In a recent article, Hahn [Hahn, S. (2008). The convergence of fictitious play in games with strategic complementarities. Economics Letters 99, 2, 304-306] claims to prove convergence of fictitious play in games with strategic complementarities. I show here that the proof is flawed and convergence remains an open question.Fictitious play; Strategic complementarities; Supermodular games

Berger, Ulrich

English

Research Papers in Economics

The convergence of ﬁctitious play in games
with strategic complementarities: A Comment
Ulrich Berger
WU Vienna, Department of Economics, Augasse 2-6, 1090 Wien, Austria
Abstract
In a recent article, Hahn [Hahn, S. (2008). The convergence of ﬁctitious play in
games with strategic complementarities. Economics Letters 99, 2, 304-306] claims
to prove convergence of ﬁctitious play in games with strategic complementarities. I
show here that the proof is ﬂawed and convergence remains an open question.
JEL classiﬁcation: C72, D83.
Key words: Fictitious play; Strategic complementarities; Supermodular games
1 Introduction
Fictitious play (FP) was introduced by Brown (1949, 1951) as an algorithm to
calculate the value of a two-person zero-sum game and now serves as a classical
example of myopic belief learning (see Fudenberg and Levine, 1998, or Young,
2005). A large part of the literature on FP focused on identifying classes of
games where every FP process converges (in beliefs) to equilibrium. One class
of games which has been investigated is the class of games with strategic
complementarities (GSCs), also known as supermodular games (Topkis, 1979,
Vives, 1990, Milgrom and Roberts, 1990). GSCs are interesting since they have
important economic applications (Vives, 2005).
Convergence to Nash equilibrium under the dicrete-time or the continuous-
time version of FP has been shown for GSCs under various additional assump-
tions. 1 First, convergence in GSCs with a unique equilibrium has been shown
Email address: ulrich.berger@wu.ac.at (Ulrich Berger).
1 Typically, two-person GSCs with totally ordered strategy spaces are studied.
Without a total order of strategies, strategic complementarities become almost
ubiquitous (see Echenique, 2004). Additionally, nondegeneracy of the game or a
tie-breaking rule is required, see Monderer and Sela (1996).
Preprint 3 September 2009by Milgrom and Roberts (1991). In an unpublished paper, Krishna (1992) then
studied convergence in GSCs with diminishing returns. Hahn (1999) showed
convergence in 3×3 GSCs. The stochastic version of FP was studied for GSCs
by Hofbauer and Sandholm (2002). Berger (2007) proved convergence in 3×m
and 4×4 GSCs and Berger (2008) extended Krishna’s (1992) result to games
with ordinal strategic complementarities and diminishing returns.
2 An open question
In a short note published recently in this journal, Hahn (2008) claims to prove
convergence of FP in GSCs without any further restrictions.
His main result is based on ﬁve lemmas and a corollary. Lemma 1 and Lemma
2 contain known results, while Lemma 3 and Corollary 1 follow immediately
from the Improvement Principle 2 for FP (Monderer and Sela, 1997, see also
Sela, 2000, or Berger, 2007, 2008). The proofs of Lemmas 4 and 5 each are
basically repeated proofs of special cases of the Improvement Principle. How-
ever, they both contain an error which invalidates these two lemmas and the
main result.
Lemma 4 tries to show that because of strategic complementarities, if player
1 switches from playing some strategy i to playing his largest strategy I in
an FP process, then he can not switch to playing another strategy i0 < I
unless player 2 switches before him. The last sentence of the proof of Lemma
4 concludes, 3
[...] and this leads to uI1 < ui01 which contradicts Lemma 2 (2).
But Lemma 2 (2) correctly states that in this case we have ui01 > uI1, so
contrary to Hahn’s claim there is no contradiction.
Lemma 5 tries to prove that because of strategic complementarities, if player
1 switches from playing some strategy i to playing his largest strategy I in an
FP process, then he can not switch to playing another strategy i0 < I, even
if player 2 switches from playing j to playing j0 in between. The third- and
second-to-last sentences of the proof of Lemma 5 read,
[...] together with uIj0 ≤ ui0j0 we will have (uIj0 − uij0) ≤ 0 [...]. It clearly
violates the condition of strategic complementarities.
However, in going from the ﬁrst inequality to the second, a typo occurs, and i0
2 The Improvement Principle says that whenever a player in an FP process switches
her strategy choice while her opponent does not, this player’s payoﬀ improves. Put
shortly, unilateral switches always move to better responses.
