) ⇒ (3.5).
Lemma 3.1. If a utility function u
i
satisfies (3.2) and (3.5) for each x
i
, y
i
∈ X
i
and
x
−i
, y
−i
, z
−i
∈ X
−i
, then
[y
−i
≥ x
−i
& y
i
∈ R
i
(y
−i
) & x
i
∈ R
i
(x
−i
)] ⇒ [y
i
∨ x
i
∈ R
i
(y
−i
) & y
i
∧ x
i
∈ R
i
(x
−i
)].
The statement means that R
i
(x
−i
) is a sublattice of X
i
(pick y
−i
= x
−i
) and R
i
(·) is
increasing w.r.t. the strong set order defined by Veinott (see Topkis, 1979).
Proof. Indeed, x
i
∈ R
i
(x
−i
) implies u
i
(x) ≥ u
i
(x
i
∧ y
i
, x
−i
) and u
i
(x) ≥ u
i
(y
i
, x
−i
), hence,
by (3.5), u
i
(x
i
∨ y
i
, x
−i
) ≥ u
i
(y
i
, x
−i
), hence, by (3.2), u
i
(x
i
∨ y
i
, y
−i
) ≥ u
i
(y), hence
x
i
∨ y
i
∈ R
i
(y
−i
). On the other hand, y
i
∈ R
i
(y
−i
) implies that u
i
(y) ≥ u
i
(x
i
∨ y
i
, y
−i
),
hence, by (3.2), u
i
(y
i
, x
−i
) ≥ u
i
(x
i
∨y
i
, x
−i
), hence, by (3.5), u
i
(x
i
∧y
i
, x
−i
) ≥ u
i
(x), hence
x
i
∧ y
i
∈ R
i
(x
−i
).
5