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3Compliance Pervasion and the Evolution of Norms: the Game of Deterrence Approach
Michel Rudnianski1, Huo Su2 and David Ellison3
1 CNAM-IIM & ORT FRANCE E-mail: michel.rudnianski@cnam.fr 2 CNAM-IIM & Universite Paris 1 Pantheon-Sorbonne E-mail: Huo.Su@malix.univ-paris1.fr 3 Ecole Polytechnique E-mail: davidellison@noos.fr
Keywords: compliance, culture, deterrence, evolutionary stability, game, norm, pervasion, playability, posture, strategy.
1. Introduction
It is a well known fact that multiplication of interactions within a given group of individuals generates norms and values, which in turn shape the group’s culture, and hence the behaviors prevalent within that group. Now observation of daily life shows that contradictions might occur between individual behaviors and norms prevailing within the group. Depending on their nature, these contradictions may either foster the group’ dynamic and thus pave the way toward progress, or on the opposite play a negative role in terms of consistency and hence social efficiency. Such will be the case in particular when non compliance with a given legal or regulatory corpus becomes pervasive. Two conflicting sets of norms have then been developed, social practice on the one hand, and the legal and regulatory corpus, the latter being now unfitted to the group or the society1.
Various states of development of the group can be propitious to the emergence of such contradiction. In particular, the pace of globalization and the accelerated rhythm of scientific and technological progress is known to be a source of constant change in individual behaviors as well as in the regulatory and legal corpuses supposed to regulate human societies. The growing complexity of the norms system resulting from these changes may produce inconsistencies and a poor relevance of the regulatory and legal corpus. This may in turn significantly decrease the expected individuals’ degree of compliance . This problematic is epitomized quite well by the fast developing issues pertaining to Intellectual Property Rights.
At the methodological level, cultural dimensions of groups and societies have been the subject of many studies. In particular, interest has been sharply growing for the impact of cultural factors on business management (Trompenaar & Hampden-Turner 1997, Schein 1999, Hofstede.& Hofstede 2004 ): how norms and values carried by social practices within a given culture should be confronted by management in setting its own norms and rules?
On its side, Game Theory has shown interest in cultural issues (Casson 1991, Bed-nar & Page, 2000), and modeled through an evolutionary approach the pervasion of behaviors within a given population. The existing literature contains also many
1 This lack of fitness may have two kinds of sources: either the values carried by the legal
and regulatory corpus are not adequate, or enforcement is inefficient.
papers or books dealing with economic models of Law (Blair & al 2005) and application of game theoretic tools to legal issues (Baird & al 1994).
Leaning partially on these works as well as on an elementary and quite well spread out case of Intellectual Property Rights, the present paper will propose a methodology for analyzing the conditions on the set of norms that should prevail at both levels of social practice and of legal or regulatory corpuses to make compliance pervasive.
More precisely, we shall start from a Basic Copyrights Game confronting the author of a work of mind and an end user of this work, and analyze conditions under which the end user will be compliant with the legal corpus at two different levels.
The first one can be called static / individual, since it considers a single end user facing a single author in a one shot game. Different cases will be considered based on the players perceptions of the various possible states of the world in terms of their acceptability. To take these perceptions into account as well as the standard utilities associated with the possible states of the world, we shall alternatively resort to two types of matrix games: standard Games of Nash and Games of Deterrence, which are qualitative games aiming at analyzing acceptability threshold effects. The second level of analysis will be called dynamic /collective. It will assess the evolution of behaviors within the group through using the results already obtained, in analyzing the strategies properties of a Replicator Dynamic based evolutionary game of deterrence (Ellison & Rudnianski 2006). In particular, depending on the specific structure of the matrix supporting the evolutionary game, and hence on the specific type of graph of deterrence associated with that game, it will be possible to assess pervasion of compliance with norms.
By so doing, the analysis will thus enable to point out how the impact of a specific set of norms on the individual will orientate the evolution of the group.
2. The Static / Individual Level
2.1. The Basic Copyright Game
As already mentioned in the introduction, the Basic Copyright Game describes the interaction between:
• The author of a work of mind
• an end user of the work
The end user can be:
• compliant (C)
• non compliant (NC)
When facing a violation of his / her intellectual property rights, the author can be:
• defensive (D): take the violator to court
• passive (P): do not react
The interaction can then be represented by the following matrix: where:
• a — c represents the loss of value for the author caused by infringement with no response
Author
End user
D P
С (x,a) (x,a)
NC (У,Ь) (z,c)
Fig.l. the Basic Copyrights Game
• b — c represents the loss for the author due to litigation when facing infringement of his / her intellectual property rights
• z — x represents the value of infringement for the end user
• x — y is the penalty for the end user having infringed the author’s intellectual
property rights
Now, looking at the end user’s outcomes, and considering that when facing evidence of no compliance, the court will issue a judgment against the infringer , it is then quite straightforward that z > x > y
If we then look at the author’s outcomes, we can similarly consider that:
• a > b : the author can never benefit from infringement, and even if he / she
obtains compensatory damages and the total reimbursement of litigation cost, time and efforts devoted by the author to manage the issue will never be totally compensated
• a > c
Now what remains to compare is b and c.
Discarding the possibility that b = c as improbable, we have then to consider two possible cases.
The first case, b > c, corresponds to situations where the author’s loss due to infringement is higher than the cost of winning in front of a court.
In such situations the game displays a single (pure) equilibrium: (C,D).
In other words, anticipating that he / she will be taken to court in case of infringement, a rational end user will choose to be compliant.
From the game theoretic point of view the problem is over.
The second case, b < c, corresponds to situations where damages or benefits resulting from an IPR violation of IPR are small compared to litigation costs. This happens for instance when an end user illegally downloads a song or a limited number of music albums on Internet. This case is much more spread out than the first one, and is related to the issue of norms that we wish to address here. Therefore in the sequel we shall always assume that: a > c > b Then the game displays two Nash equilibria: (C, D) and (NC, P).
These two equilibria are diametrically opposed in that sense that each one allocates an opposite strategy to each player.
To discriminate between the two equilibria, several approaches can be envisaged. One is the classical Selten’s sub-game perfectness. We shall come back to it later. Another one which we shall consider here is based on the concept of threshold that we shall now introduce.
2.2. Acceptability thresholds
In daily life, decision makers usually do not resort to optimization tools to make their choices. There are a number of reasons for that: they do not master the tools,
they lack information, time etc. Nevertheless, ’’rational” decision makers need to base their choices on some bounded rationality principles which they can use as discriminating tools. A quite spread out principle is acceptability: a decision maker will select a decision if he / she considers the outcome of that decision acceptable to him/her.
This amounts to considering that for each player there are only two possible outcomes:
• Acceptable outcome noted 1
• Unacceptable outcomes noted 0
A rational player will of course look for an acceptable outcome.
In the Basic Copyrights Game, for each player, the four possible states of the world are associated with three ordered outcomes. It follows that there are four possibilities to position the acceptability threshold of each player. A particular acceptability threshold will establish a correspondence (consistent with the outcomes ordering) between the set of possible outcomes (or the set of states of the world associated with it) and {0,1}.
