Nathan Woodbury and Dr. Sean Warnick, Department of Computer Science

Introduction

My original purpose in this project was to “explore the nature of self-enforcing policies that induce participation in coalitions within a competitive environment.” In order to do this, I proposed to use a model of a general market of many suppliers. This model was to be an extension of previous models created by Dr. Sean Warnick, Nghia Tran, and Tanja Brown ([1] and [2]).

In this model, each supplier would sell exactly one product at a price determined by that supplier. The quantity demanded of this supplier’s good was not only negatively affected by the price the supplier sets, but also positively or negatively aected by the prices set by all of the other suppliers in the market. In addition, each supplier could choose to redistribute the profit she makes in this market to other suppliers in the market with the hope that the other suppliers would change their prices in a benecial direction.

In the models previously studied, the manner in which each supplier chooses to redistribute his profit was determined a priori, meaning that no supplier had the ability to choose if and how he would cooperate with the other suppliers. The novelty of my project was to add dynamics so that each supplier could choose exactly how to redistribute his wealth in order to maximize his final profit.

Research Process

We first defined an equilibrium in this model as a set of prices such that no supplier has an incentive to deviate from her chosen price (this is Nash Equilibrium in game theory). We then sought to determine the equilibrium that the suppliers would reach without cooperation. We quickly found a closed-form equation that determines this equilibrium.

We also studied the idea of social welfare, or the sum of the profits received by all of the suppliers in this market. We were  able to find a closed-form equation that determines the maximum social welfare that can ever be achieved in any market by cooperating in any manner, as well as the unique equilibrium that leads to this maximal social welfare.

It was at this point that we began to experience diculties and failures. We quickly found that the model includes non-linear (and perhaps non-convex) feedback mechanisms where cooperation affects prices and prices aects cooperation. These feedback mechanisms make an analysis of the model extremely dicult and complex.

We experimented with several manners in which suppliers choose to cooperate. Sadly, each of these resulted in equilibria where some or all suppliers lost prots by cooperating. We then studied the connections that exist between people who decide to cooperate. This resulted in an analysis in which we related the network created by connecting cooperating people to real-world market structures such as departments and supply chains. After a time, however, we realized that this analysis of market structures was not bringing us any closer to understanding the dynamics of self-enforcing cooperation; therefore we abandoned this line of research.

By this point, I was a bit frustrated. All of our planned milestones had passed and we were not much further in the project than we were when I submitted the original proposal. We decided to experiment by reevaluating and simplifying our original model. To our delight, this finally lead us to find signicant and interesting results.

Results

Our simplied model reduced the size of the market to exactly two suppliers. As before, the quantity demanded of the first supplier’s product is negatively affected by its price and positively or negatively affected by the price of the other supplier’s product. The quantity demanded of the second player’s product, however, is only aected by its own price and not by the price of the first player’s product. Our equations that determine the non-cooperative equilibrium point in this market as well as the maximum social welfare possible remained relevant in this modied model.

We then replaced the rules of cooperation presented in the original model with a simple game consisting of the following steps:

1. Both suppliers set their prices at the point that maximizes their individual prots.
2. The rst supplier gives “favor,” or a fixed portion of her profit (which we will call k) to the second
   supplier.
3. The second supplier accepts this favor and adjusts his price in a direction benecial to the first
   supplier. We define personality as the degree of which the second supplier adjusts his price. This
   personality ranges from the second supplier not signicantly changing his price to the second
   supplier changing his price in such a way that the amount he gains from the favor equals the
   amount he loses by changing his price.
4. If the second supplier changes his price, the first supplier adjusts her price in order to maximize
   her own prot. Since this decision does not affect the second supplier, the game ends here.

We were able to prove that, if it is possible for either player to increase his or her prot by cooperating (e.g. the non-cooperative equilibrium does not maximize social welfare), then both players will increase their profits by playing this game. This result is critical in demonstrating self-enforcing cooperation because if both suppliers are guaranteed to benet from cooperating more than by not cooperating, then they have no incentive to deviate away from cooperation.

We proved that there always exists a k, which we call k, such that when the rst supplier gives k to the second player, the game results in the equilibrium that maximizes social welfare. We also prove that the existence of k is independent of the personality of the second supplier. The size of k, however, is dependent on the personality of the second supplier. In addition, the rst supplier will only choose to give k to the second supplier if the second supplier is not interested in making any additional prots from the deal.

We also showed that, in order for the second supplier to maximize his prot, he should keep roughly half of the k that he receives from the rst supplier, and use the other half to compensate for the losses he experiences by adjusting his price. Sadly, this shows that the suppliers will never reach an equilibrium that maximizes social welfare by playing this game. However, it does show that both suppliers have an incentive to play the game, and are guaranteed to increase their prot.

The mathematics of the nal model, the game, and the theorems and results of this project are included in [3] and have not been included in this paper for simplicity and brevity.

Planned Publications

These results have already been compiled into a paper [3] and formatted according to the IEEE submission guidelines and will be ready for submission before the end of this year (2012). We plan on submitting it to the 2013 IEEE Conference on Decision and Control. This work will also play a fundamental role in my Honors Thesis, which will be published at the end of 2013.

Future Work

Now that a simple model is understood, we can extend this model in two fundamental ways. First, we can allow for feedback in pricing and demand. For the two-supplier market, this implies that the quantity demanded for each supplier’s product is aected in some way by the other supplier’s product. Second, we can expand the model to allow for more than two suppliers.

Another potential direction for future work is to apply the results of this project to the controversial Ultimatum game. Without going into details, it is sucient to say that there is a divide between the theoretical and experimental results of the ultimatum game. The game modeled in this project is similar in structure to the ultimatum game, and the results may help to explain this divide.

References

1. N. Tran, and S. Warnick, “Stability Robustness Conditions for Market Power Analysis in Industrial Organization Networks,” Proceedings of the American Control Conference, Seattle, WA, June 2008.
2. T. Brown, N. Tran, and S. Warnick. “Stability Robustness Conditions for Gradient Play Dierential Games with Partial Participation in Coalitions,” Proceedings of the American Control Conference, 2009.
3. N. S. Woodbury and S. Warnick, “A Model of Rational Reciprocity Leading to Self-enforcing Cooperation in a Two-person Supplier Market,” IEEE Conference on Decision and Control, Firenze, Italy, to be published, 2013.