towards a logic of stylized facts in computational social science

8/3/2019 Towards a logic of stylized facts in computational social science

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Towards a Logic of Stylized Facts inComputational Social Science

[preliminary version; please do not cite]

Gabriel Istrate

Abstract

Can virtual interactions and computerized simulations provide a testing groundfor theories that try to account for the evolution of norms in real-life settings?This question forms the basis of the important (and so far unsolved) problem ofverification and validation in the area of Social Simulation.

In this paper I will attempt to situate this problem within the mechanism-basedapproach to Analytical Sociology [Hedstrom and Bearman, 2009]. I will advocatethe development of an agile logical framework (or combination of logical frameworks)necessary to precisely specify concepts in social simulations that enable verificationand validation.

As a pedagogical example I will discuss the dynamics of stylized facts in a ver-sion of Peyton-Youngs stochastic stability approach to selection of risk-dominantequilibria. The model is built in a bottom-up fashion, also highlighting adversarial

scheduling approach I am currently developing ([Istrate et al., 2008, Istrate, 2008,Istrate et al., 2011]), a robustness analysis that should be part of any solution tothe verification and validation problem.

1 Introduction

Can there be a logic of society ? Or, stated less ambitiously, what is the role (if any)of logical methods in describing social dynamics ? This is a question that seems tohave been asked so many times, with so many different interpretations in mind thata complete survey of this literature would not be particularly enlightening. Elster ar-gued [Elster, 1978] that logical theory can be applied not only in the formalization of

knowledge already obtained by other means, but that logic can enter in the creative andconstructive phase of scientific work (op.cit. pp. 1). He explored the role of quanti-

fied modal logic in describing social reality, with a particular focus towards developinghis method as an alternative to Hegelian dialectics. Ragin [Ragin, 1989, Ragin, 2000]

Center for the Study of Complexity, Babes-Bolyai University, Fantanele 30, cam. A-14, RO-400294,Cluj Napoca ande-Austria Research Institute, Bd. C. Coposu 4, cam. 045B, Timisoara, RO-300223,Romania. email: [email protected]

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formalized the logic of case-based comparison in comparative sociology using Booleanalgebra (the algebraic counterpart of propositional logic). Closer to present Hannan etal. [Hannan et al., 2007] (see also [Peli et al., 1994, Peli et al., 2000]) proposed a ratio-nal reconstruction of social theory (organization science in particular) using techniques

based on predicate logic. Such methods are, of course, well-established in economics,with its insistence on methodological individualism and rational behavior. To give justone example, the so-called interactive epistemology program [Aumann, 2000] is by nowa classical part of theoretical economics, and a key ingredient of a recent intriguingproposal for a common foundational grounding of all social sciences [Gintis, 2009].

The working scientist, particularly the one coming from applied areas in ComputerScience, would most likely be unimpressed by a foundational question such as the one Istated above. To such a scientist the adequacy of a method is measured by the breadthand scope of its applications. He would (rightfully) point out to the considerable impactof formal logic in areas such as Artificial Intelligence, Semantics of Natural Language orAnalytic Philosophy. Temporal logic is a particularly significant success story in the field

of formal methods - techniques such as model checking [Clarke et al., 1999] and runtimeverification [Barringer et al., 2004] lie behind eliminating errors in designing computercircuits, in writing software for technological artifacts (from remote controls and mobiledevices to airplanes) or the Mars Rover [Brat et al., 2004].

The advent of computers is increasingly impacting the Social Sciences as well: Schelling[Schelling, 1971] could conceive his celebrated segregation model using pen and paperonly. Nowadays, a popular way to study social dynamics is via (massive) agent-basedsimulations [TRANSIMS, 2011, Eubank et al., 2004, Bishop et al., 2011]. Specializedprogramming environments such as Swarm, Netlogo, Repast, CORMAS, or GAMA, andspecification languages such as MAML, SDML and FABLES have been developed; a sci-entific community has emerged, with a shared culture, scientific venues and curriculum

(e.g. [Gilbert and Troitzsch, 2005]). The social impact of advances in social simulationis increasing - often simulations serve as a consultant to (and implicitly affect) publicpolicy [Martinez, 2006, National Institute of Health, 2010].

Given that all simulation models cannot be more than incomplete abstractions of real-ity, given the nature of social dynamics, often displaying complex behavior [Sawyer, 2005]including multiple types of emergence [Gilbert, 2002], it is important to certify thesoundness of the tools we employ; we would also further like to verify the robustnessof our conclusions with respect to variations in model specification, and the generalalignment of results produced by such methods with general sociological knowledge.This issues form, of course, significant aspects of the crucial problem of verifying andvalidating social simulations [Axelrod, 1997].

