# 1 Automated Syllabus of Game Theory Papers

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# 2 Comparative statics

## 2.1 Monotone Comparative Statics

• Carefully consider the definitions of your fundamental concepts and ensure they align with established terminology, as well as thoroughly check your proofs to avoid missing critical steps.

• Consider using iterative fixed-point comparative statics to analyze the effects of parameter changes in economic systems, particularly those involving strategic complementarities, as this approach allows for unstable equilibria, divergent learning processes, and unordered perturbations.

• Consider using the i-directional set order, which is a reformulation of the Ci-flexible set order, to study monotone comparative statics in situations where traditional lattice-based approaches are not applicable due to non-standard constraint sets.

• Utilize a two-step approach to conduct a likelihood-ratio test for order restrictions on the conditional quantiles of Y given X, first constructing nonparametric estimators for the conditional quantiles and then developing a test based on your asymptotic distributions, while accounting for the presence of numerous nuisance parameters due to unknown equilibrium selection probabilities.

• Employ a novel likelihood-ratio test for order restrictions on the conditional quantiles of Y given X, rather than relying solely on traditional methods such as OLS regression, when working with multiple equilibrium models where the MCS property holds.

• Utilize ordinal conditions rather than relying solely on traditional assumptions like smoothness, linearity, or convexity when conducting comparative statics analyses, as demonstrated through the development of a theory and methods for comparative statics analysis based on ordinal conditions alone.

• Consider leveraging the power of the monotonicity theorem of Topkis (1978) to analyze complex systems involving multiple interacting components, especially when those interactions exhibit complementarities and the system can be represented as a lattice.

• Consider relaxing the assumption of antisymmetry in binary relations when conducting comparative statics analysis, allowing for a more flexible framework that can handle constrained optimization problems with nonlinear constraints.

• Consider using lattice programming techniques and flexible set orders to study comparative statics problems in constrained optimization, particularly when dealing with non-smooth, non-interior, non-convex, or non-unique solutions.

• Consider reparameterizing your optimization problems to achieve monotone comparative statics, which can be done by identifying a vector field indicating the direction of monotonicity in the parameter space and then transforming the problem using this vector field.

• Utilize the monotone comparative statics approach to generate empirical predictions that are robust to model misspecification and can be tested using ordinal information and nonparametric methods, thereby avoiding reliance on potentially unfounded technical assumptions.

• Consider using log-supermodularity as a tool for deriving comparative statics predictions, particularly when dealing with uncertain environments, as it provides a powerful framework for understanding how changes in parameters affect optimal choices.

• Consider using log-supermodularity as a tool for deriving comparative statics predictions, particularly when dealing with uncertain environments, as it provides a strong condition for monotonicity that is preserved under integration.

• Utilize the concept of “monotone comparative statics” to analyze how changes in exogenous parameters impact endogenous outcomes in models, particularly when dealing with optimization problems. This approach provides ordinal answers regarding whether an increase in a parameter leads to an increase or decrease in the decision variable, while requiring fewer assumptions compared to alternative methods like the implicit function theorem. (NA?)

## 2.2 Comparative Statics

• Consider using the provided sufficient condition and algorithm to identify a minimum threshold parameter value at which every old equilibrium becomes strictly smaller than every new equilibrium, allowing for stronger conclusions about comparative statics in games of strategic complementarity.

• Leverage the power of “robust” comparative statics to analyze large dynamic economies, which provides generalizable insights into how stationary equilibria respond to exogenous shocks and changes in the distribution of idiosyncratic shocks, without requiring extensive simulation or numerical analysis.

• Carefully consider the tradeoffs between assumptions about payoff functions and probability distributions when making comparative statics predictions in stochastic optimization problems, as single crossing properties and log-supermodularity are crucial concepts for ensuring the validity of such predictions. (NA?)

## 2.3 Distributional Comparative Statics

• Focus on the concavity or convexity of policy functions when studying distributional comparative statics, as long as they understand the conditions under which these policy functions will be concave or convex.

# 3 Game theory

## 3.1 Comparative Statics

• Carefully distinguish between “strongly competitive” and “weakly competitive” games of incomplete information, as the comparative statics predictions differ significantly between them. Specifically, in strongly competitive games, even weak types are motivated to play more aggressively in response to a stochastically higher distribution of types, whereas in weakly competitive games, weak types are discouraged and compete less aggressively in a more competitive environment.

## 3.2 Game Theory

• Carefully control and standardize experimental procedures across sites to ensure uniformity and minimize potential confounding variables when conducting cross-cultural studies.

• Consider conducting cross-cultural studies using diverse samples from various economic and cultural backgrounds to capture the variability in human behavior and challenge assumptions of universality in economic models.

