Lesson 2
Lesson 2
Prisoner’s Dilemma and the First Welfare Theorem
Prisoner’s Dilemma
In mathematical game theory, the Prisoner’s Dilemma is a fundamental example used to illustrate the concept of Nash Equilibrium. Before delving into the specifics of the Prisoner’s Dilemma, it is essential to define what we mean by a "Nash Equilibrium."
Nash Equilibrium: In game theory, a Nash Equilibrium is a situation where no player can improve their payoff by unilaterally changing their strategy, provided that all other players keep their strategies unchanged. It represents a state of mutual best responses, where each player’s strategy is optimal given the strategies of others.
The Prisoner’s Dilemma is an ideal scenario for illustrating this theoretical concept. In this realistic setup, which often mirrors real-life situations, two accomplices are apprehended and interrogated separately. The payoff matrix of their decisions is as follows:
If neither confesses: Each receives a light six-month prison sentence, which is collectively the optimal strategy.
If both confess: Each gets a two-year prison sentence.
If one confesses and the other remains silent: The confessor goes free, while the silent accomplice gets a ten-year sentence.
Despite the optimal strategy being mutual silence (non-cooperation), this does not result in a Nash Equilibrium. The reason is that one accomplice can improve their payoff by unilaterally choosing to confess, assuming the other remains silent. This creates a temptation to defect, leading both to confess, resulting in a worse collective outcome.
This dilemma underscores the problem of asymmetric information, where each accomplice is uncertain about the other's strategy. This uncertainty prevents achieving a Nash Equilibrium, both theoretically and practically.
The adage, "your friends are your friends up to the first policeman, and then they will all rat you out immediately," exemplifies this reality. First-time offenders often confess to receive lighter sentences, while more experienced criminals, such as those in the Mexican mafia, enforce severe punishments for betrayal, thereby reducing confession rates.
First Welfare Theorem
The key about the First Welfare Theorem is not what it tells us about market efficiency, but rather, and far more importantly, what it says about market inefficiency. The First Welfare Theorem is part of a much larger framework in mathematical economics known as the Arrow-Debreu framework. This formal axiomatic system proves, using strict deductive logic, that under several additional assumptions beyond symmetrically informed and unfettered exchange—such as no externalities, diminishing marginal utility of consumption, and so on—a Pareto-optimal equilibrium is certain to occur.
The First Welfare Theorem essentially states that any competitive equilibrium leads to a Pareto-efficient allocation of resources, provided that the aforementioned conditions are met. However, this ideal state is rarely achieved in reality due to various market imperfections and externalities.
As George Orwell once famously said, “All animals are equal, but some animals are more equal than others.” Similarly, while all “perfect market conditions” posited under the Arrow-Debreu framework are necessary to ensure Pareto-efficiency, violations of some conditions result in far more inefficient real-world outcomes than others. For example, if we measure Pareto-efficiency as high and increasing rates of per capita real GDP growth, violations of most assumptions, like no externalities, do not result in such inefficiencies, barring two: unfettered and symmetrically informed exchange. Any violations of these two key assumptions are certain to result in significant inefficiencies, as trade stops being mutually beneficial or Pareto-improving.
The Prisoner’s Dilemma connects to these broader economic principles by demonstrating how asymmetric information leads to suboptimal outcomes. Just as Akerlof showed in “The Market for ‘Lemons’” that asymmetric information results in inefficient outcomes in unfettered exchanges, the Prisoner’s Dilemma illustrates how asymmetric information precludes a group-efficient outcome in involuntary exchanges, such as prisoners deciding whether to confess. In game theory, symmetric information is often assumed because, without it, a group-optimal Nash Equilibrium is not achievable, either in theory or reality. This highlights the significance of symmetric information in achieving efficient market outcomes and underscores the challenges of real-world markets where perfect information is rarely available.
The Prisoner’s Dilemma serves as a powerful illustration of Nash Equilibrium and the challenges posed by asymmetric information. It highlights the limitations of achieving optimal outcomes in real-world scenarios, emphasizing the importance of information symmetry for market efficiency. The First Welfare Theorem, within the Arrow-Debreu framework, further elucidates the conditions under which markets can achieve Pareto efficiency, offering valuable insights into both market efficiency and inefficiency. The key point is that two conditions are absolutely mandatory for achieving any kind of real-world market efficiency: unfettered and symmetrically informed exchange. Any violations of these immediately result in inefficient outcomes, which is what we want, as inefficiencies are how we make money in arbitrage.
