1 Introduction
When purchasing information, one wants to be sure of the quality of the information in question. However, if one is not an expert oneself in the relevant domain, assessing the quality of information can be difficult. For the sellers of information (which we will simply refer to as “the consultants”) this provides an incentive for dishonesty. After all, gaining real expertise costs significant efforts as well as time and money. However, if the consumer of information (which we will refer to as “the client”) has difficulties assessing the quality of the provided information, then why not pretend to have a higher level of expertise than one actually has? As long as the chance that the client detects this dishonesty is low, a consultant can charge the same price for his advice, yet spend less resources on maintaining uptodate of the state of the art. Moreover, the more consultants decide to take it easy, the less likely it is that the clients will find out about it. This is because once a critical mass of consultants gives illinformed advice, it becomes more and more difficult for the clients to obtain the kind of wellinformed advice that they need in order to detect the illinformedness of the other advice. This implies that once a critical mass of low expertise consultants has become established, the incentive for consultants to take it easy will increase, since the chance of discovery has decreased.
The issue of low quality information has been studied in [8, 5]. What is new, however, is that we have now developed a software simulator that is able to assess the payoff for the consultants of either a strategy of hard work or a strategy of taking it easy when it comes to staying up to date with the state of the art. In particular, we are able to provide qualitative insight on which strategy yields the most profitable results under which circumstances.
2 Argumentation and Informedness
The aim of this section is to examine how argumentation can play a role to examine the concept of informedness, which can be seen as background theory for the remaining, more practical part of this paper.
In standard epistemic logic (S5), informedness is basically a
binary phenomenon. One either has knowledge about a proposition or one does not.
It is, however, also possible to provide a more subtle account of the
extent to which one is informed about the validity of proposition .
Suppose Alex thinks that Hortis Bank is on the brink of bankruptcy because
it has massively invested in mortgage backed securities. Bob also thinks
that Hortis is on the brink of bankruptcy because of the mortgage backed
securities. Bob has also read an interview in which the finance minister
promises that the state will support Hortis if needed. However, Bob also
knows that the liabilities of Hortis are so big that not even the state
will be able to provide significant help to avert bankruptcy. From the
perspective of formal argumentation [7], Bob has three arguments
at his disposal.
: Hortis Bank is on the brink of bankruptcy,
because of the mortgage backed securities.
: The state will save Hortis, because the finance minister promised so.
: Not even the state has the financial means to save Hortis.
Here, argument attacks , and argument attacks (see eq.
1). In most approaches to formal argumentation,
arguments and would be accepted and argument would be rejected.
(1) 
Assume that Alex has only argument to his disposal. Then it seems to regard Bob as more informed with respect to proposition (“Hortis Bank is on the brink of bankruptcy”) since he has a better knowledge of the facts relevant for this proposition and is also in a better position to defend it in the face of criticism.
The most feasible way to determine whether someone is informed on some given issue is to evaluate whether he is up to date with the relevant arguments and is able to defend his position in the face of criticism. One can say that agent is more informed than agent if it has to its disposal a larger set of relevant arguments.
We will now provide a more formal account of how the concept of informedness could be described using formal argumentation. An argumentation framework [7] is a pair where is a set of arguments and is a binary relation on . An argumentation framework can be represented as a directed graph. For instance, the argumentation framework is represented in eq. 1.
Arguments can be seen as defeasible derivations of a particular statement. These defeasible derivations can then be attacked by statements of other defeasible derivations, hence the attack relationship. Given an argumentation framework, an interesting question is what is the set (or sets) of arguments that can collectively be accepted. Although this question has traditionally been studied in terms of the various fixpoints of the characteristic function
[7], it is equally well possible to use the approach of argument labelings [2, 4, 6]. The idea is that each argument gets exactly one label (accepted, rejected, or abstained), such that the result satisfies the following constraints.
If an argument is labeled accepted then all arguments that attack it must be labeled rejected.