Thus, the four possible acceptability thresholds for each player are:
• Threshold 1: all outcomes are unacceptable
• Threshold 2: all outcomes except for the best one are unacceptable
• Threshold 3: all outcomes except for the worst one are acceptable
• Threshold 4: all outcomes are acceptable.
It follows that the Basic Copyrights Game can be associated with 16 possible
binary games, that we shall call games of deterrence for reasons that will appear
later. A game characterized by threshold i for the end user and threshold j for the author will be denoted by (i,j)
All 16 game matrices are given in the appendix.
Switching from the Basic Copyright Game to anyone of these binary games changes the player’s perspective and hence may constrain his/her choice.
2.3. Strategies playability and deterrence
Let us consider for instance the binary game such that
• for the end user, only being taken to court is unacceptable
• for the author, supposedly rich and powerful, one all states of the world are acceptable
This means that the binary game is (3,4). The corresponding game matrix is:
Strategies D and P are equivalent and always guarantee the author an acceptable outcome (i.e. whatever the end user’s choice). They will therefore be termed safe. The same goes with strategy C of the end user.
Now it is just common sense that in such a binary game, there is no reason for
Author
End user
D P
c (1,1) (1,1)
NC (0,1) (1,1)
Fig.2. Game (3,4)
which a player should not play a safe strategy. Therefore the latter will be considered playable.
By contrast resorting to strategy NC may lead to an unacceptable outcome for the end user if the author plays D. NC will therefore be termed dangerous. Moreover the end user cannot exclude that the author might play D, while on the other hand the former has a playable strategy. Therefore strategy NC should be considered not playable.
Now one should not consider that only safe strategies are playable while all dangerous strategies are not playable. Le us for instance consider game (4,2) which corresponds to a situation in which:
- all states of the world are acceptable for the end user
- the author considers that non compliance is always unacceptable
The game matrix is:
Author
End user
D P
c (1,1) (1,1)
NC (1,0) (1,0)
Fig.3. Game (4,2)
The two strategies of the end user are safe, while the two strategies of the author are dangerous. Indeed the author cannot exclude that the end user will decide to be non compliant, in which case, whatever the author’s choice, the latter outcome will be unacceptable. Nevertheless, in a game a player must play. Therefore, given the fact that his / her two strategies are equivalent, the author can play any one of them, which in turn are playable.
Now it is obvious that playability in this second example is of a different nature than playability in the first one: in the first example a strategy was playable because it was a ”good” strategy, while in the second example the author’s strategies are playable, only because the author has no better choice.
We shall therefore distinguish between those two types of playability.
The first one is will be termed positive playability while the second one will be termed playability by default. So a strategy will be termed playable if it is either positively playable or playable by default.
Now the reasoning developed in the framework of the two previous examples shows that the playability properties of a player’s given strategy might depend on the playability of the other player’s strategies.
More precisely, given a strategy $ of the end user and a strategy ^ of the author, ^ is said to be deterrent vis--vis $ iff:
1. 0 is playable (positively or by default)
2. the outcome of implementing ($,■0) is unacceptable for the end user
3. the end user has another strategy $'which is positively playable .
Thus in the first example here above, the author’s defensive strategy D is deterrent vis--vis the non compliance strategy NC of the end user.
2.4. Games of deterrence solutions and equilibria
The two concepts of playability have been introduced here above in a quite intuitive manner. To avoid any ambiguity, the above definitions can be generalized and stated more formally. Let SE (card SE = n), and Sa (card Sa = p) be the strategic sets of the end user and the author respectively.
We can then consider finite bi-matrix games (SE, Sa , Me, Ma, S) in normal form where possible outcomes are taken from the set {0,1}.
For any strategic pair ($,0) € SE x Sa
- Me ($, 0) = 1 defines an acceptable outcome for the end user
- Me ($, 0) = 0 defines an unacceptable outcome for the end user.
Similar definitions apply by analogy to the author with matrix Ma.
A strategy $ of the end user is said to be safe iff V0 € Sa, Me($, 0) = 1 A non-safe strategy is said to be dangerous.
Let J($) be an index called index of positive playability, such that :
If $ is safe then J($) = 1 If not
J($) = n t1 - (1 - ME($,0)J($))](1 - )(1 - jR)
PESa
with
jE = Yi(1 - J(0)); = n (1 - J($))
PESe 0GSa
If J($) = 1, strategy $ € SE is said to be positively playable.
If there are no positively playable strategies in SE, that is if jE = 1, all strategies $ € Se are said to be playable by default.
Similar definitions apply by analogy to strategies 0 of Sa .
A strategy s € Se U Sa is playable iff it is either positively playable or playable by default.
The system S of all J($),$ € SE,andJ(0),0 € Sa, is called the playability system of the game.
A solution of S is a consistent set
{J($1), J($2)...J($n), J(01), J(02)...J(0n)}
In the general case, there is no uniqueness of the solution.
Consider for instance game (3,3) where the court case is the only state of the world unacceptable for both players. One can easily establish that the game displays two solutions:
{J (C) = 1, J (NC) = 1,J (D) = 0, J (P) = 1}
• {J (C) = 1, J (NC) = 0, J (D) = 1, J (P) = 1}
A strategic pair ($,0) € SE x Sa is said to be an equilibrium of this game if both strategies are playable for some solution of the playability system.
In game (3,3) each solution has two equilibria:
• (C,P) and (NC,P) for the first solution
• (C,P) and (C,D) for the second solution
We see that these two solutions differ by the two strategic pairs which are Nash equilibria in the standard game.
As far as the safe strategies are concerned we see that these strategies remain safe and hence positively playable, whatever the solution under consideration.
2.5. Graphs of deterrence
Given a game of deterrence (SE, Sa, Me, Ma, S), we shall call graph of deterrence, a bipartite graph G on SE x Sa such that given ($, 0) € SE x Sa, there is an arc of origin $ (resp.0),and extremity 0 (resp.$), iff Ma($,0) = 0, (resp.ME($,0) = 0) Solving the playability system S amounts to determining playabilities of the graph vertices. Since a graph can be broken down into paths and circuits, we shall call:
- E-path (resp. A-path) a path the root of which is an element of SE (resp. of
Sa);
- r-graph, a graph, that includes neither an E-path nor an A-path.
It has been shown (Rudnianski,1991) that :
1. if G is an E-path (resp. A-path), the only positively playable strategy for the end user (resp the author) is the root, while all strategies of the author (resp. of the end user) are playable by default;
2. if G is a primary circuit, all strategies of both players are playable by default;
Moreover, it has been shown (ibid) that through appropriate cuts, it is always possible to break down the graph of deterrence into connected parts, each one being an E-path, an A-path, or a r-graph. Hence, depending on the presence of these elementary components in the graph, one can distinguish between 7 types of games : type E, type A, type r, type E-A, type E-r, type A-r, type E-A-r.
It has also been shown [ibid] that :
• if G is an E-path, the only positively playable strategy for the end user is the root, while all strategies of the author are playable by default;
• if G is a primary circuit, all strategies of both players are playable by default;
• if G is a r-graph,a solution of S satisfies :
i. for any strategy so,
J (so) = -[jE V ja] A3s € N (so) : J (s) A Vs' € N' (so) : [-J (s')]
Where N (so) (resp. N '(so)) is the set of the first strategies met when following G backward from so and belonging to the same strategic set as so (resp. to the other);
ii. on a path, the vertices positive playability is determined by the parity of their distance to the origin of the path;
iii. each player has at least one non positively playable strategy.
