Chapter 9 Puzzles about Weak Dominance
The account causal ratificationism gives of what makes a decision rational has two clauses. The first is that the decision has to make sense once it is made. That is, it has to maximize expected utility given some credal distribution that is rational once the decision is made. The second is that the decision must not be weakly dominated. That is, given the states and choices that are possible (given that the decision is made), no other option must have the status that it could do better, and couldn’t do worse. The first clause seems very intuitive to me, and is backed up by plausible principles about choices like the ABC game. The second clause seems less intuitive. But it, or something like it, is required to make sense of those same plausible principles, and so I’ve included it as part of causal ratificationism.
This chapter has four aims, corresponding to its four sections. In section 9.1 I’ll show why the ABC game supports the idea that rational choosers do not select weakly dominated options. In section 9.2 I’ll note one reason that weak dominance unintuitive; it suggests that seemingly insignificant boundaries are in fact significant. If it weren’t for the argument of the previous section, this would be enough to convince me to do without weak dominance. In section 9.3 I’ll discuss how weak dominance interacts with some hard puzzles about a simple symmetric game. I’ve had two previous attempts to say something sensible about this game [(Weatherson201x?); Weatherson201y]; hopefully third time’s the charm. Finally, in section 9.5, I’ll argue that it can be rational to choose options which are ruled out by iterated weak dominance. The argument will primarily turn on an analysis of the money-burning game [cite], and I’m going to endorse Stalnaker’s [cite] criticism of the standard treatment of that game.
9.1 Why Weak Dominance
Consider the following version of the ABC game.
Figure 9.1: A version of the ABC game that supports weak dominance.
GOTTA CLARIFY WHETHER I’M REALLY ALREADY COMMITTED TO THIS
ALSO RAISE WORRY ABOUT WHETHER WEAK DOMINANCE LEADS TO PARADOX IN VERSIONS OF ABC WHERE DEMON HAS PR 1 IN A
Given what I’ve said already about the ABC game, I’m committed to the following two claims. First, in this game the only rational choice is \(U\). That’s the only rational choice at the only information set where Chooser might move, and that determines the rational strategy. Second, a choice is rational in this game iff it is rational in the following simultaneous move game.
|
A |
B |
C |
|
|---|---|---|---|
|
U |
\(2, 2\) |
\(1, 1\) |
\(1, 0\) |
|
D |
\(2, 2\) |
\(0, 0\) |
\(0, 1\) |
From these claims, it follows that the only rational move in the game in @(tab:abc-weak-dominance-strategic) is \(U\). But that’s hard for the causal ratificationist to explain, since it seems that \(D\) is ratifiable. If Chooser believes, after choosing \(D\), that the probability that Demon will play \(A\) is 1, then playing \(D\) will maximise expected utility. Is that a rational credence to have? Maybe! After all, Chooser knows that Demon is a utility maximiser, and for Demon, \(A\) strictly dominates \(B\) and \(C\). So if causal ratificationism was just restricted to the view that choices must be ratifiable, it would not be consistent with what I’ve said about the ABC game, since it would say that \(D\) is impermissible in the extensive form version of the game, but permissible in the strategic form.
This shows that I need to add something to causal ratificationism to make it consistent with what I’ve said about the ABC game. Adding weak dominance would solve this problem. While \(D\) is ratifiable in the strategic form of the game, it is weakly dominated. So adding the weak dominance clause restores the key constraint that the two forms of the game are treated the same way. And that’s the primary reason that I added this clause to causal ratificationism.
Note that this argument is well short of a proof that weak dominance is needed. It shows something is needed, and weak dominance would be sufficient, but it doesn’t rule out weaker constraints. But I can’t see any weaker constraint that would suffice to bring these two problems back into alignment, and would be nearly as intuitive as weak dominance. So I’ve added it as the second clause.
There is one intuitive motivation for weak dominance. The core idea behind causal ratificationism is that choices should be defensible. The chooser should be able to say “This is what I’m doing, and this is why I’m doing it,” in a way that makes sense. And there is something strange about a speech defending choosing \(D\) in @(tab:abc-weak-dominance-strategic). It looks like the person who chooses \(D\) is taking a risk for which they are not receiving any compensation. And it’s irrational to take uncompensated risks. That intuition, if it is correct, generalises to all cases of choosing weakly dominated options, and so supports a general bar on choosing weakly dominated strategies.
Weak dominance has some problems though, and I’ll turn to them next.