Analysis of Inaccurate Axiomatic Assumptions about Chabad
The Chabad organization, beloved and respected by many, including yours truly, often faces axiomatic assumptions that may not accurately reflect its true nature. These assumptions can be examined to clarify the organization's actual role and the diversity within Jewish religious interpretations.
Chabad is widely recognized for its community services and religious adherence, such as observing Yom Kippur and Shabbat. These activities are integral to Jewish life and are valuable for both individual spiritual growth and communal cohesion. However, the purpose of this short write-up is to ensure that as members, we are symmetrically informed about Chabad.
Despite its non-profit status, Chabad operates similarly to a commercial organization, akin to a McDonald’s franchise, with structured territorial allocations. This analogy highlights the systematic approach Chabad takes in fulfilling its mission. However, it’s important to note that being organized and efficient does not detract from its non-profit and spiritual aims. Indeed, we are in the process of trying to obtain a franchise license to open up a new Chabad operation in Vieques, where we potentially plan on spending a part of the year, at some point in the future, should tax considerations require this.
Chabad's interpretations of the Torah are rooted in the Talmud, a central text in Judaism. However, it's essential to recognize that this represents just one of many interpretations. The Talmud, written between 200 and 400 AD, is a significant but not exclusive part of Jewish tradition.
Judaism, Christianity, and Islam all stem from the Torah but have diverged significantly over time. With 2.3 billion Christians and 1.8 billion Muslims, the number of Jews, approximately 14.7 million, indeed represents a minority. This demographic context underscores the plurality and diversity of religious beliefs.
Labeling the Talmudic interpretation as a "tiny sliver of heretics" within the broader context of Abrahamic religions is a stark but arguable perspective. It highlights the internal and external diversity of religious thought and the relative size of Jewish adherents compared to Christians and Muslims.
Chabad members, like any other group, may sometimes confuse their organizational success with theological accuracy. The analogy to "Microsoft University" graduates lacking expertise in Linux underscores the potential for insular perspectives. True knowledge about God transcends organizational boundaries and requires humility and openness to diverse interpretations.
It's crucial to differentiate between Chabad's subjective Talmudic interpretations and the objective truths about God as presented in the Torah. While Chabad’s views are deeply respected, they are one of many subjective viewpoints, and none can claim absolute authority on divine truth.
When comparing Chabad Rabbis' opinions about God with those of Jesus Christ, Albert Einstein, Baruch Spinoza, the Catholic Pope, the Dalai Lama, or Sir Roger Penrose, it becomes evident that no single group holds a monopoly on theological truth. Empirical evidence and real-world outcomes should guide our understanding and ranking of these belief systems.
Assumptions about Chabad often overlook the organization’s dual nature as both a community and commercial entity and the broader context of religious diversity. Recognizing and respecting the multiplicity of interpretations within Judaism and across other faiths fosters a more nuanced and inclusive understanding of Chabad's role and the search for divine truth. It is crucial not to conflate any single subjective religious viewpoint, especially that of a small ultra-orthodox subset of Judaism, with the objective truth about God.
The ultra-orthodox Chabad, for instance, conducts auctions during Yom Kippur that suggest increased business success in exchange for donations, such as the $9K communal Maftir Yonah prayer about a man who lived inside a fish—alleged to bring success in the real world. We posit that such donations are unlikely to enhance business profitability, regardless of the Rabbi's assurances. This practice resembles the sale of indulgences by the Catholic Church in the Middle Ages. As Marx observed, religion can sometimes serve as an opiate, particularly for those gullible enough to believe that purchasing a communal prayer will improve their business success. But beyond this bit of rent-seeking—which, to be fair, many of us are guilty of—we deeply appreciate our Rabbi and the Chabad organization. They’re truly great!
However, it is important to acknowledge that the New Testament predates the written Talmud by about 200 years and represents a heretical viewpoint within Judaism. Jesus Christ, who was also a Rabbi, could have been right, as Sir Bertrand Russell suggested, and the written Talmud could turn out to be false. In reality, the best way to get to know God is by actually solving Pascal’s wager independently, which is way outside the topic of this discussion.
