Abstract
Chapter 4 adopts a history of economic thought approach to show that Bacharach, an economist trained in mathematics, contributed to integrating many of the dimensions of Schelling’s reorientation of game theory into his work in game theory. He integrated games with another intersubjective dimension, which, similar to Schelling’s focal points, relied on both subjective and collective dimensions. We show in particular that Bacharach’s theoretical contribution suggests a potential formal solution to a new game theory based on a new form of intersubjectivity that involves integration of players’ mental states but without the difficulties inherent in standard and epistemic game theory underlined in Chap. 2. We underline Bacharach’s contribution in the form of the questions he posed and his highlighting of the limits to game theory that are rarely discussed. He formally extended standard game theory while simultaneously highlighting the porosity of some of its boundaries and the impermeability of others. We show that both of his theories, the variable frame theory and the team reasoning theory, are critical of the standard and closed-system mathematical boundaries to game theory. At the core of Bacharach’s work is an open-systems methodology.
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Notes
- 1.
Bacharach’s interdisciplinarity does not end there, later in his career he was heavily involved in the development of philosophical thinking in economics.
- 2.
Bacharach’s explicit reference to frames emerged later in his work on VFT. Reference to frames rather than perceptions however does not distort this contribution since perceptions and frames are equivalent for Bacharach (if we leave aside the formal aspect of framing developed in the VFT). Reference to frames in our context helps to demonstrate how Bacharach’s concern about the impact of perceptions in individual decision-making emerged early in his career and provides evidence of a common underlying epistemology in his work.
- 3.
Bacharach’s reference to language and to the term “conceptual” allows contributions from linguistics and philosophy of logic. We show below that Bacharach later resorted to semantic and syntactic logic to question the validity of game theoretic models and the way players’ knowledge is standardly defined in particular.
- 4.
Such critique echoes Schumpeter’s (1940) argument that a major problem in social sciences is distinguishing between the rationality of the observer and the rationality of the observed. It implies also that Bacharach was cautious about mathematization in economics. He does not accept a purely mathematized conception of economics. We discuss this concern in Sect. 4.2.2 in the context of game theory.
- 5.
Bacharach (1986, p. 189) claimed that the standard theories of belief in economics “are all deductive theories; they are all aprioristic; they are all rationalistic.”
- 6.
This applies to beliefs as formalized by subjective Bayesian probabilities for Bayesian revision, for the beliefs supporting a NE. For Bacharach (1986, p. 193) none of these accounts of individual beliefs is rational or supported by a process of rational reasoning.
- 7.
The peripheral debates that Bacharach mentions concern the refinement program and questioned the notion of strategic rationality, i.e., n why players should play a particular equilibrium.
- 8.
More specifically, the EP requires that “saying that a particular game” has a solution" implies that (i) for each player of the game there is an action α such that it is in some sense good for him to do α. Perhaps less obviously, it implies that (ii) the player is in a position to come to know that this is true of α. And it implies thirdly, at least for many writers, that (iii) for each player there is exactly one such α” (Bacharach 1987, p. 35).
- 9.
See Arena and Larrouy (2016) for more detail on those two forms of knowledge and their roles in individual decision-making.
- 10.
Bacharach’s conception of framing is very different from Kahneman and Tversky’s (1979, 2000) notion which is commonly used in economics and the “framing effect literature.” Bacharach retains from Kahneman and Tversky’s “prospect theory” (1979) the idea that individuals’ beliefs and preferences depend on their subjective descriptions (representations) of the world (Bacharach, 1986, p. 183). However, Bacharach distances himself from Kahneman and Tversky and the framing effect literature by being interested only in natural framing, i.e., in the absence of “manipulations” designed by the theorists to affect individuals’ decision making (2001, p. 4).
- 11.
For Bacharach (1987, 1994) a valid theory of games must be grounded on a metatheory of games. More specifically, a metatheoretic game theory “does not regard the game directly, but regard a feature of the game theorists’ theory of it, namely, the ‘logical closure of the information attributed to the players in this theory’s assumptions” (Bacharach, 1994, p. 8). For instance, “it allows one to focus attention on the logical relations between what players know just before they choose and the “normal form information” with which they are traditionally credited; more specifically on how they know just before they choose depends on what can and cannot be proved from their normal form information” (ibidem). In addition to axioms must specify the structure of games, they enclose: “the “set-up” (the available actions and preferences); the players’ knowledge about this and about each other; and what it is that makes actions “satisfactory” and how “satisfactoriness” relates to choice” (Bacharach, 1987, p. 17). In addition, such a theory must specify what is rational for a player to do, given her information.
- 12.
To quote Hargreaves Heap (2004, p. 31), in standard game theory, “individuals know the rules of the game: that is, they know all the possible actions and how the actions combine to yield particular payoffs for each player.”
- 13.
Bacharach uses the labels “concept,” “attribute,” and “predicate” to refer to the same semantic. However, note that “attribute” is the term generally used by psychologists in the framing literature, whereas the other terms are specific to Bacharach.
- 14.
The definition of a matching game is that it is “a pure coordination game in which there are two players with the same act-set; both get a prize if and only if both choose the same act; and the prize is the same whatever this act may be” (Bacharach & Bernasconi, 1997, p. 2).
- 15.
However, Bacharach does not make a clear distinction between these two concepts. This first ambiguity raises another, more noticeable in the 1991 paper. There is no clear account in Bacharach’s VFT about when the first unconscious phase ends.
- 16.
This matrix example is from Bacharach and Stahl (2000, p. 222).
- 17.
In this case, the player does not perceive the shape of the bottle.
- 18.
The empty frame for Bacharach is a subframe of the shape frame.
- 19.
The possible equilibrium according to this payoff matrix is discussed in Subsect. 4.3.3.
