Abstract
I argue that standard decision theories, namely causal decision theory and evidential decision theory, both are unsatisfactory. I devise a new decision theory, from which, under certain conditions, standard game theory can be derived.
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Notes
Just to be clear: Desire Reflection does not say that one cannot foreseeably change one’s tastes. For instance it is perfectly compatible with Desire Reflection that one currently prefers the taste of bananas to that of oranges and knows that tomorrow one will prefer the taste of oranges to that of bananas. What Desire Reflection rules out is that one now prefers receiving a banana for tomorrow’s lunch, but knows that tomorrow morning one will prefer receiving an orange for tomorrow’s lunch.
Transformations of the form Des⇒aDes + b, where a is some positive number and b is some number.
One might worry that loss of memory can lead to a violation of Reflection. Indeed it can, but it is not crucial to my example. One can modify the example as follows: both at an earlier time and a later time Mary is unsure of her action, and all she learns in between the two times is the contents of the boxes. Or imagine Mary has a joint bank account with Rosie, that Mary thus has desirabilities for Rosie’s actions, that Mary is not sure of Rosie’s actions, and at some time sees the contents of the boxes.
One might prefer to think of them as her expected utilities for news concerning her actions, rather than for her actions per se, but still, they exist.
I didn’t do it that way in the current example, since, once the contents of the boxes has been seen the EDT’r and the CDT’r make the same choices, so that it would no longer be an example of the CDT’r and EDT’r making different choices.
In the next section I will discuss Joyce-style Causal Decision Theory which makes use of probabilities that are “imaged” on acts, rather than probabilities of causal situations.
One might baulk at the demand that such pairs always uniquely determine an outcome. What if the world is indeterministic? Well, one can, by convention, simply beef up the notion of a situation so that any situation S conjoined with any act A uniquely determines an outcome O. Of course, it is then not plausible that what situation one is in is determined by the intrinsic state of the world at the time of one’s action. Still, one might hope that this does not harm the applicability or plausibility of Savage-style decision theory. Unfortunately, we will soon see that there is a problem as to what one takes situations to be, indeed that without a further constraint on what can count as a situation Savage-style decision theory is unsatisfactory.
I am assuming that he does not know he is a causal decision theorist, indeed that he has no credences about which bet he will take out when he calculates his causal utilities.
Can one escape this problem by adopting Jeffrey-style evidential decision theory? That depends. If one assumes that all that one learns when one has come to a decision is what one will do, and if one always updates by conditionalisation, and if even in the extreme case where one knows which act one will perform with certainty the conditional probabilities upon not doing that act are still well-defined (i.e. if one allows conditionalisation on credence 0 propositions), then one will not regret one’s decision once it is made. But if not, then even the evidential decision theorist can be in a situation where he must regret his decision as soon as he has made it. The crux here is whether the conditional credences Cr(Oi/A) can change as a result of deliberation. Note that this possibility is exactly what Jeffrey invoked when he gave his ‘ratificationism’ defense of evidential decision theory in Newcomb type cases (see Jeffrey 1983, Sect. 1.7), and what others invoked when they gave a ‘tickle’ defense of evidential decision theory in gene G type cases (see e.g. Eells 1982).
See, especially, Skyrms (1990).
A similar idea can be found in Weirich (1985).
That is to say, the most natural way in which to do this is to ‘Jeffrey-conditionalise’ relative to the action partition.
For a function to be continuous its domain and range need to have a topology. If the credences are distributed over N actions and situations, where N is finite, we can represent a credence distribution as a vector in an N-dimensional vector space, and use the natural topology of this N-dimensional vector space.
It does rule out cases where you believe something like this: a perfect predictor rewards you iff you make a non-trivial mixed decision, e.g. the predictor pays you big bucks if your final credences in your actions are not all 1 or 0, and pays you nothing if your final credences are all 1 or 0. In fact, I have only been considering cases in which you believe that what the predictor does depends on how you act, rather that on your (final) credences. I can in fact allow for predictors whose actions depend on the decision maker’s (final) credences rather than on his acts. What I cannot allow, while retaining the necessary existence of equilibria, is a discontinuous dependence of the predictors actions on one’s credences. Luckily such a discontinuous dependence of the predictors actions on one’s credences can plausibly be ruled out. For it is very plausible that the dynamical laws of our world are continuous, and no physical system whose dynamics is continuous could implement such a predictor.
A space is convex if the line between any two points in the space is in the space. A space is compact if it is closed and bounded.
In order to be able to define what it is for a credence distribution to be an equilibrium distribution, we will need to assume that even though a perfectly rational person never actually uses any update rules to update his credences during deliberation, there still are facts about what these update rules are, or rather, should be. For otherwise we do not have enough structure to define what his equilibrium credences are.
It does not matter whether the players are Deliberational Causal Decision Theorists or Deliberational Evidential Decision Theorists. The only relevant features of the situation are the players actions, and we have assumed that there are no correlations between the players acts according to their credences.
References
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Acknowledgements
I would like to thank two anonymous referees, Hans Rott, Adam Elga, and those present when I gave this paper as a talk in Regensburg for very useful comments.
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Arntzenius, F. No Regrets, or: Edith Piaf Revamps Decision Theory. Erkenn 68, 277–297 (2008). https://doi.org/10.1007/s10670-007-9084-8
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DOI: https://doi.org/10.1007/s10670-007-9084-8