Abstract
Richard Pettigrew [13, 14] defends the following theses: (1) epistemic disutility can be measured with strictly proper scoring rules (like the Brier score) and (2) at the beginning of their credal lives, rational agents ought to minimize their worst-case epistemic disutility (Minimax). This leads to a Principle of Indifference for ignorant agents. However, Pettigrew offers no argument in favour of Minimax, suggesting that the epistemic conservatism underlying it is a “normative bedrock.” Is there a way to test Minimax? In this paper, we argue that, since Pettigrew’s Minimax is impermissive, an argument against credence permissiveness constitutes an argument in favour of Minimax, and that arguments for credence permissiveness are arguments against Minimax.
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- 1.
We borrowed this example from Schoenfield [18, p. 640].
- 2.
In such a case Pria is facing a situation akin to the one an agent faces in the famous Ellsberg paradox. In our framework, however, there is no need to make a distinction between risk and uncertainty, since in all cases, Pria is maximally ignorant. Indeed, since she has no prior information, she is always dealing with uncertainty. Minimax does not in fact solve the Ellsberg paradox: it simply avoids it when it is used by a maximally ignorant agent.
- 3.
- 4.
See [3].
- 5.
The expected result is sometimes called the weighted mean result. For example, suppose that, in a fair lottery, 5 participants each have 1 chance in 5 to win a single prize of $50. In that lottery, 4 participants won’t win a prize, and 1 participant will win $50. Since (40 + 150)/5 = 10, the weighted mean value of this lottery is $10. This means that $10 is the expected prize to each participant.
- 6.
- 7.
Specifically, while every agent could be bound to a specific impermissive scoring rule and a specific impermissive decision rule, different agents could be bound to distinct incompatible scoring rules or distinct incompatible decision rules.
- 8.
What we here call a “carving” is sometimes called a partition elsewhere. The reason for this terminological choice is that, in the literature from which our paper draws, “carving” is a much more popular term.
- 9.
Decidability should be understood in its most basic sense: given an input (evidential carving), there is always an effective method for returning an output (credence assignment).
- 10.
- 11.
See [13], Sect. 4.3] for formal definitions and the importance of calibration. Decomposition is crucial to Pettigrew’s account of accuracy: it allows him to exclude uncalibrated divergence measures. Roughly, an agent has a calibrated credence X in P if, relative to a body of evidence, X is the proportion of all propositions of a certain type (P) that appear to be true. This is the distinctive feature of strictly proper scoring rules: an agent’s expected score is maximized by assigning calibrated credences in P. For example, the following improper scoring rule respects Alethic vindication, Perfectionism, Divergence Additivity and Divergence Continuity but violates Decomposition:
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1.
Relative to an agent’s credence X in P, if P is true, then the agent’s score is \((1-X);\)
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2.
Relative to an agent’s credence X in P, if P is false, then the agent’s score is \(|0-X|\).
See also [13, pp. 37–40, 48].
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1.
- 12.
Some authors have suggested that there are no distinct consistency requirements of rationality. Specifically, it is possible that process requirements of rationality, which govern how rational agents form and revise beliefs, secure consistency [11]. What matters in this paper is that inconsistent agents violated at least one rationality requirement. See also [4, sect. 9.2] on consistency requirements.
- 13.
We simply calculated the derivative of this function to find its optimum. The derivative of the function \(f(X) = 0.8 \cdot ((1-X)^{3/2}) + 0.2 \cdot ((0-X)^{3/2})\) is equal to 0 when \(X \approx 0.941\).
- 14.
The derivative of the function \(f(X) = 0.8 \cdot ((1-X)^3) + 0.2 \cdot ((0-X)^3)\) is equal to 0 when \(X \approx 0.666\).
- 15.
While we focused on two instances of improper scoring rules, the same problem would arise with any improper scoring rule. The fact that a scoring rule is not uniquely optimized when \(Z = X\) explains why an agent could end up with an inconsistent combination of attitudes. Only improper scoring rules are not uniquely optimized when \(Z = X\).
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Daoust, MK., Montminy, D. (2017). Testing Minimax for Rational Ignorant Agents. In: Baltag, A., Seligman, J., Yamada, T. (eds) Logic, Rationality, and Interaction. LORI 2017. Lecture Notes in Computer Science(), vol 10455. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55665-8_37
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