- 20.
Those postulates echo Bacharach’ (1997a, 1997b) work on investigating the sources of CK. What he called “the shared environment approach” could explain the genesis of CK. He refers to Gilbert (1981) and Clark and Marshall (1981) to support this statement. He gives the example of shared environment and explains the epistemic consequences of such an environment in the following quotation: “the carafe situation is a shared environment situation in which f is the presence of the carafe, since it seems that both you and I will recognize the (seeing the carafe) situation as one which both of us will recognize. However, this still leaves the question: what enables us to recognize it as this? An appealing answer was first given by Lewis (1969) and Schiffer (1972), based on the notion of “normality” (Bacharach, 1992a, 1992b gives an axiomatic version). A normal human will not only see the carafe, but will also see the normality of the other copresent normal; last, normality has the reflexive property that it is part of being normal to know the perceptual and epistemic capacities of normal people” (Bacharach, 1997b, p. 4).
- 21.
Some other characteristics proper to the rules of the games are also assumed to be mutual knowledge. These characteristics are: (i) the “admissibility condition,” i.e., if there is only one object which possesses a given characteristic a player must choose that object; (ii) the “preference condition,” i.e., players attach utility 1 to the fact that they choose the same object or 0 otherwise; (iii) the number of objects which possess a given characteristic, say, b; (iv) the fact that a player can pick a nonunique object at random; and (v) the fact that each player is rational in the sense in which Bacharach defines rationality in this paper (i.e., via the different principles of rational choice which we explain in Subsect. 4.3.3).
- 22.
In this respect, Janssen (2001) is critical of Bacharach’s approach which aims to respecify games to solve the problem of coordination (based on the principle of coordination) but in some games players are compelled to choose at random, which for Janssen means “choosing a “nonsolution”” (Janssen, 2001, p. 124).
- 23.
See Bacharach (1987) for this connection between the equilibrium, the solution, and the principle of rational play which must be part of the theory of games in order for that theory to be considered a theory of rationality in interaction (see also Luce & Raiffa, 1957; von Neumann & Morgenstern, 1944).
- 24.
In the VFT, conspicuousness is formalized more specifically by the availability function (see also Janssen, 2001, p. 126).
- 25.
- 26.
- 27.
This account suggests that a metatheory of framing in Bacharach’s VFT is required to explain why frames are structured in one way or another.
- 28.
In fact, Bacharach asserts “the agent’s frame “blinkers” her: it is too narrow to enable her to see a highly relevant possibility. This possibility is that her coplayer is in the second simple frame” (ibid, p. 14).
- 29.
This contribution echoes Bacharach’s later work on level-k theory (see Bacharach & Stahl, 2000).
- 30.
Bacharach explains his choice to refer to a “state” and not a “type.” He explicitly recognizes that “type” may be a confusing term: “[w]e can think of i as being one of two types in the sense of incomplete information game theory: participating, in which case she TR for M [the team], or lapsing. However, since “type” may misleadingly suggest a permanent trait, I shall speak not of i’s “type” but for her “state”” (ibid, p. 121). This position echoes the limits identified in his 1995 paper, i.e., the need to resort to a natural trait.
- 31.
- 32.
The importance of common interest for the collective triggered numerous debates in philosophy with respect to the nature of a collective (see Gold, 2018; Gold & Sugden, 2007). Common interest is the pillar of Margaret Gilbert’s account of a plural subject in philosophy. For Gilbert (1989), when people have a common interest they cease to act individually and become a plural subject acting in concert to perform an action.
- 33.
For Bacharach, S frame players have basic altruist preferences (ibid, p. 25), i.e., a specific form of individual preference.
- 34.
Bacharach (1997a, 1997b, p. 17) defines the concept of reason dominance as follow: “z strongly reason-dominates z′ if it is better than z′ in term of both group evaluation and personal evaluation in S. If z strongly reason-dominates z′ [z and z′ are two possible actions, e.g., cooperate or defect, respectively], then in cases in which personal and group evaluation are comparable, z defeats z′ on balance whatever the relative weights. z weakly reason-dominates z′ if it is better than z′ in terms of some evaluation, and not worse than z′ in either.”
- 35.
This questioning is at the center of Gilbert’s (e.g., 2003) account of collective entity. For Gilbert, if individuals agree to engage in a collective action, they become entirely committed to the collective purpose and then become a collective entity. They cease to be full-fledged individuals.
- 36.
Tan and Zizzo’s (2008) model builds on the concept of “harmony of interests” and “disharmony of interests.” In a comparative analysis, they dissociate games by showing that some games such as the Hi-Lo game show a “harmony of interests” while others such as the prisoners’ dilemma show a “disharmony of interests.” However, in contrast to Bacharach, they manage to endogenize the probability that a given player in a given game fails to TR.
- 37.
Bacharach’s challenge for Davis (2011, p. 119) is “the fusions of agency.” Davis considers Bacharach’s modus operandi to be that he “employ[s] one kind of relational conception of the individual—one in which people are single individuals in virtue of how their interaction with others makes them more than collection of multiple selves” (ibidem). This is true if we consider the fact that only one mode of reasoning at a time can be enhanced.
- 38.
There is a strong link between this social psychology theory and what we identify in Chap. 4 as the DSP literature or the mindreading and mindshaping literature.
- 39.
How and why framing entails violation of the standard conception in individual decision theory, and especially EUT, is well argued in Bacharach (2001a).
- 40.
- 41.
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Larrouy, L. (2023). Bacharach: How the Variable Frame and Team Reasoning Theories Challenge Standard Noncooperative Game Theory. In: On Coordination in Non-Cooperative Game Theory. Springer Studies in the History of Economic Thought. Springer, Cham. https://doi.org/10.1007/978-3-031-36171-5_4
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