This typology leads in turn to the Classification Theorem (ibid) :
(1) Given a game of deterrence, its playability system’s solution set is not empty.
(2) The game type defines the properties of the solution set.
It follows from (1) that every game of deterrence has an equilibrium, but the above shows that this equilibrium may not be unique.
The graph of deterrence associated with game (3,4) is:
C P D ^ NC
So (3,4) is an E-A type game.
Likewise the graph of deterrence associated with game (4,2) is:
C D ^ NC ^ P
So (4,2) is an E-type game.
Last in the case of game (3,3), the graph of deterrence is :
C P D ^ NC Hence, (3,3) is an E-A-r type game.
2.6. The effect of acceptability thresholds
It stems from the above discussions that acceptability thresholds impact the game type and hence the solutions and equilibria associated with that game. As already stated, with the Basic Copyrights Game there are 4 possible acceptability thresholds per player, and hence 16 possible games of deterrence. One can then systematize the analysis conducted above on three of these 16 games, and associate with each one its solution set. The results of this analysis can be summarized on the following 4x4 table in which:
• pp means positively playable
• pd means playable by default
• np means not playable
Now the conclusions given by the table in each of the 16 cases still need to be translated into strategic choices.
One can reasonably assume that:
• If nothing differentiates the two strategies of a player in a given game of deterrence, the player will then proceed to a random choice.
• In case of a game with several solutions, a strategy playable in all solutions should be preferred to one which is not playable in some of the solutions.
• if for any solution of the game the positive playability index of a player’s strategy is superior or equal to the positive playability index of another strategy of the same player, then the latter will select the first rather than the second.
Author
End User
1 2 3 4
1 • All strategies pd • All strategies pd • C & NC pd • P safe, D np • C & NC pd • P & D safe
2 • All strategies pd • All strategies pd • C np & NC pp or C & NC pd • P safe & D np • C & NC pd • P & D safe
3 • C safe, NC np • P & D pd • C safe & NC np • P & D pp or P & D pd • C & P safe • NC pp & D np or NC np & D pp • C safe & NC np • P & D safe
4 • C & NC safe • P & D pd • C & NC safe • P & D pd • C & NC safe • P safe & D np • All strategies safe
Fig.4. Acceptability thresholds and game solutions
Author
End User
1 2 3 4
1 • Random choice for both players • Random choice for both players • End user : random choice • Author : P • Random choice for both players
2 • Random choice for both players • Random choice for both players • End user : NC • Author : P • Random choice for both players
3 • End user : C • Author : random choice • End user : C • Author : random choice • End user : C • Author : P • End user: C • Author : random choice
4 • Random choice for both players • Random choice for both players • End user : random choice • Author : P • Random choice for both players
Fig.5. Acceptability thresholds and strategic choices
On the basis of these assumptions the players’ strategic choices are given by the table of Fig.5:
It stems straightforwardly from the above table that compliance is granted only for those end users whose acceptability threshold is 3, according to which the only unacceptable state of the world for the end user is the one corresponding to a court case.
It is noticeable that this conclusion is valid whatever the author’s acceptability threshold, and in particular even in those cases where a court case is unacceptable for the author himself / herself.
In other words, an end user whose acceptability threshold is 3 will not look at the acceptability thresholds of the author. This means in turn that from a legal point of view, how the law treats the author doesn’t matter at all. What counts is how the law treats the non compliant end user.
This observation might look weird, with regard to the Basic Copyrights Game model, since in the latter - as under many legal corpuses, one could assume that there is no disconnection in a court case within the fate of the infringer (end user) and the one of the infringement’s victim (the author). In other words, what the second gets should depend on what the first one has to pay. So it seems at first glance that the outcomes of the two parties are intertwined.
But in reality, there are two reasons at least for which the situations of the two parties can be assessed separately.
The first one, and the most common, simply stems from the fact that outcomes and utilities are two different categories. Just as two end users having two different acceptability thresholds, will have different perceptions of at least one state of the world, there is no rationale for pretending that an end user and an author will always have identical assessments of a given state of the world.
The second reason lies in the mechanism of penalties and damages, which should be considered as two different dimensions of legal provisions associated with an Intellectual Property Rights court case. One can imagine a legal corpus that will not strictly compensate the author for his / her loss due to infringement, while inflicting the infringer a penalty much higher than the loss supported by the author, with the objective to strengthen deterrence of non compliance. Whether this objective can be reached or not is an issue that will be addressed in the sequel. Identical conclusions can be drawn from the above table if one switches now from the end user to the author. Indeed, for authors which acceptability threshold is 3 (according to which the only unacceptable state of the world is the court case), being passive will be the only rational choice. The table shows that this choice doesn’t depend on the end user’s particular acceptability threshold. In other words, a frightful author will always remain passive. This will occur in particular if the end user’s acceptability threshold is 2 (all states of the world, except infringement with no court case, are unacceptable). In such case, the end user will be non compliant and the author will not react.
2.7. Acceptability thresholds and game type
We have seen that each matrix game of deterrence can be associated in a one to one mapping with a graph of deterrence, which in turn defines the game type. Let us now analyze the relation between game types and compliance.
To make compliance compulsory, it is necessary that simultaneously strategy C be playable and strategy NC be non playable. This implies that C must be positively playable. Now as compliance should occur whatever the author’s acceptability threshold, C must be safe. Hence the game type may only be one among the following four: E, E-A, E-r , E-A-r.
If NC is a root, then NC is safe and constitutes a playable alternative to C, in which case compliance is not compulsory.
If the game-type is E, NC belongs to an E-path while all the author’s strategies are playable by default.
If NC is a root, then NC is safe and constitutes a playable alternative to C, in which case compliance is not compulsory.
Let us then assume that NC is not a root. This means that NC is a successor of an author’s strategy.
Given that the interaction is modeled by a 2 x 2 game, one can easily see that there are only four possible graphs of deterrence for the above condition to occur. These
are:
1. C - P- NC - D
2. C - D- NC - P
3. C P NC
I
D
4. C — D — NC
I
P
Each of the games associated with the above graphs of deterrence displays contradictions with the characteristics of the Basic Copyrights Game. Thus:
1. In the first game, the strategic pair (C,P) implies an unacceptable outcome for the author, while for the latter non compliance with no response is acceptable
2. In the second game the strategic pair (C,P) implies an unacceptable outcome for the author, on the opposite of (NC,P). Likewise going to court is acceptable for the author in case of non compliance while being passive is not
3. In the third game, non compliance is always acceptable for the author, while compliance is always unacceptable. For the end user court case is acceptable, while infringement with no response from the author is unacceptable.
4. In the fourth game non compliance is always acceptable for the author, while compliance is always unacceptable
It follows that the case of an E-type game in which NC is not playable can be discarded.
If we then look at the three remaining candidate game types E-r, E-A-r, and E-A,we can easily show that these types do not contradict the characteristics of the Basic Copyrights Game.
