Several Iterated Prisoner’s Dilemma tournaments have generated much interest; Axelrod’s original tournaments [2, 3], two 2004 anniversary tournaments , and the Stewart and Plotkin 2012 tournament , following the discovery of zero-determinant strategies. Subsequent research has spawned a number of papers (many of which are referenced throughout this paper), but rarely are the results reproducible. Amongst well-known tournaments, in only one case is the full original source code available (Axelrod’s second tournament , in FORTRAN). In no cases is the available code well-documented, easily modifiable, or released with significant test suites.
To complicate matters further, a new strategy is often studied in isolation with opponents chosen by the creator of that strategy. Often such strategies are not sufficiently described to enable reliable recreation (in the absence of source code), with  being a notable counter-example. In some cases, strategies are revised without updates to their names or published implementations [25, 26]. As such, the results cannot be reliably replicated and therefore have not met the basic scientific criterion of falsifiability.
This paper introduces a software package: the Axelrod-Python library. The Axelrod-Python project has the following stated goals:
As stated in : “few works in social science have had the general impact of [Axelrod’s study of the evolution of cooperation]”. In 1980, Axelrod wrote two papers: [2, 3] which describe a computer tournament that has been a major influence on subsequent game theoretic work [5, 6, 7, 8, 10, 11, 12, 13, 15, 18, 23, 24, 27, 32, 33, 34, 36, 41, 42]. As described in  this work has not only had impact in mathematics but has also led to insights in biology (for example in , a real tournament where Blue Jays are the participants is described) and in particular in the study of evolution.
The tournament is based on an iterated game (see  or similar for details) where two players repeatedly play the normal form game of (1) in full knowledge of each other’s playing history to date. An excellent description of the one shot game is given in  which is paraphrased below:
Two players must choose between Cooperate (C) and Defect (D):
and the following reward matrix results from the Cartesian product of two decision vectors 〈C, D〉,
The game of (1) is called the Prisoner’s Dilemma. Specific numerical values of (R, S, T, P) = (3, 0, 5, 1) are often used in the literature [2, 3], although any satisfying the conditions in 1 will yield similar results. Axelrod’s tournaments (and further implementations of these) are sometimes referred to as Iterated Prisoner’s Dilemma (IPD) tournaments. An incomplete representative overview of published tournaments is given in Table 1.
|Year||Reference||Number of Strategies||Type||Source Code|
|1979||||13||Standard||Not immediately available|
|1979||||64||Standard||Available in FORTRAN|
|1991||||13||Noisy||Not immediately available|
|2002||||16||Wildlife||Not a computer based tournament|
|2012||||13||Standard||Not fully available|
In  a description is given of how incomplete information can be used to enhance cooperation, in a similar approach to the proof of the Folk theorem for repeated games . This aspect of incomplete information is also considered in [6, 24, 33] where “noisy” tournaments randomly flip the choice made by a given strategy. In , incomplete information is considered in the sense of a probabilistic termination of each round of the tournament.
These works investigate particular evolutionary contexts within which cooperation can evolve and persist. This can be in the context of direct interactions between strategies or population dynamics for populations of many players using a variety of strategies, which can lead to very different results. For example, in  a machine learning algorithm in a population context outperforms strategies described in  and  that are claimed to dominate any evolutionary opponent in head-to-head interactions.
Further to these evolutionary ideas, [8, 10] are examples of using machine learning techniques to evolve particular strategies. In , Axelrod describes how similar techniques are used to genetically evolve a high performing strategy from a given set of strategies. Note that in his original work, Axelrod only used a base strategy set of 12 strategies for this evolutionary study. This is noteworthy as the library now boasts over 139 strategies that are readily available for a similar analysis.
The library is written in Python (http://www.python.org/) which is a popular language in the academic community with libraries developed for a variety of uses including:
Furthermore, in  Python is described as an appropriate language for the reproduction of Iterated Prisoner’s Dilemma tournaments due to its object oriented nature and readability.
