Leadership and Voluntary Contribuitons

I read over what we have and have been thinking about it. I have not come to any definite conclusion yet.

Here is the basic problem. My taste in research is to try and write general models that provide a theoretically elegant abstract framework for thinking about a problem. So far, everything I have been able to come up with has too much structure, is too applied and is not very elegant. Attempts to produce a more elegant structure quickly devolve into models in which anything could happen and thus add little insight. I have not given up, but this is the reason for the lack of progress here.

The issue I really want to address is a more realistic explanation for voluntary contributions. The elements I see as key in this are the following:

1. I think agents contribute because of a reward they get from the act of contributing. Andreoni's warm glow is an abstract statement of this, but I think bit too abstract. It might be (a) reputational rewards (Linux, large visible donors to charities) (b)pleasure from being allowed to do good in a particular coalition (Doing a fun-run with my work-mates, getting together with my Shiner buddies to build a house for the poor, or joining an AEA committee so that I can have contact with insiders) (c) Less structured direct pleasure from the act of giving (I like to see the smile on the orphan's faces when I feed them; this is closest to "warm glow")

2. I think agents almost totally discount the private benefits they get from the public goods they produce in deciding contribution levels

3. On the other hand, my pleasure in making a contribution increase if the public good produced is one I like, or perhaps one that has objective value, or perhaps is one that people generally value.

4. Connected to this is the notion that Leadership is a key element of these projects. That is, agents do no coalesce in a vacuum. There is has to be a “seed” (as in what is at the heart of the pearl). Of course the seed here is strategic player with is own objectives. Thus:

We need to think of what motivates leaders. It could be (a) the size of the coalition (b) the average quality of the coalition members (c) the social value of the public project and so on

We also need to think about what it is about leaders that affect the size of the coalition and the value of the project. It could be (a) hard work to attract followers. This could be structured (like making specific offers to individual agents to join a network) or unstructured, (b) some inherent quality like charisma, intelligence or organizational skill (c) the choice of project type.

Last, this brings up two structural questions:

1. The first is a difficult question of how to differentiate project types. This has connections to the hedonic goods and monopolistic competition literatures. Unfortunately, both these literatures and complex and inconclusive. One could take a simple Hotelling approach and place projects on a line with agents having preferences over the line. This is a bit limiting and only really works with two projects. One could simply have abstract projects with each agent having arbitrary preferences over the set of choices offered by leaders. This is realistic, but it makes it very hard to back-out the optimal choice for leaders since describing the distribution of tastes is all but impossible as is working out expected values of choices. One might just describe a probability to any given agent will switch to the new project, but this should not really be independent of number and type of current projects offered. This is the hardest issues to me.

2. The second issue is what the basic framework is. It could be some price taking equilibrium notion, a coalitional framework and the core, or perhaps a network. All have there attractions.

Okay, this is the research problem as I see it.

In the current approach, we take the position that projects have different innate qualities, and that agents have common preferences over quality. Also, leaders don’t get to choose projects, and agents have different abilities to contribute to projects. Agents join all coalitions at some level by making some contribution. We also put in the art/science divide.

The common preference idea might be a good thing to maintain as it simplifies the problem and is a reasonable reference point.

I am not so keen on the idea that leaders can’t choose projects, that the number of projects is exogenous, and that leaders simply exist and can’t decline to lead projects.

The different ability of agents to contribute is fine, but I don’t see this as the central point. If it adds complexity, it could easily be dispensed with.

I don’t like that all agents contribute to all projects. It may be worth keeping because it makes the reaction functions differentiable, but it is not realistic and is one of the things that the paper will be criticized for.

Finally, I think the art/science divide, while not the central point. has the advantage of giving us a one parameter way of distinguishing two interesting cases and generates the potential for interesting theorems.

Thus, I am not convinced at this point that we have the right start, and this, as I said above is what is holding me up on this project.

What to do?

