Parity Argument Research Articles

Substitution with Satiation: A New Class of Utility Functions and a Complementary Pivot Algorithm

We introduce new classes of utility functions and production sets, called Leontief-free, which are applicable when goods are substitutes and utilities/production are subadditive (to model inter-good satiation). When goods are complements, the well studied Leontief utility functions do an adequate job; however, to the best of our knowledge, a similar concept for the case of goods that are substitutes was not known. For markets with these utility functions and production sets, we obtain the following results: Rational-valued equilibria, despite the fact that these utility functions and production sets are nonseparable. We prove that the problem of computing an equilibrium is PPAD-complete, where PPAD stands for Polynomial Parity Arguments on Directed Graphs. We obtain complementary pivot algorithms based on a suitable adaptation of Lemke’s classic algorithm. Our algorithms run in strongly polynomial time if the number of goods is a constant, despite the fact that the set of solutions is disconnected. Experimental verification confirms that our algorithms are practical.

Faith and Reason

AbstractA novel account of the rationality of religious belief is offered, called quasi-fideism. According to this proposal, we are neither to think of religious belief as completely immune to rational evaluation nor are we to deny that it involves fundamental commitments which are arational. Moreover, a parity argument is presented to the effect that religious belief is no different from ordinary rational belief in presupposing such fundamental arational commitments. This proposal is shown to be rooted in Wittgenstein's remarks on hinge commitments in On Certainty, remarks which it is claimed were in turn influenced by John Henry Newman's treatment of the rationality of religious belief in An Essay in Aid of a Grammar of Assent.

God, evil, and occasionalism

AbstractIn a recent paper, Alvin Plantinga defends occasionalism against an important moral objection: if God is the sole direct cause of all the suffering that results from immoral human choices, this causal role is difficult to reconcile with God's perfect goodness. Plantinga argues that this problem is no worse for occasionalism than for any of the competing views of divine causality; in particular, there is no morally relevant difference between God directly causing suffering and God indirectly causing it. First, we examine Plantinga's moral parity argument in detail and offer a critical evaluation of it. Then we provide a positive argument, based on the doctrine of doing and allowing, to show why there is a morally relevant difference between God's direct and indirect causation of suffering.

On the Epistemic Bilking Argument

The standard bilking argument is well-known and attempts to prove the impossibility of backwards causation. In this discussion note, I identify an epistemic bilking argument, which has not received sufficient attention in the literature, and indicate how best to respond to it. This response involves a parity argument based on a forwards causation case.

A Complementary Pivot Algorithm for Market Equilibrium under Separable, Piecewise-Linear Concave Utilities

Using Lemke's scheme, we give a complementary pivot algorithm for computing an equilibrium for Arrow--Debreu markets under separable, piecewise-linear concave (SPLC) utilities. Despite the polynomial parity argument on directed graphs (PPAD) completeness of this case, experiments indicate that our algorithm is practical---on randomly generated instances, the number of iterations it needs is linear in the total number of segments (i.e., pieces) in all the utility functions specified in the input. Our paper settles a number of open problems: (1) Eaves (1976) gave an LCP formulation and a Lemke-type algorithm for the linear Arrow--Debreu model. We generalize both to the SPLC case, hence settling the relevant part of his open problem. (2) Our path following algorithm for SPLC markets, together with a result of Todd (1976), gives a direct proof of membership of such markets in PPAD and settles a question of Vazirani and Yannakakis (2011). (3) We settle a question of Devanur and Kannan (2008) of obtaining a “sy...

Combinatorial Nullstellensatz Modulo Prime Powers and the Parity Argument

We present new generalizations of Olson's theorem and of a consequence of Alon's Combinatorial Nullstellensatz. These enable us to extend some of their combinatorial applications with conditions modulo primes to conditions modulo prime powers. We analyze computational search problems corresponding to these kinds of combinatorial questions and we prove that the problem of finding degree-constrained subgraphs modulo $2^d$ such as $2^d$-divisible subgraphs and the search problem corresponding to the Combinatorial Nullstellensatz over $\mathbb{F}_2$ belong to the complexity class Polynomial Parity Argument (PPA).

