Non-cooperative game theory is presented as the presumed default theory, if not the authoritative one, for the understanding of interpersonal interactions (including consideration of how cooperation may emerge even in the presence of self-seeking behavior and the absence of an ability to make binding agreements[1], with a game formally defined as a situation in which a number of individuals interact in a setting of strategic interdependence[2]). The chapters are devoted to a consideration of what might be called the internal discourse of game theory, in particular various ‘refinements’ of the concept of equilibrium which are deemed to add explanatory force. Thus, the chapters proceed from a consideration of the iterated elimination of dominated strategies, to those of Nash equilibrium, subgame perfect equilibrium, trembling hand perfect equilibrium, perfect Bayesian equilibrium and so forth (how hard it is to tell pianissimo from crescendo!).
After having encountered this battery of concepts — supported by an entire symbolic armamentarium and buttressed by suitably general existence proofs — and having learnt to apply them by doing a sufficient number of exercises, the student is left suitably impressed by the technical sophistication of game theory and eager to apply its insights in the world. However, what has she really learnt? An ‘external’ critic, willing to raise issues outside of the established frame, might introduce the following issues:
The Ultra-Calculating Conception of the Strategic Actor: Uses and Abuses
As we know the model of the homo economicus foregrounded in the textbook portrays individuals as actors seeking to maximize utility (often interpreted in a narrowly self-interested manner) or profits. The text explicitly identifies “rationality” with the possession of complete and acyclical preferences, which it takes to be the foundation of such maximization. Shaikh (2012) uses the term of “hyperrationality” to distinguish this concept of rationality in economics from a more general notion that actions and opinions should be based on reason (a broader view of rationality, as having good reasons to do what it is that one does, which we have also associated with Amartya Sen). However, the agent of game theory, perhaps more than any other agent encountered so far in the textbook, is one who is assumed to be not merely “rational” (indeed hyperrational, in the sense of Shaikh) but to be so in an even more restrictive sense which we refer to as being “ultra-calculating”. The ultra-calculating agent is assumed to be heroically forward looking (anticipating all possible combinations of actions and resulting future states), to engage in elephantine record keeping (recalling all previously known actions and states of the game), and to have unrestricted and costless computational capabilities. Further, an ultra-calculating rational agent will also assume that every other player is engaged in the same calculations, benefiting from the same computational capabilities. Agents possessing such an expansive ‘algorithmic’ capability entails that they can, inter alia:
- Build complex scenarios consisting of multiple layers (for instance, by identifying possible responses to possible responses to possible responses to their actions in order to a form a ‘complete contingent plan’ or strategy; a bedrock of game theoretic reasoning), as well as
- Form complex conjectures about the beliefs that agents hold about one another (exemplified by the assumption called common knowledge — regarding indefinitely iterated alternating layers of belief about one another’s propensity to act “rationally” — in order to identify certain strategies as unlikely to be played or to altogether exclude them).
There are many examples of conclusions that appear to depend on such assumptions concerning the presence of shared information and behavioral propensities (including the capacity to engage in heroic feats of calculation). These include the successive deletion of strictly dominated strategies or processes of backward induction as illustrated, for example, in the centipede game (treated in MWG on pp 281-282, Example 9.B.5). An example of this kind lends itself to ambivalent interpretation. On the one hand, it demonstrates the virtuoso technical prowess of game-theoretical analysis through its ability to reach theoretical conclusions based on specific assumptions. On the other hand, it can also be viewed as a reductio ad absurdum — showing the implausibility of a certain view of strategic interaction rather than the ability of game theory to suggest a determinate and plausible outcome. In the centipede game specifically, this result in striking: there is a unique subgame-perfect Nash equilibrium in which both players cut the game short, each earning only $1 as a result, when they might instead get $100 dollars by continuing it. Is this unique Nash equilibrium a reasonable prediction of what real players might play? Empirical studies[3] show that it is actually rarely observed, and that some level of cooperation is evidenced in the game when it is played. Similarly, there is evidence of cooperative behavior in finitely repeated prisoners’ games where standard reasoning determines it could not arise. If backward induction suggests that rational players will ‘defect’ at the first chance they have to do so in the centipede game, or that no ‘rational’ self-interested prisoner would do anything other than betray the other in the last round (and therefore in every round) of a finitely repeated prisoner’s dilemma, how do we explain that cooperation seems to emerge even in these very scenarios?
