An impossibility theorem for price-adjustment mechanisms

Abstract

We show that there is no discrete-time price-adjustment mechanism (any process that at each period looks at the history of prices and excess demands and updates the prices) such that for any market (a set of goods and consumers with endowments and strictly concave utilities) the price-adjustment mechanism will achieve excess demands that are at most an ϵ fraction of the total supply within a number of periods that is polynomial in the number of goods and . This holds even if one restricts markets so that excess demand functions are differentiable with derivatives bounded by a small constant. For the convergence time to the actual price equilibrium, we show by a different method a stronger result: Even in the case of three goods with a unique price equilibrium, there is no function of ϵ that bounds the number of periods needed by a price-adjustment mechanism to arrive at a set of prices that is ϵ-close to the equilibrium.

The price equilibrium existence theorems of Arrow and Debreu (1) are fundamental positive results in economics. Their proofs, however, are nonconstructive; in other words, even though it is shown that in appropriately well-behaved situations there are prices under which markets clear, no mechanism is provided for finding these prices. The search for a “price-adjustment mechanism” for finding these price equilibria started almost a century before their existence was established, when Walras proposed a price-adjustment process (a rule for updating prices based on excess demand) that he called tatônnement (2). However, tatônnement was shown by Scarf in 1960 to fail to converge for certain markets (3). Following this negative result, Scarf proposed in 1967 a family of computational procedures (4–6), which arrive at an approximate price equilibrium by “pivoting” on simplicial subdivisions of the price simplex. These works initiated a long line of subsequent research on the development of methods for the computation of equilibria and on the design of adjustment processes, alternative to tatônnement, that would converge to an equilibrium.

In 1976, Smale proposed a price-adjustment process that converges to a price equilibrium from almost any starting point (7). In 1978, Saari and Simon showed that Smale’s process is informationally optimal, in that any convergent price-adjustment process based on difference or differential equations needs the information used by Smale’s (namely, the excess demand and its derivatives at all commodities) (8). Smale’s result was later improved to a process that converges from all starting points (9, 10, 11), ostensibly settling this important problem. However, these price-adjustment processes follow a complicated sequence of segments of algebraic curves within the price simplex, and their precise complexity has not yet been called into question. It follows from our results (Theorems 1 and 2) that these procedures must involve, for some markets, an arbitrarily large number of segments.

In this past decade, equilibrium problems were studied more intensely from the computational point of view. One of the lines of investigation in this research has been to establish that several special cases of the market equilibrium problem can be solved by efficient algorithms (see, for example, ref. 12 and Chapters 5 and 6 of ref. 13). These algorithms, however, work for special cases of the problem (mostly falling within the gross-substitutability family of markets, known to behave well even under tatônnement) and, besides, they do not qualify to be called price-adjustment processes, as they find the price equilibrium by zooming into it in noniterative ways, via convex programming or combinatorial algorithms. One exception is the work of Cole and Fleischer (14), who indeed describe a price-adjustment process (within our framework, see Statement of the Results) that converges to an equilibrium in polynomial time—but again, for special kinds of markets satisfying the gross-substitutability property. On the negative side, it has been recently shown (15) that finding a price equilibrium in markets whose consumers have Leontief utilities (a class that violates gross-substitutability) is intractable—that is, it cannot be solved in polynomial time under certain widely accepted complexity assumptions. Our main result complements the one in ref. 15 with an exponential lower bound that is independent of any complexity assumptions.

In this paper we consider a very general form of discrete-time price-adjustment mechanisms, in which we start with some set of prices and, for t = 1,2,…, prices are updated in an arbitrarily sophisticated way taking into account the excess demands of the current prices and their derivatives (if they exist) plus all prices, excess demands, and derivatives observed in the past. By making our definition of a price-adjustment process so general, we render our negative results stronger. The discrete-time nature of our mechanisms is not a restrictive assumption: It is easy to see that any continuous algebraic process such as the ones employed in refs. 9 and 11 can be approximated by a discrete-time one of complexity that is polynomial in the number of segments in the path to equilibrium. We prove that any such process cannot converge to any near-equilibrium in time less than exponential in the number of goods. More precisely, we show that for any discrete-time price-adjustment process and any n and sufficiently small ϵ there is a market with n goods such that the number of adjustment steps required to reach a set of excess demands that are within a factor of ϵ away from the total supply is exponential in the number of goods, with the base of the exponent depending on ϵ.

