[[!meta title="Game Theory: a Critical Introduction"]] ## Index * Difference between meteorological and traffic-jam-style predictions: 18. * Frequent assumption (but not always) on game theory that individuals knows the rules of the game; that they even known their inner motives: 28. * Individualism, separation of structure and choice, 31. # Excerpts ## Intro What is game theory: In many respects this enthusiasm is not difficult to understand. Game theory was probably born with the publication of The Theory of Games and Economic Behaviour by John von Neumann and Oskar Morgenstern (first published in 1944 with second and third editions in 1947 and 1953). They defined a game as any interaction between agents that is governed by a set of rules specifying the possible moves for each participant and a set of outcomes for each possible combination of moves. How it can help: If game theory does make a further substantial contribution, then we believe that it is a negative one. The contribution comes through demonstrating the limits of a particular form of individualism in social science: one based exclusively on the model of persons as preference satisfiers. This model is often regarded as the direct heir of David Hume’s (the 18th century philosopher) conceptualisation of human reasoning and motivation. It is principally associated with what is known today as rational choice theory, or with the (neoclassical) economic approach to social life (see Downs, 1957, and Becker, 1976). Our main conclusion on this theme (which we will develop through the book) can be rephrased accordingly: we believe that game theory reveals the limits of ‘rational choice’ and of the (neoclassical) economic approach to life. In other words, game theory does not actually deliver Jon Elster’s ‘solid microfoundations’ for all social science; and this tells us something about the inadequacy of its chosen ‘microfoundations’. Assumptions: three key assumptions: agents are instrumentally rational (section 1.2.1); they have common knowledge of this rationality (section 1.2.2); and they know the rules of the game (section 1.2.3). These assumptions set out where game theory stands on the big questions of the sort ‘who am I, what am I doing here and how can I know about either?’. The first and third are ontological. 1 They establish what game theory takes as the material of social science: in particular, what it takes to be the essence of individuals and their relation in society. The second raises epistemological issues 2 (and in some games it is not essential for the analysis). It is concerned with what can be inferred about the beliefs which people will hold about how games will be played when they have common knowledge of their rationality. Instrumental rationality (_Homo economicus_): We spend more time discussing these assumptions than is perhaps usual in texts on game theory because we believe that the assumptions are both controversial and problematic, in their own terms, when cast as general propositions concerning interactions between individuals. This is one respect in which this is a critical introduction. The discussions of instrumental rationality and common knowledge of instrumental rationality (sections 1.2.1 and 1.2.2), in particular, are indispensable for anyone interested in game theory. In comparison section 1.2.3 will appeal more to those who are concerned with where game theory fits in to the wider debates within social [...] Individuals who are instrumentally rational have preferences over various ‘things’, e.g. bread over toast, toast and honey over bread and butter, rock over classical music, etc., and they are deemed rational because they select actions which will best satisfy those preferences. One of the virtues of this model is that very little needs to be assumed about a person’s preferences. Rationality is cast in a means-end framework with the task of selecting the most appropriate means for achieving certain ends (i.e. preference satisfaction); and for this purpose, preferences (or ‘ends’) must be coherent in only a weak sense that we must be able to talk about satisfying them more or less. Technically we must have a ‘preference ordering’ because it is only when preferences are ordered that we will be able to begin to make judgements about how different actions satisfy our preferences in different degrees. [...] Thus it appears a promisingly general model of action. For instance, it could apply to any type of player of games and not just individuals. So long as the State or the working class or the police have a consistent set of objectives/ preferences, then we could assume that it (or they) too act instrumentally so as to achieve those ends. Likewise it does not matter what ends a person pursues: they can be selfish, weird, altruistic or whatever; so long as they consistently motivate then people can still act so as to satisfy them best. An agent is "rational" in this conext when they have preference ordering" and if "they select the action that maximizes those preferences: Readers familiar with neoclassical Homo economicus will need no further introduction. This is the model found in standard introductory texts, where preferences are represented by indifference curves (or utility functions) and agents are assumed rational because they select the action which attains the highest feasible indifference curve (maximises utility). For readers who have not come across these standard texts or who have forgotten them, it is worth explaining that preferences are sometimes represented mathematically by a utility function. As a result, acting instrumentally to satisfy best one’s preferences becomes the equivalent of utility maximising behaviour. Reason and slavery: Even when we accept the Kantian argument, it is plain that reason’s guidance is liable to depend on characteristics of time and place. For example, consider the objective of ‘owning another person’. This obviously does not pass the test of the categorical imperative since all persons could not all own a person. Does this mean then we should reject slave-holding? At first glance, the answer seems to be obvious: of course, it does! But notice it will only do this if slaves are considered people. Of course we consider slaves people and this is in part why we abhor slavery, but ancient Greece did not consider slaves as people and so ancient Greeks would not have been disturbed in their practice of slavery by an application of the categorical imperative. Reason dependent on culture: