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18/ TL;DR: Adding a dash of noise (random experimentation) to learning dynamics, evolutionary reasoning can resolve (in many static games) equilibrium indeterminacy. The one with larger basin of attraction – the risk-dominant convention – tends to reign supreme in the long run.

17/ A truly remarkable scholar of the field and whose book you must read is the late William H. Sandholm.

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16/ Intellectual history: Evolutionary game theory is inspired from biology’s idea of natural selection. John Maynard Smith (w/ G.R. Price) introduced the concept of Evolutionarily Stable Strategy (ESS) in 1973, showing how certain behaviors become stable in animal conflicts.

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15/ Significance: KMR and Young bridged ideas from biology and bounded rationality to explain which social convention “wins” when everyone occasionally makes mistakes. The concept of stochastically stable equilibrium introduced here has become a key tool in economic theory.

14/ Beyond 2×2 games: Young showed the selected convention need not be the risk-dominant or Pareto-best. Nonetheless, his general algorithm can identify the stochastically stable convention in such cases. And more work has generalized the KMR techniques to general games as well.

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13/ The two papers--- KMR and Young have somewhat different setups, but they both eventually use Markov chain methods and results from perturbation theory (Freidlin–Wentzell).

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12/ Young's method involves finding the minimum “cost” (fewest simultaneous mutations) needed to escape each equilibrium; the equilibrium with the highest escape cost is selected in the long run.

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11/ Young’s result: By analyzing the “graph” of state transitions, he finds a unique stochastically stable equilibrium in many games. In 2×2 coordination games, it coincides with the risk-dominant convention – exactly as in KMR.

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10/ Even under adaptive play, multiple equilibria can persist (each is an absorbing convention if no one deviates). Young adds a small chance that players choose a non-best-response at random. Just as in KMR, these “mistakes” let some conventions prove more enduring than others.

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9/ Young introduces a boundedly rational learning rule: adaptive play. Population of players randomly matched to play a game; only remember a small sample of recent rounds; choose a best response to that. This limited memory (different for each player) prevents cyclic behavior.

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8/ Risk dominance was introduced by Harsayni and Selten (1988): (G,G) here risk dominates (H,H) because the expected payoff from playing G is higher than H when you assume the other player is playing 50-50: 1/2* 4 + 1/2* 2 > 1/2* 5 + 1/2* 0. So, (G,G) is the SSE.

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7/ KMR’s insight: In any 2×2 coordination game with 2 Nash equilibria, the risk-dominant equilibrium will prevail in the long run: it’s the equilib hardest to upset. If it takes a larger rare shock to leave Equilib A than to leave B, the system will spend most of its time at A.

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6/ KMR analyze a finite population repeatedly playing a symmetric game. Successful strategies spread (Darwinian) and each player occasionally mutates (chooses a random action) instead of the best response. This ongoing noise “drastically reduces” the set of long-run equilibria.

5/ Long-run equilib: Game never truly “freezes” at a NE, a tiny chance someone experiments. So process has a stationary distribution over states (what is played). As mutation probability → 0, distribution puts ~100% weight on a single stochastically stable equilibrium (SSE).

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4/ A dynamic approach: Instead of assuming players magically coordinate on an equilibrium, KMR/Young model how conventions emerge via a dynamic learning process with occasional random perturbations (small “mutations”). Players mostly adjust to success, but sometimes experiment.

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3/ Consider the Stag Hunt: In this coordination game, two pure Nash equilibria (NE) exist: (Hunt,Hunt) and (Gather,Gather). (Hunt,Hunt) is Pareto-best for both, but (Gather,Gather) is risk-dominant – it’s safer if you’re unsure what the other will do.

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2/ Just when you think you've understood all that Nash brought to game theory, he surprises you. This from his PhD thesis (1950), where he conceptualizes early on, how players could arrive at playing the (well, Nash) equilibrium: a population and an empirical perspective!

Rohit Lamba @rohitlamba.bsky.social · 20/07/2025

1/ Week 30: The theory paper(s) in focus this week are KMR 93 & Young 93, which provided a remarkably elegant evolutionary perspective to equilibrium selection in games. If games have multiple equilibria, which is more "reasonable", if at all? A deep problem in game theory.

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The consequences for human welfare involved in questions like these are simply staggering: Once one starts to think about them, it is hard to think about anything else. --- Bob Lucas (1988).

24/ On a personal note, I've learnt a lot about optimal taxation and particularly about the Atkinson-Stiglitz Theorem, its stark implications and its obvious practical limitations from @jacobsgoldin.bsky.social.

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23/ In practice: Atkinson-Stiglitz is a remarkable theoretical benchmark, not a blanket rule. Real-world tax systems do tax consumption (VAT, excises) and capital, because the theorem's assumptions often fail (preferences not separable, multiple sources of inequality, etc.).

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22/ Excellent recent work by Pai-@philippstrack.bsky.social, and Doligalski-Dworczak-Akbarpour- @skominers.bsky.social brings fresh perspectives on the Atkinson-Stiglitz theorem.

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21/ A-S remains a benchmark, but knowing when it fails is key for real-world tax design. These lectures by @s-stantcheva.bsky.social provide a great overview of how relaxing A-S assumptions shows when commodity or capital taxes become warranted. stantcheva.scholars.harvard.edu/sites/g/file...

