1. Subjective Probabilities and Bets The Interference Problem for the Betting Interpretation Lina Erikson and Wlodek Rabinowicz

2. Betting Interpretation (BI) of degrees of belief Degrees of belief (credences, subjective probabilities) are measures of betting dispositions (Ramsey, de Finetti) A bet on a proposition A with a non-zero stake S and a price C is fair iff the agent is willing to take each side of the bet. Existence assumption: There exists fair bets on A. (But what if the highest buying price < the lowest selling price? One might then have to work with lower and upper credences.) Constancy assumption: C/S is constant for all fair bets on A. (But what about risk aversion and decreasing value of money? Still, the assumption may be ok for small C and S)

3. BI, cont’d This constant ratio C/S is the agent’s betting rate for A. Degree of belief P(A) = the betting rate for A If P is so defined, the expected monetary value of a fair bet is 0. Which accounts for the agent’s indifference between buying such bet or selling it. The agent is also indifferent between buying/selling a fair bet and abstaining. Note: Identifying degrees of belief with betting rates does not require that beliefs be identified with dispositions to bet. Such radical behaviorism might well be rejected. Beliefs can instead be seen as inner (mental) states of the agent, which provide bases for her betting dispositions.

4. Advantages of BI - Degrees of belief become observable and measurable Which appeals to operationalists like de Finetti: “ In order to give an effective meaning to a notion – and not merely an appearance of such in a metaphysical-verbalistic sense – an operational definition is required. By this we mean a definition based on a criterion which allows us to measure it.” (de Finetti 1990, p 76) While operationalism has fallen from grace in many other fields, it has shown an amazing staying power in decision theory. - Rationality constraints on beliefs (probability axioms, conditionalization, reflection,..) can be justified by pragmatic arguments (Dutch book style)

5. Well-known problems with BI - agents with beliefs lacking manifestation in behaviour (dislike of betting, Putnam’s superspartans and Galen Strawson’s weatherwatchers) - agents with reasons to bet unconnected to degrees of belief. (betting at gun-point, etc.) - cases in which bet outcomes wouldn’t be decidable (betting on life after death) etc.

6. Here, another difficulty: The interference problem. Being placed in a betting situation might in various ways affect the agent’s degree of belief in the proposition to be betted on. An instance of a general problem for measurement. Frank Ramsey: “the proposal of a bet may inevitably alter [one’s] state of opinion; just as we could not always measure electric intensity by actually introducing a charge and seeing what force it was subject to, because the introduction of the charge would change the distribution to be measured.” (“Truth and probability”, 1926, in D. H. Mellor’s Ramsey volume,1978, p.74). This undermines the identification of the agent’s degree of belief in A with the rate at which she would be willing to bet on A.

7. Interference problem, cont’d Whether an agent would be willing to accept a bet on A does not depend on her degree of belief in A but on the degree of belief regarding A she would have in a betting situation. - Being placed in a betting situation might have evidentiary bearings on A. In addtion, - a bet on A might be expected to have causal effects on A. These two phenomena will now be considered in more detail.

8. Causal effects of bets on propositions betted on Example: I bet on the proposition that I will fall asleep tonight. (i) Trying to fall asleep makes it harder to do so. Analogies: - trying to be happy, to be friendly, to be spontaneous - self-defeatingness of the consequentialist decision procedure. (ii) Thinking about the bet I have made (worrying about a loss, or hoping to win) might influence the outcome (in one direction or another).

9. Falling asleep is not an action. With bets on our own actions, the problem is aggravated. The expected gain from winning a bet on my action might offset the reasons not to act and thus lead me to expect that I’ll perform the action. Acc. to some proponents of BI (Wolfgang Spohn, Isaac Levi), this shows that we cannot have subjective probabilities for the choices we are about to make: “Deliberation crowds out prediction.” (Levi)

10. Spohn: “The agent’s readiness to accept a bet on an act does not depend on the betting odds but only on his gain. If the gain is high enough to put this act on the top of his preference order of acts, he will accept it, and if not, not.” (1977, p. 115)

11. Explanation Let A be an action at the agent’s disposal. If the agent bets on A and wins, her net gain is G = S - C. She’ll therefore accept the bet iff G provides a sufficient incentive to act, i.e., iff the addition of G puts A at the top of her preference order, as compared with alternative options. But the size of G = S - C is not fixed by the ratio C/S. When we vary C and S while keeping C/S constant, G can increase to make bet on A attractive. One’s willingness to bet thus depends on G, and not on C/S. “The agent’s readiness to accept a bet on an act does not depend on the betting odds but only on his gain.”  there is no constant rate C/S for all fair bets on A. Conclusion: The betting rate for A doesn’t exist.

