We use “ought” in two senses, evaluative (“things are not as they ought to be”) and hortatory (“you ought to do the best thing”). The latter is future-directed, and time entered deontic logic (with stit semantics) for that reason. But time brings much in train. What if you do what is the best thing in the short run, for tomorrow for example, and it precludes important options for the day after tomorrow? Or the day after that?
This is how infinity enters: a branching time, infinitely branching possible futures, with the outcomes of our choices. Our deliberation and decision making is inevitably short-sighted, considered sub specie aeternitatis. We can only see finitely many moves ahead. But that implies a danger: how do I form a policy that does not, with equal inevitability, lead me into an ultimately unsatisfying life?
Reflecting on this I remembered F. H. Bradley’s critique of Utilitarianism.
Frances Herbert Bradley, Ethical Studies (1876)
As a student I never liked ethics until I read Bradley. As a British Idealist he was intent on bringing to light all the contradictions in our experienced world; in ethics this led him to see an unresolvable conflict between the ideals of self-sacrifice and self-realization. Fascinating … but here I just want to focus on one little point, his criticism of Utilitarianism, which focused on moral deliberation over time.
The form of Utilitarianism Bradley confronts is, more or less, what is now typically described as rational preference based decision making, cost and benefit analysis, maximizing expected value. As he sees it, this form of reasoning leads inevitably to a sort of life to be regretted, such as the life of a miser who saves to become rich but can never stop saving, or of the businessman who works to make a million but can never stop pursuing the next million. In the exquisite prose of the British Idealists, which we cannot emulate today:
Happiness, in the meaning of always a little more and always a little less, is the stone of Sisyphus and the vessel of the Danaides — it is not heaven but hell. (Essay III)
Falling into this trap seems inevitable if we pursue a quantifiable value, and if this pursuit is not subject to any constraint, whether external or internal, independent of that value. Let’s make this concrete.
The Contra Costa game
To the literature about problems with infinity in decision-making, such as the St. Petersburg Paradox and the Pasadena Game, I propose to add this one, to illustrate Bradley’s argument.
A coin will be tossed as often as we like. If you enter into the game here is what will happen. If the first toss comes up heads, you have two options. The first is to accept $10, and the game stops for you. The second option is to stay in the game. If the second toss then comes up heads you may choose between accepting $100 or staying in the game. And so on: if the n^th toss came up heads, you can choose between accepting $10^m, where m is the number of tosses that have come up heads so far; or you can stay in the game. If the toss comes up tails the game ends, and the player who stayed in ( unlike in the St Petersburg game), ends up with nothing at all.
Suppose toss N has just come up heads. If you stay in you have a 50% chance of getting the option to accept 10^(N+1), which is ten times more than what you could get now. There is also a 50% chance of getting 0. So the expectation value of staying in equals 0 plus 0.5(10)(10^N) = 5 times the value of opting out.
Thus what you ought to do (if your rule is to maximize expected value and you look only one day ahead), as long as you are in the game, at every stage, if the toss came up heads, is to stay in. And the result is that you will never get anything at all: you are living a life of anxious expectation, never able to let go of this devilish ladder of fortune, until either you are out with nothing to show for it, or you go on forever with no no payoff ever.
It may be objected here that no casino could be in a position to offer this game, it could not set a high enough price for entry. That is what we usually say about the St. Petersburg game as well. But think about the real-life analogue. The person who sets out to make a million, and remains at every stage intent on making the next million, has no reason to think that the world cannot afford to offer this possibility. The price paid is the work involved in making money, which is gladly paid.
There was a writer who traded on his readers giving some credence to there being a source with unlimited means: Blaise Pascal.
Similarity to Pascal’s Wager
Here is how Pascal might have posed the problem of the Contra Costa game for the rational unbeliever:
Everyday God says “Repent now, and you shall have eternal bliss!”
Everyday the unbeliever responds in her/his heart “I can take one more day and repent tomorrow, that has a higher value!”
(a) The game ends when s/he dies, and s/he loses.
(b) s/he lives forever. With this policy, she has the fate of Barend Fokke, the Flying Dutchman.
Either way, s/he misses out on eternal bliss.
Is there help from long-term thinking?
The unfortunate player we have just been discussing has this flaw: s/he looks only one day ahead. That it is a day, does not matter: there is a fixed unit of time such that s/he looks ahead only that unit of time, in making her rational decision on the basis of expected value.
I did not eat a chocolate bar just now because it would ruin my appetite, and make dinner much less enjoyable. So I am not that naive Utilitarian agent who just looks one minute, or one hour, ahead. I forego the immediate gain for the sake of gain over a longer period.