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20/ @josephestiglitz.bsky.social critiques 0-capital-tax view. Notes A-S separability assump is implausible. In more realistic settings (imperfect info, inheritance, etc.), taxing capital can improve equity w/ acceptable efficiency costs. [Though implementing inheritance taxes well has been hard.]

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19/ In a lovely paper, @ludwigstraub.bsky.social & @ivanwerning.bsky.social revisit Chamley-Judd & overturn the zero-capital-tax result. If people are not very willing to substitute consumption over time (low intertemporal elasticity), the optimal policy has a positive long-run capital tax!

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18/ If inequality has multiple sources, the A-S result can fail. Their model assumed only ability differences. In reality, people also differ in inherited wealth. With such inequality, taxing only labor income may not suffice—some capital taxation can be part of the optimum.

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17/ Another assumption was identical preferences. If people's tastes differ, A-S can fail. For example, if high-ability people spend more on certain luxuries, taxing those goods effectively targets them. Different preferences can thus justify commodity taxes.

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16/ If separability fails, the optimal tax mix changes. For example, if a certain good is consumed mostly during leisure, taxing it effectively taxes leisure (which otherwise escapes taxation). In such cases, differential commodity taxes can improve outcomes.

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15/ This powerful result hinges on particular assumptions. If those fail, optimal policy might require commodity or capital taxes after all. The crucial assumption was separability of consumption and labor (no good is especially a "leisure luxury"). What if that's not true?

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14/ One implication: all redistribution should be done via the income tax. Commodity taxes can then be kept simple (even uniform rates). In theory, equity goals are achieved by a well-designed progressive income tax, without needing special high taxes on luxury goods or capital.

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13/ The Atkinson-Stiglitz result was surprising. Earlier theory (Ramsey) argued for taxing goods at different rates (e.g. high taxes on inelastic or luxury goods). A-S showed that if the income tax is truly optimal, those differential commodity taxes aren't needed.

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12/ In a macro image, Chamley-Judd (1986/85) found that optimal long-run tax on capital is zero! In A-S terms, a capital tax is a tax on future consumption. If consuming tomorrow is "like" consuming today in utility, then there's no gain to taxing future consumption.

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11/ We are considering the "direct planner's problem". One could plausibly think that in the "indirect" world of prices, redistribution can be achieved by differential pricing? Since ability is unobservable, deadweight loss from doing this would outweigh the benefits.

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10/ Commodity taxes add no value for screening. If broadly everyone spends money similarly regardless of skill, taxing specific goods won't separate high vs low ability. In short, commodity consumption offers no extra benefit beyond what the income tax already achieves.

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9/ Why consider taxing specific goods? Because of screening. In the model, government can't observe ability—only earnings or spending. A high-ability person might try to act low-income to pay less tax. Taxing certain goods that high-skill folks favor could help separate types.

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8/ In full generality, the planner can tax the purchase of commodities, and it can also tax income and wages. And boom, A-S say to solve this problem you can only tax income, that is sufficient to achieve the redistributive objective.

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7/ The objective of the government/planner is to maximize a social welfare function, hence redistribution, subject to raising some minimum revenue and satisfying the constraint that ability/effort is unobservable.

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6/ Moreover, the typical Mirrleesian assumption, which is at the core of the models, is that income/output is observable, but the ability and labor effort is not.

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5/ In optimal tax models, "ability" has a specific meaning: it (mostly) means differences in labor productivity. A high ability produces more output for the same quantity of labor supply than a lower ability. A simplified way of capturing heterogeneity is earning potentials.

4/ What is weak separability? Preferences for goods are independent of labor. In plain terms: how someone trades off different commodities is the same whether they work a lot or a little. And, taxing one good vs another doesn't help identify high vs low "ability" people.

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3/ Atkinson-Stiglitz (1976) Theorem: If utility is weakly separable between labor and consumption, and the government levies an optimal income tax on earnings, then no commodity taxes are needed. The income tax alone can achieve the optimum!

2/ Policy question: Should we tax capital or specific goods (like luxury items), what they call indirect taxation, to enhance equity? Or can an income tax, what they call direct taxation, alone achieve our redistribution goals? Atkinson & Stiglitz offered a surprising answer.

Rohit Lamba @rohitlamba.bsky.social · 07/07/2025

1/ Week 28: Theory paper in focus this week is the seminal article by Anthony Atkinson and @josephestiglitz.bsky.social that is now famously referred to as the "zero commodity/ capital taxation" benchmark in public economics. It has led to much technical & policy debates. Let's find out.

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Been listening to the audiobook of this masterpiece of a biography. What a life Deng Xiaoping, and what a researcher and writer Ezra Vogel.

25/ Bottom line: sometimes crowd is wiser than any individual {wisdom of crows}, but other times the crowd is just following itself. 🤪 Understanding information cascades helps us tell when “everyone’s doing it” is a signal of quality… or just a quirk of people copying people!

24/ An excellent recent survey revisits the theoretical literature on herds & cascades It notes that cascades cause poor information aggregation, lead to fragile mass behaviors, and remain central to understanding social learning. Those 1992 papers launched a vast literature 😎

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23/ Great theory research followed: eg. In a beautiful appear, Smith & Sørensen (2000) showed if people receive richer signals (not just yes/no hints), a cascade might never completely lock in – some learning continues. Still, eventually some herd forms, and it can still be wrong.

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22/ There is also interesting research on how misinformation spreads faster via cascades. A 2018 study of 126,000 Twitter rumors found that false news consistently spreads “farther, faster, deeper, and more broadly” than true news.

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