Cooperation in an infinite-choice continuous-time prisoner's dilemma.(Special Issue: Teaching Strategic Management)

KEYWORDS: computer simulation; cooperation; experimental gaming; joint decision making; Prisoner's Dilemma.

The Prisoner's Dilemma (PD) has enjoyed widespread popularity in the social sciences over the last 35 years. As evidence, research in psychology, sociology, economics, communication, and political science has spawned literally thousands of studies using the PD or variations on the PD. The dilemma's popularity can be attributed to its versatility as a game theoretic vehicle to study conflict. In this article, we introduce a variation of the PD in which the players have an infinite number of choices and make those choices continuously rather than at discrete points in time.

After reviewing related attempts to expand the generalizability of the PD framework, we construct a true infinite-choice, continuous-time PD. Then we look at some preliminary data showing the types of cooperation patterns. Finally, we discuss the implication of these data for the PD paradigm in particular and the study of cooperation and conflict in general.

The Classic PD

The principle behind the PD game is to show how cooperative or competitive choices can influence the result of two people or groups making a decision (Ashmore, 1987). The PD thus embodies the tension between collective and individual interests (Tutzauer, Feeley, & Young, 1995). In the classical PD, each participant must make a choice between two actions, typically labeled cooperation and defection. The game is structured so that regardless of what the opponent does, defecting is more profitable than cooperating. The dilemma is that if both of the players defect, then both are worse off than if they had cooperated (see Table 1).

TABLE 1: Payoff Matrix of the Classical Prisoner's Dilemma

 
                                Y's Choices 
 
X's Choices           Cooperate            Defect 
 
Cooperate               (3,3)                (0,5) 
Defect                  (5,0)                (1,1) 

The matrix depicted in Table 1 is the one used by Axelrod (1980a, 1980b) in his computer tournaments of the iterated PD. Although other payoff matrices exist that embody the structural elements of the PD, Axelrod's work has been so influential that we might call Table 1 the canonical PD payoff matrix.(1)

As Table 1 shows, the classical PD quite admirably captures the tension between individual and group rationality; nonetheless, certain other facets of the game limit its generalizability to everyday conflict situations. First, in the classical PD, players have but two options, and although some conflicts can be so characterized (e.g., launch your missiles or do not), in most situations the participants have a variety of options from which to choose. For example, a country may devote all of its wealth to the building of arms, none of its wealth, or any proportion in between. Thus, although there may be bounds to the permissible actions available to the players (all of the wealth or none of it), there nonetheless is an infinite number of choices within the bounds (Tutzauer et al., 1995).

In a similar manner, the conflict typically takes place on a temporal continuum rather than at discrete points in time. The standard PD paradigm today involves the so-called iterated PD, in which the game is played not once, but many times (see, e.g., Axelrod, 1980a, 1980b). Although an iterated PD is certainly more flexible than a one-shot PD, again the continuous nature of the interaction is ignored. Consider a verbal dispute between coworkers. The choices of what to say and how to say it are not made in discrete rounds; rather, the disputants make their choices at every moment--that is to say, continuously.

This article will construct and investigate an infinite-choice, continuous-time PD. After reviewing related attempts in the literature, the usefulness of such an approach is illustrated by conducting an experiment whereby subjects are offered bounded infinite-choice over a continuous 5-minute time period.

Previous Extensions of the PD

Calls to expand the number of choices in the PD are not novel, but there has been surprisingly little empirical or theoretical work. To (1988) conducted a computer tournament using a 5 x 5 payoff matrix. As in tournaments involving the classical PD (Axelrod, 1980a, 1980b), the strategy tit for tat (TFT), which cooperates on the first move and matches the opponent's previous moves thereafter, performed well in To's tournament. Because the payoff matrix was expanded to five choices each, however, some variants of TFT were also possible (e.g., cooperating at one degree less than the opponent). These strategies also performed well; one of them even beat TFT.

Andreoni (1988) offered participants in a study 50 tokens to invest in a private good or a public good. Subjects played 10 trials of the game in either a strategy or a learning condition. Results indicated that subjects invested in private goods (i.e., defected) rather than the public good (i.e., cooperation) as the game progressed.

In a study examining the benefits of using relaxed versus restrictive accounting systems, Kollock (1993) pitted strategies using varying degrees of cooperation (from 0 to 10 points) against each other. He also manipulated noise in the environment--that is, the degree to which there were errors in the perception of the opponent's behavior. Results showed that varying degrees of cooperation gave more relaxed accounting strategies (i.e., more forgiving strategies) an advantage in many environments. By contrast, the more restrictive accounting systems (e.g., TFT) were disadvantageous in a cycling or noisy environment.

Even more choices were used by Fleishman (1988), who constructed an n-person social dilemma (e.g., the tragedy of the commons) in which each player could contribute up to 100 "resource units" (p. 168) to a common pool. The expansion to 100 units gave subjects greater flexibility in choosing cooperation. Participants played nine trials of the game in groups of four or five players. Fleishman found an interaction between decision frame (whether framed as giving to the common pool or taking from it) and the …