A short note on concepts in rational choice theory
Microeconomic theory is probably the most elaborate and well-structured theory in the social sciences. Microeconomic theory is basically a theory of rational choice by individual actors. Microeconomic theory in its original form includes consumer-theory, producer-theory and theory on how consumers and producers interact in markets. However, the core concepts underlying this theory can and has also been used to study a broader range of phenomena. This is due generality of the core assumptions underlying the theory, and I will here speak generally about the theory of rational choice. Not only consumers and producers, but also voters, politicians, lobbyists social classes and states in international affairs have been studied under the same core assumptions and by using similar concepts as those used in microeconomic theory in a narrower sense. That being said, the core rational choice assumptions are too general to be used fruitfully in social science research without puttingadditional information about social structure and actors into the particular model of interest. I will here dwell mostly on the general core concepts and assumptions, but from time to time illustrate how rational choice analysts put more flesh on the skeleton of the general framework in concrete research.[1]
Thin and instrumental rationality
A “deceptively simple sentence” that summarizes the theory of rational choice is according to Jon Elster: “When faced with several courses of action, people usually do what they believe is likely to have the best overall outcome” (Elster, 1989:22). It is perhaps best to think of rational choice not as one theory, but rather a more general theoretical framework or a family of related theories. However, the notion of instrumental rationality, that individual actors choose those actions that maximize the satisfaction of their goals, is common for this family. Rationality does not say anything particular about the value of the goals actors pursue; rationality is therefore thin in general. Rationality, as used in rational choice theory, relates to how actors use the means and strategies at their disposal to follow these goals. Rationality is instrumental. However, in concrete research settings analysts feed certain preference structures into their models, and ideally, these preference-structures should resemble central elements of the goals and aims that the real world actors (which the model is supposed to capture) actually have. Politicians might for example be motivated by holding office, grabbing economic rents, increasing their power over society or by promoting specific ideologies. However, the rational choice framework as such does not come with a predisposed list of what the goals should be. Rational choice theory does not require that the actors are self-interested either. It is possible for the actors to have altruistic or sociopathic preferences, even though self-interested actors are most common in the literature.A basic feature of rational choice theory is that actions are chosen because of intended consequences.Actions arenot valuablein their own right, which can be contrasted for example with Kant’s Categorical Imperative.
Requirements on preferences
There are however some requirements made on the preferences of rational actors. In order for the theory to work, preferences must satisfy at least two criteria; completeness and transitivity. Completeness requires that the actor must be able to rank all possible outcomes. That is, if we have outcomes x and y, we have either that x>y, x=y (indifference) or x<y. > and < indicate a strict preference (better than or worse than), whereas ≥ and ≤ indicate a weak preference (better than or equal to and worse than or equal to). Transitivity requires that if x>y and y>z x>z. If an apple is preferred to a banana and a banana is preferred to a kiwi, an apple must be preferred to a kiwi.
These assumptions (together with reflexitivity; an outcome cannot be strictly preferred over itself) are sufficient to utilize a rational choice framework, but as mentioned, in practical research settingsone often puts more structure on the preferences of actors.
Utility functions
If we in addition to the assumptions above assume that preferences are continuous, Gerard Debreu (1959) has shown that there exists at least one utility function that can be constructed to represent these preferences. We can for example have a utility function that depends on income, or on probability of remaining in office, or over location on a policy-dimension (left-right, where extreme left =0 and extreme right=1.)
Figure 1: Hypothetical utility function, U(x), where x is income
Figure 2: Hypothetical utility function, U(x), for centrist party, where x is value on left-right dimension
The utility functions, as long as there is no uncertainty involved, can be interpreted as ordinal. An ordinal measurement level implies that we can only draw information about ranking, and not relative size, from numbers. That is, if U(1) =2 (the utility of having 1 unit is 2) and U(2)= 6, we cannot with ordinal utility functions say that the utility of receiving two units (for example Euros) is three times the utility of receiving one. The only thing we can say is that the utility of receiving two is higher than the utility of receiving one Euro. If we had a cardinal utility function however, the numbers would have contained information about relative size in addition to rank.
