The title paraphrases Rawls, "Political power rapidly accumulates and becomes unequal; and making use of the coercive apparatus of the state and its law, those who gain the advantage can often assure themselves of a favored position"– keep this in mind that he is describing a western liberal democracy as you read. Let me preface this by saying I am not a mathematician, I'm merely a social critic, an analyst of critical theory, so excusing any nuanced misunderstanding of the mathematical rigor of actual late mathematician John Nash's work, I'd like to emphasize the importance of Nash's work culturally, geopolitically, and in particular in this depiction of Nash, as both a man and a mathematician, a historical and dialectically significant material sense.
The scene in the film "A Beautiful Mind", adapted from a biography of the life of Nash, that depicts the Nash equilibrium, an early piece of Nash's legacy, is not an accurate portrayal of the concept in its intention, but that it can be repackaged accessibly while missing the larger point of its purpose is the problem with the real world applications of Nash's work at RAND Corporation to begin with:
The scene itself depicts mathematician John Nash and his colleagues competing for the attention of a group of women at a bar; Nash's approach is to ignore the most attractive woman, assuming that his colleagues will also ignore her and instead focus on the less attractive women. This strategy is presented as an example of the Nash equilibrium, where each player's strategy is optimized given the strategies of the other players, and the outcome may be suboptimal but possible to strategize for, all participants being rational, and assuming self-interest on the part of one of the participants (in this case, Nash, leaving the most attractive woman for himself)
However, this portrayal is inaccurate because the Nash equilibrium is not necessarily the best outcome for all players; it is a situation where no player can improve their outcome by changing their strategy, given the strategies of the other players. In the bar scene, Nash's strategy may have been optimal for him, but it was not necessarily optimal for the group as a whole. It is possible that a different strategy, such as cooperation or coordination, could have resulted in a better outcome for everyone, which is to miss the point of the exercise to begin with– he's negotiating on behalf of himself in this scenario, and not his group of friends as the scene depicts the equilibrium point being, one where it's possible to persist in a strategy where they all come out winners (in Nash's case, yes, sub-optimally, for his friends).
This leads us to the second point at issue: the scene goes onto imply that the Nash equilibrium is a straightforward and easily identifiable solution to a game, where realistically identifying the Nash equilibrium in a real world scenario (you see why this might not be a reasonable exercise for war planning if you can't even assume good faith among friends, let alone theoretically non-cooperative global superpowers?) can be a complex and challenging task, perhaps infinitely so in the case of humanity writ large, especially in games (in a theory context) with many players and at least as many strategies. This requires analyzing all possible combinations of strategies and dependently determining which result in a situation where no player can improve their outcome, thus no longer arriving at suboptimal outcomes.
The scene in may have been intended to provide a simple and accessible explanation of the Nash equilibrium, but it is not an accurate or even particularly useful depiction of the concept in practice, choosing to be reductive about behavior rather than the top-level abstract commentary it was in practice; there are nuances, in practice, it would need to consider, and this example, in its brevity for the sake of accessibility, exposes its core weakness; that you cannot simply operate formulaically along lines of cooperative or not, in collective interest or not, etc. The nuance must go deeper to actually demonstrate what an equilibrium point is, where irrational players (such as the United States in the Cold War, dealing almost exclusively in bad faith with the Soviet Union by the time of the Korean War, the era in which Nash was at RAND) cannot be assumed to act cooperatively even if cooperation costs nothing (this was the entire metaphysical proposition of the Cold War; the United States was willing to burn the entire world down, as long as it took the USSR with it, which for the duration of the Cold War had been the first in each instance to de-escalate, sub-optimally for them, which the United States antagonized into a position of cooperation--coercion in this case-, something few admit to despite being blatantly obvious from even a subjective reading of the facts). It's kind of metaphysics that requires that you not read into the nature of cooperation, and that kind of logic is, perhaps, more real from the perspective of game theory in terms of telegraphing outcomes mathematically, but is no more reflective of the true nature of the players, which this presupposes would never change or be subject to forces like the j-curve of rising expectations, etc.
The problem I'm getting at is that the metaphysics would be pragmatic and functionally sound if this were a cynical comment on self-interest, which Nash himself always asserted was a factor, but it wasn't cynicism just a fact of life, in Nash's view. It's simply how the world works because, from his vantage, those were the rules the United States military was simply able to enforce on the rest of the world, where non-cooperation from the Soviet Union could only take a handful of forms, whereas cooperation through coercion could take potentially even fewer, but far more likely (especially as we exited the Stalin and to a lesser extent Khrushchev eras of Soviet superpower).