With respect to this question, our (hypothetical) working scientist could convincinglyargue that logic is in a good position to tackle this issue, and that decisive progress shouldbe relatively easy to achieve, given current state of the art. He could invoke the signifi-cant success of logic-based methods in the area of multiagent systems [Wooldridge, 2000,Wooldridge, 2002, Shoham and Leyton-Brown, 2009]. He could also point out to the in-creasingly usefulness of model checking techniques in the verification of software agents[Wooldridge et al., 2002, Lomuscio and Raimondi, 2006, Lomuscio et al., 2009], thus ver-

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ification techniques for social simulations should be available in a not-too-distant future.Yet the situation is not quite as good as optimistically described above. The above ad-

vances in software agents do not necessarily translate into corresponding advances on sim-ulating social agents. The techniques developed in this latter literature (and, more gener-

ally, the use of logic in social simulation) rely too little on existing sociological knowledgeand address to an insufficient degree the concerns of social scientists. Unsurprisingly,they have been criticized ([Edmonds, 2004], see also [Fasli, 2004, Dignum et al., 2004,Gaudou et al., 2011]) as not useful given the state of MAS and not [...] useful in ei-ther understanding or building MAS. Less formal approaches do not fare much better,though: There is no agreement what verification and validation mean in the contextof social simulations (see e.g. [Kuppers and Lenhard, 2005, Boero and Squazzoni, 2005,Windrum et al., 2007, Moss, 2008]). Methodological recommendations exist (see e.g.[Axelrod, 1997]); theoretical concepts having been developed in response to this situa-tion, such as the generative approach to social simulations [Epstein, 1999, Epstein, 2007],docking [Axtell et al., 1996] and replication [Wilensky and Rand, 2007]. They are, of

course, important components of a solution. But (as witnessed by the discussion inthe literature) they are not the whole story, and it is likely that no simple technique (orcombinations of techniques) is going to account for all aspects of verifying and validatingmultiagent models.

The goals of this paper are quite modest: I would like to advocate a particular ap-proach I am developing, the adversarial scheduling approach, a small component of asolution to the problem of verification and validation of multiagent models. My firstgoal is to situate this approach in the context of a popular recent approach in sociol-ogy, one relying on mechanism-based explanations. Second, I will use the adversarialscheduling approach as part of an illustrative example, built in a bottom-up manner, toinvestigate some of the requirements on a specification formalism for social mechanisms

and social simulations.Note that Theorem 1 is technically new (though a small extension of a result stated

in a more restricted form than needed in [Istrate et al., 2008]). The main focus of thispaper is, however, methodological rather than mathematical.

2 Mechanism-based explanations in Analytical Sociology

A popular approach in Analytical Sociology [Hedstrom and Bearman, 2009] concentrateson explaining social phenomena by recourse to social mechanisms [Hedstrom, 2005,Hedstrom and Swedberg, 2006, Demeulenaere, 2011]. There is little consensus whata social mechanism is: Hedstrom ([Hedstrom, 2005] pp. 25) compiles a list of seven

very different definitions (due to Bunge, Craver, Elster, Hedstrom and Swedberg, Lit-tle and Stinchcombe). Of these seven definitions the most useful for my purposes isdue to [Machamer et al., 2000] (also [Craver, 2001, Craver, 2006]). As paraphrased in[Hedstrom, 2005]

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mechanisms can be said to consist of entities (with their properties) and the ac-tivities that these entities engage in, either by themselves or in concert with otherentities. These activities bring about change [...]. A social mechanism, as heredefined, describes a constellation of entities and activities that are organized such

that they regularly bring about a certain type of outcome. We explain a socialphenomenon by referring to the social mechanism by which such phenomena areregularly brought about

Social mechanisms are not the only alternatives in describing evolution of social struc-ture: Hedstrom compares and contrasts them to covering-law explanations [Hempel, 1965]and statistical explanations. These alternatives are not mutually exclusive: social mecha-nisms can, for instance, be sometimes inferred from statistical considerations; or they canhave themselves involve stochastic and statistical ingredients. Social mechanisms alsocan be contrasted with theories, laws, correlations and black boxes [Schelling, 1998].

In any case, whatever social mechanisms are, they often have a complex structure;they can appear in families [Schelling, 1998], they can concatenate [Gambetta, 1998]

and be hierarchically nested [Craver, 2001]. It seems to us, therefore, that the followingstatements are indisputable:

Verifying and validating social models (including simulation models) needs to ad-dress issues pertaining to explanation and causality in social dynamics. In thisrespect statistical testing guidelines pertaining to replication such as those dis-cussed in [Axelrod, 1997], or generative explanations such as those proposed in[Epstein, 1999, Epstein, 2007] are necessary but not sufficient. On the other handsocial mechanisms, being in one acception interpretations in term of individualbehavior of a model that abstractly reproduces the phenomenon that needs ex-plaining [Schelling, 1998] naturally complete and complement these methods.

The role of social mechanisms in validating social models could be informally de-scribed as follows: simulation models should reproduce known social mechanismsthat are part of the expert knowledge in the area of concern and, of course, suggestnew ones.

Conversely, validating social simulations involves several forms of robustness anal-ysis: we want to make sure that the conclusions we arrive to and the causal mech-anisms we identify are not crucially affected by particular features of the modelspecification. In other words we want to eliminate competing mechanism-based ex-planations that would involve features not explicitly encoded into our frameworks.

In accordance with [Squazzoni, 2008] formalizing models is a prerequisite to illu-minate social mechanisms.

