There is a common joke in computing circles to announce at the beginning of a course that “The last person in the room must close the door.” Though at first blush the request seems reasonable, this is an example of an uncomputable function. The last person to enter the room has no way of knowing (computing) whether or he or she will, in fact, be the last to enter. Many believe that the calculation problems inherent in centralized systems can be solved by simply increasing the possible computing power. However, the existence and persistence of uncomputable functions highlights the limits of computation. Though Hayek addresses the calculation debate throughout his work, updating the Hayekian argument for liberalism with the modern language of computability and complexity can translate the knowledge problem and the liberal conclusions to a new generation. Computing power, no matter how capable, will never be able to appropriately manage resource allocation nor plan the economy.
Many popular science fiction novels explore the idea of an all-powerful Artificial Intelligence as the solution to economic planning problems (e.g. Asher 2013; Banks 1994; Heinlein 2014; Williamson 1990). Though many of these texts explore the potential drawbacks of such a system, the potential of computation to solve allocation problems has also percolated through the work of computational scientists, philosophers, and economists (e.g. Cockshott 1988; Cottrell et al. 2007; Cottrell and Cockshott 1993; Mirowski 2002). A key example of this phenomena is Ursula K. Le Guin’s futuristic society in The Dispossessed (2015). In this narrative, a holistic AI fully controls the society – including decisions about resource allocation, employment opportunities, and even social relationships. The core idea is compelling: one day, a computer program will know what is best for each of us and for the economy as the whole much better than we can know ourselves. The futuristic view of an omniscient artificial intelligence has captivated science fiction writers and movie makers, taking hold of the political imagination of the public. Utilizing the language of computability and explicitly discussing the limits of computational machines is an integral part of responding to this popular critique of the Hayekian knowledge problem and can assist in the contextualization of the Hayekian argument for liberalism.
The allure of computational solutions to economic problems is not only apparent in the public fascination with futuristic AI. Over reliance on computational solutions is more subtly reflected in the modern political debate as an insistence on utilizing science and computing methods in evidence-based policy making (Brooker 2001; Nutley et al. 2000; Practice 2006; Sutherland et al. 2004). Evidence-based policy requires that politicians and statisticians quantify the outcomes of a policy, typically in the form of risk analysis or cost-benefit analysis (Head 2010). Cost-benefit analysis, risk analysis, and all other optimization policies require making assumptions about the goals and potential costs of policy decisions and often are subject to probability neglect (Saltelli and Giampietro 2017; Sunstein 2003). These assumptions often eliminate alternative models of possible outcomes from the policy decision making process (Biesta 2007). In this way, the scientific method applied to policy may actually limit the consideration of the full set of options (Pawson 2006). For example, environmental conservation policy decisions are often based on subjective evaluations of ecosystem or species values, that are used in high-power forecasting. Though the forecasts are often striking, they are based on shaky assumptions and may not always be describing what is claimed. Additionally, the models require such high degrees of simplification, especially when considering more complex interactions within an ecosystem, that the results may be rendered relatively meaningless in the face of the complexity of the environmental system. The models, however, allow for quantitative storytelling that edge out other subjective readings of the political and environmental situation (Feinstein and Horwitz 1997). Modulating the insistence on evidence-based policy with an honest assessment of the limits of computation and the Hayekian idea of limited knowledge is key to fully considering the scope of policy options— especially liberal policy alternatives.
Hayek distinguishes himself from the other leading economists of his age through his insistence that the economy must be understood as a process that emerges from the bottom up through the interactions of decentralized agents possessing unique local knowledge (Hayek 1945, 1948, 1964). One of Hayek’s guiding questions was “How can fragments of knowledge existing in different minds bring about results which, to be brought about deliberately, would require a knowledge on the part of the directing mind which no single person can possess?” (Hayek 1948). In “The Use of Knowledge in Society,” Hayek claims that the challenges of decentralized knowledge render a central planner unable to rival the efficiency of the market (Hayek 1945). The emergent nature of the market, encapsulating the knowledge of each of the members of a society, will always outperform a single agent knowing a fraction of the relevant knowledge.
Hayek originally wrote “The Use of Knowledge in Society” as a rebuttal to Oskar Lange (Lange 1936, 1937), who developed an argument in favor of market socialism. Lange stated that there are only three kinds of data necessary to perform national economic planning: individual preferences, knowledge about available resources, and prices. Lange argued prices could be calculated from the other two. His basic model suggested that a trial and error process could find the equilibrium prices of goods quickly and efficiently. The assumptions of Lange’s market socialism persist in shaping the priors of the general public towards economic theory: if we know what people want and what we have, we should be able to calculate the best prices. The argument against economic planning is key to the Hayekian argument for liberalism, however, the argument must be restated to reflect persistent public misconceptions in the face of advancing in computational capabilities.
