The Mathematical Basis of the Catastrophic Nature of Identity Politics

Illustrating tribal extinction (black curve) when mistrust is allowed to evolve (other curves)

This is something I observed nearly two years ago now. I didn’t quite believe it. There is a familiar phenomenon in algebraic work. You may have checked it three times but still feel there is something wrong. The best course of action in these cases is to leave the thing alone for several months and then come back to it having largely forgotten the details. It is amazing how often an error which was previously invisible can become glaringly obvious later. I followed this time-honoured approach. But the result remained stubbornly unchanged, no blunder was discovered. I now believe it to be correct and my account, complete with mathematical details, is here.

Of what do I regale you?

I speak of identity politics, of which feminism is the prime example and prototype.

The academic presentation rather obscures my intended application: deliberately so, to distance the mathematics from the controversy. Here I am more forthright about its possible interpretation in the world of identity politics.  

The defining essence of identity politics is that primacy is placed upon an individual’s identity group membership rather than upon their own individual merits. This is directly contrary to my own credo. Indeed, I regard it as a form of moral corruption. However, that is opinion. The remarkable thing about my mathematical analysis is that it shows the adoption of an identity political stance necessarily leads to catastrophe.

Mathematical analysis can only be applied to a model, and a sufficiently idealised and simplified model at that. The real, messy world of people is not directly amenable to mathematical analysis. Actually, the real, messy world of inanimate matter is not directly amenable to mathematical analysis either, though the success of physics has led us to believe that it is. In truth, physics is only a model of the world. The lesson we can draw from physics, though, is the power of simplification. By neglecting complicating features, the essential nature of the system can be exposed. Simplification makes a mathematical analysis of the system feasible.

This is an important point. Many people might be inclined to argue that, by neglecting so much of the complicated reality of a system, one is merely left with a fiction that bears no relationship to the true behaviour.

Newton’s first and second laws of motion are now so familiar that it is easy to forget what a conceptual leap they represented. To posit that a body will continue in a state of uniform motion unless acted upon by a force was revolutionary. It is contrary to one’s daily experience that, unless you continually push something, it tends to stop moving. What Newton did (standing, of course, on the shoulders of giants like Galileo) was to understand that neglecting forces of resistance was the key. Because resistance forces are absent in planetary motion, Newton’s schema was strikingly successful in astronomy. The simplified model was mathematically tractable in this application, and provided the key to understanding all motion, including cases where resistance forces are important – by recognition of that very fact.

People do tend to underestimate the power of mathematical analysis. Of course, mathematical analysis tends to be tractable only when deployed in the context of a model which is sufficiently simplified. At least, this is the case when the analysis is purely algebraic, rather than computational. Computer models can be monstrously complex (which does not necessarily make them more accurate – but let’s not get into climate change).

The key to understanding is to ignore those features which complicate but do not enlighten, and thus to expose to clear view those (few) truly essential features upon which the qualitative nature of the system’s behaviour truly depend.

I repeat: the defining essence of identity politics is that primacy is placed upon an individual’s identity group membership rather than upon their own individual merits. This, then, is the key feature to be modelled, to the exclusion of other complicating features. To exclude complicating features, “primacy” is replaced in the model with an absolute. My model starts with the simplifying assumption that individuals are not recognisable at all, only their identity group membership is apparent to others. (In my analysis, identity groups are referred to as “tribes”).

The model is based on game theory. I need to back-track a little to explain the relevance of game theory in sociology, specifically in the emergence, and stability, of very large scale cooperative human societies. Deep breath…

It is not hard to understand cooperation, even altruism, arising in kinship groups. The basis of such behaviour is genetic. Paradoxically (and rather pleasingly) the selfishness of genes is responsible for the dramatically unselfish behaviour of related individuals, who behave towards each other with genuine altruism, even sometimes being willing to sacrifice themselves for others. But humans, uniquely, form extremely large social groups of genetically unrelated individuals. Not only that, but human societies demand an extraordinary degree of cooperation, supporting a high degree of specialisation and interdependence. How such cooperation between genetically distant individuals arises, and is stabilised, has been (and continues to be) the subject of much academic analysis. One of the tools that has been used in such analyses is game theory, specifically in the form of evolutionary game theory.