Wittgenstein suggests that if you want to know why people act in the way that they do, then ultimately you are often forced in a somewhat circular fashion to say that such actions are part of the practices of the society in which those persons find themselves. In other words, it is the fact that people behave in a particular way in society which supplies the reason for the individual person to act: or, if you like, actions often supply their own reasons. This is shorthand description rather than explanation of Wittgenstein’s argument, but it serves to make the connection to an influential body of psychological theory which makes a rather similar point. Cognitive dissonance and free market proponents: Festinger’s (1957) cognitive dissonance theory proposes a model where reason works to ‘rationalise’ action rather than guide it. The point is that we often seem to have no reason for acting the way that we do. For instance, we may recognise one reason for acting in a particular way, but we can equally recognise the pull of a reason for acting in a contrary fashion. Alternatively, we may simply see no reason for acting one way rather than another. In such circumstances, Festinger suggests that we experience psychological distress. It comes from the dissonance between our self-image as individuals who are authors of our own action and our manifest lack of reason for acting. It is like a crisis of self-respect and we seek to remove it by creating reasons. In short we often rationalise our actions ex post rather than reason ex ante to take them as the instrumental model suggests. [...] Research has shown that people seek out and read advertisements for the brand of car they have just bought. Indeed, to return us to economics, it is precisely this insight which has been at the heart of one of the Austrian and other critiques of the central planning system when it is argued that planning can never substitute for the market because it presupposes information regarding preferences which is in part created in markets when consumers choose. Infinite regress of the economics of information acquisiton (i.e learning, eg. from a secret service): Actually most game theorists seem to agree on one aspect of the problem of belief formation in the social world: how to update beliefs in the presence of new information. They assume agents will use Bayes’s rule. This is explained in Box 1.6. We note there some difficulties with transplanting a technique from the natural sciences to the social world which are related to the observation we have just made. We focus here on a slightly different problem. Bayes provides a rule for updating, but where do the original (prior) expectations come from? Or to put the question in a different way: in the absence of evidence, how do agents form probability assessments governing events like the behaviour of others? There are two approaches in the economics literature. One responds by suggesting that people do not just passively have expectations. They do not just wait for information to fall from trees. Instead they make a conscious decision over how much information to look for. Of course, one must have started from somewhere, but this is less important than the fact that the acquisition of information will have transformed these original ‘prejudices’. The crucial question, on this account, then becomes: what determines the amount of effort agents put into looking for information? This is deceptively easy to answer in a manner consistent with instrumental rationality. The instrumentally rational agent will keep on acquiring information to the point where the last bit of search effort costs her or him in utility terms the same amount as the amount of utility he or she expects to get from the information gained by this last bit of effort. The reason is simple. As long as a little bit more effort is likely to give the agent more utility than it costs, then it will be adding to the sum of utilities which the agent is seeking to maximise. [...] This looks promising and entirely consistent with the definition of instrumentally rational behaviour. But it begs the question of how the agent knows how to evaluate the potential utility gains from a bit more information _prior to gaining that information_. Perhaps he or she has formulated expectations of the value of a little bit more information and can act on that. But then the problem has been elevated to a higher level rather than solved. How did he or she acquire that expectation about the value of information? ‘By acquiring information about the value of information up to the point where the marginal benefits of this (second-order) information were equal to the costs’, is the obvious answer. But the moment it is offered, we have the beginnings of an infinite regress as we ask the same question of how the agent knows the value of this second-order information. To prevent this infinite regress, we must be guided by something _in addition_ to instrumental calculation. But this means that the paradigm of instrumentally rational choices is incomplete. The only alternative would be to assume that the individual _knows_ the benefits that he or she can expect on average from a little more search (i.e. the expected marginal benefits) because he or she knows the full information set. But then there is no problem of how much information to acquire because the person knows everything! Infinite recursion of the common knowledge (CKR): If you want to form an expectation about what somebody does, what could be more natural than to model what determines their behaviour and then use the model to predict what they will do in the circumstances that interest you? You could assume the person is an idiot or a robot or whatever, but most of the time you will be playing games with people who are instrumentally rational like yourself and so it will make sense to model your opponent as instrumentally rational. This is the idea that is built into the analysis of games to cover how players form expectations. We assume that there is common knowledge of rationality held by the players. It is at once both a simple and complex approach to the problem of expectation formation. The complication arises because with common knowledge of rationality I know that you are instrumentally rational and since you are rational and know that I am rational you will also know that I know that you are rational and since I know that you are rational and that you know that I am rational I will also know that you know that I know