In the current age, the imagined single agent planner is no longer a government planner, but often a god-like Artificial Intelligence that can contain the relevant knowledge and perform the Lange calculations for prices and allocation. Updating the Hayekian argument for liberalism requires a serious treatment of the abilities and limits of computational machines. One relevant concept in this discussion is computability theory, a branch of computer science and mathematical logic also known as recursion theory (Bălţătescu and Prisecaru 2009; Casti 1997). Computable functions, according to the Church-Turing thesis, are those that can be calculated using any imagined machine (mechanical or physical calculation device) if there were unlimited amounts of time and storage space (Copeland 2007). Computable functions are thus restricted to only those functions for which an algorithm can be created (Casti 1997; Copeland 2007). In this case, an algorithm could be imagined as a list of rules that a person with unlimited time and an unlimited supply of pens and paper could follow through to find a solution. Under this framework, if a function or system does not follow an algorithm it is uncomputable. Another aspect of computability is known as Cantor’s diagonal argument (Ewald 2005; Lawvere 1969; Murphy 2006). Published in 1891 by Georg Cantor, the diagonal argument is a mathematical proof that uncountably infinite sets exist, and that they cannot have correspondence one-to-one with the set of infinite natural numbers (Ewald 2005). Such uncountable sets cannot be utilized in an algorithm because the algorithm could not be calculated in an infinite amount of time.
Computability in the economic realm deals primarily with the question of whether the rational allocation of resources can be solved in the framework of computability theory (Bălţătescu and Prisecaru 2009; Bartholo et al. 2009). Murphy’s extension of the knowledge problem in the calculation debate is based on Cantor’s diagonal argument (Murphy 2006). Murphy’s thesis is that if the central planner (which may be a supercomputer) hopes to mimic or outperform the market, it would need to consider not only an infinite number of prices, but an “uncountably infinite number of prices.” Murphy’s argument has three main assertions: 1) computation over an infinite uncountable domain is, in principle, impossible; 2) the central planning unit must consider an infinite number of prices; which leads to 3) a central plan is not merely impossible but uncomputable even in theory.
The Murphy argument, when taken together with the Hayekian knowledge problem, argues against not only the possibility of a computational machine discovering appropriate prices but also other aspects of policy making. This is because Hayek recognizes that the knowledge within an economic and social system is inherently complex (Hayek 1964). In “The Theory of Complex Phenomena,” Hayek defines complexity as “the minimum number of elements of which an instance of the pattern must consist in order to exhibit all the characteristic attributes of that class of patterns in question” (Hayek 1964). In biological and social phenomena, the degree of complexity means that there are infinite initial conditions necessary to forecast the possible end states of complex systems (Anderson et al. 1988; Bălţătescu and Prisecaru 2009). Over the past few years, the similarities between complexity science and Austrian economics have been progressively pointed out (Barbieri 2013; Koppl 2000, 2009; Rosser 1999; Vaughn 1999; Vriend 2002). Hayek’s research specifically is often highlighted as essential to understanding current frontiers of complexity (Gaus 2006; Kilpatrick 2001). One of the key insights of Hayek on complexity and knowledge is that economic behavior is not only complex but uncomputably complex (Bălţătescu and Prisecaru 2009; Hayek 1953; Kilpatrick 2001).
Applying this insight means that economic planners are dealing with uncomputability and even unlimited artificial intelligences would be unable to effectively set prices (Bartholo et al. 2009; Velupillai 2017). However, this insight also extends to evidence-based policy. Because knowledge is distributed and the relevant initial conditions of policy are uncountably infinite, the mathematical and computational methods necessary for policy making will never be able to fully anticipate policy outcomes (Feinstein and Horwitz 1997; Pawson 2006). In the age of computing, it is important for proponents of Hayekian liberalism to highlight the limits of computation and to understand the applications of complexity and computability theory in policy making.
Updating the Hayekian argument for liberalism in the face of rapidly expanding simulation and calculation capabilities requires that liberals explore the edges and bounds of computability. Recognizing the Hayekian knowledge problem requires considering human agents as imperfect algorithms with uncountably infinite packages of information. Together, this leads to an understanding of collective human action and an economy that is not only difficult to conceptualize but fundamentally uncomputable. The language of computation enables proponents of Hayekian liberalism to effectively harness the unique context, language, and concepts of the computing age to effectively argue against the socialist central planners – both man and machine (Hayek 2013).
And, if you are the last person to read this essay, please close the door.
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