The reason why game theory is an appropriate tool with which to explore human social behaviour is that it provides an environment within which the natural tension between competition and cooperation can be analysed. The archetypical case is the Prisoners’ Dilemma. Two prisoners are questioned separately. If both refuse to talk, they will both be convicted but receive only a modest sentence (say 2 years) for lack of evidence on all charges. If both talk, on the other hand, they will both be convicted and get a longer sentence (say 5 years). But if one talks and the other does not, the prisoner who talks will get the lightest possible sentence (say 1 year), whereas the prisoner who does not talk takes the bulk of the blame and gets 10 years. Prisoner A is better off talking whatever prisoner B does. This inexorable logic leads to both talking and hence getting 5 years each. But, paradoxically, if only they’d both kept schtum they’d both have got only 2 years.

This is the natural tension between cooperation and competition. So long as you trust each other, both parties cooperating is mutually beneficial. Both mistrusting each other is mutually disbeneficial. Societies based on cooperation are always open to be exploited by freeloaders who are treacherous. So why don’t societies collapse into a chaos of mistrusting non-cooperation?

The answer is that, unlike the Prisoners’ Dilemma scenario, the ‘game’ is not a one-off. We all interact repeatedly with others, including interacting many times with the same individuals. In the jargon, this is called an “iterated game”, i.e., the game is played many times over between each pair of players. Experiments with computer models and with humans show that tit-for-tat behaviours tend to do best (though it is important to realise that there is no such thing as the “best strategy” – the best strategy will depend upon what everyone else is doing). Tit-for-tat means that you behave towards a given individual in the same way he last behaved towards you. If he turned Queen’s evidence on you last time, you do the same to him now. But if he kept quiet last time, you repay the compliment now. Tit-for-tat might be called “firm-but-fair”. The optimal players want to cooperate if possible, but take an intolerant stance towards those who cheat. It is no accident that this conforms to the way most people think.

In sociological application, the “game” can be any interaction between individuals whose outcomes can be measured by some “payoff”, which might be positive (beneficial) or negative (detrimental). It is not necessary to specify the nature of the interaction. It might be trade, for example.

In its evolutionary form, the aggregate payoff to a group can be used to define the group’s evolutionary success, i.e., the payoff is reinterpreted as “fitness” in the Darwinian sense. A population is considered as comprising a number of different groups. At the end of each game “round”, the reproduction rate of the group is taken as proportional to their aggregate payoff. Successful groups therefore become more prevalent in the overall population, whilst unsuccessful groups diminish.

Large scale cooperative societies evolve, and are stabilised, through the four Rs: recognition, reputation, reciprocation and retribution. Recognition of individuals is necessary in order to be able to associate them with their past behaviour (reputation). If they were nice last time, you are nice now; if they reneged last time, you do so now (reciprocation). Retribution is partly refusal to cooperate with those who have not done so themselves in the past. But retribution can also be a society-wide phenomenon whereby a person’s reputation goes before them: “don’t trust that guy, he cheated me”. Ultimately, retribution may involve legal sanction. Sufficiently harsh retribution is important to supress the rampant spread of free-loaders.

These are the basic ingredients of reciprocal altruism. It is not really altruism at all; it is enlightened self-interest. Reciprocation may be direct, in the sense that A favouring B is reciprocated by B favouring A; or it may be indirect, in the sense that A favours B, who favours C, who favours D, and so on, with A hoping to be favoured in the end.

In practice, no one performs game theoretical optimisation when deciding how to act. Instead we are possessed of social emotions and intuitions which provide a shortcut. Thus shame, blame and guilt play an important part in guiding behaviour whose role is to promote cooperative behaviours whilst also promoting a social tendency to punish those who do not do so.