There are two distinct notions of approximating an equilibrium. In the previous paragraph we discussed one of them, namely finding prices which approximately clear the market. Note, however, that such an “approximate equilibrium” may not be near any true equilibrium. The second notion requires that the prices obtained be ϵ-close to an actual price equilibrium of the market; clearly, any price-adjustment mechanism that converges to an equilibrium (such as those in refs. 7, 9, 10, and 11) must eventually get ϵ-close to it, by definition. This notion of approximation usually begets more difficult problems (compare, e.g., ref. 16 with ref. 17). For the convergence to an actual equilibrium, we can show a stronger result by a different technique: For any price-adjustment process and any function of ϵ, however steep, there are markets with three goods and three consumers and with a unique price equilibrium, such that the number of periods required for the prices to get ϵ-close to the price equilibrium grows faster than the given function.

Technically, our results rely on lower bounds for finding Brouwer fixed points in the n-dimensional simplex that are variants of previously known such results for the hypercube (18, 19). We transfer these lower bounds to the price equilibrium problem via the theorems of Uzawa (20) (reducing Brouwer fixpoints to price equilibria) and Debreu, Mantel, and Sonnenshein (21, 22, 23) (showing that all well-behaved excess demand functions are realizable through endowments and utility functions), and by adjusting them to the simplex, as opposed to the hypercube, and to differentiable functions with bounded derivatives, as opposed to piecewise linear Lipschitz-continuous functions.

Statement of the Results

We start by reviewing equilibrium theory notation. ℜ₊ denotes the nonnegative real numbers, and Δ_n denotes the n - 1-dimensional simplex. A market consists of n goods and m consumers. Each consumer i has an endowment of goods and a strictly concave utility function u_i: . The sum of the endowments is the supply vector assumed without loss of generality to be (1,…,1). A (normalized) price vector is an element of Δ_n. Given a price vector π, consumer i generates demand vector x_i(π) equal to arg max u_i(x_i(π)): π^T × x_i ≤ π^Te_i (the arg max is unique due to strict concavity). The excess demand for price vector π is . It is well known that the excess demand satisfies Walras’s law π^TZ(π) = 0, and that also Z_j(π)≥0 whenever π_j = 0.

We shall next define a discrete-time price-adjustment mechanism as any process, however complex, which starts at some initial price vector π₀ and generates a sequence of price vectors π_t, t = 1,2,…, where each π_t depends on the history of prices π₀,…,π_t-1 and the corresponding excess demands Z(π₀),…,Z(π_t-1). That is, we allow the mechanism to be any function F: ∪_i≥0[Δ_n × ℜⁿ]ⁱ↦Δ_n. If Z is differentiable (as it will be in our constructions), we allow the mechanism to depend also on the values of the derivatives. We say that such a process converges to an ϵ-equilibrium after T steps, for some ϵ > 0, if there is a t ≤ T such that |Z_j(π_t)| ≤ ϵ for all goods j. We say that F converges to within ϵ of an equilibrium after T steps if there is a t ≤ T and a price equilibrium π^∗ such that |π_t - π^∗| ≤ ϵ (the latter requirement is stronger for Lipschitz-continuous excess demand functions such as those considered here).

Our main results are these:

Theorem 1.

For any discrete-time price-adjustment mechanism F, any n≥4, and any positive , there is a market with n goods and n consumers and with excess demand function that is differentiable with bounded derivatives, such that F fails to converge to an ϵ-equilibrium after steps.

Theorem 2.

For any discrete-time price-adjustment mechanism F, any function T: [0,1]↦Z, and any , there is a market with three goods and continuously differentiable excess demand function with bounded derivatives, which has a unique price equilibrium, such that F does not converge within ϵ of the equilibrium after T(ϵ) steps.

Proof of Theorem 1

Because it is known that any excess demand function over m goods that satisfies Walras’s law can be expressed as the excess demand function of m consumers, for appropriate utilities and endowments (21–23), to establish Theorem 1 it suffices to show that there is an excess demand function Z: Δ_n↦ℜⁿ satisfying Walras’s law such that F behaves as stated in the theorem.