As a consequence of these statements I believe that the development of formal meansto specify social simulations, in particularly social mechanisms, is worthwhile. I do nothave in mind the development of yet another formalism of the sort that [Edmonds, 2004]labeled as belonging to the philosophical approach to logic. Indeed, I do not search

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for the one true logic of society (nor do I believe in the existence of a single suchformalism). Instead, I take a pragmatic stance. To me logic is primarily a tool forprecise specificationof simulation settings and results (rather than an inference tool; thevery development of computational simulations obviates to a certain extent the need for

inference). Precisely specifying system properties was the great enabler for the successfulapplication of model checking techniques in formal verification. I think social simulationcan share this philosophy and (ultimately) this success.

3 A note on mathematical modeling versus agent-basedsimulations

The main competitor (and alternative) to agent-based simulations is, of course, math-ematical modeling, mainly in the form of game theory. There is some tension betweenthe two areas: in some cases [Binmore, 1998] game theorists argue that the area of socialsimulation seems to pay too little attention to all the insights and techniques developedin game theory, pointing out to subtle errors in some famous simulation models thatcould have been avoided by appeal to standard game-theoretic results.

To be fair, mathematical theories of rational behavior have problems (other than theirintrinsic difficulty) that preclude a better adoption of game-theoretic techniques. Deci-sion and game theory have developed as primarily mathematical theories; accordingly,their main models were defined with mathematical tractability as primary focus. It isnot surprising that they are often inadequate into a computational setting: it is, for in-stance, hard to represent common knowledge of rationality (needed to justify the choiceof Nash equilibria) on the computer in a form that would enable a reasoning agent to effi-ciently deal with it. A more serious objection is that classical game-theoretic frameworksare often incapable to accurately capture important aspects of real-life scenarios. Toappeal to a powerful analogy, though Turing machines are universal, in the sense thatthey can represent any computational process (if we believe the Church-Turing thesis),in practice we do not use Turing machines for programming but special programminglanguages that are more expressive/better suited for this task. In a similar way, thoughin principle one can specify a decision theoretic situation using the standard framework,the combinatorial explosion associated with the translation process will render strategicdecision making impracticable.

I view the choice of one technique or the other as subject to tradeoffs that ultimatelydetermine the proper one to choose in a given context or other. Agent-based simulationsare naturally more expressive. On the other hand mathematical models produce morereliable and transparent results, and are naturally better suited for a bottom-up approachto social dynamics such as the one I develop in the next sections. The reason is thatthey generally have fewer tunable parameters than an agent-based model. Mechanism-based explanations (the sort of causal connections envisioned by [Hedstrom, 2005]) canoften be easier deduced by reverse-engineering a mathematical proof, rather thanfrom analyzing a simulation model. This is why the model I develop is mathematical.Naturally, I hope that some of the observations and techniques I develop would ultimately

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be useful in a computational setting as well.

4 Adversarial scheduling in a nutshell

According to [Istrate et al., 2011], the adversarial approach to validation of models ofsocial dynamics, be them mathematical or computational, is specified by the followingprinciples:

1. Start with a base case result P, under a particular scheduling model, often ran-dom scheduling.

2. Identify several structural properties of the scheduling model that impact the validityof P. Ideally, these properties should be selected by a careful examination of theproof of P, which should reveal their importance.

3. Of these properties identify those that (alone or in combinations) are necessary(sufficient) for the validity of P. Correspondingly, those that can be abstractedout without affecting the validity of P.

4. The process outlined so far can be continued by recursively applying steps (i)-(iii).In the process we may need to reformulate the original statement in a way thatmakes it hold under larger classes of schedulers, thus making it more robust. Theprecise reformulation(s) normally arise from inspecting the cases when the proofof P fails in an adversarial setting.

One should not expect that a program such as the one described above can always berealized. Instead, as previously explained, I argue for the development of a multitude of

approaches and technical tools that enable the analysis of social systems. This fact onlyparallels the situation in sociology: the area gradually adopted a middle-range approachto theorizing, preferring it to a grand-theory vision. It is only natural that tools built tosimulate social dynamics should reflect this situation. Further, middle-range theorizingis compatible [Hedstrom and Udehn, 2009] with a mechanism-based approach.

Nor is adversarial scheduling analysis to be expected to be implemented exactly asdescribed. Instead, it can be just one ingredient of incrementally constructing more andmore expressive models. Indeed, this is the situation with the (pedagogical) examplediscussed below.

5 Adversarial Scheduling: A Bottom-Up ExampleIn this section I will discuss a simple example of adversarial scheduling. It is a bottom-upmodel: similar to Sugarscape [Axtell and Epstein, 1996] I start with a very basic frame-work and then gradually add more realistic features. My concern is how the properties ofthe model change as new features are added, how this ties in with the mechanism-basedapproach and related issues, and what the requirements of a language for expressing

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stylized facts in this setting should be. Note that such an exercise is not completelynew: [Edmonds, 2003] discusses the requirements for an ideal simulation languagein the context of a SDML implementation of Schellings segregation model (see also[Gaudou et al., 2011]). In contrast to this work I start from an even less specific vantage

point: the primary goal of this paper is to observe what kind of logical tools seem to benecessary to formally specify and analyze the developed model.