9.2 The Boundaries of Games
The definition of weak dominance involves quantifying over possible ways the world might be. In the cases that have been most central to this book, those states have been moves made by another agent, typically a demon. So we could say, equivalently, that the definition of weak dominance involves quantifying over moves available to the other player. Indeed, the usual definition of weak dominance in game theory does quantify over moves available to the other player. A consequence of that is that the boundary between the available and the unavailable moves becomes very significant. That’s a problem, because often this boundary seems to be insignificant.
The importance of this boundary is a key difference between the two clauses in causal ratificationism. The definition of expected utility maximization also makes an appeal to the boundary between possible and impossible states. But whether one treats a state as impossible, or as possible but with probability zero, doesn’t affect the expected utility of an action. The difference between treating a state as impossible, and as possible but with probability zero, does matter to determining which states are weakly dominant.
It’s easier to think about this boundary with some concrete examples. So let’s get away from the simple games that have been the focus of this book, and imagine that Chooser is playing chess with Demon. In this game, Chooser has the black pieces. The game starts in a fairly traditional manner: 1. e4 e5, 2. Nf3.
Now it is Chooser’s turn to move, and they are thinking about Qh4. This is obviously a bad move, but I want us to think for a minute about why it is a bad move.
The causal ratificationist says that when evaluating a move, one thing to check is whether it is weakly dominated. To do that, one must know what the possible states are, i.e., what the possible moves for Demon are. Now pretty clearly Qh4 is not going to be utility maximizing, so the answer to this question won’t affect what the causal ratificationist ultimately says about whether one should play Qh4. But it is important to figure out just what the theory says, so it is important to figure out just what these ‘possible moves’ are.
In particular, which of the following two ‘moves’ should be in the domain of quantification when we are asking whether a choice by Chooser is weakly dominated?
- Demon moves Ng5.
- Demon knocks over the board and walks away.
There is an important sense in which 1 is in the domain, and 2 is not. The rules of chess allow for Ng5, and they do not allow for knocking the board over. When I appealed to weak dominance earlier, I assumed that all and only the moves specified in the rules of the game went into the domain of quantification, and I suspect most readers went along with that. The same principle here would include Ng5, and exclude knocking the board over, as possible moves.
This division is hard to motivate from a traditional philosophical perspective. There is no relevant epistemic difference between the two options. Given very weak assumptions, Chooser knows that Demon will take neither of these options. There is no relevant difference between the options in terms of ability. Demon can move Ng5, and can also (given some assumptions about Demon’s corporal form) knock the board over. The difference between the options is essentially a legal one. Ng5 is a legal move, and knocking the board over is not. That matters when one is playing a formal game. But game theory isn’t just meant to be about formal games like chess. It is meant to include human interactions, notably including wars, that don’t have these kinds of clear rules. And in those settings, it is unclear what counts as a possible move, and hence unclear what counts as a weakly dominant move.
Let’s bring this all back to decision theory involving demons. Chooser is playing the following game. They have to choose \(U\) or \(D\); if they choose something else, they get nothing. Demon is very good at predicting them, and will either turn to the left, if they predict \(U\), or turn to the right, if they predict \(D\). The turn will be after Chooser writes down their choice, but before it is revealed. If Chooser writes \(U\), they get 1 whichever way Demon chooses, but if Chooser writes \(D\), they get 1 if Demon turns right, and 0 if Demon turns left. So far, this looks like a case where weak dominance matters. The expected return of each move is 1, but \(U\) weakly dominates \(D\). Let’s add something to the game though. Demon will turn one way or the other; Chooser knows that. But if Demon instead drinks a beer, Chooser will get 1 if they have chosen \(D\), and 0 if they have chosen \(U\). Demon won’t drink a beer; Demon is allergic to beer, and there isn’t any beer around. But these are the payoffs if beer drinking happens. Which it won’t. Question: Does \(U\) still weakly dominate \(D\)? And this suggests a further question: Is \(D\) permissible?
The problem for using weak dominance in a theory of choice, and it’s a problem wherever weak dominance is used, not just in causal ratificationism, is this. Given that Demon definitely won’t drink beer, it seems like it shouldn’t matter whether we exclude beer from the decision table, or include it while marking it as maximally unlikely to happen. But given that weak dominance is one of our constraints, it turns out this makes all the difference in the world. That seems unfortunate.
Here is how I think we should deal with this problem. I’ve said deal with rather than solve advisedly. It is a problem; it’s why I’m not happy having to include a weak dominance clause. But there are ways to deal with it.