12. ’Bets as incentives’-argument generalized (Rabinowicz 2002) Spohn’s argument applies even if the agent is not quite certain that she will perform action A if she bets on it. It’s enough for the acceptability of a bet on A that G is sufficiently high to compensate for this uncertainty.  The argument applies not only to actions subject to one’s current choice, but also to one’s actions in the future. Expectation of gains from one’s current bets should provide incentives for one’s future choices.  The agent has no betting rates for future actions either. Nor for events that can be affected by her future actions. To save the betting interpretation, one would have to accept that the agent lacks degrees of belief for a very wide range of propositions. This isn’t appealing.

13. Possible solution (Rabinowicz 2002): Identify degrees of belief with betting rates for purely hypothetical bets that, after having been made, would immediately be forgotten by the agent. Forgotten bets wouldn’t provide the agent with incentives for future actions. But: The more counterfactual idealization is required for BI to work, the more problematic becomes the claim that BI accounts for what degrees of belief are. More on idealization later.

14. Evidentiary bearings of betting situations Different aspects of the betting situation can sometimes carry information that bears on the proposition on which the bet is to be made. Information can be carried by any of the following facts: (i) someone offers you a bet (ii) you accept a bet offer (iii) you make a bet offer (iv) your bet offer is accepted

15. Case (i): Information is carried by an offer of a bet on proposition A Does he know something I don’t know? My probability for A decreases. I decline a bet that is offered at a rate that matches my ex ante probability. But: Isn’t it enough that I am willing to accept a bet at a rate that matches my ex post probability? No. BI is supposed to account for all subjective probabilities – ex post and ex ante. Finkish dispositions, that disappear as soon as the triggering condition materializes? A rather unappealing view.

16. Change in probabilities isn’t even necessary. ”Forgery” (Bradley&Leitgeb 2006) A fair coin is flipped. I am offered a bet on Heads at a rate ½. But I know that, iff Heads, the bet will use false money (both my notes and the bookie’s will be exchanged for fakes). No way I would accept a bet like this! A simpler alternative: A bet offer such that the bookie only ex post will disclose whether the agent is to be the buyer or the seller.

17. Case (ii): Information carried by my acceptance of a bet offer Example: The proposition to be betted on: I am not a gambler (A) My degree of belief in A is high. But when I am offered a bet on A, I accept it, to my surprise. This reveals something about myself: My probability for A decreases. So, it seems I should take the evidential bearings of my action into account when deciding whether to accept the bet or not. Case (iii): Information carried by my bet offer Similar to case (ii). Case (iv): Information carried by my bet offer being accepted Similar to case (i).

18. In these four kinds of cases, betting or getting an opportunity to bet would have evidentiary (but not causal) bearings on the proposition betted on. This evidentiary effect explains why the rate at which I would be willing to bet on A doesn’t match my current degree of belief in A. ________________________________________________ But, here we need to stop and reconsider. Cases (ii) and (iii) are problematic. Our diagnosis of these cases goes against causal decision theory (CDT).

19. Causal vs evidential decision theory On CDT, unlike as on EDT, purely evidentiary (i.e. non- causal) bearings of actions we decide on should be disregarded. Thus, CDT would not justify decisions on bets that take into account purely evidentiary bearings of the act of betting. (as in the example of a bet on the proposition that I am not a gambler). But is CDT unassailable?

20. Andy Egan (2007): A strategy for generating counter- examples to CDT from counter-examples to EDT One standard form of counter-examples to EDT: If X were performed, it would probably be caused by C, where C – if present - would also cause some bad outcome O. X would therefore increase probability for O, by increasing probability for C, but these evidentiary bearings of X should be disregarded, since X has no causal effect on O. Now, change the set-up: Assume that C, if present, wouldn’t itself cause the bad O, but it would instead enable X to cause O. Despite this potential causal connection between X and O, CDT might still recommend X if, prior to performing X, C is very improbable. This recommendation might seem counter-intuitive.

21. Psychopath button I want to make the world free from psychopaths and I am nearly certain I am not one of them. If I press a button, this will kill all the psychopaths. But I find it very probable that only a psychopath would do such a thing. X – pressing the button, C – I am a psychopath, O – I die. Note: The case of betting on my not being a gambler is similar. C – betting on my not being a gambler, C – I am a gambler, O – I lose the bet.