5.1 The baseline scenario

The baseline scenario is very simple: n agents are in one of two states, A and B. Eachagent derives an utility u() from being in one of states A and B. Agents are scheduledat random: when scheduled each agent changes its state according to the best-responsedynamics. That is, it will change its state to the one that gives him the highest utility.

It is not hard to see what the stylized fact should be for this dynamics:

S0: Eventually every agent willplay strategy A.

This intuitive fact is so simple that apparently not much can be inferred from it.Notice first, though that a natural formalization of this stylized fact uses quantifiers (eventually) N (next) and A (globally) from temporal logic. In fact we need touse a probabilistic version of this quantifier, a.s., eventually with probability 1 o(1)(almost surely). Indeed, since the scheduler is probabilistic there is a small chancethat it will go forever without scheduling a certain node x, thus precluding stabilization.Almost certainly though this is not the case.

Conclusion 1: The logical formalism for stylized facts should be able torepresent concepts of probabilistic temporal logic.

How would we represent the above stylized fact in a pre-formal notation ? Assumethat S refers to the scheduler of the process, that somehow we are able to specify theRandom Scheduler and that the theory has equality. The suggested preformal notationfor this stylized fact is displayed in Figure 1. It is (perhaps not surprisingly) reminis-cent of the Extended Kamp Notation formalism [Barwise and Cooper, 1993] in SituationTheory [Devlin, 1991, Barwise, 1989, Barwise and Moss, 1996]. A box represents con-texts (situations), with the internal box corresponding to a situation and the outer box

specifying the fact that the inner situation supports the infon in the outer box.Specifically, in our case that the simulation process, codified as the situation s

in which infons Equal(S, RandomScheduler) and x|Agent(x) : [Schedules(S, x) NEqual(State(x), A)] hold, supports infon = a.s.A[x|Agent(x) : Equal(State(x), A)].

Why is it that we are suggesting a situation-theoretic approach, and not simply writea logical implication ? An answer is that in general the semantics of the implicationabove is not that of logical implication: the support of an infon may be derived from

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Equal(S, RandomScheduler)

x|Agent(x) : [Schedules(S, x) NEqual(State(x), A)]. . .

|=[a.s.A(x|Agent(x) : Equal(State(x), A))]

Figure 1: Pre-formal notation for the baseline stylized fact

a situation that specifies the simulation by running it, not by logically inferring thenecessary truth of the conclusion.

There is another good reason for considering a theory, such as situations theory, thattakes contexts as first-class objects: suppose that instead of the simple stylized fact dis-played above we would intend to represent a statement such as X causes Y to happen.Causality is, obviously, a subtle concept, difficult to capture exactly. One simple idea

pertaining to formalizing it [Pearl, 2000] is that of interventions: a situation supportsa given infon, and changing something in the situation makes the result no longer hold.A natural representation of such interventions would lead to a multiple nesting ofboxes, such as the one employed in the EKN formalism.

Note as well an important difference with respect to situation theory: classicallysituations are sets ofinfons. This representation is naturally well-suited for representingsnapshots of a social process. In contrast, we want to model entire processes. Thereforewe will need a formalism that allows the represented infons to evolve dynamically.

Conclusion 2: The specification language for stylized facts is likely to require adynamical version of situation theory.

Let me point out here that there exist executable formalisms that allow the representa-tion of changing elements. Gurevichs abstract state machines [Borger, 2010] (originallydeveloped under the name of evolving algebras)) have developed into a rich seman-tic specification method. Abstract state machines have already been applied to spec-ifying social phenomena (e.g. [Brantingham et al., 2005b, Brantingham et al., 2005a,Brantingham et al., 2009]).

It is easy to represent predicate logic into ASM: their mathematical specification isbased on sets of (evolving) partial functions. A state variable (e.g. the state of an agentx, State(x) in the previous stylized fact) can naturally be represented as an evolvingpartial functions. Similarly, predicates are naturally partial functions, whose codomains

encode truth values.The one difference with the semantics of ASM lies in the fact that we would like

to represent situations as first-class objects. This is the reason that the formalizationof situation theory is based on the theory of non well-founded sets [Aczel, 1988], setsthat satisfy the so-called anti-foundation axiom and are able to contain themselves aselements. In contrast, the ground sets of ASM are (usual) finite sets. We would thushave to modify the definition of ASM to employ finite non well-founded sets.

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5.2 Making the scheduler adversarial

Assume that instead of the random scheduler we employ a general scheduler. This meansthat we would consider instead a family of situations obtained by removing from sany reference to the random scheduler. Lets assume that we try to logically derive the

conclusion of the stylized fact from the premises, perhaps employing some automatedreasoning program.

This attempt immediately exposes the fact that the specification of the stylized factin Figure 1 is incomplete: there is no way to derive from the conclusions of the infonsin the specification of the situation (which employs the temporal quantifier next) thesupported conclusion (employing the quantifiers eventually (a.s.) and all).

