A model of abstract cooperation in games of uncertainty.

• Author: Bavli, Hillel

1. INTRODUCTION

   Sir Ronald Fisher remarked that it was "Darwin's chief contribution, not only to Biology but to the whole of natural science, to have brought to light a process by which contingencies a priori improbable, are given, in the process of time, an increasing probability, until it is their non-occurrence rather than their occurrence which becomes highly improbable." (1) The idea that evolutionary processes naturally propel a state of affairs toward a higher, perhaps more complex or advanced, state of affairs is one that may extend to any context characterized by a dynamic time frame, including oligopoly models of repeated Prisoner's Dilemma.

   I argue that, contrary to the popular assertion that coordinated pricing necessarily requires voluntary coordination, (2) oligopoly markets may evolve to a state of cooperation--one of collective profit maximization--absent a conscious state of coordination among the players, or even knowledge of such cooperation.

   Professor Donald Turner, in his seminal treatise on the definition of "agreement" under the Sherman Act, (3) touches upon the idea that oligopoly may naturally precipitate parallel non-competitive pricing that may reasonably be considered individual conduct, but stops short of asserting that cooperative equilibria may result without any form of conscious commitment to coordinate prices. Turner argues that oligopoly markets are defined by their interdependent nature and that each player will rationally and naturally calculate the consequences of its price decisions with regard to the expected reactions of its competitors. (4) This explanation does not go far enough.

   While it is true that cooperation is a natural consequence of the interdependent nature of oligopoly markets, it is not necessarily a result of conduct based on conscious regard of future reaction by competitors. While Turner proposes a theory of cooperation based on forward-looking consideration of future reaction, and similarly, George Stigler presents a theory of cooperation based on fear of detection and retaliation, (5) I propose a theory of evolution to cooperation based on the progression of consequences from previous actions.

   Specifically, George Stigler "reasoned that 'oligopolists wish to collude to maximize joint profits' but 'if any member of any agreement can secretly violate it, he will gain larger profits than by conforming to it,' so a model of oligopoly should focus on the 'problem of policing a collusive agreement.'" (6)

   Many economic and legal models determine a firm's ability to detect cheating by analyzing the quality and quantity of information exchanged among firms. (7) For example, industry trade associations are often accused of existing for the sole purpose of facilitating tacit collusion. Stigler's model goes a step further: it allows participants in a collusive arrangement to infer that a rival is secretly cutting prices, and defecting from the arrangement, if they unexpectedly lose many old customers or unexpectedly gain few new customers. (8)

   This paper extends Stigler's model from a theory of tacit interaction and sustained collusion based on a player's ability to detect other players' defections to a theory of independent action based on a player's natural tendency to implement payoff-maximizing strategies by comparing previous performance to current performance and adjusting conduct accordingly. I propose that neither conscious coordination nor information exchange is necessary to achieve cooperative equilibria. Rather, an evolutionary process that parallels Darwinian biological evolution propels economic markets toward states of cooperative equilibrium.

2. WHEN COLLECTIVE AND INDIVIDUAL INCENTIVES CLASH: AN INTRODUCTION TO THE PRISONER'S DILEMMA

   Imagine a situation in which a troublemaker is asked to report to the high school principal's office for investigation of his involvement in a prank. Earlier that morning, the student, Jim, along with one other student, John, released a chicken into the men's restroom. The principal sat the students in separate rooms and spoke to each individually. He informed Jim of the following: Jim, as well as John, was being investigated for pulling a prank earlier that morning, in which he had released a chicken in the men's restroom. Jim had been seen with John entering the men's bathroom during first period classes holding a large brown sack. Neither Jim nor John had followed school protocol, which required that they report to the school office to get a hall pass anytime they left class for any reason, even to use the restroom. For this infringement, the principal had authority to suspend the students for one month. For releasing the chicken in the restroom, the principal had authority to suspend them for eighteen months. The principal, however, lacked sufficient evidence to condemn Jim or John as the culprits of the chicken prank.