To this end, we establish a lemma concerning the difficulty of approximating Brouwer fixpoints. Recall that Brouwer’s theorem states that any continuous function ϕ: Δ_n↦Δ_n has a fixpoint. A black box algorithm for finding approximate Brouwer fixpoints is any process that produces a sequence of points x₀,x₁,… in Δ_n, where the point x_t+1 depends on x₀,…,x_t and ϕ(x₀),…,ϕ(x_t). That is, a black box algorithm is any function B: . If the function ϕ is differentiable (as it will be in our constructions) then the algorithm receives also the values of the partial derivatives of ϕ at every point in the constructed sequence and can use this information also in determining the next point. Intuitively, B “queries” points x₀,… in order to find out the corresponding values of ϕ and its derivatives, and it decides the next point to query based on the complete history of previous results. We say that B converges to an ϵ-fixpoint in t steps for an ϵ > 0 if |ϕ(x_t) - x_t| ≤ ϵ (where |v| denotes the Euclidean length of a vector v).

Lemma 1.

For any black box algorithm B, any integer n≥4, and any , there is a continuous differentiable function ϕ: Δ_n↦Δ_n with bounded derivatives such that B fails to converge to an ϵ-fixpoint in steps.

Theorem 1 follows from the above lemma via Uzawa’s transformation (20), whereby from any function ϕ from the simplex to itself one can obtain an excess demand function obeying Walras’s law via the transformation Z(p) = ϕ(p) - (p^Tϕ(p)/p^Tp)·p. The price equilibria of this function are precisely the fixpoints of ϕ, and Z satisfies, besides Walras’s law, the stated differentiability and continuity conditions, assuming that ϕ does.

To translate the accuracy of approximation between ϕ(p) and Z(p), we need the following fact:

Lemma 2.

For ϕ(p) and Z(p) as above, and .

Proof of Lemma 2:

Write Δ = ϕ(p) - p. Then Z(p) = ϕ(p) - (p^Tϕ(p)/p^Tp)·p = Δ - (p^TΔ/p^Tp)·p. Thus, , where θ is the angle between vectors Δ and p. The vector Δ is constrained to have coordinates adding up to zero, and p is constrained to have nonnegative coordinates summing to 1. It is easy to see that sin θ is minimized (equivalently, maximized) when p is the vector (1,0,…,0), and Δ is proportional to the vector , which gives after a simple calculation . Because , it follows also that .

It is now clear that Theorem 1 follows from Lemmas 1 and 2. We thus turn to the proof of Lemma 1.

Proof of Lemma 1:

We shall start with a high-level description of function ϕ. In almost the whole volume of the simplex the function is , that is, it decreases its x₁ coordinate by ϵ^′, where ϵ^′ = 10ϵ. As a result, such a generic point is definitely not the sought ϵ-approximate fixpoint. There are three classes of exceptions: (i) the points along a certain path Π that we will construct; (ii) the points that are within L₂ distance δ of the path Π, where δ = 12ϵ; and (iii) the points with x₁ < δ.

The first class of exceptions, and the one that is most central to our construction, is a path Π, a one-dimensional continuous and differentiable non-self-intersecting curve starting from the barycenter of the face x₁ = 0, continuing to the point (where K is defined below), and then on to other points in the interior of the simplex. We next define the structure of the path and the function ϕ on points along this path. The intention is that ϕ will have a unique fixpoint that is precisely the end x^∗ of this path, and the only ϵ-fixpoints of the function are points that are very close to x^∗.

Let K be the largest integer multiple of n - 1 that is ≤ 1/6δ. Consider the subset Δ_n[K] of Δ_n consisting of b and all points whose coordinates are positive integer multiples of , and consider the graph Γ_n[K] whose set of vertices is Δ_n[K], and whose set of edges consists of the edge (b,b^′) and all edges joining two vertices whose difference has two nonzero components, one that is and one that is . Instead of constructing the path Π, we shall construct a path in this graph, and Π will be simply with all intermediate vertices approximated smoothly by an arc of a circle of radius 2δ for differentiability. We think of the path Π as directed away from its starting point b. The value of ϕ(x) = y at any point x∈Π of the path is the point y that is a distance ϵ^′ from x along the tangent of Π at x, directed the same way as the path, i.e., away from b. The only exception to this are the points in the last segment that are within a distance ≤ δ from the end x^∗ of the path (the fixpoint of the function), for which the displacement ϕ(x) - x decreases gradually to 0 as follows: , where ψ[z],z∈[0,1] is the smooth interpolating function ψ[z] = -2z³ + 3z². Note that ψ(0) = 0, ψ(1) = 1, and ψ(z) is continuously differentiable with derivatives 0 at z = 0 and z = 1.