The easy explanation is that we have not included into the specification of ourmodel all state-change axioms: to infer that eventually some property is goingto be true from the fact that it will be (sometimes) true at the next state we needto make sure that once an agent gets to state A it will never change its state again.The reader might have recognized the well-known problem of axiomatizing change inArtificial Intelligence better known as the Frame Problem [Shanahan, 2004].

Conclusion 2: The logical formalism for stylized facts is subject to dealingwith the Frame Problem.

[Edmonds, 2010] argues that the frame problem can often be eliminated by a properincorporation of the context heuristic. We do not share this view. Contexts as em-ployed in social simulations are often incomplete specifications of reality. Instead, to-gether with [Nakashima et al., 1997], we believe that it is causality as a heuristic thatcan often achieve this task. Ultimately though the frame problem needs to be addressedthrough action languages such as the situation calculus [Reiter, 2001] (confusingly differ-ent from Barwises situation theory). What variants of the situation calculus tend notto have (and this needs to be added to any practical specification formalism) is situationtheorys view of contexts as first-class objects. The envisioned logical formalism wouldprobably need to consist of a certain mixing of the two frameworks.

Returning to our example one can ask: Is the baseline stylized fact true when replacingthe random scheduler by an adversarial one ? Of course not: to give an agent x thechance to update its state the scheduler has to schedule it. An agent is fair if it satisfiesthis condition. Fairness could be expressed formally as

Fair(S) x|Agent(x) : [a.s.Schedules(S, x)]

Note that to be able to assert that Fair(RandomScheduler) holds we need to employ

the probabilistic quantifier a.s. and not its deterministic counterpart.Fairness is a necessary condition that a scheduler leads to stabilization. Is it sufficient ?

It is easily informally seen that it is indeed so. Proving it formally provides another hint:even if we use the state change axiom in Figure 1 we are only to infer that x|Agent(x) :[a.s.AEqual(State(x), A)]. To be able to commute quantifiers and a.s. we need tocritically employ the finiteness of the agent society.

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Conclusion 3: The logical framework needs to take into account the factthat some characteristics of the system (e.g. the number of agents) are finite.

Finally note that the mechanism that allowed us to identify fairness as a necessary

condition for eventual stabilization is the familiar backward chaining. For further connec-tions between backwards chaining and situation theory see [Nakashima and Tutiya, 1991].

5.3 Convergence time

The stylized fact in Figure 1 is omitting one important aspect: quantifying the number ofsteps needed to reach the steady state. Under random scheduling this time is, of course,a random variable. One could study experimentally the expected time of convergenceassuming the most unfavorable situation, that when all agents start in state B. Inthis particular case a mathematical result is simple explanation for the nature of theconvergence time:

THEOREM [COUPON COLLECTOR PROBLEM:]Let c R be a constant, let n 1 and let m = n ln n + cn. Denote by X the randomvariable defined as the minimum number of balls that has to be thrown uniformly atrandom into n bins so that each bin contains at least one ball. Then

limn

Pr[X > m] = 1 eec

.

The result applies directly, and shows that for any monotone unbounded functionf(n), with probability 1 o(1) the convergence time is at most n log n + f(n). Canthis be generalized if the scheduler is less fair than a random scheduler The answer is

of course affirmative and relies on defining a g(n)-fair scheduler (for a certain monotonefunction g()) as one that is guaranteed with probability 1o(1) to schedule at least onceevery single agent in a consecutive sequence of g(n) steps. Clearly a g(n)-fair schedulerleads to stabilization in at most g(n) steps.

Conclusion 3: For deterministic individual choice the (amount of) fairnessdetermines convergence (time).

5.4 Adding randomness to individual choice

Modify now the baseline scenario by making each agent x only choose the better strategyA with probability 1 x for some x c > 0, c being a prespecified constant. Formallythis is accomplished by replacing the update axiom x|Agent(x) : [Schedules(S, x) NEqual(State(x), A)] by its probabilistic counterpart:

x|Agent(x) : [Schedules(S, x) NPrEqual(State(x), A, 1 x)]

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(where PrEqual is the probabilistic extension of the equality predicate with obvioussemantics)

In this case the previous stylized fact breaks down: one can no longer guarantee thateveryone eventually plays A simultaneously. Instead what we can guarantee is a weaker,

probabilistic fact:

S: Eventually it will hold that for every agent x, its state will be A withprobability 1 x and B with probability x.

The above stylized fact still specifies a fixed point property, though a weaker onethan the one in the deterministic case: the distribution of the system state is (eventually)a product distribution, with each agent being A independently with probability A. Thishas testable implications: for instance, the proportion of agents playing strategy B isclose (with high probability) to Avgx[x]. But in any case, the weaker fixed pointproperty includes the old one as a special case: it is obtained when all x 0.

However, in spite of this fact, by extending the nature of the dynamics we are introduc-ing a qualitatively new fact: moving from a deterministic to a probabilistic update rulemakes possible the existence of an adversarial scheduler that is able to forever precludeprobabilistic self-stabilization.