In the first instance, causal ratificationism is a theory of decision not in real world decision problems, but in the kind of formalized problems we discuss in decision theory and game theory textbooks. Solving a real world problem is a two-step process. First, the real world problem has to be translated into a formal problem. Causal ratificationism has something to say about this—it says that the states must be causally independent of the options—but it doesn’t have a lot to say. Second, the formal problem has to be solved. Causal ratificationism has a lot to say about this stage, and I’ve been spelling out what it has to say at some length in this book.
Usually, there are multiple acceptable ways to translate a real world problem into a formal problem. Sometimes, the different translations will have different solutions. In those cases, it is indeterminate what is rational to do. Arguably, that’s the case in the case I described a few paragraphs back, about turning and beer-drinking. It’s acceptable to model this as a problem where Demon’s only options are to turn left or to turn right. On that model, the only rational move is \(U\). It’s also acceptable to model it as a problem where Demon could (but won’t) drink beer. On that model, both \(U\) and \(D\) are rationally permissible. So it is determinately true that \(U\) is rationally permissible, but indeterminate whether \(D\) is rationally permissible. That is a plausible enough response to the problem, I think.
I said earlier that to solve a real world problem, one has to do two things: translate the problem into a formal problem, and solve the formal problem. There is a way that might be misleading. It might suggest that the two steps are analytically distinct from each other, and that the plausibility of an answer to one problem should be assessed independently of the answer to the other. That’s not right. Whether a formal model of a real world problem is acceptable is a function, in part, of whether the solutions to the formal problem are reasonable solutions to the real world problem. It’s perfectly sensible to reason as follows. If it is clear that \(D\) is an unreasonable move in the real world game (I don’t think this is clear, but someone might), then it follows that modeling the game as a formal game where Demon has three possible moves—turn left, turn right, drink beer—is to model the game incorrectly. There is no such thing as an absolutely correct or incorrect model of a game, or of any other real world situation; the quality of a model is dependent on what the model is being used for. And this also helps us avoid some of the most troublesome consequences of the fact that weak dominance is dependent on just how a game is modeled. Sometimes, the fact that rational choices are not weakly dominated is evidence for or against modeling the game a particular way.
9.3 Three Kinds of Demon
Set up red green game and properly cite
Note this is my third try at it
First demon, limit prob. This one is easy
Second demon, zero prob but possible. Answer one this can’t happen. But this takes us into infinitesimal territory and I’m not going there. Answer two, the speech sounds bad. Don’t take uncompensated risks.
Third demon, can not fail. Then not in fact weak dominance post choice bc alternative is not in fact possible.
Same goes for symmetric humans but third is really impossible.
9.4 Benefits of Weak Dominance
Chooser and Demon pick an integer between 0 and 1,000,000.
Demon gets 1 if same, 0 if otherwise, and Demon is very good
Chooser gets lower of two numbers unless they pick the same positive number, in which case Chooser gets that minus two.
Only ratifiable outcome is 0,0
But that’s weakly dominated by just about everything else, and so should be rejected.
Weak dominance helps here.
Question: What equilibria are there? Feels like there should be a mixed strategy one.
Note to self: Think about case where game is capped at 10. Is there an equilibrium where Chooser plays 1/2 9, 1/2 10.
Need expected return of 9 and 10 to be the same.
Demon plays 9 with prob x, 10 with prob 1-x.
If Chooser plays 9, gets 7x + 10(1-x) = 10 - 3x.
If Chooser plays 10, gets 9x + 8(1-x) = 8 + x
Haha, it’s 50/50.
Another sort of benefit. Think about the following three option case.
PU PM PD U 0 5 2 M 5 0 2 D 2 2 2
Is D acceptable? No, I say, it’s weakly dominated by 1/2 U, 1/2 M. That’s good enough. Ideally, Chooser would play that. Even if they can’t play it, they should.
The same helps if you take the earlier problem, and say that if 0, 0, Chooser gets payout epsilon. It’s still weakly dominated by this mixture. Oh that’s not right - Hmm come back to this.
9.5 Iterated Weak Dominance
Sometimes thought that if WD then committed to IWD. Cite HH and Stalnaker in reply.
Three objections 1. Order effects 2. Gets unintuitive results 3. Nothing incoherent about speeches that violate
Start with Bonanno on order effects
Do strategic version of money burning and show what IWD leads to Note that nothing wrong with all H speech This might get complicated
Now do dynamic version Really absurd that having an untaken option can make a difference, when others know you won’t take it But does HH mean that you think demon is stupid? No it means that you think demon might follow up stupid with stupid Also do this using counterfactuals maybe