Escaping Unrealism: Between Scylla and Charybdis
There are two ways of addressing the embarrassment of unrealism. The first approach is to adopt still greater unrealism in the description of the setting but in such a way as to engineer the required result — while maintaining the premise of the ultra-calculating approach to rationality. For instance, one can assume that the game is infinitely repeated, thus creating the possibility that strategies involving retaliation for non-cooperative behavior can be used to sustain cooperation (or more precisely, its behavioral equivalent, since it is typically supposed that individuals are motivated only by self-seeking considerations). Although this approach generates more realistic empirical consequences it does so by making the description of the setting even less realistic.
The second approach is to adopt greater realism in the description of human agency (i.e. relaxing the premise of an ultra-calculating agent) while maintaining the formal description of the setting. For instance, one may allow for the possibility that individuals are motivated by considerations of fairness, or by adherence to social norms, or allow for more extended conceptions of rationality (such as those involving Smithian enlightened self-interest or Kantian regard for moral law — discussed further below) as a way of directly introducing the foundations of cooperative behavior. Dropping the assumption of common knowledge, and introducing uncertainty in the mind of at least one player about the ‘rationality’ of other players, may lead to accounts of the process of learning in which agents develop knowledge about each others’ actions and beliefs over time (See e.g. Bicchieri 1993) leading to strategies different from those which would otherwise be chosen.
While both approaches can appear to rescue the initial framework from its fatal incapacity to generate plausible conclusions, the latter is decidedly more attractive, as it does away with an assumption (the ultra-calculating approach) that is patently inconsistent with knowledge of ourselves and of others.
Distinct from the problem of the ultra-calculating approach generating determinate but unconvincing results is that of its generating insufficiently determinate results. We can consider, for example, the introduction (in relation to MWG’s discussion of the centipede game on p282) of the concept of trembling-hand perfect Nash equilibrium in order to exclude strategies which would not be played if one’s opponent were to make small mistakes or act in a fashion which was ‘slightly’ irrational. Embracing the possibility of departures from ultra-calculating rationality becomes here a tool for narrowing the range of predicted outcomes.
The extraordinary range of equilibria which is a generic feature of many games (captured for instance in a specific context by the ‘folk theorem’ accommodating for a wide range of possible behavior, see MWG p404[4]), as well as the lack of robustness of the predicted outcomes of games to the way in which the strategic interaction is characterized (an example is provided by the radically different outcomes encountered when inter-firm competition is characterized in terms of quantity-based Cournot competition or price-based Bertrand competition) are among the features which have dispatched the idea that game theory could have very much predictive content. If one can predict (almost) anything then one has predicted nothing.
We can also think of instances in which computational complexity might render the prescribed strategy (even if theoretically computable) completely irrelevant as a practical matter. For instance, if the game of chess is treated as finite (ending if a specified number of moves of a given type occur[5]) it possesses a ‘solution’ in the sense that there exists a (weakly) dominant strategy which necessarily leads to a win or a draw (i.e. either white can force a win or a draw, black can force a win or a draw, or both sides can force a draw – see Aumann (1989), Eichberger (1993), or Hart (1992)). This is a consequence of the proposition described in MWG as Zermelo’s theorem[6] (Prop. 9.B.1 on p272): every finite game of perfect information has a pure strategy Nash equilibrium that can be derived through backward induction. The fact that in chess such a strategy has neither been found nor is likely to be found, even with the aid of the most powerful of computers (which, of course, is exactly what makes the game exciting and worth playing), seems to provide sufficient evidence that the ultra-calculating approach is not a good assumption. It is perhaps unsurprising that the world’s best chess players spend much of their effort on psychic preparations or on efforts to undermine the psychic state of their opponent, and that chess is often thought of as involving a heavy element of intuition which cannot be reduced to brute-force computation (see e.g., here). The surprising conclusion that at least one player in chess has a (weakly) dominant strategy is not especially useful in description, explanation, prediction or prescription. Analogously, in less deterministic contexts than chess, the choice of heuristic decision-making rules (as opposed to the use of computational methods) may be crucial to describing and explaining the choices of actors (and perhaps even to prescribing for them)[7].