Indeed, consider the following scheduling strategy:

1. Consider agents according to a fixed permutation.

2. Given agent x schedule it as many times as needed until it plays strategy B. Thengo to the next agent in the list.

This scheduler may even have fairness properties that are no worse than those of therandom scheduler: Indeed, the expected number of times in the first n ln(n) steps that

the scheduled agent is turned to B is at least cn log n. Now invoking simple concentrationresults, more precisely a version of the Chernoff bound can establish the above claim.

Conclusion 4: For stochastic individual choice scheduler fairness does not(on its own) determine convergence.

The previous example is, however, instructive: the adaptive scheduler constructedas counterexample crucially depends on the system state. Suppose instead that thescheduler is specified as a random walk on a given influence network G on the nagents. Then it is easy to see that at the (stopping time) Tcover(G) when the randomwalk has covered all the nodes the following is true: every node x is A with probability1 x and B with probability x independently of the other nodes. In other words thesystem is in the required product distribution.

Conclusion 5: For stochastic individual choice scheduler non-adaptivenessdetermines convergence (time).

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strategies A BA a,a c,dB d,c b,b

Figure 2: Coordination games: payoff matrix

5.5 Adding network interaction

Lets now move beyond the simple scenario when agents act independently and introducestrategic interaction. For the moment in a very limited form. Specifically, we assumethat agents are grouped in pairs, and play a two-person game whose strategies are Aand B (their states). When scheduled, an agent updates its state (strategy) to be thebest-response against the choice of its neighbor.

From the standpoint of mathematical specification, all we have done is extended theclass of dynamics under consideration by modifying the update axiom

x|Agent(x) : [Schedules(S, x) NPrEqual(State(x), A, 1 x)].

The class of dynamics under consideration properly includes those that were consid-ered before, obtained under the case where decision in the two-player game is actuallyindependent of the partners actions. It is readily seen that this extension is too large:for some games the stabilization of the system to the all A configuration. But it iseasy to prove that the dynamics will eventually stabilize, assuming the game is sym-metric and the best-response strategy is not indifferent between A and B. On the otherhand the stable state might not be unique: this is the case, for instance, of coordinationgames, in which A (B) are the best response against a similar strategy.

Moving from a single neighbor to multiple neighbors (i.e. to a general network) is

simple: a scheduled agent simply plays an independent game against each of its neighborsand sums up all his profits in such games. As discussed before, it makes sense toconsider symmetric coordination games only. Such games can be specified by the tablein Figure 2. To be a coordination game we have to require that a > d and b > c.Peyton-Young [Young, 2001, Young, 1993] further considered the case of so-called risk-dominant equilibria. This further requires that a d > b c > 0. Intuitively thiscondition expresses the fact that A is preferred to B when playing against the mixedstrategy 0.5A + 0.5B.

In this setup employing a best-response dynamics can still lead to multiple equilibria ina path-dependent manner: both all A and all B are fixed points for the best-responsedynamics. Peyton-Youngs fundamental insight was that considering a noisy version of

the best-response dynamics (called logit dynamics and formally specified below) and analternate notion of convergence (called stochastic stability) can solve the equilibriumselection problem. Formally

Definition 1 The logit update rule is specified as follows: if agenti is the one to update,x is the joint profile of agents strategies, and z A, B is the candidate new state, then

p(xi z|x) ei(z,xi),

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where i(z, xi) is the total payoff obtained by player i by playing strategy z againstits neighbors that play strategy profile xi and > 0 is a parameter (called inversetemperature).

Define now the fundamental concept of a stochastically stable state for dynamics de-scribed by a Markov chain:

Definition 2 Consider a Markov process P0 defined on a finite state space . For each > 0, define a Markov process P on . P is a regular perturbed Markov process ifall of the following conditions hold.

P is irreducible for every > 0.

For every x, y ,lim>0

Pxy = P0xy.

If Pxy > 0 then there exists r(m) > 0, the resistance of transition m = (x y),

such that as 0, Pxy = (r(m)).

Let be the (unique) stationary distribution of P. A state S is a stochasticallystable strategy if lim0

(S) > 0.

Peyton Youngs framework for the diffusion of norms can be recast into the frameworkof Definition 2. Indeed, let = exp(). Then as , 0. His result has thefollowing statement:

Proposition: The unique stochastically stable state for the logit dynamicsunder random scheduling in risk-dominant games is the state A in which everyplayer plays the risk-dominant equilibrium A.

Indeed, the stationary distribution of the perturbed chain corresponding to the logitrule is the Gibbs distribution

P r() =eH()

Z

with H the potential function of the game on state and Z is the partition function.For the distribution will only put positive weight on states maximizing potentialenergy H, in this case state A

It is, perhaps, fitting to explain the particular choice of the logit rule (and the associ-ated stationary Gibbs distribution) using the case of uncoupled agents, developed above.As we saw, for probabilistic choice under random scheduling the distribution of theagents state is a product distribution. This fact partially precludes overly complex be-

havior, by disentangling the individual payoffs into contribution of each individual two-player game. Not entirely: the normalizing factor in the Gibbs distribution, the partition

function is hard to compute (technically: #P complete [Moore and Mertens, 2011])in many cases. Indeed, many versions of the Ising model display phase transitions. Butin a sense the appeal to the logit rule ensures the best generalization of the stylized factthat the stationary distribution is a product distribution we can hope for (perhaps), inthe case of coupled agents.