Next, we describe the value of ϕ at points x with x₁≥δ that are in the “tube” of radius δ around Π, that is, at points x whose distance from x to Π, , satisfies r(x) ≤ δ. It is easy to see that r(x) is well-defined, and so is the projection of x on Π, x_Π (the y that realizes the inf above). We subdivide the interior of the tube into three parts: the inner subtube consists of points with r(x) ≤ δ/3, the middle subtube contains the points with δ/3 < r(x) ≤ 2δ/3, and the outer subtube has the remaining points with 2δ/3 < r(x) ≤ δ. We will define first ϕ(x) for points that lie on the boundaries of the subtubes, and then extend to the interior of the subtubes. If r(x) = δ/3, then the displacement ϕ(x) - x of x is a vector of length ϵ^′ pointing toward x_Π. If r(x) = 2δ/3, then the displacement ϕ(x) - x is the opposite of the displacement of x_Π, unless x_Π belongs to the last segment of length δ of the path near the fixpoint x^∗. In the latter case, if x_Π = x^∗, then the displacement is the vector of length ϵ^′ that points toward x_Π;otherwise, . We have already defined the function on the outer boundary of the tube [points with r(x) = δ]: They have the default displacement .

If x is a point in the interior of the inner subtube, then the displacement ϕ(x) - x is defined as a convex combination of the displacements at the projection x_Π on the path and at the point z where the ray from x_Π to x intersects the boundary of the inner subtube (i.e., the point on the ray at distance δ/3 from x_Π), with the proportions of the combination defined in a nonlinear way so that we have differentiability:

[1]

If x is in the middle or the outer subtube, then we interpolate between the two points y, z in which the ray from x_Π to x intersects the boundaries of the subtube that contains x, using the analogous formula . Note that if x is close to the x₁ = δ plane, then the points z and/or y may lie below the plane (thus their value is not defined yet); in this case we use in the above formulas the same values for the displacements at y, z as those for points above the plane on the boundaries of the subtubes. For example, if x is in the inner subtube, then ϕ(x) is given by Eq. 1 with ϵ^′(x_Π - x)/|(x_Π - x)| in place of (ϕ(z) - z).

Lastly, we define the function for points x with x₁ < δ. We define it first for points on the facet x₁ = 0. Any point x = (0,x₂,…,x_n) that is at distance δ/3 or more away from the line (b,b^′) is mapped to , i.e., the point on the facet that is a length ϵ^′ away in the direction of b. For points x on the facet that are at distance 0 < r(x) < δ/3 from the line (b,b^′), we define .

For points x = (x₁,x₂,…,x_n) with 0 < x₁ < δ, consider the line through x parallel to the line (b,b^′) and let x⁰, x^δ be the points at which it intersects the planes x₁ = 0 and x₁ = δ, respectively. Note that x⁰ = (0,x₂ + x₁,x₃,…,x_n) belongs to the simplex Δ_n, and its displacement is already defined. The point x^δ = (δ,x₂ + x₁ - δ,x₃,…,x_n) also belongs to the simplex Δ_n (and its displacement is already defined) unless x₂ + x₁ < δ; in this case, we consider x^δ as having the default displacement in the following formula. We interpolate between the displacements of the two points: Define . This completes our description of ϕ.

It is not hard to verify that the conditions on ϕ required for the theorem are satisfied by our choices in its definition:

Lemma 3.

The function ϕ as defined above maps Δ_n to Δ_n, for n≥4, and is continuous and differentiable with derivatives bounded by 10. Its only fixpoint is the endpoint x^∗ of the path Π, and all the ϵ-fixpoints are within distance δ of x^∗.

Proof of Lemma 3:

The fact that ϕ maps Δ_n to Δ_n is obvious for all points x except perhaps for the points with 0 < x₁ < δ. For these points, it is clear that all coordinates of ϕ(x) except possibly the first one are nonnegative and that the sum is 1. The only thing to check is that the first coordinate is also nonnegative; this follows from the fact that for z∈[0,1] and ; therefore, . Continuity and differentiability follow from the properties of ψ at 0,1.

Regarding the ϵ-fixpoints, consider all the points x that are more than δ away from the end of the path x^∗. Clearly there are no ϵ-fixpoints in the space with the default displacement. For points with x₁≥δ that belong to an inner or a middle subtube, the displacement is a convex combination of two perpendicular vectors of length ϵ^′; therefore, it has length at least . In an outer subtube, the displacement is a convex combination of and a vector of length ϵ^′ that has at most three nonzero coordinates that sum to 0; it is easy to check that for n≥4 the combination must have length more than .

