Actual Causation and Simple Voting Scenarios[Note 3. I would like to thank Peter Spirtes, Christopher Hitchcock, ...]

Several prominent, contemporary theories of actual causation maintain that in order for something to count as an actual cause (in the circumstances) of some known effect, the potential cause must be a difference-maker with respect to the effect in some restricted range of circumstances. Although the theories disagree about how to restrict the range of circumstances that must be considered in deciding whether something counts as an actual cause of a known effect, the theories agree that at least some counterfactual circumstances must be considered. I argue that the theories are still too permissive in the range of counterfactual circumstances they admit for consideration, and I present simple counter-examples that make use of this overpermissiveness. Harold enjoyed playing basketball. Knowing that he had a heart condition, Harold asked himself what effect playing basketball would (likely) have on his heart. The question Harold asked himself is about the singular effect of a given singular cause. If Harold had been a statistician, he might have estimated the expected effect his playing basketball would have on the condition of his heart by testing causal models on data drawn from a large population of diverse people. Since he was not a statistician, Harold thought about his situation as best he could without data or models. After deliberating, Harold decided to go ahead and play basketball with some friends. While playing, he collapsed from a heart attack. Harold was rushed to the emergency room, where the doctors saved his life. While he was recovering, Harold’s wife, Helen, asked the doctors, “Was Harold’s heart attack caused by his playing basketball?” Helen’s question is not like Harold’s question about the singular effect of a given singular cause; rather, Helen’s question is about the singular cause of a given singular effect. In Harold’s case, the (potential) cause is known, but the effect of applying it is uncertain. In Helen’s case, the effect is known, but the (actual) cause of that effect is uncertain. Structural equation models were developed (primarily by twentieth-century statisticians and econometricians) in order to answer questions like the one Harold asked himself about the effects of causes. By contrast, philosophers, lawyers, and historians have typically been interested in questions like the one Helen asked the doctors about the causes of effects. Recently, structural equation models have been adapted by Pearl (2000), Hitchcock (2001), Woodward (2003), Halpern and Pearl (2005), Glymour and Wimberly (2007), and Hall (2007) in order to answer questions about causes of effects. Such accounts are called theories of actual causation. In order to answer questions about the causes of effects, an adequate theory of actual causation must solve two problems: (1) a metaphysical problem and (2) a logical, epistemological, or inferential problem. The metaphysical problem is to specify in some detail what it is for one thing to be an actual cause of another. The metaphysician aims to identify what we should call the actual cause(s) of some thing. In carrying out her task, the metaphysician assumes that she has perfect information about the world. The inferential problem is to specify in some detail how we can come to know that one thing actually causes another. The two problems are clearly related. In the limit of total information, the inferential problem collapses into the metaphysical problem. Progress on the inferential problem requires having some account of the metaphysical problem, though progress may be made on the inferential side without having a complete account on the metaphysical side. The project in the present paper is metaphysical, though there is a tight connection to the inferential problem. The aim of philosophical theories of actual causation is to reduce inferences about actual causation to inferences about causal structure plus the application of a (metaphysical) definition. I argue that several promising, contemporary theories of actual causation are defective. In order to show their defects, I apply these theories to some quite ordinary voting scenarios and note that the theories say rather strange things about them. Two prima facie examples of defects in the theories are these: (1) in all simple-majority elections that allow abstentions, the theories of actual causation under consideration count every abstention as an actual cause of the winning candidate’s victory, regardless of whether the election is closely contested; and (2) in all simple-plurality elections involving three or more candidates, every theory of actual causation under consideration counts every vote as an actual cause of the winning candidate’s victory, regardless of how the votes are actually distributed among the candidates. Why do the theories say these things? The theories of actual causation under consideration accommodate the individualist intuition that when some outcome is over-determined by two or more occurrences, each occurrence is a cause of the outcome. In order to accommodate the individualist intuition, the theories have to take account of facts about difference-making in the actual circumstances and in counterfactual scenarios. However, the theories are too permissive about the range of counterfactual scenarios they consider. The theories do not have the resources to block enough counterfactual scenarios from consideration (or at least they do not have the resources to block the right ones). Every theory of actual causation under consideration begins with a structural equation model, which represents a collection of structural causation relations, and then adds something in order to represent the actual causation relations. Hitchcock, Woodward, and Halpern and Pearl add the values that the variables in the model take on in the actual circumstances. Hall adds both the values that the variables in the model take on in the actual circumstances and also a designation of some values of the variables as defaults for the model. I argue that these added constraints are not enough: we need to know the conditional default values of the variables in the model and possibly much else besides. Here is how I will proceed. In Section 1, I distinguish between structural causation and actual causation. I briefly review some necessary technical machinery and set out two closely related examples. In Section 2, I describe theories of actual causation due to Hitchcock, Woodward, Halpern and Pearl, and Hall. In Section 3, I discuss the application of those theories to three simple voting scenarios: twocandidate, simple-majority elections without abstentions, two-candidate, simple-majority elections with abstentions, and three-candidate, simple-plurality elections without abstentions. (The results easily generalize to all simple-plurality elections with and without abstentions.) I argue from examples that the theories cannot be correct as they stand. Finally, in Section 4, I speculate about why the theories fail and how they might be repaired. 1. Structural Causation and Actual Causation In this section, I will begin with a description of a simple case of early pre-emption and close with a case of over-determination. I will use the first case to introduce some necessary technical details and to distinguish between structural causation and actual causation. Let U, called the universe (of discourse) or population, denote an arbitrary set of units, ui. Units might be people, states, universities, actions, events, processes, or anything else one might be interested in. A random variable (or simply a variable) is a measurable function from U into the real numbers. Random variables typically represent properties of units, and the value of a variable X for ui, denoted X(ui) = x, represents the result of a measurement of the property represented by X taken with respect to the unit ui. For example, a random variable might represent height in meters or annual operating budget in dollars. A unit may be regarded as having (or being) a collection of measurable properties. Whenever a variable takes a unit as its argument, the variable indicates which property value the unit has. Suppose U is a collection of assassinations carried out by a pair of marksmen, Ralph and Lauren, working in tandem. Each unit u is a single assassination. Sometimes Ralph takes the lead and Lauren acts as backup. Sometimes Lauren takes the lead and Ralph acts as backup. Suppose that for each unit in the population, whoever takes the lead is successful. Let the variable R(·) represent Ralph’s action such that for all u, if Ralph shoots, then R(u) = 1 and if Ralph does not shoot, then R(u) = 0. Similarly, let the variable L(·) represent Lauren’s action. Moreover, let the variable V(·) represent the state of the victim such that for all u, if the victim is alive, then V(u) = 1 and if the victim is dead, then V(u) = 0. For present purposes, suppose that both Ralph and Lauren are perfect marksmen, so that if either one shoots, the victim dies. Hence, for each u, one may write V(u) = R(u) + L(u), where ‘+’ is the Boolean OR. A structural equation model (SEM) is a collection of equations in which (1) the independent variables in a given equation are interpreted as causes of the dependent variable in that equation and (2) the dependent variable in one (or more) of the equations may appear as an independent variable in one or more of the equations in the model. Let denote an arbitrary SEM, where V is an ordered set (or vector) of random variables and F is a set of equations involving the variables in V. A structural causation relation is a relation between random variables, which are just measurable functions. As the name suggests, a structural equation model represents a collection of structural causation relations. Another way of thinking about what an SEM represents is in terms of possible experiments. An idealized experiment consists in manipulating some causal