The Complexity of Approximating a Bethe Equilibrium

This paper resolves a common complexity issue in the Bethe approximation of statistical physics and the belief propagation (BP) algorithm of artificial intelligence. The Bethe approximation and the BP algorithm are heuristic methods for estimating the partition function and marginal probabilities in graphical models, respectively. The computational complexity of the Bethe approximation is decided by the number of operations required to solve a set of nonlinear equations, the so-called Bethe equation. Although the BP algorithm was inspired and developed independently, Yedidia, Freeman, and Weiss showed that the BP algorithm solves the Bethe equation if it converges (however, it often does not). This naturally motivates the following question to understand limitations and empirical successes of the Bethe and BP methods: is the Bethe equation computationally easy to solve? We present a message-passing algorithm solving the Bethe equation in a polynomial number of operations for general binary graphical models of n variables, where the maximum degree in the underlying graph is O(logn). Equivalently, it finds a stationary point of the Bethe free energy function. Our algorithm can be used as an alternative to BP fixing its convergence issue and is the first fully polynomial-time approximation scheme for the BP fixed-point computation in such a large class of graphical models, whereas the approximate fixed-point computation is known to be polynomial parity arguments on directed graphs (PPAD-)hard in general. We believe that our technique is of broader interest to understand the computational complexity of the cavity method in statistical physics.

Gröbner bases for finite-temperature quantum computing and their complexity

Following the recent approach of using order domains to construct Gröbner bases from general projective varieties, we examine the parity and time-reversal arguments relating to the Wightman axioms of quantum field theory and propose that the definition of associativity in these axioms should be introduced a posteriori to the cluster property in order to generalize the anyon conjecture for quantum computing to indefinite metrics. We then show that this modification, which we define via ideal quotients, does not admit a faithful representation of the Braid group, because the generalized twisted inner automorphisms that we use to reintroduce associativity are only parity invariant for the prime spectra of the exterior algebra. We then use a coordinate prescription for the quantum deformations of toric varieties to show how a faithful representation of the Braid group can be reconstructed and argue that for a degree reverse lexicographic (monomial) ordered Gröbner basis, the complexity class of this problem is bounded quantum polynomial.

Cognitive extension: the parity argument, functionalism, and the mark of the cognitive

During the past decade, the so-called “hypothesis of cognitive extension,” according to which the material vehicles of some cognitive processes are spatially distributed over the brain and the extracranial parts of the body and the world, has received lots of attention, both favourable and unfavourable. The debate has largely focussed on three related issues: (1) the role of parity considerations, (2) the role of functionalism, and (3) the importance of a mark of the cognitive. This paper critically assesses these issues and their interconnections. Section 1 provides a brief introduction. Section 2 argues that some of the most prominent objections against the appeal to parity considerations fail. Section 3 shows that such considerations are nevertheless unsuitable as an argument for cognitive extension. First, the actual argumentative burden is carried by an underlying commitment to functionalism, not by the parity considerations themselves. Second, in the absence of an independently motivated mark of the cognitive, the argument based on parity considerations does not get off the ground, but given such a mark, it is superfluous. Section 4 argues that a similar dilemma arises for the attempt to defend cognitive extension by a general appeal to functionalism. Unless it can be independently settled what it is for a process to be cognitive, functionalism itself will be undermined by the possibility of cognitive extension. Like parity considerations, functionalism is thus either unable to support cognitive extension or superfluous. Hence, nothing short of the specification of an appropriate mark of the cognitive that can be fulfilled not only by intracranial but also by extended processes will do as an argument for cognitive extension.

Settling the complexity of computing two-player Nash equilibria

We prove that Bimatrix, the problem of finding a Nash equilibrium in a two-player game, is complete for the complexity class PPAD (Polynomial Parity Argument, Directed version) introduced by Papadimitriou in 1991. Our result, building upon the work of Daskalakis et al. [2006a] on the complexity of four-player Nash equilibria, settles a long standing open problem in algorithmic game theory. It also serves as a starting point for a series of results concerning the complexity of two-player Nash equilibria. In particular, we prove the following theorems: —Bimatrix does not have a fully polynomial-time approximation scheme unless every problem in PPAD is solvable in polynomial time. —The smoothed complexity of the classic Lemke-Howson algorithm and, in fact, of any algorithm for Bimatrix is not polynomial unless every problem in PPAD is solvable in randomized polynomial time. Our results also have a complexity implication in mathematical economics: —Arrow-Debreu market equilibria are PPAD -hard to compute.

