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# Ability and Diversity of Skills

Communicated by *Notices* Associate Editor William McCallum

## 1. Introduction

The aim of this paper is to build a simple model of problem solving, both by single agents and by teams. We realize that our model is crude and of course far from universal. Yet the results we get seem to us quite illuminating, and show the importance of both ability and diversity of skills.

The question of how to measure effectiveness of problem solving by individuals and (even more importantly) by teams, and how to choose the best individual/team, has been a subject of a lot of research. We can give as examples papers GJIBHP2KIKR, and the literature cited there. We do not address explicitly the problem of choosing a team, but our findings may serve as the basis for further research in that direction (in cases where it seems that our model may be applicable).

For a single agent, or a team of agents, we try to measure the probability of success as a function of the difficulty of a problem (or rather the easiness of the problem, measured by a variable the larger ; the easier the problem). In Section ,3 we show that the probability of success is concave as a function of .

In Section 4 we show that in our model for a single agent specialization is better than versatility.Footnote^{1} We also show that comparing agents is difficult. In most cases, for a given agent and chosen values of there can be another agent, who is better at solving problems with easiness , for those chosen values, but worse at solving problems with all other values of .

In Section 5 we consider teams of agents. We get what can be considered the main result of the paper: whenever our model can be applied, both abilities of the team members and the diversity of skills in the team matter. If any of those increases, so does the probability of success. For simplicity, we consider teams with two members, but it is clear that similar results should hold for larger teams. Interestingly, there is an example where for easier problems ability is more important, but for more difficult problems diversity is more important.

In Section 6 we show how our model can be applied to a situation where the agents are trying to defend an organization against an attack. In this application, diversity is even more important than for general problem solving.

Some of our ideas came from studying the model of L. Hong and S.E. Page HP1. Our model is much simpler, and can be easily investigated both by pure mathematical means and by computational means. Moreover, we avoid the main deficiency of the Hong–Page model, where high-ability teams consist basically of clones of the same agent (and as a result, ability excludes diversity).Footnote^{2}

## 2. Preliminary Model

If an agent will be trying to solve problems that are not known in advance, her expected performance can be measured by an average over various possible problems.

Our first, preliminary model is as follows. An agent has some set of skills. This set is a subset of An immediate problem is represented by a subset . of of cardinality An agent can make progress if the intersection . is nonempty. Clearly, the difficulty of the problem is measured by problems with smaller ; are more difficult.

When we want to measure the ability of an agent, we average performance of an agent over all problems of a given difficulty (that is, with a given cardinality The result clearly does not depend on a concrete set ). of skills, but only on its cardinality (the skillfulness of the agent).

For given it is easy to compute the probability of making progress. If this probability is , If . out of all , possible sets only result in failure. Therefore the probability of success (that is, making progress) is

In Figure 1, we can see how the probability of success varies with the difficulty of the problem, for agents with various numbers of skills. If the problem is easy, the skillfulness of an agent does not matter much (provided the agent has some minimal number of skills). However, for difficult problems it matters a lot.

## 3. Main Model

The preliminary model is very crude, because for each skill an agent either has it or does not. However, one should allow an agent to have partial skills. Then becomes a *strength function* If . then the probability that the agent can make progress using skill number , is We assume that those probabilities for different . are independent. This means that it is easier to use in computations the *weakness function* where , is the constant function 1. Then the probability of failure for a given agent and given problem is equal to the product of the numbers over .

Often instead of speaking of the strength and weakness functions we will speak of the *strength and weakness vectors* and .

For a given the sum of , over all sets of cardinality is equal to

where denotes the cardinality of Observe that this number is equal to the coefficient for the polynomial .

of Thus, the average probability . of failure over all sets of cardinality is equal to this coefficient divided by Note that . is the coefficient of for the polynomial This in particular means that if an agent has no skills (so . her probability of failure is 1 no matter what. ),

Clearly, if is a permutation of the set then Therefore we may assume that . Sometimes, if we do not want to make this assumption, we will say that . is a permutation of .

Of course, if takes only values 0 and 1, we get the previous model.

Let us investigate some basic properties of the function .

## 4. Specialization and Versatility

We would like to be able to measure the skillfulness of an agent in our model. There may be various ways of doing this, and as we will see in Theorem 5, we cannot expect to find a perfect one. We will settle on what seems the most natural way of doing it, by defining it to be where , is the strength function of the agent.

Within our model, one of the first questions that comes to mind is what is the best distribution of strengths given the skillfulness of an agent. The agent can be more specialized or more versatile. We will show that in our model specialization is better than versatility.

The simplest case is when we have two agents, one with and for and the other one with , and for The first agent is more versatile and the second one more specialized. We have .

and

Thus, for and , for Let us make exact computations. .

The coefficient of for the polynomial is the same as the coefficient of for the polynomial that is, , Thus, .

To get the coefficient of for the polynomial we have additionally to add the coefficient of , for the polynomial so ,

This means that while the graph of the probability of success as a function of lies on the straight line from to for it lies on a parabola from ,