Here are three ideas:

(A) Let’s think of a network. One question is how much contribution do agents make to a project. Suppose the network is asymmetric in the sense that leaders spend effort contacting agents (maybe this costs different leaders different amounts) and this generates an outgoing link. Agents in return may decline the link back or may form a link of some size which they choose. The key is that we equate the size of the link with the size of contribution and so put voluntary contributions into a network context. It is unclear if we will learn about asymmetric networks using public goods ideas or learn about public goods using asymmetric networks. Note that the probability of linking back and the size of link-back could depend on the effort of the leader, the quality of the leader or project, the number of current link-backs and so on. At first, we could study this in without competitive projects. We might find that a leader should try to link with the most likely strong supporters first since this generates the most net linkage for least cost and persuades the agents asked later to link more strongly. We also might observe that the leader could put in a minimum link size price. This might extract rent from agents who would accept the all or nothing offer if it yielded net benefit and so would agree to go beyond the linkage point where MB=MC of contribution

(B) I don't know a lot about this, but what if we thought of leaders as staking out a position and deciding on how hard to work and then give agents a well defined utility that causes them to joint a coalition depending on its quality, the number of members, the nature of alternative projects and so on, but in a discrete decision theoretical way. Then agents are drawn randomly and make choices. We could find stable outcomes through a process called "simulated annealing". This plays off the "seed pearl" idea. The notion comes from metallurgy and numerically simulates how atoms fall into solid lower energy crystalline states as the liquid metal cools. The slower their rate of cooling, the more optimal (or crystallize) the final state. There a paper by Scott Page and coauthor published in the AER about 10 years ago on local public goods and voting on this. If you can do and like simulations, this might be a way to go, otherwise, forget it.

(C) We might try to improve on our existing model in some way. One thing that occurs right away is taking the total contribution level out of the “science” value function. This muddles it with art and does not gain us anything. Maybe we can get rid of the credit/reputation thing as well. I am not sure what to replace this with. At this point, you think we can show existence, right.? We don’t have a good welfare measure, so welfare theorems are out. What sort of comparative static problems do you see me answer?

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Describe Leadership Home here.

By Kung

John:

It seems that you want an all-in-one model that can address broad ideas about projects and leaders. But what I see here is the possibility of a series of papers, one by one, addressing aspects of projects. I try to relate your ideas with specific modeling language and see if I understand what you said.

1. Rewards

Suppose agent i contribution xi and the project has a set S of contribution members.

(a) Reputation

In a simple static setting, it is the same as getting utility from total contributions, with utility function u(sum xi). In a dynamic setting, reputation may bring gains in the next period.

(b) Coalition members: use u(S)

Each of, and combinations of the above can be a separate model and we can only separate mathematically different objects. For example, agent may care about the actual social benefit not project size if we use u(f(sum xi)) where the production function of the project output is f(sum xi).

2. Rewards for leaders

(a) Reputation: eg. coalition size or project size

(b) Member quality

If we want endogenous leaders, then the value for leader should be no different from other agents. Why they become leaders must be answered by some parameters, for example, having low cost for efforts, high value for public goods, low patience. If we want a separate set of players as potential leaders, than we can follow the IO literature with firms a leaders and consumer as followers. This may seems very easy as in the Hotelling literature.

3. Function of a leader

Here we use a separate set of leaders from common contributors.

(a) Invite members

This seems like a proposal game in coalition bargaining literature where players propose payoffs and invite members into coalitions.

(b) Organizational quality

This means low cost or high return from efforts

This also relates to personal traits (parameters). If I am particular good in programming (low cost) and I benefit more (high return) than others from Lynux, I will start it first.

4. Modeling choices

4.1. Space for projects

I actually do not see it a difficult choice. First, we cannot use preference ordering since we need to compare cost and benefit; this is cardinal. So, just from the parameters, there can be a structure for projects. For example, there is ranking of projects for a leader in terms of expected net benefit from initiation. He will choose the one with highest return.

4.2. Equilibrium concepts

This may follow directly from our choice of models. And of course there are different versions of cores, SPE, (Baysian) NE,..

Therefore, by the combination of the above 4, we should have at 3x3x3… kinds of models. They may give us different results, although some results may not be qualitatively different.

