Basics of Game Theory for Bioinformatics.

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Basics of Game Theory for Bioinformatics
Fioravante Patrone
DIPTEM, University of Genoa, patrone@diptem.unige.it
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
?eld.
1Introduction
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

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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.
2Game 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),
where:
- 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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We introduce here a couple of very simple (but quite relevant) examples
of games in strategic form. The ?rst one is the so-called ?prisoner's dilemma?:
푁 = {퐼,퐼퐼}, 푋퐼= 푋퐼퐼= {퐶,푁퐶}, and 푓퐼,푓퐼퐼are described in table 1 (where,
e.g., 푓퐼(퐶,푁퐶) = 4,푓퐼퐼(퐶,푁퐶) = 1.
Table 1. Prisoner's dilemma
퐼∖퐼퐼
퐶
푁퐶
퐶 푁퐶
(2,2)
(1,4)
(4,1)
(3,3)
The description of a game by means of a table like table 1 is quite common
in the case in which there are two players, with ?nite strategy sets (the table is
often referred as a ?bimatrix?, since cells contain two numbers, the payo?s for
each of the players).
A second example is the ?battle of the sexes?, which is given in table 2.
Table 2. Battle of the sexes
퐼∖퐼퐼
푇
퐵
퐿푅
(2,1)
(0,0)
(0,0)
(1,2)
It is important to stress the fact that the standard assumption behind a
game modeled in strategic form is that each player chooses his strategy without
knowing the choices of the other players (just think of a sealed bid auction).
The relevant question is: what is a ?solution? for a game in strategic form?
Before answering this question, it is important to notice that game theory can
be seen both from a prescriptive (or normative) and a descriptive (or positive)
point of view, these ?dual? points of view are often found in models for the social
sciences. That is: a ?solution? can be seen either as:
- what players should do (prescriptive point of view)
- what we expect that players (will) do (descriptive point of view)
The basic idea of solution for strategic games is the Nash equilibrium, which
can ?nd justi?cations based both on normative and descriptive bases. The formal
de?nition is as follows.
Given a game 퐺, a strategy pro?le ¯ 푥 ∈ 푋 is a Nash equilibrium if:
- for every 푖 ∈ 푁, 푓푖(¯ 푥) ≥ 푓푖(푥푖, ¯ 푥−푖) for all 푥푖∈ 푋푖(here 푥−푖denotes the set of
the remaining ?coordinates? of 푥, after having deleted the 푖 − 푡ℎ).
An essential condition behind the use of the Nash equilibrium as a solution for
games in strategic form brings us to consider the so-called ?institutional setting?
under which the game is played: we refer here to the fact that players cannot sign

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binding agreements. This assumption is fundamental in considering the game as
a non cooperative game: if players can sign binding agreements, we say that we
have a cooperative game. Namely, if we assume that agreements are not binding,
one can see the de?nition of Nash equilibrium as some kind of stability condi-
tion for the agreement ¯ 푥: the condition that de?nes a Nash equilibrium amounts
to say that no player has an incentive to unilaterally deviate from the agreement.
What are the Nash equilibria for the two examples introduced before? For the
prisoner's dilemma, the unique Nash equilibrium is (퐶,퐶), which yields a result
that is not e?cient. This surprising fact (players are rational and intelligent!)
has been the reason for a lot of interest about this game, which is by far the
?most famous? one in GT.
For the battle of the sexes there are two Nash equilibria: (푇,퐿) and (퐵,푅).
Not only the battle of the sexes provides an example in which the Nash equilib-
rium is not unique, but it has the additional feature that players have opposite
preferences on these two equilibria: player 퐼 prefers (푇,퐿), while player 퐼퐼 prefers
(퐵,푅). But the most interesting point is that the battle of the sexes shows quite
clearly that a Nash equilibrium is a pro?le of strategies, and that it is (generally)
nonsense to speak of equilibrium strategies for the players: which would be the
equilibrium strategies for player 퐼?
It has to be stressed that the model of a game in strategic form can be used
in principle to represent both cooperative and non-cooperative games, but de
facto it is essentially used only for non-cooperative games. The same thing can
be said of the second basic model used in GT: games in extensive form. We
shall not discuss this model, which is useful to describe interactive situations
in which the ?timing? of the choices made by players is relevant (chess, poker,
English auctions): we shall just mention the fact that it can be considered as
a natural extension of a ?decision tree?. It is also important the fact that a
game in extensive form can be transformed in a canonical way into a game in
strategic form: this fact (already emphasized in von Neumann (1928)) is relevant
in extending the scope of games in strategic form, going beyond the immediate
and trivial interpretation (players decide ?contamporarily?). This is obtained by
means of the idea of ?strategy? as an action plan, allowing to use games in
strategic form o describe also situations in which players are called to make
choices that can be based on the observation of previous choices made by other
players.
So far, we have almost left out the cooperative games. Even if the strategic
and extensive form models can be used, the most common model for cooperative
games is given by the so-called games in ?characteristic form? (also said ?in coali-
tional form?). Such a model provides a description of the interactive situation
which is much less detailed than those provided by the other two models. Apart
from this, it is a model which is ?outcome oriented?, since it displays the ?best?
results that can be achieved (possibly using binding agreements) by coalitions,
i.e., group of players.