Aside from the implausibility of the conception of the agent as a relentless calculating machine, the narrowness of the conception of the agent’s motivations proves another severe limitation. In game theory, the agent is generally assumed to be motivated only by the payoffs realized in the game. If the agent is instead assumed to give any importance at all to moral or expressive concerns relating to the nature of an action (e.g. on whether it is compatible with a certain sense of integrity or personal identity) rather than focusing exclusively upon the outcome to which it gives rise, this can lead to a very different analysis (see e.g. Rabin (1994)). It is often possible to interpret whatever a person appears to be maximizing as that person’s goal (not always: As Sen’s work on menu dependence (Sen 1997, 2002b) has shown, some very reasonable behaviors, and in particular that of acting in accordance with social norms, may not even be compatible with the maximization of a utility function, as it may violate WARP, a necessary condition for such maximization). However, there appears to be little explanatory value to such an approach. Sen (1987, 2002a) undertakes a tripartite classification of the aspects of behavior attributed to the utility maximizing homo economicus (self-centered welfare, self-welfare goals, and self-goal choice). Relaxing any of these behavioral assumptions, or characterizing them in specific ways, may lead to very different conclusions as to what is to be expected in situations of strategic interdependence. Ultimately, a substantive diagnosis of what specific elements of reasoning, psychology and environment guide an individual’s actions is likely to be needed to generate a contentful explanation (on which see also Sen (1977)).
Importantly, individuals may adhere to social norms (which may lead to some recognition being given to other people’s goals, or simply to acting in accordance with relevant contextual requirements). Norms of honesty, promise keeping, and reciprocity, to name just a few, play a critical role in social dynamics. Although there has been a literature attempting to show how the emergence of norms as a result of repetitive social interactions could lend itself to a game-theoretical account[8], there are strong reasons to cast doubt on the ultimate explanatory value of an approach in which norms are viewed as the extended consequence of goal-seeking instrumental behavior (see e.g. Elster (1989, 2009)). Whatever the origins of such norms, if individuals (at least some) act in accordance with them, social interactions and outcomes will be influenced accordingly, as recognized in the game theoretic literature on co-ordination games, as well as on the impact of some players choosing “irrational” out-of-equilibirum play. Moreover, social norms may be activated in sophisticated ways that are contingent on the behavior and orientations of others, as well as on the nature of available strategies and payoffs, and failing to take account of their empirical salience can led to consequent errors in explanation and prediction (i.e. in the very ‘heartand’ of game theory’s claims to relevance: see e.g. Bicchieri (2006), Bowles and Gintis (2013) and Ostrom (1990)).
In the social context, the presence of such procedural considerations can lead to outcomes which differ greatly from those conventionally highlighted in game theory, and as noted above, shed light on the dynamics of cooperative behavior. To take an obvious and insufficiently recognized example, we may consider a version of the prisoner’s dilemma that involves actions with a symmetric private cost and public benefit (such as not littering). Whereas the prisoner’s dilemma in its conventional rendition involves the seemingly inescapable conclusion that all agents will ‘defect’ if they are ‘rational’, one may predict that they will act in a fashion which is exactly the contrary if the agents are assumed instead to be Kantians following the categorical imperative (doing as they would will others to do if they were all to act in accordance with a universal law). Do economists really wish to argue that Kant’s conception of the agent is one that is insufficiently rational? One can have good reasons to do what one does without acting in a fashion that is narrowly instrumental. In particular, acting in accordance with procedural moral criteria may be viewed as perfectly reasonable (Sen (1987); on the lengthy history of such arguments see also Tuck (2008)).