With these metaphysics, actual cooperation is not possible, and therefore a mutually beneficial one is therefore not possible– this isn't a fault of the theorem, but of how it's used rhetorically, socially, in its flattened state to explain geopolitics; a mutual benefit is not, for example, the United States deciding that you get to live, that's a consequence of capitulation, not mutually assured destruction forcing mutual deescalation, as was the case in the Cuban Missile Crisis, for example, where Khrushchev first deescalated without capitulating in any real way that his successors would have absolutely done so, and more familiarly in the arc of American geopolitics. But why the importance on true cooperation?
In a Nash equilibrium, each player is optimizing their own payoff based on the assumption that the other player's strategy is fixed (often in favor of self-interest contra potential sacrifice for a mutually beneficial outcome) despite the fact that, in many instances (games), players can benefit from cooperating and coordinating their strategies to achieve a better mutual, not necessarily shared, outcome. In the classic prisoner's dilemma game, both players can achieve a higher payoff if they cooperate and choose the option that benefits them both, rather than choosing the option that maximizes their own individual payoff– it could be assumed that individual interest wills out if both prisoners assume the other will rat, if they do not rat first, opting not to protect each other from a (potentially false) allegation; cooperation leads to a mutually beneficial outcome (not being directly implicated by the other), even if that outcome could potentially be individually suboptimal (you might still go to trial and be found guilty rather than going free because of the informing on the other prisoner). The character of these decisions matters metaphysically, is my point, and a Nash equilibrium (correctly) is not designed to necessarily take into account that level of nuance, because that's not the level game theory operates on, and nor should it, but so often it does when bearing out potential military application, even into the present.
In general, cooperation in a Nash equilibrium can lead to mutually beneficial outcomes and can increase the chances of reaching a stable equilibrium. It can also help build trust and establish a positive relationship between the players, which can be useful in future interactions, this is (theoretically) true of real world diplomacy, in a world where the players (nation-states, blocs of geopolitical power, what have you) operate in good faith, which again, has not been proven to be the case, and perhaps what further stimulated Nash's own mental illness. This is a common response to Western individualist thought; you represent your nation, but your nation does not necessarily stand with you, it might as long as you prove beneficial, but it does not invest in you as you are expected to invest yourself in it– you are not the state, and the state sees no part of itself in you in a western liberal democracy. This is something John Nash learned upon his dismissal from RAND– he was valuable enough to the government to spare him from being drafted into the Korean War, but was not valuable enough to subject him to due process for an allegation that he was a homosexual (something he didn't necessarily deny, but was summarily terminated upon an arrest where he disputed the facts to, both, the police and his superiors at RAND).
The western liberal democratic tendency toward postmodern individualism emphasizes the importance of individual rights and autonomy. This emphasis on individualism can sometimes lead to a lack of social cohesion and make it difficult for individuals to work cooperatively with others. In the cases of someone like Bobby Fischer and likewise Nash, their mental instability made it challenging to work cooperatively with their peers, and this was by design, to keep them desperate and fearful of social order, but ultimately expendable to the same. Let's just take a look at what I think are fair, more or less objection-less biographies of these men:
Fischer was a highly talented chess player who became the youngest grandmaster in history in 1958. In 1972, he played a historic chess match against Soviet champion Boris Spassky. Fischer's erratic behavior during the match led some to speculate that he was suffering from mental instability. During this time, Fischer was also known for his anti-American and anti-Semitic statements, which led the US government to view him with suspicion. In fact, the US government tried to use Fischer as a propaganda tool to promote American values and win the Cold War, but he refused to cooperate. Ultimately, Fischer's mental health deteriorated, and he became a recluse in later life.
Nash, on the other hand, was less a propaganda tool as, again, a rank-and-file mathematician for the RAND Corporation (a conglomerate war planner, essentially) during the Cold War. Nash's work on game theory and its application to military strategy was highly influential, and he was awarded the Nobel Prize in Economics in 1994. Nash suffered from schizophrenia, which led to delusions and hallucinations. Nash's mental illness made it difficult for him to work with his colleagues at RAND, and he eventually left the organization. Nash's mental health improved later in life far away from prominence to the US military, and he was able to resume his work in mathematics, pursuing more abstract realms (such as an ideal currency, among other ideas still discussed in niche economic and technological circles today).