   The principal asked that Jim testify regarding John's involvement in the prank. Jim's testimony would provide enough evidence to allow the principal to suspend John for 18 months, and in return, Jim would receive no punishment at all. If, however, John also testified against Jim, each student would be condemned to a twelve-month suspension (their respective eighteen-month suspensions would be downgraded to twelve-month suspensions as reward for their testimonies). If Jim refused to testify but John testified against Jim, Jim would receive an eighteen-month suspension and John would receive no punishment. Finally, if neither of the students testified against the other, the principal would have sufficient evidence only to suspend each of them for one month for failing to obtain a hall pass, but he would lack evidence to suspend them as the chicken prank culprits.

   The foregoing example illustrates a well-known and much analyzed phenomenon dubbed the "Prisoner's Dilemma." (9) The Prisoner's Dilemma occurs when individual incentives and collective incentives clash. Assuming that Jim wished only to minimize the severity of his own punishment, (10) and was indifferent to the severity of John's punishment, his optimal strategy was to testify against John regardless of whether he believed John would testify against him. (11) If John testified, Jim's optimal decision was to testify since a twelve-month suspension is less severe than an eighteen-month suspension. If John refused to testify, Jim's optimal decision was to testify anyway, since no punishment at all is less severe than a one-month suspension, and since Jim was indifferent to the severity of John's punishment.

   Thus, the dilemma is clear: Jim and John, each strategizing individually and in their respective self-interests, would testify against each other and consequently each receive a twelve-month suspension. Had they, however, each strategized to minimize the aggregate severity of their respective punishments--had they "cooperated"--each refusing to testify against the other, they would have achieved the optimal collective outcome, a mere one-month suspension each.

   The Prisoner's Dilemma is "common in everything from personal relations to international relations." (12) It applies to interactions between bacteria, individuals, nations, and corporations (13) and is the foundation of "many of the best-developed models of important political, social, and economic processes." (14) Oligopoly markets are characterized by interdependence among market suppliers ("firms"): each firm's profits are products of the decisions made by other firms in the market. Oligopoly market structures thus embody the elements of the Prisoner's Dilemma and are among the most examined of such contexts. (15)

   Firms can maximize industry profits by behaving like a single monopolist and then dividing the profits among themselves. (16) The problem is that each firm has the temptation to cheat; (17) each firm individually maximizes its profit in any round of decision-making by expanding its output beyond the agreed-upon level (or, alternatively, by decreasing its price below the agreed-upon monopoly price).

   Thus, once again, a dilemma unfolds: each firm, acting in its individual self-interest, chooses to overproduce (or underprice). (18) A competitive, profit-minimizing price emerges. Had the firms cooperated by foregoing the opportunity to overproduce (or undercut the others' prices), a monopolistic, profit-maximizing price would have surfaced.

   The situations, or "games," described above are such that "players" each simultaneously make a single decision. Absent a mechanism of enforcing cooperation, the Prisoner's Dilemma holds true and the players forego the collectively optimal outcome. No player has reason to trust that the others will forego acting for the individual good for the sake of the collective good. A different outcome results, however, if the game is played repeatedly by the same players. (19)

   "Repeated play allows players to respond to each other's actions, and so each player must consider the reactions of his opponents in making his decision." (20) Repeated play provides a critical enforcement mechanism: If a player cheats, or "defects" from a collusive arrangement in one period, the other players can "punish" him by defecting in future periods. (21) "In a repeated game, each player has the opportunity to establish a reputation for cooperation, and thereby encourage the other player to do the same." (22) The presumption, of course, is that players have the ability to detect when other players are cheating. Therefore, models of cooperation often focus on players' ability to detect defection.

3. GAME THEORY: A CONCISE REVIEW

   Game theory is a "collection of tools for predicting outcomes (23) for a group of interacting agents, where an action of a single agent directly affects the payoffs (welfare or profits) of other participating agents." (24) Game theory is particularly useful in situations where the number of players is small and the decisions of each...