In the region of points x with 0 ≤ x₁ < δ, consider the tube around the edge (b,b^′) with its subtubes as extending to the facet x₁ = 0. For points x in the inner subtube, the displacement is a convex combination of three vectors of length ϵ^′ that lie on the same two-dimensional plane, namely, the plane that contains x and the line (b,b^′). The three vectors are along (b,b^′), a vector v₂ perpendicular to v₁ and directed toward (b,b^′), and a vector v₃ parallel to x₁ = 0, also directed toward (b,b^′). Because the first coordinate of v₃ is 0, the projection of v₃ on (b,b^′) is less than , and it follows then with a little algebra that the length of the combination must be at least . The points in the middle subtube are similar. In the outer subtube, the displacement is a convex combination of three vectors: , , and a vector u₃ of length ϵ^′ on the x₁ = 0 face. If the coefficient of u₃ in the convex combination is at most 0.6, then, because of the first coordinate, the combination has length at least ; if the coefficient is more than 0.6, then the projection of the combination on the x₁ = 0 plane has length at least . The argument for points outside the tube is similar.

It remains to define the path . This is done as follows: We start with being just the edge (b,b^′), and we run the black box algorithm B on the function ϕ (as defined so far) for T steps (where T is the time bound in the statement of the lemma).

If B queries a point x that is within the tube of the path already defined, the response is the correct value of ϕ on that point calculated as above. Also, if it queries a point that is not within a distance δ of any edge of Γ_n[K], then the response is the value of ϕ on x, i.e., the generic value if x₁≥δ, and the appropriate value ϕ(x) if x₁ < δ (note that the value for such points does not depend on the path); intuitively, these are queries “wasted” by B. Thus, the only interesting case is when B queries a point x that is within a distance δ of an edge of the graph Γ_n[K] that is not on the path defined so far. Define a canonical query sequence to be one that queries only vertices of Γ_n[K]. A query near an edge can be simulated by two queries on its endpoints (because, as it will become clear soon, when a vertex is queried the value of ϕ on all edges adjacent to it is revealed), and so by assuming canonical sequences we overestimate the number of queries by a factor of two. Therefore, we shall henceforth assume a sequence of 2T canonical queries.

In response to a canonical query to a vertex v we do one of two things (and choosing between them is the crux of the construction). Either:

1. we leave the path unchanged, in which case the response to the query is the generic value of ϕ at the point queried; or

2. we extend the path Π from its current endpoint, possibly through several intermediate vertices of Γ_n[K], until it reaches v, and a vertex v^′ adjacent to v becomes its new endpoint; in this case the response to the query is the value of ϕ calculated by interpolation as explained above for the point v in the tube of Π, taking into account that the path now continues after v to v^′.

This completes the description of the construction. The function ϕ is as described in the beginning of the proof, with the path Π being the one generated by the canonical sequence of 2T queries by B as just described. What we have to show next is that, no matter which 2T vertices are queried by B and in which order, there is a way to make 2T choices between 1 and 2 above so that Π is a legitimate simple (that is, not self-intersecting) path in Γ_n[K].

To establish this, we define the following zero-sum game, played between two players called Probe and Extend on a connected graph Γ = (V,E). The state of the game is (S,u), where S is a subset of V inducing a connected subgraph of Γ, and u∈S. Initially, the state is (V,u) for some initial node u.

At each move, say at state (S,u), Probe selects a vertex v∈S. Extend responds as follows:

1. If S - {v} is connected and v ≠ u, then Extend does nothing, and the next state becomes (S - {v},u).

2. If S - {v} is connected and v = u, then Extend chooses a vertex u^′∈S adjacent to u, and the next state becomes (S - {u},u^′).

3. If S - {v} is disconnected, then Extend chooses a connected component S^′ of S - {v} (actually, the largest connected component resulting from the deletion of v).

   1. If u∈S^′, then the new state becomes (S^′,u).
   2. Otherwise, Extend chooses a vertex v^′ adjacent to v in S^′, and the new state becomes (S^′,v^′).

The only case when none of these conditions applies is when S is a singleton {u}, at which point the game ends and Probe pays Extend an amount equal to the number of moves played.

The relationship of the Probe–Extend game to our construction is captured by this lemma:

Lemma 4.