If an argument is labeled rejected then there must be at least one argument that attacks it and is labeled accepted.

If an argument is labeled abstained then it must not be the case that all arguments that attack it are labeled rejected, and it must not be the case that there is an argument that attacks it and is labeled accepted.
A labeling is called complete iff it satisfies each of the above three constraints. As an example, the argumentation framework of eq. 1 has exactly one complete labeling, in which and are labeled accepted and is labeled rejected. In general, an argumentation framework has one or more complete labelings. Furthermore, the arguments labeled accepted in a complete labeling form a complete extension in the sense of [7]. Other standard argumentation concepts, like preferred, grounded and stable extensions can also be expressed in terms of labelings [2].
In essence, one can see a complete labeling as a reasonable position one
can take in the presence of the imperfect and conflicting information
expressed in the argumentation framework. An interesting question is
whether an argument can be accepted (that is, whether the argument
is labeled accepted in at least one complete labeling) and whether an
argument has to be accepted (that is, whether the argument is labeled
accepted in each complete labeling). These two questions can be answered
using formal discussion games [9, 11, 1, 6]. For instance,
in the argumentation framework of eq. 1, a possible
discussion would go as follows.
Proponent: Argument has to be accepted.
Opponent: But perhaps ’s attacker does not have to be rejected.
Proponent: has to be rejected because ’s attacker has to be accepted.
The precise rules which such discussions have to follow are described in
[9, 11, 1, 6]. We say that argument can be defended
iff the proponent has a winning strategy for . We say that argument
can be denied iff the opponent has a winning strategy against .
If informedness is defined as justified belief, and justified is being interpreted as defensible in a rational discussion, then formal discussion games can serve as a way to examine whether an agent is informed with respect to proposition , even in cases where one cannot directly determine the truth or falsity of in the objective world. An agent is informed on iff it has an argument for that it is able to defend in the face of criticism.
The dialectical approach to knowledge also allows for the distinction of various grades of informedness. That is, an agent can be perceived to be at least as informed as agent w.r.t. argument iff either and originally disagreed on the status of but combining their information the position of is confirmed, or and originally agreed on the status of and in every case where is able to maintain its position in the presence of criticism from agent , is also able to maintain its position in the presence of the same criticism.
When and are argumentation frameworks, we write as a shorthand for , and as a shorthand for . Formally, agent is at least as knowledgeable about argument as agent iff:

can be defended using (that is, if assumes the role of the proponent of then it has a winning strategy using the argumentation framework of ), can be denied using (that is, if assumes the role of the opponent than it has a winning strategy using the argumentation framework of ), but can be defended using , or

can be denied using , can be defended using , but can be denied , or

can be defended using and can be defended using , and for each such that can be defended using it holds that can also be defended using ,