How to type: reply to Halbach

In my paper (Paseau 2008), I noted that Fitch's argument, which purports to show that if all truths are knowable then all truths are known, can be blocked by typing knowledge. If there is not one knowledge predicate, ‘K’, but infinitely many, ‘K1’, ‘K2’, … , then the type rules prevent application of the predicate ‘Ki’ to (names of) sentences containing ‘Ki’ such as ‘p ∧¬Ki⌜p⌝’. This provides a motivated response to Fitch's argument so long as knowledge typing is itself motivated. It was the burden of my paper to explore the case that knowledge typing is as motivated as truth typing by drawing on the parallels between epistemic paradoxes generated by sentences of the kind ‘this sentence is unknown’ and semantic paradoxes generated by sentences such as ‘this sentence is untrue’. Given that typing truth is one of the acknowledged options for solving semantic paradoxes, if the parity argument succeeds it follows that epistemic typing is as well-motivated as truth typing and that the typing response to Fitch's argument is correspondingly strong.

The Complexity of Computing a Nash Equilibrium

In 1951, John F. Nash proved that every game has a Nash equilibrium [Ann. of Math. (2), 54 (1951), pp. 286–295]. His proof is nonconstructive, relying on Brouwer's fixed point theorem, thus leaving open the questions, Is there a polynomial-time algorithm for computing Nash equilibria? And is this reliance on Brouwer inherent? Many algorithms have since been proposed for finding Nash equilibria, but none known to run in polynomial time. In 1991 the complexity class PPAD (polynomial parity arguments on directed graphs), for which Brouwer's problem is complete, was introduced [C. Papadimitriou, J. Comput. System Sci., 48 (1994), pp. 489–532], motivated largely by the classification problem for Nash equilibria; but whether the Nash problem is complete for this class remained open. In this paper we resolve these questions: We show that finding a Nash equilibrium in three-player games is indeed PPAD-complete; and we do so by a reduction from Brouwer's problem, thus establishing that the two problems are computationally equivalent. Our reduction simulates a (stylized) Brouwer function by a graphical game [M. Kearns, M. Littman, and S. Singh, Graphical model for game theory, in 17th Conference in Uncertainty in Artificial Intelligence (UAI), 2001], relying on “gadgets,” graphical games performing various arithmetic and logical operations. We then show how to simulate this graphical game by a three-player game, where each of the three players is essentially a color class in a coloring of the underlying graph. Subsequent work [X. Chen and X. Deng, Setting the complexity of 2-player Nash-equilibrium, in 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2006] established, by improving our construction, that even two-player games are PPAD-complete; here we show that this result follows easily from our proof.

Exchange rate and fundamentals: the case of Brazil

Forecasting performance is tested for a broad set of empirical exchange rate models for an emerging economy with independently floating regime and inflation target monetary arrangement. Compared to the recent literature on out-of-sample exchange rate predictability, we include a more extensive set of models. We test vintage monetary models of the 1980's, exchange rate equilibrium models of the 1990's and a Taylor Rule based model. This last model assumes an endogenous monetary policy, where the Central Bank follows a Taylor rule reaction function to set interest rates. Our results show that Taylor Rule models and Behavioral Equilibrium Exchange Rate models, the last one combining productivity differentials with portfolio balance effect, have superior predictive accuracy when compared to the random walk benchmark. Some out-of-sample predictability is also obtained with parsimonious models based on uncovered interest parity arguments.

The complexity of equilibria: Hardness results for economies via a correspondence with games

We give a reduction from any two-player game to a special case of the Leontief exchange economy, with the property that the Nash equilibria of the game and the equilibria of the market are in one-to-one correspondence. Our reduction exposes a computational hurdle inherent in solving certain families of market equilibrium problems: finding an equilibrium for Leontief economies is at least as hard as finding a Nash equilibrium for two-player nonzero sum games, a problem recently proven to be P P A D -complete. As a corollary of the one-to-one correspondence, we obtain a number of hardness results for questions related to the computation of market equilibria, using results already established for games [I. Gilboa, E. Zemel, Nash and correlated equilibria: Some complexity considerations, Games and Economic Behavior 1 (1989) 80–93]. In particular, among other results, we show that it is N P -hard to say whether a particular family of Leontief exchange economies, that is guaranteed to have at least one equilibrium, has more than one equilibrium. Perhaps more importantly, we also prove that it is N P -hard to decide whether a Leontief exchange economy has an equilibrium. This fact should be contrasted against the known P P A D -completeness result of [C.H. Papadimitriou, On the complexity of the parity argument and other inefficient proofs of existence, Journal of Computer and System Sciences 48 (1994) 498–532], which holds when the problem satisfies some standard sufficient conditions that make it equivalent to the computational version of Brouwer’s Fixed Point Theorem.