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We give here the formal de?nition of a game in characteristic form. To be
more precise, we shall con?ne ourselves to the (simpler) version that is identi?ed
as the ?transferable utility? case.
A game in characteristic form with transferable utility (quickly: a TU-game)
is a couple (푁,푣), where:
- 푁 is a ?nite set (the players)
- 푣 : 풫(푁) → ℝ, with 푣(∅) = 0 is the so-called ?characteristic function?.
Of course, the key ingredient is 푣, the characteristic function, which assigns
to each subset of 푁 (usually referred as a ?coalition? of players), a real number
which has the meaning of describing (in some common scale) the best result that
can be achieved by them.
As it can be clearly seen, there is no room for ?strategies? in this model: the
basic info is the (collective) outcome that can be achieved by each coalition.
Due to the quite di?erent approach w.r.t. the model used for non-cooperative
games, it should not come as a surprise that the ?solutions? available for TU-
games obey di?erent types of conditions, compared to the Nash equilibrium.
Before going to describe the two main solution concepts for TU-games (the core
and the Shapley value), it is worth emphasizing a relevant di?erence between
cooperative and non-cooperative games for what concerns the approach to the
idea of a solution for them. Games in strategic and extensive form, used in the
non-cooperative setting, try to o?er an adequate description of the possibilities
of choice available to the players: this fact allows us to use essentially a basic
principle (that of Nash equilibrium) as the ground for solution concepts1. On
the other hand, the astounding simplicity of a TU-game has a price to be paid:
the huge amount of details which is missing in the model has to be recovered
somehow when we pass to the ?solution? issue. As we shall see, the two main
solutions that we shall describe obey di?erent lines of thought. The overall result
is that we ?nd signi?cantly di?erent groups of solution concepts: to give just
an example, the ?bargaining set?, the ?kernel?, the ?nucleolus? are three di?erent
solutions that all try to incorporate conditions2di?erent from those incorporated
in the core or the Shapley value.
Having said this, let us consider one of the two anticipated solution concepts
for a TU-game: the core. In the reminder of this section we shall assume that
the TU-games that we shall consider satisfy the superadditivity condition3:
1It must be noticed that there are other relevant solutions, apart the Nash equilib-
rium. To quote the most relevant, we can mention: subgame perfect equilibrium,
perfect equilibrium, proper equilibrium, sequential equilibrium. These di?erent solu-
tion concepts try to cope with de?ciencies of the Nash equilibrium, but are no more
that (relevant) variants of the basic idea behind the Nash equilibrium.
2The ?bargaining set? tries to take care of the possibilities of ?bargaining? among
the players, through proposals for an allocation, and the objections and counter
objections that can be made w.r.t. the proposed allocation. The ?kernel? and the
?nucleolus? were introduced having in mind approaches to ?nd allocations belonging
to the bargaining set (which is di?cult to determine).
3It is not an unavoidable condition, but it allows for more ?natural? interpretations
of formal conditions that we shall use.