The library itself is available at https://github.com/Axelrod-Python/Axelrod.
As stated in the Introduction, one of the main goals of the library is to allow for the easy contribution of strategies. Doing this requires the writing of a simple Python class (which can inherit from other predefined classes). All components of the library are automatically tested using a combination of unit, property and integration tests. These tests are run as new features are added to the library to ensure compatibility (they are also run automatically using travis-ci.org). When submitting a strategy, a simple test is required which ensures the strategy behaves as expected. Full contribution guidelines can be found in the documentation, which is also part of the library itself and is hosted using readthedocs.org. As an example, Figures 1 and 2 show the source code for the Grudger strategy as well as its corresponding test.
You can see an overview of the structure of the source code in Figure 3. This shows the parallel collection of strategies and their tests. Furthermore the underlying engine for the library is a class for tournaments which lives in the tournament.py module. This class is responsible for coordinating the play of generated matches (from the match.py module). This generation of matches is the responsibility of a match generator class (in the match_generator.py module) which is designed in such a way as to be easily modifiable to create new types of tournaments. This is described further in a tutorial in the documentation which shows how to easily create a tournament where players only play each other with probability 0.5. This will be discussed further in the reuse section of this paper.
To date the library has had contributions from 26 contributors from a variety of backgrounds which are not solely academic. These contributions have been mostly in terms of strategies. One strategy is the creation of an undergraduate mathematics student with little prior knowledge of programming. Multiple other strategies were written by a 15 year old secondary school student. Both of these students are authors of this paper. As well as these strategy contributions, vital architectural improvements to the library itself have also been contributed.
The Axelrod library runs on all major operating systems: Linux, Mac OS X and Windows.
The library is continuously tested for compatibility with Python 2.7 and the two most recent python 3 releases.
There are no specific additional system requirements.
Support is readily available in multiple forms:
The following Python libraries are required dependencies:
The names of all the contributors are not known: as these were mainly done through Github and some have not provided their name or responded to a request for further details. Here is an incomplete list:
Persistent identifier: 10.5281/zenodo.55509
Publisher: Vincent Knight
Version published: Axelrod: 1.2.0
Date published: 2016-06-13
Date published: 2015-02-16
The Axelrod library has been designed with sustainable software practices in mind. There is an extensive documentation suite: axelrod.readthedocs.org/en/latest/. Furthermore, there is a growing set of example Jupyter notebooks available here: https://github.com/Axelrod-Python/Axelrod-notebooks.
The availability of a large number of strategies makes this tool an excellent and obvious example of the benefits of open research which should positively impact the game theory community. This is evidently true already as the library has been used to study and create interesting and powerful new strategies.
Installation of the library is straightforward via standard python installation repositories (https://pypi.python.org/pypi). The package name is axelrod and can thus be installed by calling: pip install axelrod on all major operating systems (Windows, OS X and Linux).
Due to the open nature of the library the number of strategies included has grown at a fast pace, as can be seen in Figure 6.
Nevertheless, due to previous research being done in an irreproducible manner with, for example, no source code and/or vaguely described strategies, not all previous tournaments can yet be reproduced. In fact, some of the early tournaments might be impossible to reproduce as the source code is apparently forever lost. This library aims to ensure reproducibility in the future.
One tournament that is possible to reproduce is that of . The strategies used in that tournament are the following:
In parallel to the Python library, a tournament is being kept up to date that pits all available strategies against each other. Figure 9 shows the results from the full tournament which can also be seen (in full detail) here: http://axelrod-tournament.readthedocs.org/. Data sets are also available showing the plays of every match that takes place. Note that to recreate this tournament simply requires changing a single line of the code shown in Figure 4, changing:
>>> strategies = [s() for s in axelrod.demo_strategies]}
>>> strategies = [s() for s in axelrod.ordinary_strategies]}.