There are many possibilities, the following as examples. 1. A dynamic proposal game where leaders propose contribution levels and invite members to join exclusive coalitions. This is a modification from the bargaining literature (Bloch).

2. Leaders announce projects in static NE (not necessarily Hoteling) and contributors choose among the array of projects.

3. We may actually use a dynamic game setting and address everything, reputation, organization, free-rider, .. in the variations from one model. The model can have leaders offering “promises” to attract contributors.

5. Current approach

I do not see our current project addressing leaders at all. I take it as a version of the voluntary contributions model with non-pure multiple public goods (similar to Bergstrom et al).

An agent does not necessarily contribute to all projects. Someone may want to contribute negatively when MR is low and MC is high but there is a zero bound. Depending on whether the production functions of projects, Q and F, have increasing returns to scale (IRS) or DRS, the equilibrium number and size of projects (and also the number and size of contributions from each agent) vary. What I see the current approach can address is how the science and art factors affect the number and size of projects and the number and size of contributions agents make. And whether the art factor can outweigh science and “any project can be supported if in fashion.”

6. What to do

Now I see we need to find a match between the model side and the phenomenon side. There are lots of choices for models. We can either pick a model and see what phenomenon it generates, or we can pick a real world phenomenon and tailor a model to explain it. Without fixing one end, it seems too many choices. Would you decide which end to take?

(A) I may not understand fully what’s in the model from the description. Are the links real relationships of collaboration, or a metaphor? Can contributor link indirectly to the leader? Network and leaders can be a separate topic and it may or may not answer the questions you pose.

(B) Maybe in the future. I do not know what type of things he did.

(C) Existence needs some more work. The cited theorem is from Villanacci-Zenginobuz (JME 2005). They use the theorem wrong because contributions problems have a zero bound. I am working on extension to the boundary case. It should be fine, not writen down yet. This is for DRS case only and there is no theorem for IRS case. We need to start anew for IRS.

More by Kung 03/17/2007

(1) One way to extend the old model is to have a two-stage game where in the first stage people choose to be leaders (associated with cost and extra benefits), then a small group of leaders will form. In the second stage, leaders choose projects and followers contribute freely. Similarly, we can answer "how many leaders and large a project a leader leads".

Is there a real phenomenon or intuition of leaders that we should address? For example, besides who will be leaders, "can leaders bring efficiency compared with an economy without leaders," or "do leaders provide information to contributors?"

(2) Suppose we take information as the role of leaders. There can be a Bayesian game with all players having the option of being a leader (with a choice of a project). The project space correlates with the players' characteristic space in a form of "matching".

If the type of a project correlates with the type of a leader (environmental-Gore, world domination-Bush, poverty alleviation -Mother Teresa), the expected success rate/benefit of the project is higher (or the cost is lower). If there is a mismatch (poverty-Bush), then the success rate is lower. This may capture, as a reduced model, the role of leaders. This can be a one-stage (simultaneous choices of being, or not, a leader and the amount of contributions) or a two-stage game.

This model is general enough and has endogenous leaders. We can justify the stories with an existence result. Maybe a proposition of the type correlation can be proved. To make it more complicated, we may also borrow things from crowding types and clubs, and maybe a centralization mechanism for efficiency.

(3) The general setting for projects may actually be similar to clubs where people can join multiple of them. So, to formulate a general model, I suggest we break down the task as follows: (i) We may star from with endogenous club formation with leaders. We need to simplify the production side and get rid of prices. (ii) Why do we need leaders? Here we can add the role of leaders: information, organization, network connection,… The relevant equilibrium concepts are Nash or Bayesian Nash with one or two stages. (iii) I suggest to deal with the proper form of utility the last, after having a proper model. Because the utility form may not be as crucial as the role of leaders and the externality/publicness will make existence difficult.

May I suggest a structure for this project, if our taste is existence in a general model: (1) Benchmark model and main result: Role of Leaders (choosing a proper model)  Existence  Properties of equilibrium.

(2) Variations/ other issues: changing the utility form More applied models for (i) Arts vs. Sciences (ii) Self-enforcing effect of praise rings