Abstract
Consider a population of farmers who live around a lake. Each farmer engages in trade with his two adjacent neighbors. The trade is governed by a prisoner’s dilemma ‘rule of engagement.’ A farmer’s payoff is the sum of the payoffs from the two prisoner’s dilemma games played with his two neighbors. When a farmer dies, his son takes over. The son decides whether to cooperate or defect by considering the actions taken and the payoffs received by the most prosperous members of the group comprising his own father and a set of his father’s neighbors. The size of this set, which can vary, is termed the ‘span of information.’ It is shown that a larger span of information can be detrimental to the stable coexistence of cooperation and defection, and that in well-defined circumstances, a large span of information leads to an end of cooperation, whereas a small span does not. Conditions are outlined under which, when individuals’ optimization is based on the assessment of less information, the social outcome is better than when optimization is based on an assessment of, and a corresponding response to, more information.
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See, for example, Jun and Sethi (2007) who study comprehensively the impact of the structure of a neighborhood on survival and on the stability of cooperating behavior for an arbitrary number of neighbors, when imitation is the driving force behind natural selection.
In an appendix available on request, we study a span of trade that is larger than that of two immediate neighbors.
Here we assume that the defector appears in the population because of a mutation. Alternatively, a defector could possibly enter the community of farmers via migration. In such a case, the size of the population will become n + 1. The qualitative results of the analysis will hold, however.
Note that even though individuals are living along a road, we use the term ‘cluster’ to refer to a set of at least two neighboring individuals of the same type.
When we have an isolated mutant defector, the requirement of five cooperators to the left and five cooperators to the right to separate the defector from another isolated mutant defector guarantees that two cooperator clusters which are large enough to ‘recapture’ the population survive. If there was one additional isolated defector, we would only need five cooperators on one side of the additional defector to ensure the existence of a third non-vanishing cooperator cluster, because on the other side we have already required presence of five neighboring cooperators. We can conclude that in this sense, the requirement of five neighboring cooperators per isolated mutant defector is a rather stringent condition to guarantee the long-run survival of the cooperating strategy, whereas the requirement of five neighboring cooperators for at least one isolated mutant defectors is the minimal requirement for guaranteeing the survival of at least one (small) cluster of cooperators.
Nonetheless, since in generation 2 \(\bigl( {\mbox{5}\times \mbox{2}P+2\times ({T}+ P\kern.5pt )+2\times ({S} + R \kern.5pt)+5 \times \mbox{2}R =5 \times \raise0.7ex\hbox{{1}} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace}\!\lower0.4ex\hbox{{2}} +2 \times}\) \({ \mbox{1}\raise0.7ex\hbox{{1}} \!\mathord{\big/}\!\lower0.4ex\hbox{{4}} + 2 \times \raise0.7ex\hbox{{3}} \!\mathord{\big/}\!\lower0.4ex\hbox{{4}}+ 3 \times 1 \raise0.7ex\hbox{{1}} \!\mathord{\big/}\!\lower0.4ex\hbox{{2}} = 11 } \bigr)\), while in generation 1 \(\big( 2 T{\kern-.5pt}+{\kern-.5pt} 2 \times (S+R) {\kern-.5pt}+{\kern-.5pt} 9 \times 2 R=2+2\times \raise0.7ex\hbox{{3}} \!\mathord{\big/}\!\lower0.4ex\hbox{{4}}{\kern1pt} +\) \( 9\times 1\raise0.7ex\hbox{{1}} \!\mathord{\big/}\!\lower0.4ex\hbox{{2}}=17 \big)\), the population is worse off.
This result complements the results of Jun and Sethi (2007) who do not study the intersection of ‘global information’ with ‘local interaction.’
If n were an odd number, the condition would be n − 5. Unless otherwise noted, all the other conditions (and results) are valid for both even and odd numbers of farmers.
This finding is in nice congruence with the finding of Haag and Lagunoff (2006, p. 266) that ‘some [spatially less connected] designs are more conductive than others to socially desirable outcomes.’ Note, however, that a Haag-and-Luganoff-type individual is forward-looking and ‘only interacts with, and observes behavior of, those with whom he is linked’ (p. 266). Nonetheless, the analogy of the results is revealing, since Haag and Luganoff study forward-looking agents (with heterogeneous discount factors), whereas we study agents who ‘simply’ imitate past (seemingly successful) strategies.
If we were to relax the assumption that the farmers trade only with their adjacent neighbors and assume instead that they trade also with the adjacent neighbors of their adjacent neighbors, then we could show that the qualitative results reported in the paper carry over to this more general case, provided that an additional, although quite natural set of assumptions on the payoff structure is introduced. The reason for the need to make these additional assumptions is that increasing the number of interactions from two to four increases the number of the possible payoff configurations that have to be compared. An appendix displaying the case of a span of interaction of four, variable spans of information, and the associated per capita payoffs, and illustrating the conditional generalizability of the case analyzed in the paper, is available on request.
Yet another reason could be that the information that individuals marshal depreciates in distance. For example, let more distant information be considered less credible, that is, let the weight attached to information from an individual be inversely proportional to the distance that the information travels (that is, to the distance between farms). Then, information about a mutant defector will spread less aggressively. In such a case, even for large spans of information, a single defector will hardly be able to ‘take over’ the entire community.
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We are indebted to an anonymous referee for searching comments, to Luigi Orsenigo for guidance, and to Marcin Jakubek for helpful advice.
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Stark, O., Behrens, D.A. An evolutionary edge of knowing less (or: On the ‘curse’ of global information). J Evol Econ 20, 77–94 (2010). https://doi.org/10.1007/s00191-009-0137-9
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DOI: https://doi.org/10.1007/s00191-009-0137-9