Broad and narrow knowledge junction
The knowledge junction between the game theory knowledge core and the 118,083 unique references of the philosophical journals resulted into a set of 314 references, the “broad” knowledge junction. By means of manual annotation, the broad knowledge junction was subdivided in the following way: narrow knowledge junction (method-specific references), 233; philosophical references, 8; other references, 73. The number of citing articles (that is, articles published in Synthese or Philosophy of Science in which at least one of the references belonging to the narrow knowledge junction is cited) is 401. The number of citations (that is, the overall number of references made in the 401 articles to items belonging to the narrow knowledge junction) is 907.
Among the references to research areas different from game theory and philosophy, the most common are those to mathematics (especially statistics and probability theory, but also algebra, topology and mathematical logic) and economics (microeconomics, econometrics, etc.). Both cases can be reasonably accounted for: economics has obvious and close connections with game theory, while mathematics has a clear instrumental role. It is interesting that a significant part of such references are handbooks or “classical” texts. Among the cited economics works, for instance, one can find Paul Samuelson’s famous Foundations of Economic Analysis (1947), or Joseph Schumpeter’s Capitalism, Socialism and Democracy (1942). One of the few references to mathematical logic is that of Herbert Enderton’s classic textbook A Mathematical Introduction to Logic (1972). It seems that, when philosophers or game theorists cite works belonging to other disciplines, they preferably cite “reference” works, maybe because of their limited knowledge of the most up to date research, or just because the reasons for citing them are rather “generic”, so that a reference work is the most appropriate citation in that context. At a slightly different level, among the few references that strictly belong to biology one can find Richard Dawkins’s semi-popular The Selfish Gene (1976).
The references to philosophy works are rather few. This probably reflects the relative lack of interest for philosophy among game theorists, or at least among game theorists intent on their technical work and trying to publish on their specialist journals. There is a sense in which this is not surprising: one is not expecting to find many references to philosophy in—say—electrical engineering journals. Yet game theory seems to be different from electrical engineering with respect to its relationship with philosophy, which is certainly closer, as is witnessed by the reverse interest of philosophers in game theory, which is certainly more intense than their interest in electrical engineering. All this raises the rather general issue of the somewhat asymmetrical character of the relationships between philosophy and special disciplines, which—however—should be investigated from a broader point of view, by taking into account a large number of different scientific areas. We leave this for future research. It is worth noting that two among the most cited philosophic references in the knowledge junction are John Rawls’s A Theory of Justice (1971), which is cited in 9 of the 401 citing articles, and David K. Lewis’s Counterfactuals (1973), with 4 citing articles. Both of them are very important and influential books, in the fields of ethics and philosophy of language (or metaphysics) respectively, and their connections with game theory are rather evident.Footnote 9
We have also annotated both the broad and the narrow knowledge junction according to the kinds of works referred to. We distinguished references to articles, books, and handbooks (which include companions, textbooks, etc.). These are the results for the broad knowledge junction: 217 articles, 61 books, 36 handbooks. These are the results for the narrow knowledge junction: 181 articles, 44 books, 8 handbooks. What immediately stands out is the larger share of handbooks in the broad knowledge junction with respect to the narrow. This is accounted for by a phenomenon that has already been pointed out: the relatively common practice of citing “reference” works of disciplines different from both game theory and philosophy.
Empirical and formal references
The narrow knowledge junction was internally analyzed by classifying each article as “empirical” or “formal”.Footnote 10 The two categories of empirical and formal employed in the classification can be explicated as follows. Preliminarily, the two notions were “modelled” in the sense of Betti and van den Berg (2014 and 2016). A cited work is said to be formal just in case game theory is investigated or used in it as a formal discipline, i.e., as part of (applied) mathematics. In other words, the purpose of game theory in the formal sense is that of introducing, discussing or reforming some mathematical models that can be applied to rational behaviour, abstractly conceived. A work is regarded as empirical if it aims at testing a game-theoretic model empirically or if it uses some game-theoretic notions in the context of an empirical research. In other words, empirical works focus on the measurable interactions among empirical subjects such as for example psychological subjects, rather than on the mathematically modelled cooperation or competition among abstract agents. Operatively, empirical and formal references were detected in the dataset by applying some guiding instructions or annotation rules. Following this procedure, we first identified and recognized the works in which the purpose was above all empirical, i.e., the cases in which empirical data were collected and discussed in order to test a game-theoretic mathematical model or a theoretical view, whose formulation required the use of some game-theoretic notions. We labelled “formal” all the remaining game-theoretic works. Notice that for the sake of the present investigation the two notions were treated as mutually exclusive, though in principle it could be argued that they are not so.