that you are rational and so on…. This is what common knowledge of rationality means. [...] It is difficult to pin down because common knowledge of X (whatever X may be) cannot be converted into a finite phrase beginning with ‘I know…’. The best one can do is to say that if Jack and Jill have common knowledge of X then ‘Jack knows that Jill knows that Jack knows …that Jill knows that Jack knows…X’—an infinite sentence. The idea reminds one of what happens when a camera is pointing to a television screen that conveys the image recorded by the very same camera: an infinite self-reflection. Put in this way, what looked a promising assumption suddenly actually seems capable of leading you anywhere. [...] The problem of expectation formation spins hopelessly out of control. Nevertheless game theorists typically assume CKR and many of them, and certainly most people who apply game theory in economics and other disciplines Uniformity: Consistent Alignment of Beliefs (CAB), another weak assumption based on Harsanyi doctrine requiring equal information; followed by a comparison with Socract dialectics: Put informally, the notion of _consistent alignment of beliefs_ (CAB) means that no instrumentally rational person can expect another similarly rational person who has the same infor mation to develop different thought processes. Or, alternatively, that no rational person expects to be surprised by another rational person. The point is that if the other person’s thought is genuinely moving along rational lines, then since you know the person is rational and you are also rational then your thoughts about what your rational opponent might be doing will take you on the same lines as his or her own thoughts. The same thing applies to others provided they respect _your_ thoughts. So your beliefs about what your opponents will do are consistently aligned in the sense that if you actually knew their plans, you would not want to change your beliefs; and if they knew your plans they would not want to change the beliefs they hold about you and which support their own planned actions. Note that this does not mean that everything can be deterministically predicted. Reason reflecting on itself: These observations are only designed to signal possible trouble ahead and we shall examine this issue in greater detail in Chapters 2 and 3. We conclude the discussion now with a pointer to wider philosophical currents. Many decades before the appearance of game theor y, the Ger man philosophers G.F.W.Hegel and Immanuel Kant had already considered the notion of the self-conscious reflection of human reasoning on itself. Their main question was: can our reasoning faculty turn on itself and, if it can, what can it infer? Reason can certainly help persons develop ways of cultivating the land and, therefore, escape the tyranny of hunger. But can it understand how it, itself, works? In game theory we are not exactly concerned with this issue but the question of what follows from common knowledge of rationality has a similar sort of reflexive structure. When reason knowingly encounters itself in a game, does this tell us anything about what reason should expect of itself? What is revealing about the comparison between game theory and thinkers like Kant and Hegel is that, unlike them, game theory offers something settled in the form of CAB. What is a source of delight, puzzlement and uncertainty for the German philosophers is treated as a problem solved by game theory. For instance, Hegel sees reason reflecting on reason as it reflects on itself as part of the restlessness which drives human history. This means that for him there are no answers to the question of what reason demands of reason in other people outside of human history. Instead history offers a changing set of answers. Likewise Kant supplies a weak answer to the question. Rather than giving substantial advice, reason supplies a negative constraint which any principle of knowledge must satisfy if it is to be shared by a community of rational people: any rational principle of thought must be capable of being followed by all. O’Neill (1989) puts the point in the following way: [Kant] denies not only that we have access to transcendent meta- physical truths, such as the claims of rational theology, but also that reason has intrinsic or transcendent vindication, or is given in consciousness. He does not deify reason. The only route by which we can vindicate certain ways of thinking and acting, and claim that those ways have authority, is by considering how we must discipline our thinking if we are to think or act at all. This disciplining leads us not to algorithms of reason, but to certain constraints on all thinking, communication and interaction among any plurality. In particular we are led to the principle of rejecting thought, act or communication that is guided by principles that others cannot adopt. (O’Neill p. 27) Summary: To summarise, game theory is avowedly Humean in orientation. [...] The second [aspect] is that game theorists seem to assume _too much_ on behalf of reason [even more than Hume did]. Giddens, Wittgenstein language games and the "organic or holistic view of the relation between action and structure" (pages 30-31): The question is ontological and it connects directly with the earlier discussion of instrumental rationality. Just as instrumental rationality is not the only ontological view of what is the essence of human rationality, there is more than one ontological view regarding the essence of social interaction. Game theory works with one view of social interaction, which meshes well with the instrumental account of human rationality; but equally there are other views (inspired by Kant, Hegel, Marx, Wittgenstein) which in turn require different models of (rational) action. State (pages 32-33): Perhaps the most famous example of this type of institutional creation comes from the early English philosopher Thomas Hobbes who suggested in Leviathan that, out of fear of each other, individuals would contract with each other to form a State. In short, they would accept the absolute power of a sovereign because the sovereign’s ability to enforce contracts enables each individual to transcend the dog- eat-dog world of the state of nature, where no one could trust anyone and life was ‘short, nasty and brutish’. Thus, the key individualist move is to draw attention to the way that structures not only constrain; they also enable (at least those who are in a position to create them). It is