Finally, the agent of game theory also has a ‘hard self’. She is a sovereign who has an unambiguous idea of her payoffs, as well as possible choice, and pursues these relentlessly (as already noted). In fact, various traditions in social ‘science’ recognize that agents are often susceptible to formation by social processes, and by small and large manipulations by others (which may or may not be intentional) that shape their identities, self-understandings and perceived interests. The work of scholars concerned with power [see e.g., from diverse perspectives, Bourdieu (1984), Butler (1997), Fanon (1952), Foucault (1977, 2009), Godelier (1986), Hegel (1807, 1977), Lukes (2005), Said (1993) and Sen (1990)] gives an idea of some of the difficulties that may be encountered if a game theoretic framework is applied to situations in which agents can be influenced to arrive at beliefs and perceived interests, through the effects of a dominating force or power structure that may be diffuse and may incorporate subtle methods such as influencing self-understanding. The actions available to dominant actors may include such possibilities as to influence the perceived payoffs or permissible strategies of others. More generally, even without appeal to the role of power, the self-understandings of agents, including their preferences, may be endogenous to the play of the game (see e.g. Piore (1995)). In such situations, identifying an agent’s interests in terms of payoffs may not be straightforward, but more pertinently still, neglecting the presence of such modes of interaction may lead to a portrayal of the game which is incorrect in terms of its description and therefore risks inaccuracy in explanation and in prediction.
Game Theory 2.0? Prolegomenon
The historical program of game theory might be thought to have been underpinned by at least three components: the predictive, the prescriptive, and the explanatory. The predictive motivation involved the idea that game theory could provide an account of the actions and outcomes likely to emerge in situations of strategic interdependence. The prescriptive motivation involved the idea that game theory could provide a guide to how to act if one sought to bring about certain outcomes (see Mirowski 2002 for discussion of the links between game theory and national security concerns, and in particular the role played by the RAND Corporation and the Cowles Commission in the elaboration of game theory during the Cold War). The explanatory motivation (perhaps the least explicitly articulated) involved the idea that game theory could provide a repertoire of concepts through which one could understand strategic interactions.
How well has game theory done in each of these respects? We have already recognized the fact that game theory generated an ‘embarrassment of riches’ in the form of too many equilibria. Perhaps this helps to explain why game theory, which was a booming field within economics (and in particular in economic theory) in the 1970s and 1980s, replacing general equilibrium theory (after its own encounter with an embarrassment of riches, in the form of the Sonnenschein-Mantel-Debreu results) as the place where promising young economic theorists went to prove themselves, has entered its own disciplinary desuetude (if jobs and monies are any indication).
If game theory has done better as a prescriptive body, it is not because success in prediction has enabled prescription (since there has been little such success) but because of the link between prescription and explanation, and the much greater success of game theory in the latter. Where game theory has succeeded most is in providing a repertoire of concepts that can be used to describe the dynamics of strategic interactions, when applied in conjunction with empirical judgment concerning the relevance of the concepts to specific cases. Such description has in turn provided the basis of more incisive approaches to explanation of observed actions and outcomes, as well as to prescription. Whereas prediction demands a high degree of ex ante determinacy, the use of game theoretic concepts in these other respects does not, replying only on ex ante and ex post usefulness of the concepts developed in order to make sense of a messy world. To the extent there are empirical regularities which deployments of game theoretic concepts can illuminate, these are, even if useful to recognize, of a rather non-specific kind (such as the idea that collective action might be harder to achieve when there are larger numbers of agents, on which see Olson (1965)). This is the relevance that game theory finds in industrial economics, in business strategy, geopolitical analyses and other fields.[9] In applied problems (from understanding macroeconomic coordination of countries to entry deterrence in oligopolies) the language of game theory has proved a valuable aid to interpretation. Where the use of game theory has also, recently, been applied in institutional design, for instance in the design of auctions or in matching algorithms (see here), this is not so much because of its predictive usefulness as because of the theoretical surety it provides that outcomes with specific desired properties (such as generating allocations which cannot be improved upon through further trade, or being difficult to manipulate through means such as the strategic reporting of preferences) can be achieved. However, human beings are ingenious at stepping outside of a specified frame and finding entirely unanticipated methods of making things work to their advantage — reshaping ostensibly well-defined strategies and payoffs and not merely recognizing or respecting them.[10] For this reason, the confidence that game theory provides general principles for institutional design may not be wholly merited.