Here is the main result of the classification. Among the 907 citations to the 233 unique cited references in our dataset, 212 (23.4%) turned out to be empirical, whereas 695 of them (76.6%) were classified as formal. The different percentages of empirical and formal references are a quite expected result: after all, game theory was born as a formal discipline, so it is not surprising that philosophers use and investigate it, above all, as a formal discipline. However, the very presence of empirical references is a relevant fact in itself, which deserves consideration.
Most cited works belonging to the formal subset are cases of game theory (in the broad sense we use in this paper, which includes social choice theory as well), conceived of as part of mathematics and applied to traditional economic questions such as the problem of social choice in competitive conditions, the calculation of preferences and the issue of equilibrium. Examples of this kind of references are the classical works by John Nash: “The Bargaining Problem” (1950), “Non-Cooperative Games” (1951), and “Equilibrium Points in N-Person Games” (1950). Different kinds of examples are “Independence, Rationality, and Social Choice” by Charles R. Plott (1973); Gerard Debreu, “Continuity Properties of Paretian Utility” (1964); or Amartya K. Sen, “Choice Functions and Revealed Preference” (1971). A relevant and well-represented sub-topic concerns Arrow theorems. Kenneth Arrow occurs in this dataset both as an author (see for example his Social Choice and Individual Values, 1951; Essays in Risk Bearing, 1971; Social Choice and Multicriterion Decision-Making, 1986; “Rational Choice Functions and Orderings”, 1959, and “Existence of an Equilibrium for a Competitive Economy”, coauthored with Debreu in 1954) and as a subject of investigation (see, for instance, Christian List and Philip Pettit, “Aggregating Sets of Judgments: An Impossibility Result”, 2002, or Mark Allen Satterthwaite, “Strategy-Proofness and Arrow’s Conditions: Existence and Correspondence Theorems for Voting Procedures and Social Welfare Functions”, 1975). Yet another sub-topic is decision theory, represented for example by Duncan Black, “On the Rationale of Group Decision-Making” (1948), or Faruk Gul and Wolfgang Pesendorfer, “Temptation and Self-Control” (2003). Within this subgroup, a frequently investigated topic is Condorcet jury theorem: see for example “Information, Aggregation, Rationality, and the Condorcet Jury Theorem” by David Austen-Smith and Jeffrey S. Banks (1996), or “Condorcet Social Choice Functions” by Peter C. Fishburn (1977).
The empirical works that are cited by philosophers in Synthese and Philosophy of Science in the considered period try to complement mathematical game theory with frameworks and especially results taken from more empirically oriented disciplines such as social psychology, cognitive sciences, or biology. Usually, such works present some empirical, above all behavioural or psychological, data, in order to test experimentally a theorem, a model or just a view belonging to or stemming from more mathematically oriented game theory. As some examples can easily show, social psychology and cognitive sciences are the most representative cases. “The Framing of Decisions and the Psychology of Choice” (1981) by Amos Tversky and Daniel Kahneman ultimately belongs to the cognitive sciences, arguing that “the psychological principles that govern the perception of decision problems and the evaluation of probabilities and outcomes produce predictable shifts of preference when the same problem is framed in different ways” (Tversky & Kahneman, 1981, p. 453); similarly, “Thinking Through Uncertainty: Nonconsequential Reasoning and Choice” (1992) by Tversky and Eldar Shafir focuses on cognitive reactions under uncertainty conditions, i.e., on cases in which, as it often happens, people do not consider appropriately each of the relevant branches of a decision tree. Other slightly different though related cases are for example “An Experimental Analysis of Ultimatum Bargaining” (1982) by Werner Güth et al., an empirical study of behaviour in games; “Behavioral Game Theory: Experiments in Strategic Interaction” (2003) by Colin F. Camerer and “Learning in Extensive-Form Games: Experimental Data and Simple Dynamic Models in the Intermediate Term” by Alvin Roth and Ido Erev, two different attempts to complement game theory with psychology. A relevant subgroup is paradigmatically represented by John Maynard Smith’s Evolution and the Theory of Games (1982), in which the interaction is between theory of games and evolutionary theory; similarly, Jack Hishleifer studied “Economics from a Biological Viewpoint” (1977). Yet other kinds of empirical application and testing of game theory concern economics (e.g., “Social Preferences Reveal About the Real World?”, 2017, by Steven D. Levitt and John A. List, which tests a game-theoretic model using laboratory experiments in economics) and the problem of voting (as it happens, for example, in “Standard Voting Power Indexes Do Not Work: An Empirical Analysis” by Andrew Gelman et al., 2004). As to politics, Game Theory and Politics (1975) by Steven J. Brams uses plenty of real-life examples to show how game theory can explain and elucidate complex political situations.