On the whole, hence compulsory compliance can occur only in the framework of E-r, E-A-r, and E-A type games.2
Now, one can generalize the mapping already realized in section 2.5 above between
2 This doesn’t mean that as soon as the game type is one of the three above compliance is compulsory. Consider for instance E-A-r. The only Basic Copyrights game of this
three of the 16 possible games of deterrence deriving from the Basic Copyrights Game and their graphs of deterrence, and hence their game type.
This will enable to assess how acceptability thresholds affect game types, and then in turn how these game types can be related to compliance or non compliance.
One can easily show that this mapping is the one given in the table hereunder:
Author
End User
1 2 3 4
1 • T-type • T-type • A-T-type • A-type
2 • T-type • T-type • A-T-type • A-type
3 • E-T-type • E-T-type • E-A-T-type • E-A-type
4 • E-type • E-type • E-A-type • E-A-type
Fig.6. Acceptability thresholds and game type
So we see here again that acceptability threshold 3 for the end user implies three possible game types E-r , E-A-r , and E-A which occurrence depends on the acceptability thresholds of the author:
• for thresholds 1 and 2 of the latter the game type is E-r
• for threshold 3 the game type is E-A-r
• for threshold 4 the game type is E-A.
Observation of the table shows that:
• All 7 possible types of games are represented.
• Game types E-r and E-A-r occur only when the end user’s threshold is 3.
• Game type E-A may occur when the end user’s threshold is 3 or 4.
Now one may ask to what extent the conclusions here above depend on the specific assumption made on the Basic Copyrights Game, namely that b < c.
So let us momentarily assume that the opposite is true, i.e. that b > c, and look again for conditions under which compliance of the end user is compulsory.
The reasoning made here above can be developed again.
First, for compliance to take place whatever the author’s acceptability threshold,
type is game (3,3) which displays two solutions: one for which NC is not playable and
one for which NC is positively playable. In the latter case, compliance of course is not
compulsory, while it is in the first one.
it is necessary that C be safe, which implies game types E, E-r , E-A-r , and E-A. Secondly, examination of the four possible games associated with game type E leads to the same conclusion: each of these games contradicts some characteristics of the Basic Copyrights Game and hence all should be discarded.
So what is left to examine is the distribution of game types according to the author’s acceptability threshold.
It is straightforward that for author’s thresholds 1 and 4, for which all outcomes are equivalent, nothing is changed. One can also easily see that nothing is changed for threshold 2, since in that case only the best outcome a is acceptable. The only change will occur for author’s threshold 3. In this case the game type is E-A just like in the case where b < c. But the difference with the latter case is that now the game displays a single solution for which NC is not playable.
2.8. From acceptability thresholds to postures
If compliance is compulsory for those end users whose acceptability threshold is
3, nevertheless the problem of ensuring compliance is not yet solved. Indeed acceptability threshold is conveyed by the player’s perception of the various states of the world. This perception can a priori be considered as ”built in” the player, and therefore not susceptible to be changed, at least in the interaction framework envisaged till now.
More precisely a player’s perception includes several dimensions: the norms and values of the society including those which are carried by the legal and regulatory corpuses, possibly those of the group to which he / she belongs (in the present case end users or authors), his/her personal values, previous experience and personal situation.
It follows that a change in acceptability threshold at the individual level may happen only if some factors associated with one of the dimensions mentioned here above are modified.
This could be the case for instance if for the end user the work of mind’s value is small, or on the opposite big, or if a change occurs in the legal or regulatory corpus, making penalties for infringement smaller or bigger.
At the individual level we may nevertheless modify the interaction framework in order to reintroduce the author as a tool which may help to change the end user’s perception. More precisely, we shall successively consider two possibilities:
- Mere communication
- Strategic postures
By mere communication we mean that the author’s acceptability threshold is static, and that he / she will try to communicate this threshold to the end user, in the hope that the latter’s perception will change if necessary, and eventually be defined by acceptability threshold 3.
To understand how such mechanism could possibly work, we first have to come back to the concept of acceptability. Till now it has been considered that the acceptability of each state of the world was assessed independently, i.e. according to criteria which did not take other states of the world into account.
Now in real life, this may often not be the case. Facing two unpleasant states of the world resulting each one from a different decision, an actor could either consider both as unacceptable and then select his/her decision randomly (as it has been the
case for the author in game (4,2)), or reintroduce a hierarchy between those two states of the world, by considering that the least unpleasant state becomes relatively acceptable while the most unpleasant remains unacceptable.
The same thing can be said if instead of considering two unpleasant states of the world, the actor would on the opposite consider two appealing ones. He/she could either take his/her decision randomly as it was the case for the author in game (3,4) or order the two sates of the world, and consider that the most appealing is acceptable while the least appealing is unacceptable.
At first glance, this approach seems to contradict the process followed till now, and which lead through a dichotomization of the set of states of the world, from the cardinal utilities of the Basic Copyrights Game to the corresponding cluster of games of deterrence. But in reality there is no contradiction for two related reasons. First, utilities ordering can be achieved through a repeated dichotomization process, by which at each stage one takes away the most (un)pleasant element(s) from the set of states of the world. This amounts to enrich the existing ordering. The process is repeated till no distinction can be made between the remaining states of the world. Secondly in the present case where we consider acceptability thresholds, the above developments have shown that the particular positioning of a player’s acceptability threshold impacts his /her outcome in the Basic Copyrights Game, which is after all what he/she will get after interaction with the other player has taken place.
In that respect, for the end user, selecting acceptability threshold 3 will result in an outcome of x, which is the second best, while selecting acceptability threshold 1 or 4 for instance generates a risk of getting y which is the worst outcome, as soon as the author’s acceptability threshold is not 3.
So if we suppose momentarily that the end user is risk averse, and if the acceptability threshold of the author is not 3, it is in the interest of the latter to let the end user know about it.
Now nothing guarantees that such communication is efficient, for instance in the case where the end user is not risk averse. It might then be necessary to go one step further and consider that the players acceptability thresholds are not simply perceptions, but strategic tools which they can use in order to defend their interests. For this reason acceptability thresholds will now become postures. Such postures are of current use in many areas of real life, where they can support threats or negotiations, in the field of international relation as well as in the fields of business and social affairs.
Consider for instance an end user who looks for optimizing his/her expected value in the Basic Copyrights Game. What posture should he/she adopt?