At this point the purely game-theoretic development of social behaviour is incomplete. Some authors believe that competition between groups is necessary to evolve the positive social emotions which promote a high degree of intra-group cooperation. From this perspective, such cultural adaptation is seen as the origin of “moral systems enforced by systems of sanctions and rewards” which act upon the individual through evolved social emotions and behaviours, like shaming and guilt.

Some people might object to this picture by citing our tendency to tip in restaurants even when we know we will never go there again. Well, we might be inclined to do so in order to avoid the waitress running after us down the street shouting “mean bastard!” (shaming). Or it might be because we know we will feel bad if we don’t tip. We will feel bad (guilt) because of the social emotions which have evolved to promote cooperation and social cohesion. Such cases may give the appearance of genuine altruism, rather than reciprocal altruism. But this is because of the mechanism by which reciprocal altruism is instantiated, namely via the social emotions which are persistent. Matt Ridley has expressed it thus: “genuine goodness is the price we pay for having moral sentiments – those sentiments being valuable because of the opportunities they open in other circumstances.” Another good Ridley quote on the importance of the social emotions in promoting cooperative behaviour is, “Emotion rather than reason (is) the wellspring of human motivation. The desire to escape or avoid guilt…is a human universal, common to all people in all cultures”.

However, there is a snag, and a big one. Boyd and Richerson observe that sufficient punishment can stabilise any behaviour, not just that which is socially beneficial: “If everybody agrees that individuals must do X, and punish those who do not do X, then X will be evolutionarily stable as long as the costs of being punished exceed the costs of doing X. It is irrelevant whether X benefits the group or is socially destructive”. This means that stable, cooperative societies need not be nice: they can be tyrannies. Well, this is hardly surprising. History is full of long-lived stable societies which, though cooperative, were based on slavery.  

So, that’s the background. Do note the crucial role played by recognition of the individual in the formation of reciprocal altruism and hence cooperative societies. The question I asked myself is what happens if extreme Identity Politics reigns so that individuals are not recognised – only group membership is recognised? Strategies like tit-for-tat are no longer possible, nor is any other strategy which relies on recognising the individual.

However, game plays need not be deterministic. A member of tribe A may not play the same way against a member of tribe B on every occasion. Instead, the most general possible strategy is defined by the set of probabilities for how any member of tribe A will play against a member of tribe B, defined for all tribes and all possible game plays. Individual plays are stochastic (random) providing only that the frequencies over large numbers of plays are consistent with these probabilities. But the strategy, even the stochastic strategy, cannot depend upon individuals since individuals cannot be distinguished beyond their group membership. So past plays of an individual cannot be known and individuals have no reputation beyond the reputation embodied in their group’s stochastic behaviour.

The game is defined by a payoff matrix, i.e., the payoff to each player for any pair of plays by the two actors. The game considered can be more general than just of Prisoners’ Dilemma type. My results hold for any possible two-player, binary play, symmetrical game (see the paper for details). The characteristic of these games is that one play corresponds to cooperation whilst the alternative corresponds to reneging, or treachery. Thus, in the game strategy the probability of a member of tribe A reneging against a member of tribe B can be dubbed a “mistrust” parameter. (Similarly, the probability of playing the cooperative play is a trust parameter). Note that two members of the same tribe may also play in either a trusting or a mistrusting manner.  

Assuming some game parameters (the payoff matrix and the assumed play strategies defined by the set of trust/mistrust parameters) the analysis shows under what conditions the evolution of the group populations leads to a stable equilibrium with all groups extant. Considering just two identity groups, for some parameter choices one group becomes extinct. However, for many parameter choices the two groups evolve so that each attains an unchanging equilibrium population fraction. So far, so normal – for evolutionary game theory.

The novel bit is to ask what happens if inter-group (and, for that matter, intra-group) trust/mistrust parameters are also permitted to evolve? It is natural to assume they would. The trust or mistrust an individual feels for another individual is something that can change according to experience. This is the basis of strategies like tit-for-tat when individuals are recognisable and prior outcomes remembered. The analogue in this case, where individuals are not recognisable, is that tribes as a whole will be prepared to increase or decrease their mistrust if doing so leads to an increased payoff. An increased average payoff for a group (tribe) leads to an increased prevalence of that tribe in the population.

The analysis can partly be done with pure algebra. What I could not do with pure algebra I did with exhaustive numerical computation. All possible game parameters were covered. The stunning result is this: tribes are driven to extinction until just one is left – always.

It doesn’t stop there. When only one tribe remains, it is unstable to schism. Thus, if, within the now-monoculture of just one tribe, there arises a ‘sect’ with an ever-so-slightly different set of trust/mistrust parameters, the situation is unstable. Either the sect is quickly obliterated, or the sect quickly obliterates the rest of the tribe. In fact, this instability against schism is present even before the other tribe(s) become extinct.

My interpretation of this is that identity politics (at least in this extreme incarnation) drives the ultimate intolerance. It is in the interests of a given group to be highly intolerant (mistrustful) of another. The intolerance even extends to those within the same identity group. The tiniest infringement of the group’s belief system will make you an outcast, an apostate to be destroyed.

Do I need to labour (!) how familiar these behaviours are? The feminist movement has schismed into the TERFs and the SJWs and the Intersectionalists. One false word on Twitter and the mob will descend upon even the previously most staunch zealot. The endless and escalating feeding upon synthetic virtue is the neurotic obsession which arises from the struggle to keep within the ever-narrowing strictures of the dominant group. But the dynamics, as revealed by the mathematics, are that the limits of acceptance will become narrower and narrower, and the tightrope walk ultimately impossible to negotiate.

There is a school of thought abroad at present that “the most intolerant wins”. This school of thought observes that the best way to counter an intolerant ideological opponent is to be more intolerant of them than they are of you. This view is both right and wrong. It is right in the sense that my game theoretical analysis confirms this opinion. To win the game and be the group which survives, you need to be more mistrusting – more intolerant – than they are. But more profoundly this is wrong, because the real lesson of my analysis is that identity politics is inevitably catastrophic. To win is to lose. To adopt a winning strategy is to enjoin a game which is itself corrupting. What needs to be destroyed is not the opponent, but the identity political mindset itself, the moral corruption which is the eradication of the individual.

You will now understand my original reticence to believe this result. It aligns so well with my prior belief system that I was naturally concerned that I might be falling prey to confirmation bias. But the mathematics is what it is.

Readers will understandably be cautious about believing grand universal claims on the basis of such a preposterously simple model. Quite right too. But note my earlier remarks about the power of simplification. Make the right simplifying idealisation and what results is not misleading but an illumination of features which a myriad of confusing contingent details have hitherto obscured.

11 thoughts on “The Mathematical Basis of the Catastrophic Nature of Identity Politics

  1. RudolphRigger

    Hi William,

    One of my research interests is quantum game theory – I’m more of a quantum theory person than a game theoretician. Your work here interests me and I’d like to try out a few things by extending the model to include quantum states and interactions. Of course, this won’t be relevant to the “identity politics” aspect you’ve highlighted, but I think the extension might be of academic interest.

    If you’d like to discuss this further please feel free to get in touch.

    1. William Collins Post author

      I am currently completing a book on QM, though I confess I have no idea what quantum game theory is. The work linked in this piece is now published.

  2. Keith

    In the introduction to your article you excuse mathematics, or yourself, or perhaps people in general for not being able to describe (human) nature in mathematical terms correctly. I came up with a dictum a few years ago after spending some time working in research and secondly getting involved in the atheism/religion argument. It is this:

    All models are wrong, some are are useful.

    After Goethe destroyed Russell’s attempt to build a comprehensive provable basis for mathematics in Principia Mathematica and Newton’s work was undermined by Einstein and finally Derrida’s virus of deconstruction decimates society around us, the only safe position is to assume that the knowledge that we hold is probably wrong. I think (another of my dicta) we are all stupid, we are unlikely to ever ‘get’ the world around us.

    It turns out that some of the models we use to describe the world and predict how it behaves, even ones we know to be flawed like Newtons laws of motion, are wrong, have proved to be useful. Look at the world around you that we built. We did it! We did it without fully understanding anything about, for example, a simple brick: it’s absolute weight, dimensions, load bearing capacity, chemical makeup, … It turns out that we know very little about bricks but we do manage to pile them up into some quite fantastic structures.

    Once we understand that our broken models are useful it is very liberating. Firstly (an perhaps most importantly right now) we can send Derrida scurrying back off to his well earned grave… by agreeing with him. Yes we can deconstruct Shakespeare and find that he was a racist pig, but his work was beautiful and stands proudly today, warts and all. Religion, particularly Christianity, has proved a very valuable foundation of modern society so we should not be so quick to discard the baby with the bathwater just because of the trivial matter that God might not exist. It allows us to become doers rather than thinkers or, as I like to say, procrastinators.

    1. William Collins Post author

      Indeed. My brain is a small thing, and what capabilities it has were forged by survival and reproduction in primitive environments. To expect this organ to be capable of perceiving the absolute verities is groundless hubris. That it can do as well as it does is rather remarkable. (PS: I think you meant Kurt Gödel not Goethe).

    2. Mike Bell

      I agree about ‘usefulness’ rather than ‘truth’.
      Reality is infinitely complex and, consequently, unknowable. However, we start to see patterns or generalisations in our observations. We use these patterns to generate theories or explanations.
      As such, all theories are, as you say, wrong, and should only be chosen on the basis of their usefulness – their ability to explain and predict.
      In choosing between competing theories we can either test the facts or use something like Occam’s Razor -choosing the simplest explanation available. This makes all theories and notions temporary.

  3. NMEN

    The undeniably popular Sargon of Akkad recently uploaded a short video about this year’s implosion of the “Pride Parade” in Edmonton, Canada. He illustrates, in an accessible way for the public, exactly what happens when two factions of far left Identity Politics (in this case, LGBTQIAPPMSNBCCNN+ activists and extremists from Black Lives Matter) collide.

    Inevitably, as they grow and become more numerous, the spaces between these factions in the real world will only shrink, resulting in these collisions becoming more and more frequent – and everyone, and everything, else, will be injured and damaged in the crossfire.

  4. Douglas Milnes

    “My interpretation of this is that identity politics (at least in this extreme incarnation) drives the ultimate intolerance.”
    I don’t think you really needed a mathematical model to tell you this :-). Identity Politics is an intolerant social system, which harshly excludes any ‘out’ group as well as deals harshly with anyone who might have thought they were ‘in’ but dares to be slightly different. You have reported in the past on things like the Model Law on Tolerance, which is nothing other than blatant intolerance and a desire to criminalise anyone with opposing political views.

    Unfortunately, Lukács and Horkheimer (or was it Adorno .. no matter) both saw the weakness of Identity Politics (though neither called it that) which you name: it can be nullified by removing the mindset itself. This is, as I understand it, a large reason why Cultural Marxism has the built-in process of changing identities and (following from Gramsci’s political philosophies) the ever-changing vocabulary.

    In the USA, particularly, it is easy to see that the dominant victim identity has moved from coloured people, to women, to LGBTQQIAAP and is poised to move on again within a generation – maybe to Hispanics or possibly mothers (even the MRM might fall for the latter). This keeps self-interested parties vying for the victim position, with an end to Victim Politics seemingly not in their immediate interest.

    Meanwhile, the shifting sands of language and just what things like ‘equality’, ‘tolerance’, ‘freedom’, ‘violence’ and much more actually mean, all help to keep the picture confused and make the target of destroying the mentality harder. To fight Identity Politics one needs to a certain extent to insist on one’s own language and group identity: which creates the very kind of identity group that is being sought to diminish.

