Basics of Game Theory for Bioinformatics
DIPTEM, University of Genoa, email@example.com
Abstract. In this ?tutorial? it is o?ered a quick introduction to game
theory and to some suggested readings on the subject. It is also consid-
ered a small set of game theoretical applications in the bioinformatics
It could be considered strange the fact that game theory (we shall abbreviate it
as GT) is used in the ?eld of bioinformatics. After all, not only GT was created to
model economic problems, but also its foundational assumptions are very close
relatives of those made in neoclassical economics: basically, the assumption that
?players? are ?rational? and intelligent decision makers (usually assumed to be
human beings). If we deal with viruses, or genes, it is not so clear whether these
basic assumptions retain any meaning, and we could proceed further, wondering
whether there are ?rational decision makers? around.
We shall see that there are some grounds for such an extension in the scope
of game theory, but at the same time we acknowledge that there is another
reason (almost opposite) to use game theory in the ?eld of bioinformatics. This
second reason can be found in the fact that game theory can be seen as ?math +
intended interpretation?. Of course, if we discard the ?intended interpretation?,
we are left only with mathematics: by its very nature, math is ?context free?, so
that we are authorized to use all of the mathematics that has been developed
in and for game theory, having in mind whatever ?intended meaning? we would
like to focus on (a relevant example of this ?de-contextualization? is o?ered in
Moretti and Patrone (2008), about the so-called Shapley value). This switch in
the interpretation of the mathematical tools needs sound justi?cations, if it has
to be considered a serious scienti?c contribution, but this can be done, as it has
been done also in other very di?erent ?elds of mathematics.
So, we shall try to emphasize a bit these two approaches to the subject of
this tutorial. Since we do not assume any previous knowledge of game theory,
we shall start with a very quick sketch of the basics of game theory (section
3), at least to set the possibility of using its language. Since game theory is a
subject quite extended in width and depth, for the reader interested to go further
(maybe considering to apply GT on its own) we cannot do anything better than
providing suggestions for further readings. For this reason, in section 3 we shall
provide a concise guide to the main relevant literature, especially to GT books.
We shall then move to illustrate some of the GT applications in the ?eld of
bioinformatics. Due to our personal contributions to the ?eld, we shall mainly
stress the applications of a basic concept and tool for ?cooperative games?: the
so-called Shapley value. This will be the subject of the ?rst subsection of section
4. The second subsection will be a very quick tour touching some of the diverse
ways in which GT has been used in bioinformatics.
2 Game theory: its basics
There is no doubt about the date of birth of GT: it is 1944, year in which Theory
of games and economic behavior, by John von Neumann and Oskar Morgenstern,
appeared. That book built the language of the discipline, its basic models (still
used today, with the improvement given by Kuhn (1953) to the de?nition of
games in extensive form), and strongly suggested that GT should be the adequate
mathematical tool to model economic phenomena. Of course, there was some
GT before 1944 (extremely important is the so-called ?minimax theorem? of von
Neumann, appeared in 1928), but that book set the stage on which many actors
have been playing since then.
From the ?birth? of GT it has been accumulated a lot of models, results,
applications, foundational deepening. After more than 60 years, GT is really
a non disposable tool in many areas of economic theory (consumers' behavior,
theory of the ?rm, industrial organization, auctions, public goods, etc.), and has
spread its scope to other social sciences (politics, sociology, law, anthropology,
etc.), and much beyond.
The core situation that GT models is a situation in which individuals ?strate-
gically interact?. The way in which interaction is meant is the following: there is
a set of individuals, each of whom has to make a choice from a set of available
actions; it will be the full set of choices made by each individual to determine
the result (or outcome). An important point is that individuals generally have
di?erent preferences on the ?nal outcomes. Moreover, interaction is said to be
?strategic?: it is important that individuals are aware of this interactive situation,
so that they are urged to analyze it, and for that it will be important to know
the structure of the interactive situation, the information available, and also the
amount of intelligence of the players.
The form which is most used to model such a situation is the so-called ?strate-
gic form?. It is quite easy to describe: it is a tuple: 퐺 = (푁,(푋푖)푖∈푁,(푓푖)푖∈푁),
- 푁 is the (?nite) set of players
- 푋푖is the set of actions (usually said ?strategies?) available to player 푖 ∈ 푁
- 푓푖: 푋 → ℝ, where 푋 =∏
lowing: given a so-called ?strategy pro?le?, i.e. 푥 ∈ 푋, it will determine an out-
come ℎ(푥); 푓푖(푥) is the personal evaluation of the outcome by player 푖, measured
on some scale. Using a terminology which is standard in neoclassical economics,
one can see 푓푖as the composition of the function ℎ with the ?utility function? 푢푖
of player 푖 (푢푖is de?ned on the set of outcomes).
푖∈푁푋푖, are the ?payo?s?.
Many things should be said about the payo?s, but the main point is the fol-
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