the fact that they enable which persuades individuals consciously (as in State formation) or unconsciously (in the case of those which are generated spontaneously) to build them. To bring out this point and see how it connects with the earlier discussion of the relation between action and structure it may be helpful to contrast Hobbes with Rousseau. Hobbes has the State emerging from a contract between individuals because it serves the interests of those individuals. Rousseau also talked of a social contract between individuals, but he did not speak this individualist language. For him, the political (democratic) process was not a mere means of ser ving persons’ interests by satisfying their preferences. It was also a process which changed people’s preferences. People were socialised, if you like, and democracy helped to create a new human being, more tolerant, less selfish, better educated and capable of cherishing the new values of the era of Enlightenment. By contrast, Hobbes’ men and women were the same people before and after the contract which created the State. 4 Game theory as justification of individualism (pages 32-33), which reminds the discussion made by Dany-Robert Dufour in La Cite Perverse; it is also noted that the State is considered a "collective action agency": Where do structures come from when they are separate from actions? An ambitious response which distinguishes methodological individualists of all types is that the structures are merely the deposits of previous interactions (potentially understood, of course, as games). This answer may seem to threaten an infinite regress in the sense that the structures of the previous interaction must also be explained and so on. But, the individualist will want to claim that ultimately all social str uctures spring from interactions between some set of asocial individuals; this is why it is ‘individualist’. [...] Returning to game theory’s potential contribution, we can see that, in so far as individuals are modelled as Humean agents, game theory is well placed to help assess the claims of methodological individualists. After all, game theory purports to analyse social interaction between individuals who, as Hume argued, have passions and a reason to serve them. Thus game theory should enable us to examine the claim that, beginning from a situation with no institutions (or structures), the self-interested behaviour of these instrumentally rational agents will either bring about institutions or fuel their evolution. An examination of the explanatory power of game theory in such settings is one way of testing the individualist claims. In fact, as we shall see in subsequent chapters, the recurring difficulty [...] Suppose we take the methodological individualist route and see institutions as the deposits of previous interactions between individuals. Individualists are not bound to find that the institutions which emerge in this way are fair or just. Indeed, in practice, many institutions reflect the fact that they were created by one group of people and then imposed on other groups. All that any methodological individualist is committed to is being able to find the origin of institutions in the acts of individuals qua individuals. The political theory of liberal individualism goes a stage further and tries to pass judgement on the legitimacy of particular institutions. Institutions in this view are to be regarded as legitimate in so far as all individuals who are governed by them would have broadly ‘agreed’ to their creation. Naturally, much will turn on how ‘agreement’ is to be judged because people in desperate situations will often ‘agree’ to the most desperate of outcomes. Thus there are disputes over what constitutes the appropriate reference point (the equivalent to Hobbes’s state of nature) for judging whether people would have agreed to such and such an arrangement. We set aside a host of further problems which emerge the moment one steps outside liberal individualist premises and casts doubt over whether people’s preferences have been autonomously chosen. Game theory has little to contribute to this aspect of the dispute. However, it does make two significant contributions to the discussions in liberal individualism with respect to how we might judge ‘agreement’. Prisioner's dilemma and the hobbesian argument for the creation of a State (pages 36-37): resolution would require a higher State in the next upper level of recursion: Finally there is the prisoners’ dilemma game (to which we have dedicated the whole of Chapter 5 and much of Chapter 6). Recall the time when there were still two superpowers each of which would like to dominate the other, if possible. They each faced a choice between arming and disarming. When both arm or both disarm, neither is able to dominate the other. Since arming is costly, when both decide to arm this is plainly worse than when both decide to disarm. However, since we have assumed each would like to dominate the other, it is possible that the best outcome for each party is when that party arms and the other disarms since although this is costly it allows the arming side to dominate the other. These preferences are reflected in the ‘arbitrary’ utility pay-offs depicted in Figure 1.4. Game theory makes a rather stark prediction in this game: both players will arm (the reasons will be given later). It is a paradoxical result because each does what is in their own interest and yet their actions are collectively self- defeating in the sense that mutual armament is plainly worse than the alternative of mutual disarmament which was available to them (pay-off 1 for utility pay-offs depicted in Figure 1.4. Game theory makes a rather stark prediction in this game: both players will arm (the reasons will be given later). It is a paradoxical result because each does what is in their own interest and yet their actions are collectively self- defeating in the sense that mutual armament is plainly worse than the alternative of mutual disarmament which was available to them (pay-off 1 for each rather than 2). The existence of this type of interaction together with the inference that both will arm has provided one of the strongest arguments for the creation of a State. This is, in effect, Thomas Hobbes’s argument in Leviathan. And since our players here are themselves States, both countries should agree to submit to the authority of a higher State which will enforce an agreement to disar m (an argument for a strong, independent, United Nations?).