Utility functions can take on different forms and researchers often use simple mathematical functions to represent the preferences of actors. The simplest alternative is the linear utility function where U(x) = a + bx. If a is equal to 0 and b is equal to 1, U(x)=x. Another alternative is using a version of U(x)= a+bxc. We can for example have that a is equal to 0, b are equal to 1, and c equal to 1/ 2 U(x)=x1/2=√x. Another common version is U(x)=ln(x); the natural logarithm of x.
However, actors often care about more than one thing, and it is therefore common to model a utility function with more than one objective. Let me illustrate a utility function where an actor has two objectives x and y. Then U = U(x,y), and we can for example choose the simple, linear version U(x,y) = x+y to represent preferences, or the common Cobb-Douglass form U(x,y)= xαy1-α. Let us take an example. We have a political party that competes in two district, and the utility of the party depends on votes gathered in district one (x) and district two (y). The party receives 500 votes in the first district and 700 votes in the second district. The linear utility function would then yield a utility:
U(500, 700) = 500 + 700 = 1200
When it comes to the Cobb-Douglass version, let us now make a twist and say that the party values votes in district 1 more than votes in district two, and the α is therefore larger than ½. Let us set α=0,7, which implies that 1- α=0,3.
This gives us:
U(500, 700) = 5000,7 * 7000,3 ≈553
The choice of a specific utility function is not completely arbitrary however, and the mathematical structure depends to a certain extent on what more assumptions we make about actor preferences other than the very general ones presented above. I will come back to this issue, but first we need to introduce some mathematics.
Elementary calculus
Most Norwegian students learn how to take the derivative of a function (differentiation) in high school, but unfortunately, not much time is spent on this important topic. MA-Political science students and many researchers have often not encountered differentiation in many years. Since differentiation is so central to rational choice-based theory, and utilized in many political-economic studies, we will have to indulge in a crash-course.
The derivative of a function canbe thought of as the slope of the function. It is most straightforward to think about the derivative of a linear function, for example Y= 3 + 2X. The slope of the function in this case is always 2, as the function Y increases with 2 units as X increases with 1 unit. When it comes to more complex functions, the derivative of a function is not necessarily constant. Take the utility function in Figure 1. The derivative of the function is always positive, as the function increases over the whole interval. However, the derivative decreases as the x-values become larger, because the slope becomes less and less steep. In formal terms: U’(x)>0 (The derivative of U with respect to x is positive). And: U’(xb) <U’(xa) if xb>xa. (The derivative of U at point Xb is smaller than at point xa if xb is larger than Xa). The derivative of the function is decreasing as x gets bigger. The ‘ denotes that we are referring to the derivative of the U function with respect to x. Alternatively, we can write this expression as dU/dx.
Now take the utility function in Figure 2. The derivative of the function, U’(x) is positive before the maximum point and negative after the maximum point, since the function is first sloping upwards and then downwards. Here we come to an important point. What is the value of the derivative at the maximum point? Consider Figure 3below:
Figure 3: The derivative of a function in maximum
The straight line on top of the function is the slope at the maximum point. Since the line is flat, the slope is equal to zero. This means that at the maximum point of a function, the derivative of a function is equal to zero (the same is true for minimum points). This is extremely important in rational choice considerations. What will it say to make the best choice for an individual? It can be interpreted as maximizing utility (or optimizing). Therefore, when we search for the optimal action, we take the derivative of a utility function and set it equal to zero. That is, we find the x-value that satisfies the equation U’(x)=0.
We now introduce some general derivation rules for differentfunctions:
Derivative of powers
f(x) = xr f’(x) =rxr-1
Example:f(x) = x3 f’(x) =3x3-1=3x2
Derivative of powers, extended
f(x) = axr f’(x) =r*axr-1
Example:f(x) = 2x4 f’(x) =4*2x4-1=8x3
Remember that x1=x and x0=1, such that f(x) =2x f’(x) =1*2X0=2. Similarly, the derivative of f(x)=25x is equal to 25.