The current winning strategy is new to the research literature: Looker Up. This is a strategy that maps a given set of states to actions. The state space is defined generically by m, n so as to map states to actions as shown in (2).
The example of (2) is an incomplete illustration of the mapping for m = 8, n = 2. Intuitively, this state space uses the initial plays of the opponent to gain some information about its intentions whilst still taking into account the recent play. The actual winning strategy is an instance of the framework for m = n = 2 for which a particle swarm algorithm has been used to train it. The second placed strategy was trained with an evolutionary algorithm . In  experiments are described that evaluate how the second placed strategy behaves in environments other than those in which it was trained and it continues to perform strongly.
There are various other insights that have been gained from ongoing open research on the library, details can be found in . These include:
This paper has presented a game theoretic software package that aims to address reproducibility of research into the Iterated Prisoner’s Dilemma. The open nature of the development of the library has lead rapidly to the inclusion of many well known strategies, many novel strategies, and new and recapitulated insights.
The capabilities of the library mentioned above are not at all comprehensive, a list of the current abilities include:
These capabilities are constantly being updated.
The authors would like to thank all contributors. Also, they thank Robert Axelrod himself for his well wishes with the library.
The authors declare that they have no competing interests.
Astropy Collaboration et al. (2013). “Astropy: A community Python package for astronomy”. Astronomy and Astrophysics Oct. 2013558: A33.DOI: https://doi.org/10.1051/0004-6361/201322068 arXiv: 1307.6212 [as-tro-ph.IM].
Axelrod, R (1980). “Effective Choice in the Prisoner’s Dilemma”. Journal of Conflict Resolution 24(1): 3–25, DOI: https://doi.org/10.1177/002200278002400301
Axelrod, R (1980). “More Effective Choice in the Prisoner’s Dilemma”. Journal of Conflict Resolution 24(3): 379–403, 0022-0027DOI: https://doi.org/10.1177/002200278002400301
Banks, J S and Sundaram, R K (1990). “Repeated games, finite automata, and complexity”. Games and Economic Behavior 2(2): 97–117, 08998256DOI: https://doi.org/10.1016/0899-8256(90)90024-O
Bendor, J, Kramer, R M and Stout, S (1991). “When in doubt …: Cooperation in a noisy prisoner’s dilemma”. Journal of Conflict Resolution 35(4): 691–719, 0022-0027DOI: https://doi.org/10.1177/0022002791035004007
Boyd, R and Lorberbaum, J P (1987). “No pure strategy is evolutionarily stable in the repeated Prisoner’s Dilemma game”. Nature 327: 58–59, 0028-0836DOI: https://doi.org/10.1038/327058a0
Chellapilla, K and Fogel, D B (1999). “Evolution, neural networks, games, and intelligence”. Proceedings of the IEEE : 1471–1496, 87(9)00189219DOI: https://doi.org/10.1109/5.784222
Crick, T et al. (2014). ““Share and Enjoy”: Publishing Useful and Usable Scientific Models”. arXiv: 1409.0367.
David, B F (1993). “Evolving Behaviors in the Iterated Prisoner’s Dilemma”. Evol. Comput : 77–97, 1(1)1063-6560DOI: https://doi.org/10.1162/evco.19220.127.116.11
Doebeli, M and Hauert, C (2005). “Models of cooperation based on the Prisoner’s Dilemma and the Snowdrift game”. Ecology Letters : 748–766, 8(7)1461023XDOI: https://doi.org/10.1111/j.1461-0248.2005.00773.x
Ellison, G (1994). “Cooperation in the prisoner’s dilemma with anonymous random matching”. Review of Economic Studies (567): 588.61(3)00346527DOI: https://doi.org/10.2307/2297904
Gotts, N, Polhill, J and Law, A (2003). “Agent-based simulation in the study of social dilemmas”. Artificial Intelligence Review 19: 3–92, 0269-2821DOI: https://doi.org/10.1023/A:1022120928602
Harper, M (2015). Marc Harper Codes, http://marcharper.codes/2015-11-17/ipd2.html
Hilbe, C, Nowak, M A and Traulsen, A (2013). “Adaptive Dynamics of Extortion and Compliance”. PLoS ONE 8(11): e77886.1932-6203DOI: https://doi.org/10.1371/journal.pone.0077886
Hong, N P C et al. (2015). “Top Tips to Make Your Research Irreproducible”. : 5–6. arXiv: 1504.00062.
