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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.

1 Introduction

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.

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),

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.