If the value of the Probe–Extend game with initial state (Γ_n[K] - {b},b^′) is V, then for any black box algorithm B for any sequence of V canonical queries by B there are responses that define a simple path in Γ_n[K].

Proof of Lemma 4:

We extend the path Π according to an optimal strategy of Extend starting in the game with initial position (Γ_n[K] - {b},b^′). We interpret the (canonical) queries by B as moves of Probe. The inductive assumption is that, if the game is at state (S,u), then we have a valid path from b to u, and we can extend this path to any point in S. This is certainly true at the initial state. If B queries a vertex v∈S, then we think of this as the Probe choosing vertex v (if B queries a vertex v outside S, then this is a wasted query and will not happen at optimal play). If S remains connected after the removal of v, then we do not extend the path, or we extend it by one edge if v = u. Otherwise, assume that the deletion of v decomposes S into at least two connected components. All of these components are adjacent to v, and one contains u. We choose one of these components (the precise way we choose it will become relevant when we lower bound the value of the game; it will be the largest component). If it is a component that contains u, then again we do not extend the path. But if it is a component that does not contain u, then we extend the path through the component that contains u until it reaches v, and then one more step to a vertex v^′ adjacent to v in this component, and this is the new path . The new state of the game is (S^′,v^′), where S^′ is the chosen component.

Because we can respond to queries by B by simulating any strategy of Extend, by choosing the optimum strategy we conclude that we can define a simple path , and thus a function ϕ, for any sequence of V queries.

The following now completes the proof of Lemma 1 and the theorem:

Lemma 5.

The value of the Probe–Extend game with initial state (Γ_n[K] - {b},b^′) is at least .

Proof of Lemma 5:

We need to describe a strategy by Extend that achieves this. Consider the subgraph of Γ_n[K] consisting of all vectors (x₁,…,x_n) with . This subgraph, denoted by H, is essentially a hypercube grid in n - 1 dimensions with nodes, containing b and b^′. We can assume that the game is played on H instead of Γ_n[K]; in other words, we allow Probe to start by querying all nodes not in H without counting these moves in the payoff, which obviously decreases V. Consider the first time during the game on H at which the number of nodes |S| in the state of the game becomes less than , and let P be the set of nodes that Probe has selected up to this point. It is straightforward to verify that there is a set A of vertices of H - P, possibly disconnected, of cardinality between and , such that for every vertex a∈A all vertices adjacent to a are in A∪P. It follows from standard isoperimetric results for the hypercube grid (24) that .

Lemma 1, as well as Theorem 1, now follow.

Proof of Theorem 2

We will show the theorem working again with the Brouwer fixpoint problem. Let B be any black box fixpoint algorithm on Δ₃ and suppose, for the sake of contradiction, that there is a function T(ϵ) such that for any ϵ > 0, after T(ϵ) steps B is within ϵ of a fixpoint (the metric is not important). Thus, after T(ϵ) steps, the price vector p = (p₁,p₂,p₃) satisfies for some fixpoint p^∗. For concreteness, take ϵ = 0.39 (we could fix it to any positive number less than 1/2). Let m = T(0.39). We will define a family of continuously differentiable functions from Δ₃ to itself, such that all the functions have a unique fixpoint, and there are two functions in that have the same values and derivatives at all the points probed by the algorithm B, and the fixpoints of the two functions differ in the first (actually, and the third) coordinate by 0.8—a quantity greater than 2ϵ—thus establishing the theorem. The Lipschitz constants of the functions in can be made as close to 1 as we wish.

Let d = 1/2^m+5 and δ = d/2 = 1/2^m+6. For each a∈[0,0.8] and b∈[d,0.1 - d] we have a function with fixpoint (a,b,1 - a - b). It suffices to describe the first two coordinates of any function f because the third coordinate is implied: f₃(p) = 1 - f₁(p) - f₂(p) for all p∈Δ₃. Recall the function ψ(z) = -2z³ + 3z² on the interval [0, 1] from the previous section (Proof of Theorem 1).

We partition the simplex Δ₃ into regions. Fig. 1 shows the partition on the p₁–p₂ plane. The function f^a,b for a point p = (p₁,p₂,p₃)∈Δ₃ is defined as follows.