The derivative of a constant is equal to zero
f(x)=5 f’(x)=0
The derivative of sums and differences
f(x) + g(x) differentiated is equal to f’(x) + g’(x) and f(x) - g(x) differentiated is equal to f’(x) - g’(x)
This means that the derivative of 2x2+5 is equal to the derivative of 2x2 plus the derivative of 5. One differentiates the sum piecewise. The derivative of the function can be expressed as:
(2x2+5)’ or d(2x2+5)/dx, and it is equal to 4x+0=4x, following the rules above.
At this point in time, you might want to go back and read this section one more time. Alternatively, you might want to look up a text book in mathematics, such as Sydsæter (2000) for Norwegian students or Gill (2006).
I will stop the discussion on derivatives so far, and go back to the interpretational aspects, but here are some of the most used rules for differentiation. Even if you don’t remember or understand these, you can look them up if needed when reading articles or books that use differentiation:
f(x) = ex f’(x) = ex
f(x) = ln(x) f’(x) = 1/x
Product rule: f(x)*g(x) derivated is equal to f’(x)*g(x)+f(x)*g’(x)
Quotient rule: f(x)/g(x) derivated is equal to (f’(x)*g(x) - f(x)*g’(x))/g(x)2
Chain rule: f(g(x)) derivated is equal to f’(g(x))*g’(x)
Example of chain rule f(g(x))= (2x+3)3. g(x) is here 2x+3. Following the chain rule, differentiation gives us: d(f(g(x))/dx= 3(2x+3)2 *g’(x)= 3(2x+3)2 *2 =6(2x+3)2
How to interpret U’(x): marginal utility
Remember that the derivative of a function in a point is equal to the slope of the function in that point. The derivative of the utility function is often called marginal utility, and can be interpreted as the increase in utility when you increase x marginally (sometimes interpreted as one unit): How much extra utility does the actor get from one extra unit of consumption, or one extra vote.
Consider U(x) = 8x-x2, which could be the utility functionof beers on a Monday night student party
U’(x)=8 -2x, and this is the marginal utility function.
What is the marginal utility of an extra beer if you already have had one beer?
U’(1)=8-2=6.
When should the rational actor stop drinking beers? The answer is: When she has maximized her utility function. Remember that this requires that U’(x)=0, that is the marginal utility of beer is equal to zero. (There are some other conditions that need to be satisfied as well called second-order conditions, but never mind these here.)
How do we find the optimum? Simply by setting U’(x)=8-2x=0
This implies that the optimal amount of beers is 4.
What if we have a function that we would like to minimize, for example a cost function, C(x)? The same logic applies. The bottom of the cost function is found where the derivative, C’(x)=0, that is where the marginal cost is equal to zero. If for example C(x)= x2-4x+16C’(x)= 2x-4
C’(x)=2x-4=0 x=2
What if we have an action, x, say intensity of guerrilla warfare, which bears with it both gains, U(x), and costs, C(x)? Then the rational guerilla would like to optimize the expression U(x) – C(x). Let U(x)=x2 and C(x)=2x
We use the differentiation of a difference presented above and set the total expression equal to zero:
U’(x) – C’(x) = 0 d(x2)/dx –d(2x)/dx=0 2x-2=0 x=1
X=1 is the optimal intensity of guerilla warfare in this example.