To answer that question it is necessary to associate with each pair of acceptability thresholds the pair of expected utilities that both players would get in the Basic Copyrights Game. This will define a postures meta-game which matrix is:
Michel Rudnianski, Huo Su and David Ellison Author
End User
1 2 3 4
1 • (x/2 + y/A + • (x/2 + y/A + • {x/2 + z/2, • (x/2 + y/A +
z/4, a/2 + z/4, a/2 + a/2 + c/2) z/4, a/2 +
6/4 + c/4) 6/4 + c/4) 6/4 + c/4)
2 • (x/2 + y/A + • (x/2 + y/A + • (z,c) • (x/2 + y/A +
z/4, a/2 + z/4, a/2 + z/4, a/2 +
6/4 + c/4) 6/4 + c/4) 6/4 + c/4)
3 • (x, a) • (x, a) • (x, a) • (a:, a)
4 • (x/2 + y/A + • (x/2 + y/A + • {x/2 + z/2, • (x/2 + y/A +
z/4, a/2 + z/A,a/2 + a/2 + c/2) zj4, a/2 +
6/4 + c/4) 6/4 + c/4) 6/4 + c/4)
Fig.7. The postures meta-game
Four cases must a priori be distinguished depending on the relative utilities :
• For the end user the relative utilities of:
- the expected utility of compliance: x
- the expected utility of non compliance: y/2 + z/2
• For the author the relative utilities of:
- the expected utility of a random choice by the two players: a/2+b/4+c/4
- the utility associated with an infringement followed by no judicial procedure: c
Let us first consider the end user. Given the assumption already made in the Basic Copyrights Game, one can reasonably infer that the net value z — x generated by infringement is smaller than the loss x-y generated by the penalties and litigation costs that the end user would have to pay if a court case is to take place. In other words: 2x > y + z.
It then stems from this condition and the general assumptions made on the players utilities (z > x > y and a > c > b) that the strategic pairs (3,1), (3,2) and (3,4) are Nash equilibria.
Let us then switch to the author.
If the author’s utility associated with an infringement followed by no reaction is greater than his / her expected utility associated with a random choice by the two players (in other words if c > a/2 + b/4 + c/4), then the strategic pair (2,3) is also a Nash equilibrium, and hence the postures meta-game displays four Nash equilibria: (2,3), (3,1), (3,2) and (3,4).
If c < a/2 + b/4 + c/4, then the postures meta-game displays only three Nash equilibria: (3,1), (3,2) and (3,4)
Now, coming back to Figure 5, these results can be interpreted as follows.
If the author’s utility associated with an infringement followed by no reaction is greater than his / her expected utility associated with a random choice by the two players, then the end user might adopt as a Nash strategy either posture 2 or posture 3.
In the first case Figure 5 shows that the Nash equilibrium of the postures metagame corresponds to the strategic pair such that the end user infringes and the author remains passive.
On the opposite, if the author’s utility associated with an infringement followed by no reaction is smaller than his / her expected utility associated with a random choice by the two players, the end user will necessarily adopt posture 3 as a Nash strategy. The Nash equilibrium in the postures meta-game then corresponds to the strategic pair such that the end user remains compliant while the author plays randomly.
Last, one can see on the matrix that postures 1 and 4 of the end users are equivalent and - weakly - dominated by posture 2.
Likewise postures 1, 2 and 4 of the author are equivalent.
2.9. The Basic Copyrights Game and the Chain Store Paradox
Selten’s Chain Store Paradox deals with a chain store that faces successively the threat of entry of q potential competitors in n different cities. In each city, the potential competitor can either enter the market (strategy E) or not enter (strategy NE), while the chain store can accept (strategy A) the entry or refuse it (strategy R).
At the level of a single city, if the potential competitor doesn’t enter whether the chain store accepts or refuses the result is the same: the competitor keeps the whole market in the city under consideration.
If the potential competitor enters the market and the chain store accepts this entry, both parties share the existing city market. If the chain store refuses the entry, then both parties engage a costly price war.
The game can be represented by the following matrix:
Chain Store
R A
NE (0,2) (0,2)
E (-1,-1) (1,1)
Fig.8. The Chain Store Paradox
The matrix game displays two Nash equilibria: (NE, R) and (E,A)
The issue dealt with by the model is twofold:
1) Is is possible to find a refinement of the Nash equilibrium that enables a preference order between the two equilibria?
2) By refusing the entry of competitors in the early stages of the game, can the chain store build a reputation effect that will deter subsequent potential competitors to enter?
Through introducing the concept of sub-game perfectness and using backward induction, Selten shows that there is only one sub-game perfect equilibrium (E,A), and no reputation effect can deter potential competitors to enter the market.
Now, the analysis developed above shows that the Chain Store Paradox model is identical to the Basic Copyrights Game.
The chain store and the competitor in the former model correspond respectively to the author and the end user in the latter. Likewise the chain store strategies A and R correspond to the author’s strategies P and D respectively, while the competitor’s strategies E and NE correspond to the end user’s strategies C and NC respectively.
Moreover the matrix structure is exactly the same in the two games, and the Nash equilibria in the Chain Store Paradox correspond exactly to the Nash equilibria in the Basic Copyrights Game.
Nevertheless the conclusions obtained about the Basic Copyrights Game through the process developed above, do not coincide entirely with the conclusions obtained by Selten.
Sometimes deterrence doesn’t work, but sometimes it does. All depends on the players’ acceptability thresholds (which are not taken into account in the Chain Store Paradox which considers only utilities). Applying the results of the Basic Copyrights Game analysis leads to the conclusion in the Chain Store Paradox that if the competitor’s acceptability threshold is 3, he / she will not enter the market. This conclusion just meets common sense since it means that if price war is the only state which is unacceptable for the competitor, the latter, by deciding not to enter can guarantee himself / herself an acceptable outcome.
On the whole, one can consider that introducing acceptability thresholds through players’ perceptions or postures enables to find solutions that may eliminate the paradox.
3. The Dynamic / Collective Case
3.1. From perceptions and postures to evolution of norms and values
The Basic Copyrights Game can be considered as a metaphor of interactions between two groups of individuals within a given population: the group of end users on the one hand, and the group of authors on the other.
Although it is obvious that inside a group two individuals may differ to the point that they will have different perceptions or adopt different postures, one may nevertheless consider that sometimes individuals’ behaviors are driven by the norms and values of the society to which these individuals belong.
These norms and values, which define what is considered acceptable and what is not, are a product of the society’s culture, but they are also strongly and directly influenced by the legal and regulatory corpuses that proceed - partially at least
- from that culture. In that respect and with regard to the issue of compliance pervasion, one can consider that the two players of the Basic Copyrights Game play asymmetric roles, in the sense that authors may represent also the interests of the whole society. The legal and regulatory corpus through encouraging or refraining the authors to go to court in case of infringement will therefore shape the contours of what the society considers as a fair balance between pervasion of innovation on the one hand and pervasion of compliance on the other (see Blair and al).
If we now look at the individual level, the norms and values (and more particularly the acceptability thresholds which are considered here) that command behaviors have different sources, which act as many layers of the individual’s culture: norms and values of the whole society, or of the particular group to which the individual belongs (end users or authors), experience of the individual . . .
As the aim of the present paper is to introduce a methodology, in the sequel we shall not dwell into details, and just consider for the sake of simplicity that the perception or the posture of a given party (end user or author) is a feature of the group to which this party belongs.
On this basis, we shall successively analyze how compliance pervasion evolves for various pairs of acceptability thresholds, and then analyze in a particular case how norms and values conveyed by these acceptability thresholds can themselves evolve.
3.2. Norms and evolution of behaviors
To model the evolution of compliance in a simple way we shall resort to the classical Replicator Dynamic.