1. Region 1 contains the points p with p₁≥0.9; in this region f^a,b(p) = (p₁ - δ,p₂,p₃ + δ).

2. Region 2 consists of the points p with p₁ ≤ 0.8 and p₂≥b + d; in this region f^a,b(p) = (p₁,p₂ - δ,p₃ + δ).

3. Region 3 consists of the points p with p₁ ≤ 0.8 and p₂ ≤ b - d; in this region f^a,b(p) = (p₁,p₂ + δ,p₃ - δ).

4. Region 4 consists of the points p with p₁ ≤ 0.8 and b - d < p₂ < b + d. In this region, . If p₂ ≤ b then , and if p₂ > b then . In either case, .

5. Region 5 consists of the points p with 0.8 < p₁ < 0.9. Note that in this case the point q = (0.8,p₂,0.2 - p₂) is in region 2, 3, or 4, and the point q^′ = (0.9,p₂,0.1 - p₂) (if p₂ ≤ 0.1) is in region 1. Let ϕ_i be the formula of the function for region i = 1,2,3,4 as defined above; thus, for example ϕ₁(p) = (p₁ - δ,p₂,p₃ + δ). We define f^a,b(p) for region 5 to be the (nonlinear) convex combination f^a,b(p) = ψ(|p₁ - 0.8|/0.1)ϕ₁(p) + (1 - ψ(|p₁ - 0.8|/0.1)ϕ_i(p)), where i is the index of the region that contains the point q = (0.8,p₂,0.2 - p₂). Thus, for example, if p₂≥b + d, then i = 2, and thus f^a,b(p) = ψ(|p₁ - 0.8|/0.1)ϕ₁(p) + (1 - ψ(|p₁ - 0.8|/0.1))ϕ₂(p) = (p₁ - ψ(|p₁ - 0.8|/0.1))δ,p₂ - (1 - ψ(|p₁ - 0.8|/0.1)δ,p₃ + δ).

Fig. 1.

Regions of the function in the proof of Theorem 2.

It is easy to check that f^a,b(p) maps every point p∈Δ₃ to a point in Δ₃. The only points that require perhaps some attention are the points p with 0.8 < p₁ < 0.9 and 0.1 ≤ p₂ ≤ 0.2. Note that in this case, the point (0.8,p₂,0.2 - p₂) is in region 2, and in particular its second coordinate is decreased by the function, whereas the first coordinate stays the same. Thus, f^a,b(p) = p₁ - δψ(|p₁ - 0.8|/0.1),p₂ - δ(1 - ψ(|p₁ - 0.8|/0.1)),p₃ + δ), and hence, f^a,b(p)∈Δ₃.

Clearly, regions 1, 2, 3, and 5 do not have any fixpoint. In region 4, implies ψ(|b - p₂|/d) = 0, and hence, p₂ = b. Furthermore, implies p₁ = a. Thus, the only fixpoint is (a,b,1 - a - b).

It is straightforward to check that f^a,b is continuously differentiable at all points. For example, consider region 4. It is made up of two subregions: the part 4a where p₂ ≤ b and the part 4b where p₂ > b. On the boundary, p₂ = b, between the two subregions, the function as defined from both sides is f^a,b(p) = (p₁ + 0.1(a - p₁),p₂,p₃ - 0.1(a - p₁)), and the derivatives of the function from both sides agree because the derivative of ψ(z) at z = 0 is 0. At the boundary of region 4 with region 3, p₁ = b - d, the formulas for the two regions agree [because ψ(1) = 1] and the derivatives also agree [because ψ^′(1) = 0]. Similarly, it can be verified that the same holds within all the regions and at all the boundaries between regions. The functions are clearly Lipschitz-continuous with a small Lipschitz constant (it can be made arbitrarily close to 1).

If I⊆[d,0.1 - d] is an interval, let be the subclass of functions in that includes all functions f^a,b with b∈I. When the algorithm B requests the value of the function and its derivatives adaptively at a sequence of points, we will specify adversarially the values that are returned so that after m steps, there is a nonempty interval I such that all functions in are consistent with all the values returned to the algorithm in the m steps. Thus, for any b∈I, the two functions f^.8,b and f^0,b agree in all the values obtained by B, but their fixpoints are 0.8 apart in coordinate 1 (and coordinate 3), so no matter which point the algorithm produces, it is more than 0.39 far from the fixpoint of one of the functions.