3 Note that there is an inconsistency in Hahn’s notation, starting in Deﬁnition 2:
Whenever strategies i0 or j0 appear as a subscript, they are written as i0 and j0,
respectively. For the reader’s convenience I use i0 and j0 here, even in verbatim
quotes.
2becomes i. After correcting this typo, the violation of the condition of strategic
complementarities disappears.
3 Conclusion
Hahn’s main Theorem rests on the idea that once a player plays her highest
strategy, she is bound to play it forever and therefore FP converges. But the
proof of this claim is invalid since it relies on Lemmas 4 and 5, both of which
are ﬂawed. Indeed, Berger (2007, p. 256) provides an example to argue that
the Improvement Principle alone is insuﬃcient to predict convergence of FP
in GSCs of dimension 4×5 or higher.
Krishna (1992) conjectured that FP always converges in GSCs, and this be-
lieve is shared by many researchers in the ﬁeld. As we have shown, Hahn’s
attempt to prove the conjecture failed. But no counterexample is known ei-
ther. Convergence of FP in GSCs remains an open question.
References
[1] Berger, U. (2007). Two more classes of games with the continuous-time ﬁctitious
play property. Games and Economic Behavior, 60, 247-261.
[2] Berger, U. (2008). Learning in games with strategic complementarities revisited.
Journal of Economic Theory 143, 292-301.
[3] Brown, G. W. (1949). Some notes on computation of games solutions. Report
P-78, The Rand Corporation.
[4] Brown, G. W. (1951). Iterative solution of games by ﬁctitious play. In:
Koopmans, T. C., (ed.). Activity Analysis of Production and Allocation. Wiley:
New York.
[5] Echenique, F. (2004). A characterization of strategic complementarities. Games
and Economic Behavior 46, 325-347.
[6] Fudenberg, D. and Levine, D. K. (1998). The theory of learning in games. MIT
Press: Cambridge, MA.
[7] Hahn, S. (1999). The convergence of Fictitious Play in 3×3 games with strategic
complementarities. Economics Letters 64, 57-60.
[8] Hahn, S. (2008). The convergence of ﬁctitious play in games with strategic
complementarities. Economics Letters 99, 304-306.
[9] Hofbauer, J. and Sandholm, W. (2002). On the global convergence of stochastic
ﬁctitious play. Econometrica 70, 2265-2294.
[10] Krishna, V. (1992). Learning in games with strategic complementarities. HBS
Working Paper 92-073, Harvard University.
3[11] Milgrom, P. and Roberts, J. (1990). Rationalizability, learning, and equilibrium
in games with strategic complementarities. Econometrica 58, 1255-1277.
[12] Milgrom, P. and Roberts, J. (1991). Adaptive and sophisticated learning in
normal form games. Games and Economic Behavior 3, 82-100.
[13] Monderer, D. and Sela, A. (1996). A 2×2 game without the ﬁctitious play
property. Games and Economic Behavior 14, 144-148.
[14] Monderer, D. and Sela, A. (1997). Fictitious play and no-cycling conditions.
Mimeo, The Technion.
[15] Sela, A. (2000). Fictitious play in 2x3 games. Games and Economic Behavior
31, 152-162.
[16] Topkis, D. M. (1979). Equilibrium points in nonzero-sum n-person submodular
games. SIAM Journal of Control and Optimization 17, 773-787.
[17] Vives, X. (1990). Nash equilibrium with strategic complementarities. Journal
of Mathematical Economics 19, 305-321.
[18] Vives, X. (2005). Complementarities and games: New developments. Journal of
Economic Literature 43, 437-479.
[19] Young, P. (2005). Strategic learning and its limits. Oxford University Press:
Oxford, UK.
4

A 2×2 game without the ﬁctitious play property.

A characterization of strategic complementarities.

Adaptive and sophisticated learning in normal form games.

Complementarities and games: New developments.

Equilibrium points in nonzero-sum n-person submodular games.

Fictitious play and no-cycling conditions. Mimeo, The Technion.

Fictitious play in 2x3 games.

Iterative solution of games by ﬁctitious play.

Learning in games with strategic complementarities revisited.

Learning in games with strategic complementarities.

Nash equilibrium with strategic complementarities.

On the global convergence of stochastic ﬁctitious play.

Some notes on computation of games solutions.

Strategic learning and its limits.

The convergence of Fictitious Play in 3×3 games with strategic complementarities.

The convergence of ﬁctitious play in games with strategic complementarities.

The theory of learning in games.

Two more classes of games with the continuous-time ﬁctitious play property.

The convergence of fictitious play in games with strategic complementarities: A Comment

The convergence of fictitious play in games with strategic complementarities: A Comment

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