can be denied using and can be denied using , and for each such that can be denied using it holds that can be denied using .
Naturally, it follows that if then is at least as informed w.r.t. each argument in as .
In the example mentioned earlier (eq. 1) Alex has access only to argument , and Bob has access to arguments , and . Suppose a third person (Charles) has access only to arguments and . Then we say that Bob is more informed than Alex w.r.t. argument because Bob can maintain his position on (accepted) while facing criticism from Charles, where Alex cannot. more controversial consequence is that Charles is also more informed than Alex w.r.t. argument , even though from the global perspective, Charles has the “wrong” position on argument (rejected instead of accepted). This is compensated by the fact that Bob, in his turn, is more informed than Charles w.r.t. argument . As an analogy, it would be fair to consider Newton as more informed than his predecessors, even though his work has later been attacked by more advanced theories.
3 Simulation
We developed a softwaresimulator in order to better understand the impact of a consultant’s informedness on his profit. The simulator has the objective to reveal when it pays off for a consultant to be illinformed, i.e. less wellinformed as is the state of the art. If illinformedness is more profitable for a consultant than being wellinformed, the spread of outdated and possibly wrong facts or arguments is preprogrammed. Of course, such a situation would be harmful to our society, and so a study of its causes seems to be important.
In the following, we describe the simulator (Sect. 3.1), show results (Sect. 3.2) and discuss these (Sect. 3.3).
3.1 The Simulator
In the first part of this section, we describe the clientconsultant scenario that we aim to simulate. We then detail the considered argumentation framework. Two different strategies of the consultants concerning the acquisition of newly available information are defined afterwards, one representing wellinformed consultants and the other representing illinformed consultants. Finally, we specify the procedure of how clients select their consultants.
3.1.1 ClientConsultant Scenario
The scenario is basically defined by a set of clients, a set of consultants, and a finite number of rounds in each of which the clients seek advice from the consultants. In each round, consultants acquire new information, either from other sources (researchers, analysts, etc.), or do investigations on their own. We model this acquisition of information simply as fixed costs per piece of information; in our case, this information comes in the form of arguments as we will see later. The more clients request a certain consultant’s advice, the lower his price for a consultation can be. Note that this actually explains why the intermediary role of consultants exists.
Clients are free to choose their consultant, and so select in each round the consultant that they think is currently the most appropriate one. In our simulations, this selection is based on price and reputation of the consultants as it will be detailed later.
3.1.2 Argumentation Framework
We consider the following argumentation structure consisting of many arguments:
(2) 
Here, any argument (for ) defeats its predecessor argument . If is even, then all arguments where is even, are “in”, and all other arguments are “out”. If
is odd, it is the other way around.
At the beginning of a simulation, only argument is known to the consultants and only this argument is known in the whole society, i.e. it represents the “state of the art”. To model the discovery/emergence of new information, we make a certain number of new arguments available to the consultants in each round. This represents the evolution of the state of the art. The number of new arguments per round will be fixed for a simulation and is denoted by . We assume that the consultants extend their already known chain of arguments with new arguments always in a seamless manner, i.e. without gaps. This assumption was made in order to be in line with argument games (such as described in [9, 11, 1]) where each uttered argument is a reaction to a previously uttered argument, thus satisfying the property of relevance [3].
For a better understanding, the following shows the structure of the chain of arguments at any round of the simulation ( must hold):
(3) 
Consultants generally want to provide the least amount of information needed for a consultation, because this way they can give more consultations. However, consultants generally want to give good advice at the same time, to increase their reputation. We assume that consultants believe that the latest argument they know is the most justified one, because it is closest to the current state of the art. As a consequence, in order to give good advice, they only consult arguments that are compliant to the latest argument they possess, i.e. arguments whose index has the same parity as the latest argument they know. To provide the least amount of information at the same time, a rational consultant acts as follows: he provides the client with two arguments, if the latest argument known to the client is of the same parity as the latest argument known to the consultant, and with one argument otherwise. The latest argument known to the client is updated accordingly.
To keep things simple, the cost of an argument is set to a constant . So, to get the knowledge about argument , the consultant has an overall expense of (recall that arguments can only be acquired in a row). We write to denote the total number of arguments acquired by a specific consultant (where ). Thus, we model the expenses of a consultant as:
(4) 
3.1.3 Strategies of the Consultants
We consider two types of consultants that deal differently with newly available arguments:
 wellinformed (wi):

wiconsultants buy arguments as soon as these become available, since they want to be always uptodate.
 illinformed (ii):