22. While Egan’s examples are intuitively appealing, their appeal should be resisted. We should abide by CDT. Suggestion: The intuitions Egan exploits have a source in deeply rooted but basically questionable elements of our practice of decision making: (i) Maximining: We avoid acts that might cause disasters, however minute is this risk. By abstaining to press the psychopath button I avoid the minute risk of being killed. (ii) Quasi-magics: We tend to behave as though our acts might cause the states on which they evidentially bear, even if we know that the states in question are causally act- independent. Pressing the button  I am a psychopath

23. Shafir & Tversky (1992) on ”quasi-magical thinking” - One-boxing is a majority response (65%) among subjects confronted with Newcomb Problem One box  The box is non-empty. Another label for this phenomenon: ”wishful acting”. - In a one-shot PD played against an anonymous opponent: Opponent is known to have defected  3% cooperate. Opponent known to have cooperated  16% cooperate. The move of the opponent is unknown  37% cooperate. Thus, cooperation to a large extent is not explainable by feelings of fairness or altruism, but by … wishful acting? I cooperate  My opponent cooperates ________________________________________________________ To conclude: Since we stand by CDT, we must reject counterexamples to BI that appeal to the evidentiary bearings of the acts of betting (i.e. cases of type (ii) and (iii)).

24. Another such spurious counter-example to BI: Sleeping Beauty Beauty is put to sleep on Sunday. A fair coin is flipped. Heads  B. will be awakened just once, on Monday. Tails  B. will be awakened twice, on Monday and Tuesday, with the memory of first awakening erased (no causal links). At an awakening, she won’t be told what day it is. She will be offered a bet on Heads, the same bet each time. She knows all this. She is awakened and receives a bet offer. Accepting the bet would increase B’s probability that she would accept it on another subjectively indistinguishable occasion. She would have two such occasions if it’s a losing bet, and just one if it’s a winning bet. Thus, accepting the bet would have negative purely evidentiary bearings - relevant for the decision as to whether to bet on EDT, but not on CDT. On EDT, the rate at which she is willing to bet doesn’t match her degree of belief in Heads. On CDT, it does. (cf. Arntzenius 2002, Bovens & Rabinowicz 2010)

25. Provisional conclusions: - Legitimate counter-examples to BI have to do with two interfering factors: (a) the evidential bearings on A of the opportunity to bet on A; (b) the expected causal effects on A of a bet on A. - A bet b on A would be acceptable to the agent if and only if the rate of the bet equals P(bet b on A   A/opportunity for bet b on A) causal effect evidentiary bearings - Only if none of the interfering factors (a) and (b) is present, this degree of belief is reducible to P(A).

26. Going counterfactual A possible line of response: Decision theory involves idealizations. Problems with BI that arise for actual people in actual situations need not be equally serious when we consider agents viewed as theoretical constructs or bets viewed as purely hypothetical constructions.

27. One suggestion along these lines (Sebastian Enqvist, private communication) My degrees of belief at a time t aren’t given by the bets I am at willing to make at t, but by the bets my hypothetical alter ego, who is just like me up to t, would be willing to make at t, in her hypothetical world, about the actual world. - This takes care of the expected causal effects of the bet on the proposition betted on. (No causal interactions between different possible worlds.) - But the problem of the evidentiary bearings of the opportunity to bet still remains. (Does the offer of the bet suggest that the offerer possesses superior knowledge?)

28. The basic problem with the counterfactual proposals The more idealized we go, the less sense it makes to look to bets for an interpretation of degrees of belief. We might possibly succeed in constructing hypothetical betting rates that would correspond to the agent’s actual degrees of belief. But it is our independent understanding of these degrees of belief that guides the hypothetical constructions. Therefore, the former are not really illuminated by the latter.

29. Other alternatives? Measurement version of BI: Degrees of belief aren’t identical with betting rates, but can still be measured by the latter. That measurement methods sometimes don’t work is not a serious objection, if they work in most cases to which they are applied. Jeffrey’s analogy: Thermometers only work within certain temperature range, but within that range they are reliable. But the analogy is not quite right. While some objections to BI focus on particular classes of propositions (about future actions, etc), other objections apply, in principle, to every proposition (evidentiary bearings of betting opportunities).

30. But if BI is rejected, what are the degrees of belief? Functionalism: Beliefs are inner, mental states of the person, with a double role: truth-tracking and action-guidance. An input-output approach. Input side: truth tracking. Explains the common idea of the direction of fit: Beliefs are shaped to fit the world, while for desires it is the world that is shaped to fit the desires. BI only looks to the output side (action guidance). We also need to look to the role of truth tracking. (cf. Joyce 1998)