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6 Selection of risk-dominant equilibria under adversarialscheduling

It is easy to extend the argument in the previous sections to the case of the logit dynamics

to produce a maximally fair adversarial scheduler that forever precludes the system fromgetting too close to state A: scheduler S simply schedules nodes into a fixed periodicorder, repeating each node until it plays strategy B (which happens at each step withconstant probability; see full details in [Istrate et al., 2008]).

Conclusion 6: Scheduler fairness does not play a role in risk-dominant equi-librium selection.

Extend the previous example of a nonadaptive scheduler to the case where the nextscheduled node may depend on the scheduling history but not on the system state. Therandom scheduler certainly belongs to this class. To make the system Markovian, thenext scheduled node should naturally depend on the last scheduled node but not the

outcome. This motivates the following definition:

Definition 3 A weakly nonadaptive scheduler is specified as a Markov chain M on{A, B}N {1, 2, . . . , N }. Informally, the first component specifies system state, whilethe second component indicates the last nodes to have been scheduled. I assume that the

following two conditions hold:

1. If some transition with second components x y has positive probability in Mthen for any state s {A, B}N the transition (s, x) (s, y) (where s is obtained

from s by making a best-response move at y) has positive probability (we say thatx y is feasible).

2. If x y is feasible in M , then y x is feasible as well.

The following theorem is a slight extension of a result proved in [Istrate et al., 2008]:

Theorem 1 The set S = {(w, x)|w = VA} is the set of stochastically stable states forrisk-dominant equilibrium selection under a weakly nonadaptive scheduler.

Informally, the scheduled nodes can be described by a random walk on a graph on [n].Crucially though (and this was not true for the model stated in [Istrate et al., 2008])the actual transition probabilities may depend on state x. This makes the stationarydistribution of the resulting chain not necessarily factor out as a product distribution1.

Conclusion 7: Scheduler adaptiveness plays a role in precluding risk-dominantequilibrium selection.

1In [Istrate et al., 2008] it was claimed that the weaker model in that paper is general enough toaccomplish this requirement. The claim was, however, incorrect: The stationary distribution in thatpaper factors out, the first component is simply the Gibbs distribution, the second one depends onthe influence network only, and the main result in that paper, while correct, has a simpler proof.

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What about convergence time ? A recent breakthrough [Montanari and Saberi, 2009,Montanari and Saberi, 2010] has completely elucidated the convergence time of the logitdynamics in the case of random scheduling, showing that its magnitude is dependent ona structural parameter of the network of strategic interactions called tilted cutwidth.

In the case of nonadaptive adversarial scheduling we have two networks in play: a net-work G which governs strategic interaction, and another one H that specifies scheduling.As observed in [Istrate et al., 2008] when considering nonadaptive scheduling network Hreally matters: the result cannot be only dependent on structural properties of G only(as it was the case in [Montanari and Saberi, 2009]). Can we bound the convergence inthis case as well ? We do not give full details here (as they are technically complicatedand not central to our argument), but one can show that certain structural parametersof the pair of networks (G, H) determine convergence time in this case as well. This willbe formally proved in subsequent work.

7 Further discussions

7.1 Adversarial scheduling, covering laws, compositionality

In a certain sense the adversarial scheduling approach, even though it leads to an identi-fication of (some) of the factors that underscore the validity of a model provide excellentexamples of covering laws. Thus:

1. I extended the context of the baseline result from the random scheduler to anarbitrarily fair one. Fairness in scheduling provides an explanation for stabilizationthat acts as a covering law, both in the deterministic and probabilistic settings.

2. Going from a deterministic to a probabilistic choice introduces a larger class of

models that modifies the nature of the stylized fact, making it a limit case of amore complex one. It allows corresponding extension of the notion of fairness butintroduces qualitatively new behavior in the system, visible by the existence ofan adaptive scheduler precluding stabilization. In this sense going to the baselinemodel is some sort of discontinuous limit of the general case.

3. The logic of stylized facts of this sort is reminiscent of the theory of phase transi-tions (though we werent able to locate formal examples of phase transitions informal logic that would be directly relevant).

4. Adding strategic interaction (in a form of a network) does not seem to significantly

alter the logic of stylized facts, at least with respect to scheduling. The newbehavior seems to arise from the effect of the network on the convergence propertiesof the dynamics.

5. However, the fixed point evolves from the single state A to a product distribu-tion, to the Gibbs distribution, to a distribution that has two coupled components.

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Appendix

In this appendix I prove the main result (Theorem 1) outlined in the paper.

Definition 4 A tree rooted at node j is a set T of edges such that for any state w = jthere exists a unique (directed) path from w to j. The resistance of a rooted tree T isthe sum of resistances of all edges in T.

I will use the following characterization of stochastic stability:

Proposition 1 Let P be a regular perturbed Markov process, and for each > 0 let

be the unique stationary distribution of P. Then lim0 = 0 exists, and 0 is astationary distribution of P0. The stochastically stable states are precisely those statesz such that there exists a tree rooted at z of minimal resistance (among all rooted trees).