But what is that longer period? If it is, say 2 hours, or 2 days, then I am after all that naive agent, with a different time scale, focused on gain in a finite future period. In the Contra Costa game this will not keep me from continuing forever: I will not take today’s prize. Suppose I reflect on the possibilities for the next two tosses of the coin, when the first N tosses have come up heads. The probability is 0.25 that I can get two more heads, and can then accept 10^(N + 2). There is also a 0.75 chance that I will gain zero. So the expected value of that scenario is 0.25(100)(10^N), or 25 times what I can get now. Thinking farther ahead increases the temptation, the incentive, to stay in the game.
What if I am an ideal long term thinker, who does not set a finite limit on his long term evaluations? The probability is zero that the coin will always come up heads. This is relevant for those ideal long term thinkers who take themselves to be immortal: they will rationally refuse to play. But these are either deceived (if all men are mortal) or at least negligibly rare.
Escape from the game: not by cost-benefit analysis
There may be an easy advice to give to the player: Do look beyond the immediate future! Choose some N, and decide not to go farther, no matter what. That is, decide that you will take the money at some stage either before or when there are N heads in a row.
But how is this choice to be made? The expectation value when you make this choice to be N, is 0/2 + 10/4 + 100/8 + … +10^N/(N+1). That is less than for the choice of N+1. So if you choose N, you are not choosing to maximize expected value. It goes against the principle of rational cost-benefit analysis.
The tree of possible futures in this game is a finitely branching tree which has many finite branches and an infinite branch. The latter, the fate of Bradley’s naive but immortal agent, is clearly to be avoided (we ought not to do that!). But a choice among the others on the basis of their value is not possible: for every value seen there, there is one with greater value.
There is no question that we must applaud those who at some point take their winnings and rest content. (“Take the money and run!”, Woody Allen — and how did that work for you?) We must applaud them although that choice is not ratifiable by value-preference based decision making. So if there is to be an escape, we have to tell the agent to bring with her some constraint of her own, which overrides the maxim to maximize expected value. What could that be?
Escape from the game: projects and goals
What follows is a suggestion, that I think deserves to be explored when developing deontic logic. It is not my suggestion, but one I heard long ago. (Perhaps only in conversation, I am not sure.)
Glenn Shafer proposed, at one time, that practical reasoning should be conceived of as in the first instance goal-directed. Shafer was pointing to a fact about the phenomenology of practical deliberation: it is not realistic to depict us as cost-and-benefitters, we set our goals and deliberate only within the span of possibilities left open by our goals.
(What about the goal-setting activity, we may ask? A specific goal may be set as the outcome of a deliberation, which took place within the limits set by previous goals. There is no beginning, we are thrown into a life in which we find basic goals already set. Il n’y a pas dehors du … )
I am saving for a holiday in Hawaii next winter. To have that holiday is my goal. The actual value to me of this holiday, if it happens, will depend on many factors (weather, companions, …) but these factors do not figure in the identity of the goal. (They might have figured, in some way, in my goal-setting).
This goal then constrains what I ought to do meanwhile. Among choices I face along the way, it will delete options that conflict with my going to Hawaii next winter. And if my choices get to the point where only one more move is necessary (getting on the plane …) it will prevent me from self-sabotage. For it would be self-sabotage (given the goal I have) to be looking around at that point and considering alternatives, however amazingly attractive they may be just then.
And it is part of the concept of having a goal that there is a stopping point: when the goal is reached, it is not foregone in favor of a suddenly glimpsed pretty face or chance at a bit of easy money.
So in the Contra Costa game this could be the goal of gaining $1000 (perhaps exactly what I need to pay off my loans, or to buy my wife a necklace for her birthday). That implies that I will accept $1000 and end the game if the coin comes up heads three times in a row. I may lose of course, but what I will not do after three heads in a row is to take a chance on getting $10,000. True, I would have a 50% chance of getting 10 times more, but I have reached my goal, rest content, and do not start on a life of Sisyphus.
There are of course less strictly fashioned constraints: we could call this one a Shafer Goal, and agree that there are also defeasible goals, that would need more sophisticated modeling.
What is crucially important is to recognize the necessary independence, autonomy, externality of the constraint (even though set by the actor herself). For if the choice of constraint itself has to be based on value-preference expectation reasoning, we have not escaped the game at all, we have just found ourselves in the same game on another level.
If the constraint takes the form of a goal I set myself, this must be modeled as an independent imperative or default rule, inserted at a different place. It must be, that is to say, another heart’s command not assimilable in value-preference based reasoning.
Bas van Fraassen's philosophy blog 