Uncertainty and expected utility
In most interesting social science questions, there are uncertain outcomes related to taking a specific action. How does rational choice theory deal with such uncertainty? First, under uncertainty, the analyst will have to move from an ordinal interpretation of the utility function to a cardinal, but I will not go deeper into this issue. If actors are not under thick uncertainty, alá the Rawlsian veil of ignorance, but are able to make predictions on probabilities of different outcomes, the rational choice framework can easily be applied to situations of uncertainty. Actors are assumed to maximize expected utility (EU(x)), given their beliefs about probabilities of outcome. Moreover, given certain specific assumptions (look up a microeconomic textbook), Von Neumann and Morgenstern showed that maximizing expected utility is equivalent to maximizing the following expression:
EU(x) = p1U(x1) + p2U(x2) +……+pnU(xn)
In this case, there are n possible outcomes, which are assigned n different probabilities, given that a certain action is taken. Let us take an example by considering a decision on whether to go to war or not. If the actor does not go to war, he receives utility =2 for certain, that is with a probability =1. However, if the actor goes to war, there are three different outcomes, victory, stalemate and loss. The utility of victory is 10, the utility of stalemate is 0 and the utility of loss is -20. Is it rational for the actor to go to war? This depends on the probability beliefs the actor holds. If the estimated probability of victory is 0,7, of stalemate is 0,1 and of loss is 0,2, we can easily find the expected utility of going to war for the actor:
EU = 0,7*10 + 0,1*0 + 0,2*-20 = 7+0-4=3
The expected utility of going to war is 3 and the expected utility (actually a certain utility) of not going to war is 2. The rational actor chooses the action with the highest expected utility, namely war.The main, general assumption underlying rational choice analysis under uncertainty is that actors maximize expected utility, given their beliefs on probabilities of outcomes for specific actions. A qualifying statement is that actors must, when they are allowed to update their beliefs after gathering information from observations (for example in dynamic models with more than one time period), form beliefs after the so-called “Bayes’ rule”. Look up any book on game theory to read more on rational belief formation. Here is one simple example: You know that you are facing either a weak or a strong opponent in a war game, and the weak opponent never goes to war. Initially, you believe that you face a weak opponent with a 0,4 probability and a strong with 0,6 probability. If you observe, in the first round that the opponent goes to war, you should update your probability beliefs. Since weak opponent never go to war, the probability of facing a weak opponent has to be 0! Then you know that the probability of facing a strong opponent is equal to 1. You have successfully used Bayes’ rule to update your beliefs!
The expected utility framework allows the incorporation of risk aversion among the actors. The risk aversion can be incorporated into the utility function, and one does therefore not have to assume risk neutrality. Interested readers can look up a textbook in microeconomics.
Optimization over time
In many rational choice models, there is a time-structure, and thereby thenumber of time periods is larger than one. How do rational actors respond to such time structures? The common way to model such dynamic settings is by presenting a utility function that is additively separable over periods; that is, total utility can be presented as a sum of utilities gained in each period. However, most actors are impatient and would rather take an immediate prize now than the same prize in the future, and we incorporate this trait by discounting the utility in future periods. We then need a discount factor, β, which tells us how much the actor values future gains (one period ahead) relative to immediate gains. β is in the interval between 0 and 1, and the closer it is to 1, the more patient the actor. Let subscript t be time period, and we start in period 0. We now get that total utility, U, is given by:
U(xt) = u(x0) + βu(x1) + β2u(x2) + …..+βnu(xn) =∑βtu(xt)
The∑ , sigma,means nothing other than the sum of the components. It is used regularly as a summation sign, so you should get acquainted with it. Usually, the first time period is written below the sigma (t=0 here), and the last period in the sum is written on top of the sigma (n here). Notice that as the t becomes very large, βt goes towards zero, and this means that pay-offs far into the future is almost neglected by the rational actors.
One special and very important case is when we have an infinite stream of payoffs, that is t ∞. If we have short time periods or long lived actors (states?), an infinite horizon might function well as an approximation to the situation, even though no one or nothing lives forever. We can actually calculate a finite discounted pay-off for the actor, even though we do it over an infinite time horizon. The extra assumption needed for the formula below to be correct is that the size of the x is equal in each period. The result draws on the mathematical theory of geometric series (look up a textbook in mathematics for the proof). We have that: