Before I can answer your question it is crucial first to recognize that "justification" is an act arising from a situation and offered up to an audience. Political discussions with market libertarians often seem to proceed as if political assessments are a matter of offering up logical justifications to a universe of "neutral rationality" amounting to agreement in advance on a host of market libertarian assumptions about what physical force is, what it means to be an owner of property, what it means to be a responsible subject, and so on. This matters, because all of these questions are actually under contestation, indeed they are often the very questions at hand in debates with market libertarians in the first place.1. Left and Right, Back to Basics
So, when you ask "when is the initiation of physical force justified?" my first question is: To whom is the justification in your question presumably offered? The answer whether an act of force is justified (let alone whether it was an act of force at all, or the true initiation of the force in question rather than a response to some prior force) will no doubt differ if it is the perpetrator, the victim, a judge, a jury, some measure of popular opinion, or a community offering up the justification or hearing it.
One of the great quandaries of political theory and institutional practice seeking to provide some measure of legible and hence sustainable justice and order to the diversity of stakeholders to a polity is, after all, finding ways of ensuring that minorities are heard and protected in majoritarian orders, or that the losing parties in court decisions and election contests will remain invested in the ongoing life of the system of judgment and representation which has failed to go their way in a particular instance. Periodic elections with the widest possible franchise (circumscribed by the no taxation without representation principle for one thing), and the widest possible eligibility for office, jury trials, court appeals, impeachment provisions, popular referenda, universal rights resistant to majority opinions (especially rights to public assembly and critical expression), and civil disobedience premised on breaking laws and then suffering their penalties to call attention to their injustice as a spur to changing them are all indispensable institutions and practices that have been hacked over many centuries in the fraught circuitous heartbreaking process of the ongoing democratization of the state to which I and so many others are still devoted.
(As I never tire of insisting to anarchists and others: I want to democratize the state, not to smash it.)
"Justification" is a discourse, it consists of a host of practices, each with a history, norms, problems, ritual and infrastructural affordances. To keep this is mind is to be thinking about justifications politically, rather than deploying a term as if it were neutral in the service of a stealthy political agenda, whether one is conscious of this or not: Remember, there is nothing more natural than to accept some or many of the norms and forms of the status quo unconsciously or uncritically, even when one is in the process of intervening in the status quo elsewhere, indeed, the great political advantage of authority is that it tends to operate unconsciously and with an incredible resilience such that resistance to its terms requires intense and ongoing vigilance.
And so, to return to your specific question, "when is the initiation of physical force justified?" I will answer with a question: And who decides at what point "the" force is really initiated? "The initiation of force" is of course, again, a phrase familiar especially in market libertarian political theory (such as it is). When, as in so many market libertarian formulations, say, a contract is figured as the quintessential voluntary transaction -- and the one into which initiations of force are usually imagined to interfere, usually by caricatures of government agents or criminal thugs -- to what extent does it matter whether a contract is articulated by misinformation, unequal access to knowledge, duress, the threat of poverty? In any society that makes no universal provision for basic healthcare, education, income, retirement will not every transaction be stratified by structural inequities, unconscious biases, threatening precarities, unreliable information exchange that render the initiation of violence not only in the violation but in the constitution of the contractarian scene itself? Brecht famously joked, which is the bigger crime, robbing a bank or founding one? The force of his point does not depend on our inability confidently to answer his question.
Here the market ideologues merely provide extreme illustrations of a more general dilemma: How does the legibility of violence itself require a circumscription of possibility that is a violence? Is there an initiation of violence prior to the perception of an initiation of violence enabling its perception as such? I can only guess that upon hearing me insist that the scene of a legible consenting to the terms of everyday commerce and contract requires first the universal provision of equitable recourse to law, basic healthcare, lifelong education, secure retirement, and a basic guaranteed income to ensure those everyday commercial and contractarian transaction are not duressed by inequity, misinformation, bias, fraud, and threat to livelihood (and hence actually consensual, meaningfully voluntary), every self-respecting conservative and market libertarian will protest forthwith: And just who will be paying for all this welfare, all this healthcare, all this education, all this legal apparatus, and the money to support all this idleness and what amounts to a permanent strike fund for workers to hold over employers heads? You speak of volunteerism and consent, but are the talented, hard-working, meritorious few (just for fun we'll pretend for a moment this sort of phrase doesn't always end up meaning "white dudes") from whom the money for this public largess will no doubt be taxed at rates incomparably more steeply progressive than they are today imagined to be volunteering and consenting to this state of affairs? Aren't you hoping to exploit the few in order to, as you would put it, protect the many from being exploited by the few?
Again, what matters in this formulation is that it naturalizes the status quo when those are the very terms that are under contestion. The simplest answer to the question, Who is supposed to pay to prop up the many as you would do? is exactly the same people who at present pay to prop up the few, that is to say: We the People.
To put it simply, the wealth of the few is a collective accomplishment of the many from which the few preferentially benefit. Inherent in the protest that the wealth of the rich few is appropriated unjustly to make greater provision for the welfare of the many is the pretense that the wealth of the rich few was earned entirely by the rich few only to be vulnerable to expropriation by the many through the agency of majoritarian government. But it would be far more correct to say that the wealth of the rich few is produced through the collaboration of the many and enabled through the agency of government laws and investments and affordances, and then appropriated disproportionately to the rich few through effort, indeed, but also through corruption, deception, mystification, privilege, inertia, cronyism, fraud, and dumb luck.
It is presently We the People who contribute our labor and sustain the too corrupt, captured regulatory and representative apparatuses of government to maintain the fortunes of the rich few. To educate, agitate, organize, and legislate in ways that shifts We the People from our present collaboration in the terms of our own exploitation to the more democratic provision of a consensual scene to sustainable equity-in-diversity is mostly a matter of getting majorities to act in their best interests to solve shared problems and maintain political forms capable of such shared problem-solving. Such a shift only looks like the many stealing from the few if one has decided in advance that the few who presently have and keep the most have and keep it because they really are incomparably more talented and worthy than the many, and really earned and keep it in isolation from their fellows and without the support of the social world. Of course, such claims are patently absurd. It is only the prior naturalization of the terms of the status quo that insulate them from the consideration in the face of which their absurdity becomes instantly manifest. And to return to the initial question again, to ask the question "when is the initiation of physical force justified" is usually asked from a vantage into which a host of assumptions about who has earned what without a question of force entering into that consideration but which is then simply naturally neutrally "there" "available" to be stolen from by the initiation only then and there of the force of the taxman or the idle or the mob, hence the answer is usually available in the asking of the question, and the answer is usually one that will ultimately conduce to the interest of incumbency.
By way of conclusion, let me broaden the terms of the discussion a bit more: It is a commonplace in both everyday and philosophical parlance to declare persuasion, the space of discourse, the outside of violence, but since arguments so easily do or threaten violences the truth of that insight is tricky to say the least. As a teacher of rhetoric, I happen to think an attention to the traffic between literality and figurality, between the truths that arise from following but also from breaking the rules of language, provides a beginning of an answer to these quandaries. That is a very long discussion. Let me recommend a few of my prior posts to provide a sense of the way that discussion would go:
2. Eight Propositions on Taxes
3. Nonviolent Revolution As the Democratization of the State
4. Arendt, Fanon, King on Violence
5. Rhetoric and Nonviolence
6. The "Mixed Economy" Isn't A Mix, It Is "Ideal" Capitalism and Socialism That Are Mixed Up