The states in S are obviously reachable from one another by zero-resistance moves, soit is enough to consider one state y S and prove that it is stochastically stable.

To do so all we need to do is show that y is the root of a tree of minimal resistance.

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0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1X

Y

Figure 3: Decomposition of edges of tree T

Indeed, consider another state x S and let T be a minimum potential tree rootedat x. We will transform T into a tree T rooted at y having potential less or equal to thepotential of the tree rooted at x, and strictly smaller in case x is not a state having allits first-component labels equal to A. Let

y,x = (x0, i0) (x1, i1) . . . (xk, ik) (xk+1, ik+1) . . . (xr, ir)

be the path from y to x in T (that is (x0, i0) = y, (xr, ir) = x). To define thetransformation from T to T we will view the set of edges of T as partitioned into subsetsof edges corresponding to paths as follows (see Figure 3).

1. The set of edges of path y,x.

2. The set of edges of the subtree rooted at y.

3. Edges of tree components (perhaps consisting of a single node) rooted at a nodeof y,x, other than y (but possibly being x).

The transformation goes as follows:

1. Instead of path y,x we add path x,y from x to y defined as follows:

x,y = (xr, ir) (xr1, ir) (xr2, ir1) . . . (x0, i1) (x0, i0).

2. Rooted trees of type (2) are preserved.

3. The transformation is more complicated for the third type of edges, and we explainit in detail here. Suppose that (xk, ik) is the connection pointof the path.

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Case 1: xk+1 = xk.

Then the point (xk, ik) = (xk+1, ik) belongs to path x,y as well, so one can justmap the rooted tree into itself.

Case 2: xk+1 = xk and the move (xk, ik) (xk+1, ik+1) has positive resistance.

In this case, since in configuration xk and scheduled node ik+1 we have a choiceof either moving to xk+1 or staying in xk, it follows that the move (xk, ik) (xk, ik+1) has zero resistance. Therefore we can replace the subtree P with P =P {(xk, ik) (xk, ik+1)}.

Case 3: xk+1 = xk and the move (xk, ik) (xk+1, ik+1) has zero resistance.

Let j be the smallest integer such that either xk+j+1 = xk+j or xk+j+1 = xk+j andthe move (xk+j , ik+j) (xk+j+1, ik+j+1) has positive resistance.

In this case, one can replace P by P {(xk, ik) (xk+1, ik+1), (xk+1, ik+1) . . . (xk+j , ik+j)} without increasing its total resistance. The new tree now fallsinto one of the cases 1 or 2, and we can map it in the way outlined there.

It is easy to see that no two sets associated with paths in P intersect on an edge havingpositive resistance. The union of the paths of all the sets is a directed associated graphW rooted at y, that contains a rooted tree T of potential no larger than the potentialof W.

Since the resistances of tree components of type (2) and (3) are equal to their associatedpaths, to compare the potentials of T and W it is enough to compare the resistances ofpaths y,x and x,y.

I come now to a fundamental property of the game G: it is a potential game, andthe resistance r(m) of a move m = (a1, j1) (a2, j2) only depends on the values ofthe potential function at three points: a1, a2 and a3, where a3 is the state obtained byassigning node j2 the value not assigned by move to a2. Specifically, r(m) > 0 if either

1. (a2) < (a1), in which case r(m) =

(a1) (a2), or

2. a2 = a1 and (a3) >

(a1), in which case r(m) = (a3)

(a1).

In other words, the resistance of a move is positive in the following two cases.

1. The move leads to a decrease of the value of the potential function. In this case

the resistance is equal to the difference of potentials.

2. The move corresponds to keeping the current state (thus not modifying the valueof the potential function), but the alternative move would have increased the po-tential. In this case the resistance is equal to the value of this latter increase.

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Y

X

potential

4

32

1

Figure 4: Resistance of edges on a path between two nodes X and Y.

Let us now compare the resistances of paths y,x and x,y. First, the two paths containno edges of infinite resistance, since they correspond to possible moves under Markovchain dynamics P. If we discount second components, the two paths correspond to asingle sequence of states Z connecting x0 to xr, more precisely to traversingZ in oppositedirections. (The last move in x,y has zero resistance and can thus be discounted.)

Resistant moves of type (2) are taken into account by both traversals, and contributethe same resistance value to both paths. So, to compare the resistances of the two pathsit is enough to compare resistance of moves of type (1). But moves of type (1) of positiveresistance are those that lead to a decrease in the potential function. But decreasing

potential in one direction corresponds to increasing it in the other (therefore such moveshave zero resistance in the opposite direction).

An illustration of the two types of moves is given in Figure 4 where the path betweenX and Y goes through four other nodes, labeled 1 to 4. The relative height of each nodecorresponds to the value of the potential function at that node. Nodes 2 and 3 haveequal potential, so the transition between 2 and 3 contributes an equal amount to theresistance of paths in both directions. The other dashed lines correspond to transitionsof positive resistance, while solid lines correspond to transitions of zero resistance.

The conclusion of this argument is that r(y,x) r(x,y) = (x) (y) 0, and

r(y,x) r(x,y) > 0 unless x is state A, which is what we had to show.

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