Hunter, J D (2007). “Matplotlib: A 2D graphics environment”. Computing In Science & Engineering : 90–95, 9(3)DOI: https://doi.org/10.1109/MCSE.2007.55
Jones, M (2015). Evolving strategies for an Iterated Prisoner’s Dilemma tournament, http://mojones.net/evolving-strategies-for-an-iterated-prisoners-dilemma-tournament.html
Kraines, D and Kraines, V (1989). “Pavlov and the prisoner’s dilemma”. Theory and Decision 26(1): 47–79, 00405833DOI: https://doi.org/10.1007/BF00134056
Lee, C, Harper, M and Fryer, D (2015). “The Art of War: Beyond Memory-one Strategies in Population Games”. Plos One 10(3): e0120625.1932-6203DOI: https://doi.org/10.1371/journal.pone.0120625
Li, J (2007). “How to design a strategy to win an IPD tournament”. The iterated prisoners dilemma 20: 89–104, DOI: https://doi.org/10.1142/9789812770684_0004
Li, J, Hingston, P and Kendall, G (2011). “Engineering design of strategies for winning iterated prisoner’s dilemma competitions”. Computational Intelligence and AI in Games, IEEE Transactions on 3(4): 348–360, DOI: https://doi.org/10.1109/tciaig.2011.2166268
Lorberbaum, J P (1994). “No strategy is evolutionarily stable in the repeated Prisoner’s Dilemma game”. Journal of Theoretical Biology 168(2): 117–130, DOI: https://doi.org/10.1006/jtbi.1994.1092
Maclver, D R (2016). Hypothesis 3.0.3, https://github.com/DRMacIver/hypothesis
Maschler, M, Solan, E and Zamir, S (2013). Game theory. Cambridge University Press, 10039781107005488DOI: https://doi.org/10.1017/CBO9780511794216
Molander, P (1985). “The optimal level of generosity in a selfish, uncertain environment”. The Journal of Conflict Resolution : 611–618, 29(4)0022-0027DOI: https://doi.org/10.1177/0022002785029004004
Press, W H and Dyson, F J (2012). “Iterated Prisoner’s Dilemma contains strategies that dominate any evolutionary opponent”. Proceedings of the National Academy of Sciences 109(26): 10409–10413, 0027-8424DOI: https://doi.org/10.1073/pnas.1206569109
Prlik, A and Procter, J B (2012). “Ten Simple Rules for the Open Development of Scientific Software”. PLoS Computational Biology 8(12): e1002802.1553-7358DOI: https://doi.org/10.1371/journal.pcbi.1002802
Sandve, G K et al. (2013). “Ten Simple Rules for Reproducible Computational Research”. PLoS Computational Biology 9(10): 1–4, 1553734XDOI: https://doi.org/10.1371/journal.pcbi.1003285
Stephens, D W, McLinn, C M and Stevens, J R (2002). “Discounting and reciprocity in an Iterated Prisoner’s Dilemma.”. Science New York, N.Y.: 298(5601): 2216–2218, 00368075DOI: https://doi.org/10.1126/science.1078498
Stewart, A J and Plotkin, J B (2012). “Extortion and cooperation in the Prisoner’s Dilemma”. Proceedings of the National Academy of Sciences 109(26): 10134–10135, 0027-8424DOI: https://doi.org/10.1073/pnas.1208087109
The Sage Developers (). Sage Mathematics Software (Version 7.0), http://www.sagemath.org 7.0.