One can easily establish that the system of equations of the latter associated with the Basic Copyrights Game is:
(1 — )[x — z + (z — y)]
$D' = $D (1 — $D )(1 — )(b — c)
Where:
• represents the proportion of the end users who are compliant
• 0D represents the proportion of the authors who are defensive
One can notice that:
• The proportion of end users who are compliant remains constant if one of the following conditions is satisfied:
- The end users group is totally homogeneous (i.e. all end users are either compliant or non compliant)
- The proportion of defensive authors equals the ratio between the value of infringement for the end user and the sum of that value and of the penalty if there is a court case
Furthermore compliance increases as soon as the end users group is not homogeneous and the above ratio is positive.
• The proportion of authors who are defensive remains constant if one of the following conditions is satisfied:
- The authors group is totally homogeneous
- All end users are compliant
- The authors do not either gain or lose anything in litigation (it has already been mentioned that such case is quite improbable)
Furthermore, under the conditions stated in section 2.1 (c > b), if the author’s group is not totally homogeneous, the proportion of defensive authors decreases.
If we then take acceptability thresholds into consideration by switching to the game of deterrence representation, the following table gives the corresponding system of equations for each pair of norms:
Author
End User
1 2 3 4
1 • 0'c/0c = 0 • 9'D/9D = 0 • 0'c/0c = 0 • 9'D/9D = 0 • 0'c/0c = 0 • 9'd/9d = -(1-0D)(1- 9c) • 0'c/0c = 0 • 9'd/9d = 0
2 • 9'c/9c = -(1-0D)(1- ec) • 0'D/0D = 0 • 9'c/9c = -(1-0D)(1- ec) • 0'D/0D = 0 • 9'c/9c = -(1-0D)(1-9c) • 9'd/9d = — (1 —6»z>)(l — 9c) • 0'c/9c = -(1-0D)(1- 9 c) • 9'd/9d = 0
3 • 9'c/9c = 9d(1 — 8c) • 6'd/6d = 0 • 9'c/9c = 9d(1 — 6 c) • 6'd/6d = 0 • 9'c/9c = 9d(1 — 0 c) • 9'd/9d = -{1-9d){1-9c) • 0'c/9c = 9d(1 — 0 c) • 9'd/9d = 0
4 • 0'c/0c = 0 • 9'd/9d = 0 • 0'c/0c = 0 • 9'D/9D = 0 • O'c/Qc = 0 • 9'd/9d = -(1-0D)(1- 9c) • 0'c/0c = 0 • 9'd/9d = 0
Fig.9. Norms and the Replicator Dynamic
The above table shows that the 16 pairs of norms and the corresponding games can be gathered in six categories associated each one with a specific dynamic of the parties’ behaviors / strategies:
• Category 1 includes games (1,1), (1,2), (1,4), (4,1), (4,2), (4,4).
Whatever the game in this category, the initial profile of each one of the two groups (end users and authors) remains the same: there is no evolution. As far as games (1,1), (1,4), (4,1) and (4,4) are concerned, the interpretation is quite obvious. In each one of these games, all four possible states of the world are considered equivalent by the end users’ group as well as by the authors’ group. So there is no rationale to change behaviors and hence no evolution.
If we then look at games (1,2) and (4,2), while end users still consider all possible states of the world as equivalent, this is no more the case for the authors, who consider that there are two acceptable states of the world, namely those associated with end users’ compliance. So, one could expect some evolution in the authors’ behavior. That this evolution doesn’t occur can be explained by the fact that in both games by the two strategies of the authors are equivalent, and hence there is no rationale for an author to change.
• Category 2 includes games (1,3) and (4,3).
For the reasons just mentioned, in both games the end users have no reason to modify their behaviors whatever change may occur in the authors’ behaviors. Now as far as the latter are concerned, the above table shows that two main cases should be distinguished:
- If the authors’ group is totally homogeneous, or if all end users are compliant, then the corresponding equation shows there is no evolution in the authors’ behavior.
* This conclusion can be understood quite straightforwardly in the case where all end users are compliant: indeed as the users’ group profile remains the same whatever the authors’ group profile, and both authors’ strategies lead to the same outcome in this case, there is no rationale for the authors to change.
* when the authors’ group is totally homogeneous, the conclusion stems in particular from the features of the Replicator Dynamic according to which species’ reproduction requires that the species
exists 3 .
- If the authors’ group is not totally homogeneous and the end users’ group is not totally compliant, the above equations show that the proportion of defensive authors decreases with time, which can be understood since the passive strategy is -weakly- dominant in both games.
• Category 3 includes games (2,1), (2,2), (2,4).
In those three games the authors’ profile remains constant which stems from
3 This particular feature may be considered a weakness of the Replicator Dynamic, since it cannot take into account evolutions during which new species appear. Nevertheless, as it suffices that one unit of a different species be present in the group scrutinized, one can reasonably assume there is a high probability that such condition is satisfied when the group under consideration is large, for instance at the national level which is the one at which one should envisage the impact of legal corpuses.
the fact that in these three games authors consider equally unacceptable (games
(2,1) and (2,2)) or acceptable (game (2,4)) the consequences of being passive or defensive when facing an infringement of their intellectual property rights.
As far as the end users are concerned, conclusions are similar to those obtained for the authors in category 2. In particular, if the end user’s group is not homogeneous and not all authors are defensive, the proportion of compliant end users decreases
• Category 4 includes games (3,1), (3,2), (3,4).
The profile of the authors’ group remains constant for the same reasons than in category 3.
Regarding the end users, again two main cases should be considered:
- In the first one, either the end users’ group is totally homogeneous, or all authors are passive, in which case the end users’ group profile remains constant.
If the end users’ group is totally homogeneous, the fact that no change occurs stems from the features of the Replicator Dynamic, while if all authors are passive the conclusion stems from the fact that either end user’s strategy leads to the same outcome
- If the end users’ group is not totally homogeneous, and not all authors are passive, the proportion of compliant end users increases, which can be explained by the fact that for the three games under consideration the compliance strategy is -weakly- dominant.
• Category 5 is comprised of a single game: (2,3).
If the end users’ group is totally homogeneous or if all authors are defensive, the end users’ group profile remains constant, which in the latter case can be explained by the fact that both end users strategies lead to the same outcome. If not, the proportion of compliant end users decreases, which can be explained by the fact that compliance is a -weakly - dominated strategy.
Likewise if the authors’ group is totally homogeneous or if all end users are compliant, the authors’ group profile remains constant, which in the latter case stems from the fact that both authors’ strategies lead to the same outcome.
If not, the proportion of defensive authors decreases, which can be explained by the fact that being defensive is a - weakly - dominant strategy.
So in this game the evolution leads to a situation where authors will be passive while end users will be non compliant.
• Category 6 is comprised of a single game: (3,3).
For the end users, the situation is the same than in category 4, while for the authors it is the same than in category 5.
This implies an evolution toward compliance for the end users and toward being defensive for the authors.
On the whole, discarding for the sake of simplicity the case of initially homogeneous groups, the following table summarizes how norms impact the evolution of behaviors:
Author
End User
1 2 3 4
1 • Both profiles constant • Both profiles constant • Users profile constant • Defense decreases • Both profiles constant
2 • Compliance decreases • Authors profile constant • Compliance decreases • Authors profile constant • Compliance decreases • Defense decreases • Compliance decreases • Authors profile constant
3 • Compliance increases • Authors profile constant • Compliance increases • Authors profile constant • Compliance increases • Defense decreases • Compliance increases • Authors profile constant
4 • Both profiles constant • Both profiles constant • Users profile constant • Defense decreases • Both profiles constant
Fig.10. The impact of norms on behaviors evolution
We see that evolution toward compliance is fostered when end users adopt the set of norms 3, while on the opposite it is unfavoured when the latter adopt the set of norms 2. Sets of norms 1 and 4 are neutral with respect to that evolution.