Also, there is no evidence whatsoever that donating (or not donating) money to Chabad, or anyone else, will have any impact on any real-life outcomes. In reality, if you want God’s help, you have to talk and pray to God by yourself. If you think that any third-party intermediary—such as your Rabbi—either knows how to pray or is better able to do so than you, you are wrong. If you want God’s help, ask for it yourself, and you will be surprised by the answer once you solve Pascal’s wager and understand why Pascal was right and the intellectuals who never took up the wager were wrong—being blinded by false assumptions about probability and hypothesis testing, as we explain later.
However, in mathecon, everything we say in the rest of this paper is going to hold true in reality and is not conditional upon the existence of God. Everything we say in this paper is conditional on a specific set of assumptions, from which all our theorems are logically deduced, guaranteeing with absolute certainty that any mathematical economics theory will hold true in reality, unless one of the underlying axioms turns out to be false. This is a key difference between a hypothesis and a theorem in applied mathematics, being the next and last part of this second mathematical economics lesson.
Use Value and Proving Theorems in Mathematics
In any real-world economy, most goods and services available for sale have not only an exchange value – as measured by market price in non-barter economies – but also a subjective use value: how much personal benefit, or utility, the end user derives after purchasing them. This use value is subjective and can differ drastically between individuals. For example, the use value of shoes or a winter coat is to keep your feet safe and yourself warm during the winter. However, the use value of Louboutin shoes or a sable coat is primarily to flaunt wealth rather than to protect feet or keep the wearer warm. These functions are secondary; the primary use value is derived from showing off.
The question then becomes, what is the use value of proving theorems in mathematics? Why is there a $1 million reward for proving that the Riemann Hypothesis is true? The reason is that theorems, to an end user—an applied mathematician in this case—are more valuable than hypotheses, which are far less useful in applied mathematics. Why is this the case? In any formal system—all mathematics falls under this classification—theorems are derived from axioms using rules of inference where no logical claims, either theorems or axioms, contradict each other. Using logical deduction, we show that theorems hold true universally, conditional on the axioms holding true.
In reality, if a hypothesis, such as the Riemann Hypothesis, turns out to be false, it could turn out to be false either because it is false in theory or in practice. However, since theorems cannot turn out to be false in theory, if a theorem (like Bell’s Inequality) does not hold true in reality, we know with absolute certainty that one of the underlying axioms does not hold true in reality. For example, the reason Bell’s Inequality does not hold true in reality is that it is derived using the axiom of separation in ZF set theory, which posits that any set can be split into two distinct subsets. This assumption is false in reality when set elements are entangled photons, where the word “entangled” refers to the fact that such particles are inseparable in reality.
In applied mathematics, as distinct from theoretical mathematics, there are not one but two truths, as truth in reality is inevitably dually defined: a logical claim can be true in theory and false in reality, or vice versa. Forgetting that truth in reality is always dually defined (“one-true” objective reality vs. “multiple-truths” subjective theories) causes theory-induced blindness when we use a known-to-be-flawed theory in reality—logically deduced from a false axiom—as per Kahneman’s example of Bernoulli’s theory. Failing to clearly distinguish between claims that are true only in theory and those that are true not only in theory but also in reality leads to misunderstandings and misinterpretations. More importantly, any theoretical claim, including theorems that are universally true in theory, can be false in reality. This is exemplified by the claim: “2 + 2 = 4,” which, despite being universally true in theoretical mathematics, is not universally true in applied mathematics.
So, which claims can we be absolutely certain could never turn out to be false? Those whose truth—real-world accuracy—is independently verifiable. This limits our set of assertions in whose truth we can be absolutely certain to empirical facts, like IBM’s price last Wednesday, or the fact that the Earth is round and rotates around the bigger and heavier Sun. These assertions could never turn out to be false, just as mathematical proof of the Pythagorean theorem could never turn out to be false, as the truth of both claims is independently verifiable. This is not true of axioms, which are in mathematics accepted on account of being self-evidently true to the end users of any applied mathematics theory, and axioms, such as the Euclidean axiom that the shortest distance between two points is a straight line, can therefore always turn out to be false, unlike logical deduction and evidence-based claims, which cannot turn out to be false, in either theory or reality.
Since our counterparts in trade do not hold their axioms or theorems secret, in order to make money, we have to exclude from competing theories any claims that could turn out to be false, and the resulting theorems, whatever they are, become maximally likely to hold true in reality, simply on account of being the theory that is least likely to be falsified, ever, on account of relying on a strict subset of assumptions shared by all competing explanations.