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Martin’s Maximum++ implies Woodin’s axiom ()

Asperó and Schindler have just published a foundational paper, Martin’s Maximum++ implies Woodin’s axiom (), Annals of Mathematics, 193(3), 793-835. It is a fascinating result in mathematical logic with implications to reconcile two approaches to the continuum hypothesis.

It has broader implications about the limits of axiomatic systems and the ability of mathematics to solve particular problems, and by extension, about our human endeavor to uncover higher universal truths, not only material, but also the intellectual concept of knowability.

Recall that countably infinite unions (can also be viewed as countably infinite “sums”) of countably infinite sets are still countable. This is somewhat mind-boggling, because adding (countably) infinitely many (countable) infinities still gives the same number at the end.

If we denote the cardinal number of the natural numbers by N0, then we have N0+N0+…+N0 = N0 even with infinitely many addends. This is the type of result that leads to gasps of incredulity at first encounter. Hilbert’s hotel provides a somewhat intuitive explanation.

On the other hand, the cardinality of the continuum of real numbers (denoted c) is strictly larger than that of the natural numbers and (given the above) strictly larger than the (countably) infinite sum of sets of infinitely many natural numbers. Thus, N0 < c.

Set theory also shows how to construct the first uncountable ordinal; its cardinal number is N1. As the reals are also uncountable, c cannot be less than N1. Therefore, we conclude that N0 < N1 <= c.

Cantor’s continuum hypothesis states that N1 = c, but he could not prove it. As the famous results of Godel and Cohen show, the continuum hypothesis can neither be disproved nor proved in the standard Zermelo-Frankel axioms of set theory, even if we add the axiom of choice.

So, we can add the continuum hypothesis as another axiom of set theory and develop mathematics and our understanding of truth taking this axiom as given.

Or, we can add some other axiom that is reasonable and deduce whether the continuum hypothesis is true or false in the extended axiomatic system. As the real numbers are a foundational object in our world, it’d be better to deduce their cardinality rather than assume it.

One such axiom is Martin’s Maximum++. This axiom implies that the continuum hypothesis is false. That is, N1 < c, and moreover, c = N2, the cardinal number for the second uncountable ordinal. As you can imagine, N2 is considerably larger than N1.

The second is Woodin’s axiom (*). This is based on a different concept and also implies that the continuum hypothesis is false and that c = N2.

Two different methods with different applications, seemingly at odds with each other, but with the same implication for the continuum hypothesis. Asperó and Schindler reconcile their seeming opposition by proving that Martin’s Maximum++ implies Woodin’s axiom ().

As with things at this level of abstraction, this does not mean that we must adopt Martin’s Maximum++ or Woodin’s axiom (*). We can develop our universe of truth and provability without these axioms. But there is a resolution of at least one seemingly contradictory puzzle.

So invigorating to be taken back to topics in math that I first studied with wide-eyed wonder under the guidance of the brilliant Dr. Marina Ratner. Thankful for the knowledge gained.

July 15, 2021 | Tweet | In

Control and spread of contagion in networks

Pleased to release new research on “Control and spread of contagion in networks,” studying an important and increasingly central topic in current society. This is joint work with John Higgins. An article on this research by KU news is here.

We study proliferation of an action in a network of connected individuals. Proliferation occurs due to person-to-person spread based on network connectivity and due to viral effects based on similar activity by others in different parts of the network.

We extend the model of a network coordination game in Morris (2000) and Jackson (2008) to include a new, flexible, and tractable formulation of aggregated virality. The case of no virality is naturally subsumed in the extended model.

Virality is the notion that each player's decision depends not just on decisions of their neighbors but also on an aggregate based on decisions taken by others in the network. This is different from a common definition in the network science literature, where an event is viral if the number of times it spreads using person-to-person interaction exceeds a threshold. We term this depth of contagion and include it in our analysis as well.

We present new algorithms that characterize equilibrium depth of contagion and when contagion occurs from a given initial set to the entire network. The algorithms are computationally tractable and applied to study contagion in scale-free networks with 1,000 players.

Our research provides insight into the design of policies that may help to control or spread contagion in networks.

A larger starting set increases depth of contagion. Virality amplifies this effect and brings contagion closer to a type of singularity (a narrow interval of starting set sizes below which there is no contagion and above which contagion occurs to the entire network).

Therefore, in networks where virality is a common feature, it is easier to curtail spread of misinformation by nipping it in the bud. Waiting for things to play out will add to the likelihood of a considerably larger spread of contagion.

In particular, for misinformation, policies that restrict spread of incorrect or misleading information and/or identify and limit accounts designed mainly to spread misinformation help to curtail spread of misinformation.

For infectious diseases, policies that contain the spread when infections are low and/or restrict cross-network travel and non-neighbor interaction curtail the spread of disease. This provides a theoretical justification for mandates such as masks, social distancing, and quarantine.

Taking the reverse actions increases the spread of contagion.

The model and analysis are flexible to apply to many additional situations, as described in the paper. I hope this research is valuable to other researchers within and outside economics.