If we then turn toward the authors, we see that sets of norms 1, 2 and 4 are neutral with respect to evolution toward a defensive attitude, while set of norms 3 is favorable to an evolution toward a passive attitude.
It follows that if non compliance and defensiveness are considered as representing a form of aggressive behaviour, the game is asymmetric: on the one hand non compliance may increase, be constant or decrease, while on the other defensiveness can only be constant or decrease. The reason for such an asymmetry stems from the features of the Basic Copyrights Game: if the end user may benefit or lose from being non compliant with respect to an initial situation where he / she was compliant, the author can never benefit from being defensive with respect to the same initial situation.
Moreover, the above table shows that compliance evolution doesn’t depend on the set of norms of the authors group, and likewise that evolution of defensiveness doesn’t depend on the norms prevailing within the users group.
These conclusions are fully consistent with the ones found in the analysis of the individual / static case, of which they can be considered an extension.
3.3. The norms meta-game
The concept of posture introduced in the analysis of the individual / static case was used to model the consequences of the end user’s and author’s acceptability thresholds. The same process can be developed with respect to norms. But now the outcomes associated with a pair of norms are not the outcomes of the Basic Copyrights Game associated with the particular game of deterrence which these norms define, but the outcomes of the Basic Copyrights Game that would result from the evolution of behaviors triggered by the set of norms under consideration. To define such a norms’ meta-game, we need to complete the determination of those outcomes. For a real life problem the extra information required would be the initial profile of each one of the two groups. Indeed, in the case where the profile of one or of the two groups remains constant, this profile will be the basis on which the corresponding Basic Copyrights Game outcomes will be assessed. These profiles should derive from statistical studies.
Now as in the framework of this paper no such statistical study is available, we shall simply discard the cases where the groups are totally homogeneous, and therefore enable to focus on ’’proper” evolution.
On this basis, we can associate with each pair of norms a strategic pair, with which we shall in turn associate with a pair of outcomes. Thus under the above assumption, the table of strategic pairs is the following:
Now, just like in the individual / static case one can think of the two groups (end users and authors) as two abstract players interacting in a game in which the strategic sets are the norms sets. If at first glance such identification might look a little weird, it can nevertheless be justified on two grounds.
The first one is the evolutionary framework. We shall come back to that in the next section. The second one pertains more directly to the design of a legal corpus. The law maker should be interested in simulating the impact of laws on the society, for
Author
End User
1 2 3 4
1 • Random choice for both players • Random choice for both players • End user: random choice • Author: P • Random choice for both players
2 • End user: NC • Author: random choice • End user: NC • Author: random choice • End user: NC • Author: P • End user: NC • Author: random choice
3 • End user: C • Author: random choice • End user: C • Author: random choice • End user: C • Author: P • End user: C • Author: random choice
4 • Random choice for both players • Random choice for both players • End user: random choice • Author: P • Random choice for both players
Fig.11. Norms and long term behaviors
which these laws are set up, and in particular to know the optimal pair of norm sets for a given legal corpus.
The norms’ meta-game can thus be represented by the matrix of Fig.12:
Author
End User
1 2 3 4
1 {x/2 + y/4 + z/4, a/2 + 6/4 + c/4) {x/2 + y/4 + z/4, a/2 + 6/4 + c/4) {x/2 + z/2, a/2 + c/2) {x/2 + J//4 + z/4, a/2+ 6/4 + c/4)
2 {y/2 + z/2, 6/2 + c/2) {y/2 + z/2, 6/2 + c/2) (.z,c) {y/2 + z/2, 6/2 + c/2)
3 (x, a) (x, a) (x, a) {x, a)
4 {x/2 + y/4 + z/4, a/2 + 6/4 + c/4) {x/2 + y/4 + z/4, a/2 + 6/4 + c/4) {x/2 + z/2, a/2 + c/2) {x/2 + y/4 + z/4, a/2+ 6/4 + c/4)
Fig.12. The norms meta-game
We see that except for the row corresponding to the set of norms 2 for the end user, this matrix is the same than the matrix of the postures meta-game.
The two games display the same properties as far as Nash equilibria are concerned, except that in the norms’ meta-game (2,3) is a Nash equilibrium whatever the relative values for the authors of litigation cost and loss caused y infringement (provided of course that c > b).
This means that with the assumptions of the Basic Copyrights Game, in the long run these values do not impact the choice of norms. This is quite a significant information for the law maker when designing the system of penalties and compensatory damages that should apply to copyrights issues.
Another social benefit of the norms’ meta-game is that it shows that at equilibrium, only two end users’ sets of norms matter: the sets 2 and 3. On the opposite, any of the four norms sets pertaining to authors can be an element of an equilibrium, but norms 1, 2 and 4 are equivalent strategies for the norms’ meta-game.
3.4. The dynamics of norms
In the previous sections it was assumed that while the parties behaviors could evolve, the corresponding evolutions took place within the framework of fixed norms and values for the end users as well as for the authors. Even if the norms and values could be the object of a choice by the concerned parties as in the norms’ meta-game, once the choice has been made, the norms were there to stay.
If there is no doubt that this framework is relevant for relating behaviors evolutions, with norms of the end users and the authors, one may nevertheless question the realism of a norms ”fixity principle”.
After all, if a set of norms is less efficient than another, in that sense that by adopting the latter a given group would do better than by adopting the former, within an evolutionary context it might be difficult to justify why the group under consideration would stick to the first one.
On the opposite, one can reasonably assume that changes will affect the group’s set of norms, in a way that members of the group will possibly improve their efficiency. To develop an extensive study of norms’ dynamic is out of scope.
We may nevertheless focus on a quite significant case, based on the results of the norms’ meta-game.
Let us thus consider the following sets of norms:
• For the end users group, the sets 2 and 3 which are the only ones for which a Nash equilibrium may occur in the norms’ meta-game.
• For the authors group, as sets 1, 2 and 4 are equivalent in the norms’ metagame, we can select one of them, say set 2 which is an element of equilibrium
(3,2), and add set 3 which is an element of equilibrium (3,2).
There is another justification for deleting norms’ sets 1 and 4 of both groups: with respect to the Basic Copyrights Game, neither is discriminatory, i.e. they do not offer the group who has adopted one of these norms’ sets a way to order the possible states of the world, and hence to select a particular strategy.
We can then consider the following reduced 2x2 game: Keeping in mind the general
Author
End user
2 3
2 (y/2 + z/2, 6/2 + c/2) (.z,c)
3 (x, a) (x, a)
Fig.13. The norms reduced meta-game
assumptions made above:
• z > x > y
• a > c > b
• 2x > y + z
the Nash equilibria of this reduced game, i.e. (2,3) and (3,2) are Nash equilibria in the non reduced meta-game, and they furthermore represent all outcomes pairs that may occur in the latter.