iiconsultants only buy arguments as to appear knowledgeable to the clients. That is, as soon as they notice that a client is as informed as they are, or even better informed, they buy a number of new arguments such that they know one argument more than this client.
Clearly, iiconsultants can offer their consultations at a lower price. However, the reputation of a consultant decreases with each consultation where the client turns out to be as informed as the consultant – and this happens more often to iiconsultants.
The turnover of a consultant is defined by the sum of prices the consultant was paid. Let be the multiset enumerating all prices paid by clients up to a certain round. So, represents the consultations where the consultant actually was better informed than the client, and thus was paid. Then the turnover up to this point is defined as:
(5) 
Finally, the profit of a consultant up to a certain round is the difference between his turnover and his expenses so far:
(6) 
3.1.4 Selection of a Consultant
In our simulations, clients rate consultants according to two criteria: the consultant ’s current reputation and price . The two criteria are explained in more detail later in this section. For now, it suffices to know that they are normalized to . To make both parameters “positive”, the price will be expressed in the form of cheapness , i.e., . This way, both a high cheapness and a high reputation characterize a good consultant. A parameter defines which of the two criteria a client thinks is more important. Consequently, a client chooses consultant
with a probability proportional to:
(7) 
A high favors the choice of cheaper consultants, while a low favors the choice of more reputable consultants.
Price
Let be the profit margin of a consultant, with ; then for example represents a typical profit margin of . Using a certain profit margin, a consultant computes his current price :
(8) 
Here,
is a simple estimate to provide cost recovery, where
models the expenses (see eq. 4), and is the number of successful consultations so far (see eq. 5). Still, no client would choose a consultant that is more expensive than the acquisition of the information itself. Hence, we limit the price to the cost of the two arguments that a consultant advises at most ().Finally, we normalize the prices of all consultants to :
(9) 
Reputation
In our simulator, clients use a reputation system [10] to share their experience about consultants. This allows clients to better estimate the trustworthiness of the consultants and thus to better select their future consultants. We assume perfect conditions for the reputation system, since this will make it even harder for iiconsultants to hold their ground. These perfect conditions consist of:

honest reporting of the clients,

all clients have the same idea of how to fuse the experiences with consultants, and so a global reputation score can be computed, and

total information sharing, i.e., every client shares his experience with every other client.
The reputation system should be as simple as possible, and the only requirement we have is that it measures a consultant’s performance relatively to the performance of other consultants. We consider two different reputation systems. A very basic system (R1), and a more sophisticated one (R2).
Reputation System R1
In the reputation system R1, a client is satisfied with a consultation if the consultant is better informed than the client. This is motivated by the fact that clients cannot question the correctness of the consultants arguments since they are less well informed. So, any consultation where the consultant is better informed than the client counts as a positive experience with that consultant. Consultations where this is not the case count as negative experiences. The clients share their experience and maintain for each consultant a global counter of positive experiences, and a global counter of negative experiences. Then an intermediate reputation score is computed as follows:
(10) 
To bring these scores into , we normalize the intermediate values of all consultants and get the final global reputation score :
(11) 
This basic reputation score has the desired property: the overall performance of a consultant is measured relatively to the performance of other consultants.
Reputation System R2
In this reputation system, a client reactively decides about positive and negative experiences. Each client assumes that the latest argument he was advised on by a consultant is correct. Then for all consultants that advised him with a conflicting argument in the past (i.e., where the parity of the argument’s index is different), each client increases the global value , and for all consultants that advised a compatible argument, is increased. As in R1, if the consultant is not more informed than the client, is additionally increased. Given and , the computation of is the same as in R1.
3.2 Results
On the following pages we show simulation results to compare the profit of wiconsultants and ii
consultants for different parameters. We conducted the experiments by computing the mean and standard deviation of the consultants’ profit over
simulation runs. Each simulation consisted of rounds. We used clients and consultants.We varied the following parameters:

the number of arguments becoming available each round (),

the fraction of iiconsultants (),

the profit margin (), and

the factor that regulates the importance of price and reputation in the clients’ consultant selection process (, shown on the xaxis).
The figures on the left depict simulations with , those on the right depict simulations with .
Furthermore, we conducted all experiments for the two different reputation systems R1 and R2.
At the end of this section, we provide an analysis of the shown results.
(Note: For making it easier to compare the different figures, we start on the next page.)
3.2.1 Using R1

Information rate :

Fraction of iiconsultants :