Another interesting aspect concerns the distribution over time of formal and empirical references. Two kinds of data, both represented in Figs. 2 and 3 below, deserve interpretation.
It clearly turns out that philosophers initially dealt with formal game theory, whereas reference to empirical works became relevant only later, arguably from the mid-2000s. That is accounted for partly by the fact that empirical applications or testing of game theory occurred with some delay with respect to the development of game theory as a formal discipline, and partly by the fact that the reception of these studies by philosophers may have added to the delay.Footnote 11 The same phenomena may also account for that the fact that the cited references belonging to the formal subset and those belonging to the empirical subset have a slightly different average age (the number of years between the publication of a work and its citation): 31.5 years for formal references, and 26.9 for the empirical ones.
To analyse the internal structure of the citing papers corpus, we employed the techniques of bibliographic coupling and cluster detection via the Louvain algorithm, as explained in the Methodology section.
The resulting network is visualized in Fig. 4. Nodes represent the 401 papers citing the narrow knowledge junction associated with game theory, whereas links represent the bibliographic coupling links between them. The thickness of the links is proportional to the strength of the bibliographic coupling, i.e., the number of shared references normalized via the Jaccard coefficient. Nodes are positioned on the map based on their bibliographic coupling similarity, so that papers with many common references are placed closer on the map. Their size is proportional to the number of connections they have with other nodes in the network (i.e., the node degree) and their colour indicates the community to which they are attributed by the community detection algorithm. Labels were attributed to clusters based on the data presented below (Table 2).
The Louvain algorithm partitions the network into 7 main communities (plus 1 minor community containing only 1 article). Table 2 shows, for each community and in general, the top five most cited references in each cluster (note that we consider all the references in the bibliographies of the articles, not only those present in the knowledge junction). Table 3 shows the most frequent terms extracted from the abstracts of the papers in each cluster. Terms are defined as n-grams that consist exclusively of nouns and adjectives and that end with a noun (e.g., “game theory”, “text mining”, “network analysis”). Occurrences were calculated by counting the number of abstracts that mentioned each term, independently of whether the term occurred multiple times in the same abstract.Footnote 12 A threshold of 5 occurrences was set for inclusion. The set of terms was further refined by including only the 60% most relevant terms according to VOSviewer relevance score (van Eck and Waltman 2018).Footnote 13 To complete the clusters’ profile, Table 4 presents several statistics associated with them.
Here is a brief illustration of the labels that we attached to the clusters, based on the analysis of the most cited references of each cluster (Table 2) and terms occurring in abstracts (Table 3).
Papers in the dark grey cluster—the largest in the dataset—refers to the issue of uncertainty and rational behaviour: in it we find a mix of empirical and formal references, and the works by Kahneman and Tversky are the most cited ones. Some probability is present—not only Savage’s Foundations of Statistics, but even Keynes’s Treatise of Probability. This set of papers uses some formal methods—logic, probability theory, game theory—together with empirical results obtained in social psychology and cognitive sciences. Apart from the most commonly used terms, which occur in every cluster and are to a large extent irrelevant, the most frequently used terms in this cluster are ‘behaviour’, ‘person’, and ‘uncertainty’.
The green cluster concerns evolutionary game theory. The most cited author is Brian Skyrms, with a special focus on his works on evolutionary game theory and in particular on Signals (Skyrms, 2010). The most frequently used terms are ‘signal’ (the single most frequent term), ‘agent’, ‘cooperation’, ‘emergence’, ‘equilibrium’, ‘information’ and ‘convention’ (a word that clearly refers to the work of David Lewis, as confirmed by the references). Here again there is a mix of formal and empirical references. The story can be approximately reconstructed as follows (Ross, 2019). Initially, biologists and statisticians who worked on the theory of evolution started to use game-theoretic instruments. Later, after game theory had acquired a recognized status, game theorists themselves used in turn tools borrowed from evolutionary theory, extending their application from biology to the social sciences (this process structurally resembles the so-called “reverse imperialism” in the history of economics and social sciences; for the notion of “reverse imperialism” see for instance Davis, 2010).