So what is to be considered now is the case of protracted interactions between the groups, such that in each group individuals have at each stage of the dynamic of interaction, the possibility to adopt one of the two sets of norms considered here above.
To conduct the analysis we shall once again resort to the Replicator Dynamic model 4 , which will now be based on the game matrix here above.
To avoid confusion with the analysis conducted above, let us denote by:
• 02 the proportion of the end users who adopt the norms set 2
• 03 the proportion of the authors who adopt the norms set 3
One can show that the equations of the Replicator Dynamic write:
• 02 = (1/2)02(1 - 02)[(z - x) - (x - y) + 03[(z - x) + (x - y)]]
• 03 = (1/2)0203(1 - 03)(c - b)
Let us first consider the evolution of 02.
As for the dynamic of behaviors, if the group of end users is totally homogeneous (all end users choose 02 or all choose 03), it will remain so.
If not, the first equation here above shows that the evolution of the set of norms adopted by the end users depends on two main factors:
• The respective values of unpunished infringement on the one hand, and of the penalties and litigation cost for such infringement on the other
• The proportion of authors adopting norms’ set 3.
If the value of infringement for the end user is bigger than or equal to penalties and litigation cost, the above equation shows that the proportion of end users adopting norms’ set 2 will continuously increase, which after all meets common sense. It means in particular that in this case, the proportion of authors adopting norms’ set
3 doesn’t matter.
In other words if in the legal corpus the penalties and / or litigation costs are smaller than the value of unpunished infringement, the group of end users will consider that compliance is as unacceptable as court case, and will not take into consideration the distribution of norms inside the authors’ group.
In the opposite case, i.e. when the value of infringement for the end user is smaller than the penalties and litigation cost, the evolution of norms within the end users’ group depends on the proportion of authors adopting norms’ set 3.
If 03 < [(x - y) - (z - x)]/[(x - y) + (z - x)], the proportion of end users adopting norms’ set 2 decreases.
If 03 > [(x - y) - (z - x)]/[(x - y) + (z - x)], the proportion of end users adopting norms’ set 2 increases.
4 As already noticed, this model may present drawbacks, but displays the advantage to be simple enough for not generating intricacies that would blur the presentation of the methodology that the present paper wishes to propose.
Let us now switch to the authors’ group.
Once again the authors’ group is totally homogeneous, it will remain so for ever.
If not, the second equation of the Replicator Dynamic shows that if 02 > 0, 03 will always increase. This means in turn that if at initial time, neither group is totally homogeneous, the evolution of norms for the two groups will be the following:
• The end users will eventually adopt norms’ set 2
• The authors will eventually adopt norms’ set 3.
Of course, whether the direction of evolution of the end users norms will always remain the same or will switch from decrease to increase depends on the initial value of 03. If the initial value of the latter is small enough, 02 will first decrease till it reaches a minimum for 03 = [(x - y) - (z - x)]/[(x - y) + (z - x)], and then increase.
It follows that the interaction will eventually lead the end users to be non compliant and the authors to remain passive in case of infringement to their copyrights.
This means that out of the two possible Nash equilibria in the reduced meta-game, evolution will favor (NC,P).
This pessimistic conclusion (as far as compliance is concerned) stems from the assumptions of the Basic Copyrights Game according to which c > b 5 .
More precisely, as long as the authors are better off by not reacting to an infringement of their intellectual property rights, the severity of the penalties inflicted to non compliant end users is useless in the long run. To prevent the occurrence of such a situation there are theoretically two possibilities.
The first one consists in a significant decrease of litigation costs. But such a possibility doesn’t seem very realistic given lawyers’ usual fees, and the fact that the judicial system usually wants to prevent proliferation of court cases that would follow such a decrease. Moreover even if these obstacles were overcome, the decrease of litigation costs would also benefit end users, and thus accelerate evolution toward non compliance.
The second possibility is a closer coupling between outcomes of the two parties resulting from a court case, through ensuring that an increased penalty for the non compliant end user benefits the author till the point that b becomes greater than c. In this case the initial proportion of end users adopting norms’ set 2 may initially increase if the proportion of authors adopting norms’ set 3 is high enough, then decrease as well as 03, in which case the equilibrium eventually selected will be (03, 02). As a consequence 50% of the authors will be defensive, while all end users will be compliant.
4. Conclusions
Leaning on the example of the Basic Copyrights Game, the present paper has proposed a methodology for analyzing compliance issues at two levels.
The static / individual level, has considered that the player’s view of the states of the world does not entirely coincide with (cardinal or ordinal) utilities, but focuses on how acceptable the players consider these states. This is done through
5 It can be noticed that if we go back to the bridge established between the Basic Copyrights Game and Selten’s Chain Paradox, the above conclusions meet the ones obtained by Selten, according to which the competitor will enter and the monopolist will accept that entry
associating the original quantitative game with a set of games of deterrence, each one corresponding to a specific pair of acceptability thresholds.
These thresholds can be considered either as fixed norms of the players or possible postures which the latter can use to build a reputation promoting their interests. A parallel between the Basic Copyrights Game and Selten’s Chain Store Paradox, has shown that at the static/ individual level, deterrence may occur.
The second level, called dynamic/collective, has analyzed how the behaviours of two interacting populations (end users and authors) evolve with norms. Associating with each pair of norms the behaviours resulting from evolution, has enabled to build a game in which the two populations’ strategies are the norms these populations adopt. An evolutionary process based on behaviours efficiencies has been generated which leads to the selection of particular sets of norms.
It has been shown that in the case under consideration the level of penalties inflicted to non compliant end users cannot by itself generate compliance pervasion. This pervasion also requires that the author victim of an infringement should get a share of these penalties important enough, to keep credible the threat that he / she would react in case of an infringement of his / her rights.
More generally, starting from a simple case, the model has proposed elements of a methodology paving the way to a comprehensive game theoretic tool for analyzing the interaction between evolution of norms and pervasion of behaviours in a variety of social, economic, legal or political contexts.
Many thanks to Pr Helene Bestougeff from Unversite de Marne la Vallee and ORT France for reviewing the paper.
Appendix
1 2 3 4
1 D P D P D P D P
C (0,0) (0,0) C (0,1) (0,1) C (0,1) (0,1) C (0,1) (0,1)
NC (0,0) (0,0) NC (0,0) (0,0) NC (0,0) (0,1) NC (0,1) (0,1)
2 D p D p D p D p
C (0,0) (0,0) C (0,1) (0,1) C (0,1) (0,1) C (0,1) (0,1)
NC (0,0) (1,0) NC (0,0) (1,0) NC (0,0) (1,1) NC (0,1) (1,1)
3 D p D p D p D p
C (1,0) (1,0) C (1,1) (1,1) C (1,1) (1,1) C (1,1) (1,1)
NC (0,0) (1,0) NC (0,0) (1,0) NC (0,0) (1,1) NC (0,1) (1,1)
4 D p D p D p D p
C (1,0) (1,0) C (1,1) (1,1) C (1,1) (1,1) C (1,1) (1,1)
NC (1,0) (1,0) NC (1,0) (1,0) NC (1,0) (1,1) NC (1,1) (1,1)
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