of some intelligent computing
Emmanuel M. Tadjouddine
Department of Computer Science & Software Engineering,
Xi’an Jiaotong-Liverpool University, Suzhou, People’s Republic of China
Purpose – As agent-based systems are increasingly used to model real-life applications such as
the internet, electronic markets or disaster management scenarios, it is important to study the
computational complexity of such usually combinatorial systems with respect to some desirable
properties. The purpose of this paper is to consider two computational models: graphical games
encoding the interactions between rational and selﬁsh agents; and weighted directed acyclic graphs
(DAG) for evaluating derivatives of numerical functions. The author studies the complexity of a certain
number of search problems in both models.
Design/methodology/approach – The author’s approach is essentially theoretical, studying the
problem of verifying game-theoretic properties for graphical games representing interactions between
self-motivated and rational agents, as well as the problem of searching for an optimal elimination
ordering in a weighted DAG for evaluating derivatives of functions represented by computer programs.
Findings – A certain class of games has been identiﬁed for which Nash or Bayesian Nash equilibria
can be veriﬁed in polynomial time; then, it has been shown that verifying a dominant strategy
equilibrium is non-deterministic polynomial (NP)-complete even for normal form games. Finally, it has
been shown that the optimal vertex elimination ordering for weighted DAGs is NP-complete.
Originality/value – This paper presents a general framework for graphical games. The presented
results are novel and illustrate how modeling real-life scenarios involving intelligent agents can lead to
computationally hard problems while showing interesting cases that are tractable.
Keywords Artiﬁcial intelligence, Knowledge engineering, Game theory, Systems and control theory,
Paper type Research paper
As agent-based systems are increasingly used to model real-life applications, e.g. the
internet, electronic markets or disaster management scenarios, it is important to study
the computational complexity of such usually combinatorial systems with respect to some
desirable properties. An important niche of such systems is in the economics aspects of the
internet, wherein the system can be modelled as a game and the resulting model used
to ﬁnd out or prove some game-theoretic properties of the system (see for instance
Papadimitriou, 2001 for an enlightening discussion on the subject). This leads us to the
fundamental results in game theory among them the existence theorem due to Nash (1951)
proving that non-cooperative games with mixed strategies have a Nash equilibrium.
For cooperative games and related complexity issues (Wooldridge and Dunne, 2006).
The current issue and full text archive of this journal is available at
The author would like to thank Dr F. Guerin for useful discussions on game theory and the idea
of verifying game-theoretic properties in open multi-agent systems.
Received 17 November 2009
Revised 20 December 2010
Accepted 10 January 2011
International Journal of Intelligent
Computing and Cybernetics
Vol. 4 No. 2, 2011
q Emerald Group Publishing Limited