June 18, 2021 | Tweet | In

Interdisciplinary application of game theory to COVID-19 vaccination decisions

Nice to see an interdisciplinary application of game theory to COVID-19 vaccination decisions in “Prioritising COVID-19 vaccination in changing social and epidemiological landscapes,” by Jentsch, Anand, and Bauch.

The authors use a coupled model of social decisions and epidemiological contagion. Social decisions are based on evolutionary game theory (using incentives resembling prisoners’ dilemma and tragedy of the commons). The epidemiological model is age-structured SEAIR (Susceptible, Exposed, Asymptomatic Infectious, Symptomatic Infectious, Removed). Model is fitted using Bayesian particle filtering on Google mobility data and reported cases in Ontario. An intuitive summary of the results for a broader audience is available in NY Times article by Siobahn Roberts.

The research intersects biology, ecology, economics, epidemiology, environmental science, game theory, and mathematics. Looks very interesting and a nice example of eclectic interdisciplinary work. Thanks so much to the authors!

December 20, 2020 | Tweet | In

Monotone games: A unified approach to games with strategic complements and substitutes

I am delighted to announce the worldwide release of my new book “Monotone games: A unified approach to games with strategic complements and substitutes,” Palgrave Macmillan/Springer.

Games with strategic complements, games with strategic substitutes, and their combinations form the basis of numerous societal interactions governed by monotone interdependent incentives.

Environments in which decisions are governed by incentives to move in the same direction as others are classified as those with strategic complements. Canonical examples are coordinating on a bank run or technology adoption, or making a run on groceries in a pandemic.

Environments in which incentives are to move in a direction opposite to others are classified as strategic substitutes. Canonical examples are competing for market share or competing for a common good, or congestion games, or taking actions with a particular externality.

Environments in which some participants have one type of incentive and others have the other type also arise frequently. For example, penalty shooter and goal keeper in soccer, or law enforcement and law-breaking activity.

The theory of strategic complements is well-established and extensive. Indeed, some of the insights of auction theory celebrated in this year’s Nobel prize are based on the general theory of strategic complements.

The theories of strategic substitutes and of combinations of the two are newer and evolving. I am fortunate to contribute in central ways to the foundations of the newer theories. There are some similarities between the new and the old and some sharp differences as well.

The book develops anew the foundations of all three classes of games in a unified manner under the umbrella of monotone games, providing systematic connections across different cases. New results and examples are included as well.

I believe that the core principles studied in the book arise in fundamental ways in a large body of human and socioeconomic interaction with interdependent effects. There is a compelling reason for this body of knowledge to be accessible to a broader audience.

I hope this book serves to further research and applications in these areas. So, while you wait for additional election results, grab a cup of coffee and read more about herd mentality, dove-hawk strategies, and hunter-hunted dynamics. Happy reading!

November 4, 2020 | Tweet | In

Methods matter: p-hacking and publication bias in causal analysis in economics

A very nice and thought-provoking article by Brodeur, Cook, and Heyes: “Methods Matter: p-Hacking and Publication Bias in Causal Analysis in Economics,” American Economic Review 2020, 110(11), 3634-3660.

With more empirical research in economics and greater uncertainty on the part of a broader audience about the reliability of results, this paper brings a welcome look at the distribution of significance results for causal inference in a breadth of published work.

Looking at 21,740 tests in 684 articles published in Top 25 relevant journals in economics, the authors document an increase in density of z-statistic in the range determined by standard statistical significance thresholds.

There are three analyses (Randomization tests, Caliper tests, and Excess test statistics) for each of four causal inference techniques (Instrumental Variables, IV, Difference-in-Differences, DID, Randomized Control Trials, RCT, and Regression Discontinuity Design, RDD).

The broad picture that emerges is that IV and to a lesser extent DID are particularly problematic in terms of p-hacking. RCT and RDD fare better, but all four methods exhibit some discontinuity around conventional statistical thresholds when using finer windows of analysis.

A thoughtful and informative empirical meta-analysis of methods that are used widely to support the credibility revolution in empirical economics. Very well-written and accessible to a broad audience; a nice read. Thanks!

October 29, 2020 | Tweet | In

Perspectives series devoted to graduate students during summer 2020

During summer 2020's unprecedented time of uncertainty, anxiety, and restrictions on travel and social interaction, many graduate students were more isolated. As one component of collective outreach to help graduate students stay connected with research over the summer, faculty members in economics presented research seminars geared toward a general economics audience.

Combined with additional outreach activities, it is our sincere hope that this will help to alleviate academic distancing and assuage the separating effects of this pandemic. For additional information, please contact the organizer, Dr. Tarun Sabarwal. Summer schedule here.

August 2020 | Tweet | In

How market design emerged from game theory

Professors Alvin Roth and Robert Wilson have published a very readable open access article on, "How market design emerged from game theory, a mutual interview," Journal of Economic Perspectives, 33(3), Summer 2019, 118-143.

It is a pleasure to read about their thoughts on the big picture history of economic theory, game theory, and their role in the notable success of market design, especially in auctions and matching.

It is refreshing to think about complementarity between cooperative and noncooperative approaches of game theory, to consider games as embedded in larger social systems, and to consider connections between the core in games and in general equilibrium exchange economies.

A good (and relatively quick) read to cap your summer reading.

August 6, 2019 | Tweet