In the end, game theory has provided us a vocabulary, with a richer range of concepts than had previously been available (we have not discussed all of these concepts above; for instance those appearing in evolutionary game theory, which have a distinct conceptual foundation insofar as they may not rely on intentionality at all, and which incidentally find no mention in MWG). If this is a triumph — and it is, insofar as game theory involves genuine intellectual achievements, including the development of certain non-obvious concepts — it is a smaller one than had been hoped for. It also carries its own dangers. In particular, our means of understanding interpersonal relations may privilege, or even become limited to, the game-theoretic repertoire. Can we escape the tyranny of such formalism (“throw away the ladder” [11]) while deriving whatever insights it may help us to arrive at?
We are left with the following questions. What is the role of empirical investigation concerning specific contexts of strategic interaction and the attitudes as well as outcomes that prevail within them? Can a game (e.g. inter-firm competition) ever be discussed without reference to the meta-games in which it is embedded (e.g. larger institutional context of market, law, government and society) and the possibility of reshaping the meta-game as a way of influencing the outcome of the game? What role do understandings of the structures and mechanisms through which social identities, preferences and perceived interests are shaped, and the way in which these are in turn shaped by intentional and unintentional action, play in the analysis of the dynamics of interdependence? What is the relation between the dynamics of the multiple self [12] (the complex agent containing within herself distinct ideas of permissible strategies and relative payoffs) and the dynamics of multiple selves (interaction between persons)? What about non-equilibrium reasoning, which describes processes by which people react to one another instead of insisting on identifying states defined by mutually compatible reactions? [Is the cascade of actions undertaken by lemmings an ‘equilibrium’? Is this an abstruse example or does it apply to such phenomena as the formation and collapsing of asset bubbles?] Can insights that are not easy to formulate within the existing framework of game theory still find a place in the analysis of situations of strategic interdependence? Is a broader integrated perspective possible in which game theory is treated as one tool of social enquiry rather than the key? What would this imply about the appropriate relation between economics and adjoining disciplines such as sociology and psychology?
Although MWG provides a faithful introduction to game theory, it does so along decidedly conventional, and even complacent lines. The authors cannot be wholly faulted for this, as there is not very much by way of a developed alternative body of theory expressly concerned with strategic interactions.[13] Who will create it?
References
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Footnotes
[1] MWG, Part Two, p.218.
[2] MWG, Ch.7. p.219.
[3] See for example McKelvey and Palfrey (1992) and Nagel and Tang (1998).
[4] The folk theorem is introduced by MWG in the course of applying game theory to market power in Ch. 12.
[5] See e.g. http://en.wikipedia.org/wiki/Rules_of_chess . Note that a draw can be claimed, but need not be claimed, in such a circumstance.
[6] The theorem is named after mathematician Zermelo, who tried to analyze systematically (in an early article published in 1913) the question of whether there existed “winning positions” in chess, from which the other party could be unavoidably checkmated (Zermelo 1913). What is described as Zermelo’s theorem in MWG was not in fact established in Zermelo’s original article. For a modern translation of the original, and a thorough discussion of the subsequent misunderstandings of it, see Schwalbe and Walker (2001), available on www.math.harvard.edu/~elkies/FS23j.03/zermelo.pdf .
[7] For exploration of this insight in the context of individual decision-making, see Gigerenzer (2010).
[8] For references to views of social norms as equilibria of coordination games or products of evolutionary selection see e.g. http://plato.stanford.edu/entries/social-norms/.
[9] See for example Dixit and Nalebuff (1991, 2008), and others.
[10] “Anything you can do, I can do meta”: A quip attributed to the late G.A. Cohen.
[11] See 6.54 on http://people.umass.edu/phil335-klement-2/tlp/frames.html
[12] See e.g., from very different points of view, Elster (1987) and Nandy (1983).
[13] For a critical perspective, however, see Hargreaves Heap, S. and Varoufakis, Y. (2004).