The pink cluster can be labeled “philosophy and game theory”. The most cited author in this cluster is David Lewis (other frequently cited authors are the economist Thomas Schelling and the mathematician Robert Aumann, but also the moral philosopher David Gauthier); most cited references are philosophical or game-theoretic in the formal sense; the most frequently used terms are ‘model’, ‘rationality’, ‘player’, ‘principle’, ‘convention’.
The light blue cluster refers to philosophy of science (in general). Its key terms are ‘information’, ‘effect’, ‘individual’, and ‘probability’, and many terms associated with conceptual tools typically employed by analytic philosophers are also present. Its most cited authors are James Woodward, Michael Weisberg, Thomas Schelling, Daniel Hausmann, Philip Kitcher. Narrow game-theoretic references seem not to be central.
The violet cluster can be interpreted as “game theory and decision theory”. The most frequently used terms are ‘choice’, ‘order’, ‘belief’. Most references are game-theoretic in the formal sense, or formal though not game-theoretical (probability, mathematics, etc.). The most cited articles are R. Duncan Luce, Games and Decisions (1957) and Richard Jeffrey, The Logic of Decision (1965); Arrow occupies a central position too.
The yellow cluster is “logic and game theory”, or “game theory for logicians”. Key terms are ‘logic’, ‘game’, ‘game-theory’, ‘agent’, ‘player’. Most references are logical or game-theoretic in the formal sense. The most cited work is Martin J. Osborne’s handbook of game theory; other frequently cited authors are logicians such as Jaakko Hintikka or Johan van Benthem.
The orange cluster concerns Arrow theorem, which is often compared to other theorems. Christian List is the most cited author. There is much Arrow. Significant terms are ‘aggregation’, ‘belief’, ‘Arrow’, ‘theorem’. Most cited articles are formal, either game-theoretic or broadly mathematical.
Diffusion of game theory in the reference map of philosophy
As said in the Methodology, bibliographic coupling was used to produce a reference map of philosophy including all the 7976 articles published in Synthese and Philosophy of Science from 1985 to 2021. By overlaying our corpus of interest on this reference map, we can better understand where philosophical papers citing game theory are localized in the general structure of philosophy.
The map is visualized in Fig. 5. Again, each node represents a paper and links the bibliographic coupling similarity between them.Footnote 14 To simplify the visualization, the size of the nodes is fixed, i.e., it does not reflect any property of the nodes. The colour, by contrast, is based on the community to which the nodes are attributed by the Louvain algorithm. For this reference map, the algorithm finds 11 different clusters.
To better characterize and label them, we used again VOSviewer text mining utility to extract the most frequent and relevant terms from the abstracts of each cluster (Table 5).
Considering both the recurring terms and the titles of the articles in each cluster, we associated the eleven clusters with eleven philosophical sub-areas or topics. Starting from the northern part of the map and in clockwise order, they are:
Lewis, metaphysics, possible worlds
Philosophy of mathematics (rule-following considerations)
Philosophy of science (logic of justification and logic of discovery)
Philosophy of physics
Philosophy of science (realism, anti-realism)
Philosophy of science (reduction, causality)
Philosophy of biology
Philosophy of psychology and cognitive science
Decision theory and confirmation theory
Epistemic logic and rational behaviour
The game-theoretic clusters that were identified before can be then highlighted on the reference map (Fig. 6). In this way, we can better understand how the corpus of philosophical articles citing the game theory references are distributed among different philosophical sub-areas.
Two main features of Fig. 6 are worth mentioning. The first is that game theory is especially present in the following philosophical clusters: the cluster ‘philosophy of psychology and the cognitive sciences’ contains in particular references of evolutionary game theory and some empirical/cognitive investigations on uncertainty and rational behaviour; the cluster ‘decision theory and confirmation theory’ contains the game theory clusters ‘game theory and decision theory’, ‘philosophy and game theory’, ‘uncertainty and rational behaviour’, ‘Arrow theorem’; not surprisingly, the cluster ‘epistemic logic and rational behaviour’ especially includes references concerning game theory and logic. The second result is that game theory is nearly absent in many philosophical clusters, such as for example general philosophy of science, philosophy of physics, philosophy of biology. A more thorough discussion of this point will be provided in the final section.