2.3 Computational perspectives on revealed preference theory
2.3.3 Proving hardness of approximation results
The techniques used to show hardness of approximation for optimization problems are part of a seminal body of work in complexity theory covering drawing on elegant results from a diverse array of fields such as probabilistically checkable proof systems, algorithmic coding theory, algebraic and Fourier analysis methods. Below, we at- tempt to provide the barest minimum, self-contained introduction to this immensely sophisticated theory in the hope that our own results on hardness of approximation for the rationalizability problems described above are better understood. We harbor no illusion that this treatment is exhaustive and refer the interested reader to online coursework [Sud99, Tre06] and survey material [AL96, Aro98] for a more full-fledged and elaborate discussion of these techniques. The definitions we provide below are based on those provided in [AL96].
For an instance I of any optimization problem, we denote the optimal value OP T(I) to be the value of the optimal solution to an instance of the optimization problem.
Definition 2.3.1 Approximation ratio: Let P be an optimization problem and I an instance of P with sizen and optimal valueOP T(I). An algorithm forP achieves an approximation ratio α=α(n) if it produces a solution at least α·OP T(I) if P is a maximization problem, or at most OP T(I)/α if P is a minimization problem.
We say that a problem P has hardness of approximation factor α, or is inap- proximable to within α, if it is N P-hard to achieve an approximation ratio of α for P.
The core idea behind proving inapproximability results is analogous to how we go about proving hardness results: we start with an instance of a problem P that is hard to approximate and produce an instance of the target problem P0 that we are attempting to show is inapproximable. The one important property we require of
this reduction is that it should be gap-preserving. This means, that if, say, bothP, P0 are maximization problems, then the reduction should ensure that for the respective instances I, I0:
OP T(I)≥c⇒OP T(I0)≥c0 OP T(I)≤ρ·c⇒OP T(I0)≤ρ0·c0
whereρ, c, andρ0, c0 are respectively parameters of the instancesI, I0. Naturally, this means that the better parameters we can obtain in a gap-preserving reduction, the better the hardness of approximation factor would be. However, this assumes that we already have a problem with a known inapproximability result to begin with. The rest of this subsection explores obtaining these prototypical inapproximability results.
As mentioned at the beginning of the section, there is a deep connection between inapproximability and probabilistically checkable proof systems which we illuminate below, after a few preliminary definitions. Let r, q : N → N be some functions over the set of natural numbers.
Definition 2.3.2 Verifier: Given a languageL, an (r, q)verifier V is a probabilis- tic polynomial-time Turing machine which takes as input a string x of length n. V further has access to a proof string that it queries in q(n) locations which are com- puted from the input string x and a set of r(n) random bits. Finally, V accepts or rejects x after evaluating in polynomial time some function that takes as input x, the r(n) random bits and the q(n) query results.
Definition 2.3.3 Probabilistically checkable proof system: Let L be a lan- guage. An (r, q) probabilistically checkable proof system for L with completeness c and soundness s comprises an (r, q) verifier V with the following properties:
• for every input x in L, there exists a proof string Πx for which V with input x and access to Πx accepts with probability at least c,
• for an input x not in L, and every proof string Πx, V with input x and access to Πx will accept with probability at most s.
We say that the language L is in P CP(r, q) if there is an (r, q) probabilistically checkable proof system forLwith some completenesscand soundnesssbounded away from 0 and 1 respectively, and with c > s. The crucial result that heralded a new and influential approach to inapproximability was that N P was in P CP(logn,1), and was a culmination of results due to Arora and Safra [AS98], and Arora et al.
[ALM+98]:
Theorem 2.3.4 (Arora and Safra [AS98], Arora et al. [ALM+98]) N P =P CP(logn, O(1)).
As an immediate consequence to the theorem above, known as the PCP theorem, the problem of maximizing the number of satisfiable3satclauses,max-3sat, was shown in [ALM+98] to be inapproximable to within 1−for some fixed . Inapproximability results for other problems such asmax-cut,max-clique,label-cover can all be shown based on the PCP theorem and devising PCP systems with the appropriate parameters. We refer the interested reader to other sources [AL96] for a more detailed treatment.
Chapter 3
Algorithms for playing games with limited randomness
Dennis “Cutty” Wise: The game done changed.
Slim Pierce: The game’s the same. Just got more fierce.
–The Wire, Season 3.
3.1 Background
The concept of randomness plays a central role in game theory and economics. For a Nash equilibrium to exist in a finite game, we crucially must rely on the fact that the strategies in consideration also include mixed strategies, i.e., probability distributions over the set of pure strategies.
There are different interpretations of what constitutes a mixed strategy. At its simplest, we can think of a mixed strategy as exactly a probability distribution over the set of strategies played in a one-shot game, with the pay-off then tabulated in expectation. In another characterization due to Rosenthal [Ros79] and Rosenthal and Landau [RL79], each player is thought of as coming from a population and the probability weight attached to each pure strategy is the fraction of the population that will play that pure strategy. The interaction then is between players drawn uniformly at random from their respective populations, playing out their corresponding pure strategy. In the entirety of this chapter, we concern ourselves with only the first interpretation.