Teaching & Learning Guide for: Moral Realism and Moral Nonnaturalism

Teaching & Learning Guide for: Moral Realism and Moral Nonnaturalism

Quadratically Many Colorful Simplices

The colorful Carathéodory theorem asserts that if $X_1,X_2,\ldots,X_{d+1}$ are sets in ${\bf R}^d$, each containing the origin 0 in its convex hull, then there exists a set $S \subseteq X_1 \cup \cdots \cup X_{d+1}$ with $|S \cap X_i| = 1$ for all $i=1,2,\ldots,d+1$ and $0 \in conv(S)$ (we call $conv(S)$ a colorful covering simplex). Deza et al. [Discrete Comput. Geom., 35 (2006), pp. 597–615] proved that if the $X_i$ are in general position with respect to 0 (consequently, each $X_i$ has at least $d+1$ points), then there are at least $2d$ colorful covering simplices, and they constructed an example with no more than $d^2+1$ such simplices. Under the same assumption, we show that there are at least $\frac{1}{5}d(d+1)$ colorful covering simplices, thus determining the order of magnitude. A similar result was proved independently by Stephen and Thomas [http://www.arxiv.org/abs/math.CO/0512400 (2005)]. We also obtain a lower bound of $3d$ for $d \geq 3$, which is better for small d and, in particular, together with a parity argument it settles the case $d=3$, where the minimum possible number of colorful covering simplices is 10.

Responsibility Incorporated

Responsibility Incorporated

QUANTUM KALEIDOSCOPES AND BELL'S THEOREM

A quantum kaleidoscope is defined as a set of observables, or states, consisting of many different subsets that provide closely related proofs of the Bell-Kochen-Specker (BKS) and Bell nonlocality theorems. The kaleidoscopes prove the BKS theorem through a simple parity argument, which also doubles as a proof of Bell's nonlocality theorem if use is made of the right sort of entanglement. Three closely related kaleidoscopes are introduced and discussed in this paper: a 15-observable kaleidoscope, a 24-state kaleidoscope and a 60-state kaleidoscope. The close relationship of these kaleidoscopes to a configuration of 12 points and 16 lines known as Reye's configuration is pointed out. The "rotations" needed to make each kaleidoscope yield all its apparitions are laid out. The 60-state kaleidoscope, whose underlying geometrical structure is that of ten interlinked Reyes' configurations (together with their duals), possesses a total of 1120 apparitions that provide proofs of the two Bell theorems. Some applications of these kaleidoscopes to problems in quantum tomography and quantum state estimation are discussed.

Skeptical fideism in Hume's Dialogues concerning natural religion

Despite the fact that Hume is rightfully considered to be an antireligious writer, the concluding Filo's speech in the Dialogues Concerning Natural Religion contains one of the most striking proclamations of Christian fideism in the history of religious thought. In this paper, by investigating Hume's account of the relationship between religious faith and pyrrhonism, and the principal affinities between fideism and philosophical skepticism, I attempt to explain how could this happen. Hume's personal standpoint, which is different than Filo's, may be reconstructed by careful examination of the so called parity argument.

Packings and 2-packings of A-paths

Abstract We investigate the relation between packings and 2-packings of A-paths. We show a similarity to the relation between matchings and 2-matchings in graphs. The former needs parity arguments, while the latter admits a Kőnig-type characterization. The algorithmic method of proof also yields – as a by-product – a generalization of the odd ear-decomposition of a factor-critical graph. For packing non-returning A-paths there are two different natural notions of criticality, we show a decomposition for both. An odd ear-decomposition can be given for (strongly) criticals. For weakly criticals a so-called dragon-decomposition can be given – a decomposition in a specific way into a forest and some strongly criticals.