We define inductively an “interval of uncertainty” I_t = [l_t,u_t] for each step t, and we define accordingly the values returned to the queries of the algorithm B. The interval I_t has the property that the values returned in the previous steps are consistent with all the functions f^a,b with l_t + d ≤ b ≤ u_t - d. Initially, I₀ = [0,0.1]. In the general step t + 1≥1, suppose that I_t = [l_t,u_t] is the interval at the end of the previous step, and let p = (p₁,p₂,p₃) be the new point queried by the algorithm in step t + 1. If p₁≥0.9, then return (p₁ - δ,p₂,p₃ + δ) (and the identity matrix as the Jacobean of derivatives at p) and let I_t+1 = I_t. If p₁ ≤ 0.8 and p₂≥(l_t + u_t)/2, then return (p₁,p₂ - δ,p₃ + δ) (and the identity matrix for the derivatives at p) and let I_t+1 = [l_t, min(p₂,u_t)]; thus, point p is placed in region 2 of the function. If p₁ ≤ 0.8 and p₂ < (l_t + u_t)/2, then return (p₁,p₂ + δ,p₃ - δ) and let I_t+1 = [max(p₂,l_t),u_t]; thus, point p is placed in region 3 of the function. If 0.8 < p₁ < 0.9 (i.e., p is in region 5), then we distinguish again two cases: If 0.8 < p₁ < 0.9 and p₂≥(l_t + u_t)/2, then we return (p₁ - αδ,p₂ - (1 - α)δ,p₃ + δ) where α = ψ(|p₁ - 0.8|/0.1), and let I_t+1 = [l_t, min(p₂,u_t)]; thus, point p is placed in the part of region 5 where the corresponding boundary point (0.8,p₂,0.2 - p₂) belongs to region 2. If 0.8 < p₁ < 0.9 and p₂ < (l_t + u_t)/2, then we return (p₁ - αδ,p₂ + (1 - α)δ,p₃ - δ + 2αδ) where α = ψ(|p₁ - 0.8|/0.1), and let I_t+1 = [max(p₂,l_t),u_t]; thus, point p is placed in the part of region 5 where the corresponding boundary point (0.8,p₂,0.2 - p₂) belongs to region 3.

Each step divides the interval of uncertainty at most in half, thus after m steps, the interval I_m = [l_m,u_m] has length at least 0.1/2^m > 2d. Let I = [l_m + d,u_m - d], which is thus a nonempty interval, and consider the class of functions . For any b∈I and any a∈[0,0.8], the function f^a,b is consistent with all the values received by B in all the steps; note that algorithm B has not queried any point in region 4, or in the part of region 5 next to region 4. Thus, the final point p that is produced by algorithm B is at least 0.39 away in the first (and third) coordinate from the fixpoint of the function f^a,b for a = 0.8 or a = 0.01 (we can pick a > 0 so that all the coordinates of the fixpoint, and hence the prices of all the goods in the equilibrium, are positive).

The proof can be easily extended to functions that are not only once continuously differentiable, but are more generally smooth functions that have continuous derivatives of all orders. The proof is basically the same, except that instead of the specific function ψ(z) = -2z³ + 3z², use a bump function to get a smooth function ψ(z) on [0, 1] whose value ranges from 0 at z = 0 to 1 at z = 1, and whose derivatives of all orders vanish at 0 and 1.

Discussion

We showed that price equilibria in an exchange economy, even though they are guaranteed to exist by classical theorems, are inaccessible within reasonable time (polynomial in the number of goods and the inverse of the desired accuracy) by price-adjustment mechanisms, in that for any such mechanism there will be markets for which price adjustment will converge exponentially slowly to a near equilibrium, and in fact it will converge to the actual equilibrium arbitrarily slowly. We note that our lower bound in Theorem 1 is reasonably tight, in that an ϵ-equilibrium can be found exhaustively within a number of steps of the form .

It would be interesting to improve the constants in the lower bound of Theorem 1, as well as the constants in the bounds for the values of ϵ for which the lower bound holds. One possible improvement would be through a more precise analysis of the Probe–Extend game on the simplex grid itself, as opposed to through the (much smaller, but easier to analyze) hypercube approximation employed in the proof of Lemma 5.

Our proofs work essentially by constructing markets that “fool” the hypothetical price-adjustment mechanism. As is common with such constructions, these markets may seem quite unnatural. One particular objection may be that their description complexity (the amount of information needed to specify them) seems very large. One observation pertinent to this last point is that the descriptive complexity of the constructed market is precisely the descriptive complexity of the hypothetical price-adjustment mechanism it was designed to defeat.