They were both of use to the government as long as they were productive to the end of pushing a specific narrative version of events asserted as materially true, not just significant, but real and factually accepted as mainstream thought, and until such time they were unable to do this, whatever moral or public trespass they might have contained was tolerable; Fischer's mental illness manifesting in the form of virulent vocal antisemitism (Fischer was Jewish) was something many claimed was non-problematic because he posed a risk to the Soviet chess champions– only once it subsumed his ability to play the game in their service (he violated sanctions on the USSR to play a rematch against Soviet champion Spassky years later) did the State Department disavow him; his commitment to chess no longer useful to the American project.
On the role of good faith in cooperation, consider the perspective of the Soviets involved, both interpersonally in participating in what was a blatant pro-western propaganda exercise anyway, but also as a society: by contrast, the Soviet Union's core metaphysics (a term Stalin might fairly reject as a description of any part of the USSR's ideology, but I'll use here for the purposes of comparison) of collectivized incentive emphasized the importance of the collective over the individual. This cultural support of their chess champion Boris Spassky, citizen and peer, reflects this perspective-- Spassky was their representative of the Soviet Union, and his success was seen as a triumph for the collective, the Soviet people, not its government. This perspective on collectivism may have contributed to Spassky's success, as he was able to draw on the support of his country and the people who constitute it, not serve it sacrificially, as was asked of Fischer, as was committed to by Nash.
This is significant because of key metaphysical constants in how the west (and the United States' ideological core) functions philosophically, often contrary to material reality in terms of an actually rational and just form of operation: When not purposefully flattened by Western liberal democratic thought, the dialectic can be seen as a way to challenge dominant power structures and assumptions. It allows for the synthesis of opposing viewpoints and the creation of new ideas that may challenge existing power structures, whereas the Nash equilibrium may reinforce existing power structures by assuming that each player acts rationally and is motivated solely by self-interest. Again, the value is derived from the result, with or without Nash, the man. This describes a state in which no player has an incentive to change their strategy, given the strategies of the other players. In other words, it is a stable state in which all players are making optimal decisions based on their own self-interest. Again, this equilibrium is reached through a process of iterative reasoning, where each player predicts the actions of the other players and adjusts their own strategy accordingly, it again, also, assumes that players have complete information and are rational decision-makers. You are primed to use the dialectic to arrive at a place to make a cooperative movement forward using the information at hand; this is a materialist's approach, and you can, as a society, make collectively-incentivized decision– this isn't an idealism, it's reasonable governance, if your government is seeking cooperative, mutually beneficial outcomes and is acting in good diplomatic faith. As such, in this process, two opposing ideas, the thesis and the antithesis, are combined to create a new, more complex idea, the synthesis, for those unfamiliar with the dialectic (and are wondering what I'm getting at, where this comes into play once one reaches a state of equilibrium, something that, int he material realm, is something of a prerequisite to make any kind of social-collective decision, on behalf of a liberal democracy or not). This synthesis then becomes the new thesis, and the process continues. The dialectic assumes that contradictions and conflicts are inherent in society and that progress is made through the resolution of these conflicts– you are effectively proceeding from that state of equilibrium. You have two options: behave in acknowledgement of the dialectic's singular utility of handling this perpetual motion of the world's material variety set (I won't get into cyberneticism here), or handle this as a liberal democracy would, which is to force the quickest path to resolution (no matter how suboptimal that outcome is for some, as long as that outcome does not challenge power, in which case, all outcomes are positive for those, for example, on the top side of every transaction under capitalism).
This comparison--Nash and Fischer, Fischer and Spassky, Nash equilibrium and dialectical materialism, the US and the Soviet Union as players in a game theory context- is all to really muse about this question of what self-interest is actually defined as in applying game theory to real world diplomacy and geopolitical material reality, which is further complicated by social conceptions of the self as well as the self as reflected in the population: In Soviet collectivized social thought, the dialectic was used as a tool to understand the development of society and history, with the ultimate goal of achieving a classless society. The Soviet Union believed that the contradictions between the working class and the bourgeoisie could only be resolved through a socialist revolution. In contrast, western liberal democratic thought emphasizes individualism and the importance of market mechanisms in reaching efficient outcomes, such as the Nash equilibrium. While the Nash equilibrium and the dialectic have different origins and assumptions, they both offer insights into the complex nature of social interactions and decision-making. The Nash equilibrium is useful for understanding strategic behavior in situations where self-interest is paramount, while the dialectic provides a framework for understanding how social change and progress occur through the synthesis of opposing ideas.