notion6 of approximate fixed point: find x such that f (x) ≈ x. The second issue is that we want a finite description of the input to our algorithm. Naively, one can define the value of the function on a grid, but then the set is no longer convex and a fixed point might not exist at all (see, e.g., Roughgarden and Weinstein [2016]). It turns out that a better way to define the input is as an arithmetic circuit: this gives a local and easy-to-verify guarantee that the function is indeed continuous.
Admittedly, there is something dissatisfying about proving computational hardness of “circuit problems”: we are going to prove that this problem is PPAD-hard, i.e., no easier than END-OF-A-LINE—but the reason we believe in the first place that END-OF-A-LINE is intractable is that we don’t know how to gain insight to the functionality of the circuit from staring at its structure. Nevertheless, we begin with the complexity of Brouwer’s fixed point because: this is a fundamental question; it is the starting point of many of our reductions; and, as we discuss later, even the black-box hardness in this case is highly non-trivial.
We know of two algorithms for computing an approximate Brouwer fixed point: there is Scarf’s algorithm [Scarf 1967], but its running time may be exponential in the description of the function; the same holds for brute-force search.
On the complexity side, Daskalakis et al. [2009a] proved that even in three dimensions, finding an exponentially close approximation (x such that ∥f(x) – x∥∞ ≤ 2−n, where n is the size of the input) is PPAD-complete; Chen and Deng [2009] proved the same problem continues to be hard even with only two dimensions. Notice that with a constant number of dimensions, any weaker approximation desideratum becomes trivial for brute-force search. Chen et al. [2009b] showed that in n dimensions, polynomial approximations are also PPAD-complete. We will later prove that even constant approximations are PPAD-complete, i.e., there exists some absolute constant ε > 0 such that it’s hard to find an x for which ∥f(x) − x∥∞ ≤ ε; this is already a significant improvement over the existing state of the art.
We can prove an even stronger form of inapproximability for finding a Brouwer fixed point: so far, we characterized the approximate fixed point with ℓ∞-norm, which means that f(x) and x must be close in every coordinate. Our main result in this regard (Theorem 4.2) is that even if we only require that f(x) and x are close in most of the coordinates, finding such x and f(x) remains PPAD-complete.
1.1.2 Market Equilibrium
Supply and demand is central to our modern understanding of economics: when demand exceeds supply, raising the price makes it less attractive to consumers and more attractive to producers, thereby reducing demand and increasing supply. Vice versa if the supply exceeds the demand. This simple idea marks the birth of neoclassical economics [Jevons 1866, Menger 1871, Walras 1874], and continues to inspire us today when we reason about free market economies. Since the consumption and production of one good depends on other goods, we are interested in a market equilibrium: a vector of prices where the supply and demand of every good matches.
The supply and demand argument has many weak spots, one of which is Giffen goods [Marshall 1895]: Consider a poor 19th-century English family consuming 10,000 calories a day from meat and bread. They prefer to eat meat, but the bread is cheaper—so they spend their daily income on 6 loaves of bread and 4 pounds of meat (each provides 1,000 calories). What happens if the price of bread increases? They have less free money to spend on the preferred luxury good (meat), which increases their demand for the Giffen good (bread). The existence of Giffen goods in practice has been contentiously debated for over a century [Edgeworth 1909, Stigler 1947], with evidence ranging from econometric analysis of historical market data to experiments in lab rats [Battalio et al. 1991].
Despite counterintuitive issues like Giffen goods, Arrow and Debreu proved that, under quite general conditions, a market equilibrium always exists [Debreu and Arrow 1954]. In particular, in the Arrow-Debreu model, agents sell the goods they own and use the revenue to buy other goods they want; this is in contrast to Fisher markets, where agents with a monetary budget purchase from a centralized seller. Market equilibria in both models exist, but can we find them? As in the case of Nash equilibrium, this question is of particular importance because if a centralized, omniscient algorithm cannot compute an equilibrium, it is hard to expect a distributed market with selfish agents to converge to one. In the words of Kamal Jain [Jain 2004, Nisan 2009b]:
If your laptop cannot find it, neither can the market.
In the same paper, Jain also gave an algorithm for computing Arrow-Debreu’s equilibrium when consumers’ utilities are linear. Since then, there has been a long sequence of algorithms for computing or approximating market equilibria (e.g., [Birnbaum et al. 2011, Codenotti et al. 2005, Cole and Fleischer 2008, Devanur et al. 2008, Garg et al. 2015, Garg and Kapoor 2006, Jain 2004, Jain and Vazirani, 2010]), for certain models and utility functions, as well as intractability results in other settings [Chen et al. 2009a, Deng and Du 2008, Garg et al. 2017, Vazirani and Yannakakis 2011]. In particular, Chen et al. [2013] consider a setting of markets that exhibit non-monotonicity: when the price of one product increases, its demand increases. Giffen goods, described above, are one example of non-monotone markets; Chen et al. [2013] construct examples in Arrow-Debreu markets when the increased price of a product increases the revenue of the seller, who now has more available money and demands more of the same good. Chen et al. show that for essentially any class of utility function that exhibits some non-monotonicity, computing the Arrow-Debreu market equilibrium is PPAD-hard.
We extend the PPAD-hardness proof of Chen et al. [2013] and show that even approximate market equilibrium is PPAD-hard (Theorem 9.1). We note that although our inapproximability factor is stronger than that showed by Chen et al., the results are incomparable as ours only holds for the stronger notion of “tight” approximate equilibrium, by which we mean the more standard definition that bounds the two-sided error of the market equilibrium. Chen et al., in contrast, prove that even if we allow arbitrary excess supply, finding a (1/n)-approximate equilibrium is PPAD-hard. Furthermore, for the interesting case of constant elasticity of substitution (CES) utilities with parameter ρ < 0, they show that there exist markets where every (1/2)-tight equilibrium requires prices that are doubly exponentially large (and thus require an exponential-size representation). Indeed, for a general non-monotone family of utility functions, the problem of computing a (tight or not) approximate equilibrium may not belong to PPAD. Nevertheless, the important family of additively separable, concave piecewise-linear utilities is known to satisfy the non-monotone condition [Chen et al. 2013], and yet the computation of (exact) market equilibrium is in PPAD [Vazirani and Yannakakis 2011]. Therefore, we obtain as a corollary that computing an ε-tight approximate equilibrium for Arrow-Debreu markets with additively separable, concave piecewise-linear utilities is PPAD-complete.
1.1.3 A-CEEI (CourseMatch)
University courses have limited capacity, and some are more popular than others. This creates an interesting allocation problem. Imagine that each student has ordered all the possible schedules—bundles of courses—from most desirable to least desirable, and the capacities of the classes are known. What is the best way to allocate seats in courses to students? There are several desiderata for a course allocation mechanism:
Fairness. In what sense is the mechanism “fair”?
Efficiency. Are seats in courses allocated to the students who want them the most?
Feasibility. Are any courses oversubscribed?
Truthfulness. Are students motivated