    It’s not impossible to heal society from where we now find ourselves but it certainly won’t be easy, nor quick.

    1. William Collins Post author

      Of course the conclusion aligns with expectation. The power of a mathematical analysis is that it clarifies what follows with logically incontrovertible certainty from precisely what premises. The value of that is that it can circumvent pointless arguments when an outcome is certain (though, regrettably, that will never stop those who are impervious to reason).

      There is indeed a tendency to fight identity politics from the perspective of an opposing identity group. This is what I allude to as “the most intolerant wins”. But it’s unwise for the reasons the analysis uncovers – the end point is authoritarianism. What’s needed is to step outside the game & recognise the premises are themselves pernicious. In particular, the individual must be the seat of moral action and recognition.

  5. Douglas Milnes

    You make the statement that recognition needs to be of an individual. While this might be the case when modelling reputation, when modelling social outcome, recognition need not be individualistic; it can be tribal. This is where fixed features like skin, eye and hair colouring, head shape and body format play a part. So, too, do cultural matters like clothing, jewellery and most obviously speech.

    You already point out that reciprocation does not need to be to the same individual “in the sense that A favours B, who favours C, who favours D, and so on” but you missed out that this only works if A, B, C, D and so on are part of the same tribe. Or, to put it another way, when they are each recognised in turn to be identified with the same group of self-interest cooperation. Reciprocity does not happen with strangers (hence, currency for example).

    As societies have become larger and more fluid (transient/mobile) this recognition has been harder. The population limit of an ‘altruistic’ cooperation group is dependent upon the commitment required to the group*. The greater the investment made to the cooperative group, the smaller that group necessarily becomes. Hence the growth of cliques, gangs, clubs and other sub-cultures. Trust, distrust, faith and paranoia all make for individual and collective adjustments in the judgement on the level of cooperation with a group.

    [*Not quite: cooperation, to the point of conformity, is also dependent upon perceived threat. This is why governments try to govern by having an external threat: it keeps the populace malleable. If the Martians were to invade tomorrow, all 8 billion humanity would quickly become a cohesive whole.]

    To put it another way, we react more sociably with those we identify with. They don’t have to be recognised individually if we can recognise them as part of ‘our’ group.

    The part that group identity (lower case ‘i’ and without the ‘Politics’) plays is important and always has been. In medieval times, the stranger coming from the next village ten miles away would be judged on his appearance, mannerisms and speech; if these are close enough, he will be given provisional trust, if they are ‘foreign’, he will be distrusted by default. Effectively, everyone is asking whether he is ‘one of us’ or an ‘other’.

    Identity Politics is distinct from this normal phenomena in that the trust for someone in the ‘in group’ is not just a default starting position but supposedly complete and utter, while the distrust for an ‘other’ is not just a default starting position but a rigid position that may not change. This is why such vitriol is lashed out to those meant to be part of the ‘in group’ who don’t conform, such as women who reject feminism. And why a white man who gives up his life to save ten black lesbian women will still be viewed, by those steeped in Identity Politics, with disdain and hatred for being an ‘other’.

    This does not, I think, affect your computations. It means that rather than assuming that a reputation cannot be known because the individual is invisible, the reputation is irrelevant because the individual is politically unacceptable. The effect from that point on will be the same.

    1. William Collins Post author

      The reputation of a tribe as a whole is known, and can be reflected in the choices of others how to play against them. But it is only the average behaviour of the tribe which is knowable in my analysis.

  6. PrettyHowTownPrufrock

    Like the MAD policy (Mutually Assured Destruction). I remember the Film in the 1980’s, “Wargame” with Matthew Broderick. At the end of the film the computer tries to find the winning combination to launch an International Thermonuclear War. All goes blank, then, calmly the computer says: “Strange game, the only way to win is not to play. – How about a nice game of Chess?”


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