Fraction of iiconsultants

Fraction of iiconsultants


Information rate :

Fraction of iiconsultants :

Fraction of iiconsultants

Fraction of iiconsultants


Information rate :

Fraction of iiconsultants :

Fraction of iiconsultants

Fraction of iiconsultants

3.2.2 Using R2

Information rate :

Fraction of iiconsultants :

Fraction of iiconsultants

Fraction of iiconsultants


Information rate :

Fraction of iiconsultants :

Fraction of iiconsultants

Fraction of iiconsultants


Information rate :

Fraction of iiconsultants :

Fraction of iiconsultants

Fraction of iiconsultants


Information rate :

Fraction of iiconsultants :

Fraction of iiconsultants

Fraction of iiconsultants

3.3 Analysis
First of all, it is evident that in some scenarios iiconsultants have a higher profit than wiconsultants. We can conclude that under certain circumstances it is more profitable for a consultant to follow the iistrategy than being wellinformed. Some more detailed observations about the results are described in the following.
3.3.1 Using R1
One can see that if reputation is used as the only criterion for selection (), wiconsultants are always better off. As the price of the consultants gets more important, the profit of iiconsultants generally increases, whereas the profit of wiconsultants decreases. In most settings the wiconsultants’ profit decreases so much that at some point it falls below the profit of the iiconsultants. Besides, iiconsultants seem to generally benefit from a low profit margin ().
An increasing fraction of iiconsultants causes the curves of both iiconsultants and wiconsultants to have a lower slope value (have a look at the values on the yaxis to verify that). As a result, the point where the two curves intersect moves to the left, i.e. the iistrategy becomes more effective.
A varying impacts also the slope of the curves, as well as their distance. Consider for instance the variation of for otherwise fixed parameters: R1, and (figures on the right side). For , the iistrategy is the better strategy for ; for a of or , the wistrategy is always better – for very distinct, and less distinct for .
3.3.2 Using R2
The same observations as for R1 can also be made for R2. The main difference lies in the shape of the curves. In particular, the shape of the lines depends on the parity of . This can be explained as follows. If an even number of new arguments gets available each round, wiconsultants will always advice even arguments; however, if is odd, they switch each round between advising even and odd arguments. Since clients retrospectively judge consultants in R2, they will find in the odd case that wiconsultants contradict each other over time, which will never be the case in the even case. So, in the even case, clients will decrease the reputation of a wiconsultant only if their most recent consultant was illinformed and consulted an odd argument. This explains also why the profit of wiconsultants is much higher and the profit of iiconsultants is much lower in the even cases (also in comparison to R1).
Apart from these specific shapes, in the same way as for R1 we can observe that the iistrategy yields in many settings higher profits.
4 Conclusion & Future Work
Our simulations suggest that at least four factors have an impact on the profitability of being illinformed:

the clients’ balancing act of choosing a consultant according to his reputation or his price (),

the fraction of illinformed consultants in the set of consultants (),

the speed with which new information enters the system and becomes available to consultants (), and

the profit margin of the consultants ().
In our simulations, a high and a high made it generally less profitable for consultants to be wellinformed. The observation that an increasing fractions of iiconsultants increases their payoff, raises the problem of consultants that follow the crowd, even if this means to stay illinformed.
The simulated model in this work is certainly simplifying the complexity of reality. There are many ways for making it more realistic. For instance, the price computation given by eq. 8 is purely reactive; it does not exploit the fact that a lower prices would attract more clients and so could be used in a proactive manner. Still, since all consultants computed their price in the same way, it is not clear whether a more complex approach would make the iistrategy less profitable. This is subject to future research. Also, our consultants pursued two basic strategies (acquiring all arguments becoming available, and acquiring only as needed); this can be extended to more sophisticated strategies, e.g. to be a wiconsultant for a certain time to boost the reputation, and then become iiconsultant. Furthermore, the impact of publicity campaigns is completely ignored and would be an interesting continuation of our research.
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