A Group-Theoretic Approach to Rational Decision Making

Typical normative theories of rationality deliver different answers across contingent and apparently irrelevant redescriptions of a decision problem, posing a challenge for our understanding of scientific reasoning, strategic reasoning, and normative epistemology. Formally, this general phenomenon occurs in (i) Game Theory (addition of irrelevant moves, van Damme, 1989, and strategically irrelevant changes in temporal order, Kohlberg & Mertens, 1986, both alter solution equilibria); (ii) Bayesian Decision Theory (loss minimization and minimax strategies are not invariant across choice of payoff currency, Schervish et al., 2013; convergence does not obtain for natural choices of prior in a nonparametric context, Diaconis & Freedman, 1986; Belot, 2013); and (iii) Judgement Aggregation (the Doctrinal Paradox: logically equivalent redescriptions of an issue may result in different aggregated decisions, Kornhauser & Sager, 1993).

I argue that group theory provides an overarching framework for addressing these difficulties. In particular, equivalence conditions for a decision problem should be understood in terms of the set of transformations across which the relevant features of the problem remain invariant. This analysis is "group theoretic" in the sense that these transformations may be composed arbitrarily, include the identity transformation, and, if any transformation is in the set, its inverse is also in the set, i.e. the set forms a group. A set of such transformations will then partition the space of possible decision problems into equivalence classes. However, there will in general be many such sets of transformations, each preserving a different set of relevant features and, correspondingly, a different partition of decision problem space. Normative decision rules may then be categorized in terms of that group of transformations across which they deliver an invariant result. The appropriateness of applying a decision rule to a given problem should then be assessed in terms of whether that rule remains invariant across the transformations that characterize the relevant equivalence class.

The suggestion here is to apply the strategy of Felix Klein's Erlangen Program for geometry to formal theories of decision making. Klein proposed a group theoretic framework for geometry, on which different geometries are defined in terms of the transformations across which their objects remain invariant. This results in a pluralistic approach, which organizes geometries hierarchically by the restrictiveness of their permitted transformations. Adopting this program mutatis mutandis to decision theory motivates the following morals: (i) there is no absolute answer to the question of which features of a problem are relevant for a rational decision—rather, rational decisions must be relativized to an antecedent specification of the relevant features of the problem; consequently (ii) there is no univocal theory of rationality tout court, rather there are a hierarchy of distinct notions of rationality, each of which is appropriate for a different granularity of problem description.

These general points are illustrated through two examples:

1. Game Theory: the many solution concepts in game theory (Nash equilibria, perfect equilibria, proper equilibria, etc.) can be organized with respect to those features of a game they presume to be relevant, i.e. those preserved across different sets of transformations. Thus, determination of the correct solution concept is a question relativized to a determination of those features of the formal model of a game deemed relevant in a particular context. This work synthesizes and builds on the sparse tradition in game theory of examining invariance across transformations, e.g. Thompson, 1952; Selten, 1975; Bonanno, 1992; Elmes & Reny, 1994.

2. Iterated Update of Probabilities: it is well known that a natural generalization of Bayesian conditionalization, Jeffrey conditionalization, is not in general commutative. On the group theoretic perspective this is neither defect nor virtue simpliciter—rather, Jeffrey conditionalization is better understood as the appropriate rule modulo particular assumptions about the nature of the agent and of the informational problem presented to her by the world. From this perspective, Bayesian conditionalization, Jeffrey conditionalization, minimization of Kullback-Leibler divergence, and other belief update rules can be organized within a unified framework, where each is appropriate given a particular set of assumptions about both the agent and the world. These assumptions are understood in terms of invariance in informational content across sets of transformations. This work synthesizes and builds on the work of Field, 1978; Diaconis & Zabell, 1982; Leitgeb & Pettigrew, 2010, amongst others.

 

References

 

Belot (2013) "Bayesian Orgulity"

Bonanno (1992) "Set-theoretic Equivalence of Extensive-form Games"

Diaconis & Freedman (1986) "On the Consistency of Bayes Estimates"

Diaconis & Zabell (1982) "Updating Subjective Probability"

Elmes & Reny (1994) "On the Strategic Equivalence of Extensive Form Games"

Field (1978) "A Note on Jeffrey Conditionalization"

Kohlberg & Mertens (1986) "On the Strategic Stability of Equilibria"

Kornhauser & Sager (1993) "The One and the Many: Adjudication in Collegial Courts"

Leitgeb & Pettigrew (2010) "An Objective Justification of Bayesianism II"

Schervish, Seidenfeld & Kadane (2013) "The Effect of Exchange Rates on Statistical Decisions"

Selten (1975) "Reexamination of the Perfectness Concept for Equilibrium Points in Extensive Games"

Thompson (1952) "Equivalence of Games in Extensive Form"

van Damme (1989) "Stable Equilibria and Forward Induction"