## 3.3 Single Crossing Property

• Look for conditions under which an additively separable objective function satisfies the Milgrom-Shannon single crossing property, specifically when one component allows a monotone concave transformation with increasing differences and is nondecreasing in the parameter variable, and the other component exhibits increasing differences and is nonincreasing in the choice variable.

• Utilize a stronger differential version of the single crossing property and argue from first-order conditions to establish strict monotonicity in comparative statics analyses.

## 3.4 Supermodular Game

• Consider using supermodular games, which are based on monotone comparative statics and supermodular optimization, to analyze non-cooperative situations where an increase in one players strategy leads to increases in other players strategies. This approach can provide insights into the existence and stability of equilibria, and avoid assumptions associated with traditional comparative statics methods.

• Consider using monotone comparative statics methods instead of the implicit-function theorem for comparative static analysis, as they offer greater flexibility and ease of use while requiring fewer assumptions, such as differentiability, concavity, and convexity.

## 3.5 Aggregative Game

• Leverage the aggregative structure of games, where each players payoff depends on her own actions and some aggregate of all players actions, to derive robust and general comparative static results under considerably weaker conditions than traditional approaches.

## 3.6 Cheap Talk

• Consider incorporating anonymous messaging in public goods games to encourage cooperation, as the study found that anonymous exhortations significantly increased contributions compared to groups without messaging.

## 3.7 Contest Theory

• Consider applying lattice-theoretic tools to analyze games with non-monotonic best-replies, even if they were initially developed for games of strategic complements/substitutes, because these methods can still provide valuable insights about comparative statics and equilibrium properties.

## 3.8 Dynamic Games

• Carefully address unobserved heterogeneity in dynamic games through methods such as fixed effect conditional likelihood or functional differencing, as failing to do so can lead to biased estimates of structural parameters that capture both dynamic state dependence and strategic interactions among players.

## 3.9 Market Games

• Consider using the theory of supermodular optimization and games to analyze the comparative statics of strategic market games, particularly when the model involves only one good on each side of the market, as this allows for a straightforward analysis based on familiar notions from classical microeconomic theory such as normality and gross substitutes of goods.

## 3.10 Persuasion Game

• Examine the “crater property” of interim payoffs to determine if coarse-convexity shifts lead to more informative signals being chosen by senders, regardless of the prior.

## 3.11 Quasi-supermodularity

• Carefully distinguish between different types of monotonicity (i.e., type A and type B problems) and consider the appropriate single crossing conditions and quasisupermodularity assumptions required for each type to ensure valid inferences.

## 3.12 Shapley Value

• Carefully consider the implications of using either conditional or interventional value functions when interpreting Shapley values as feature importance measures, as each approach introduces distinct trade-offs and potential biases.

## 3.13 Substitutes

• Look for conditions under which the indirect strategic substitute effect does not dominate the direct parameter effect in games with strategic substitutes, as this ensures the existence of a larger equilibrium at a higher parameter value.

## References

Acemoglu, Daron, and Martin Kaae Jensen. 2009. “Aggregate Comparative Statics.” SSRN Electronic Journal. https://doi.org/10.2139/ssrn.1374641.
———. 2015. “Robust Comparative Statics in Large Dynamic Economies.” Journal of Political Economy 123 (June). https://doi.org/10.1086/680685.
Arkolakis, Costas. 2010. “Market Penetration Costs and the New Consumers Margin in International Trade.” Journal of Political Economy 118 (December). https://doi.org/10.1086/657949.
Ashworth, Scott, and Ethan Bueno de Mesquita. 2005. “Monotone Comparative Statics for Models of Politics.” American Journal of Political Science 50 (December). https://doi.org/10.1111/j.1540-5907.2006.00180.x.
Athey, S. 2002. “Monotone Comparative Statics Under Uncertainty.” The Quarterly Journal of Economics 117 (February). https://doi.org/10.1162/003355302753399481.
Balbus, Lukasz, Wojciech Olszewski, Kevin L. Reffett, and Lukasz Patryk Wozny. 2022. “Iterative Monotone Comparative Statics.” SSRN Electronic Journal. https://doi.org/10.2139/ssrn.4039543.
Barthel, Anne-Christine, and Tarun Sabarwal. 2017. “Directional Monotone Comparative Statics.” SSRN Electronic Journal. https://doi.org/10.2139/ssrn.3322667.
Baum, William M., Brian Paciotti, Peter Richerson, Mark Lubell, and Richard McElreath. 2012. “Cooperation Due to Cultural Norms, Not Individual Reputation.” Behavioural Processes 91 (September). https://doi.org/10.1016/j.beproc.2012.06.001.
Chambers, Christopher P., Federico Echenique, and Kota Saito. 2016. “Testing Theories of Financial Decision Making.” Proceedings of the National Academy of Sciences 113 (March). https://doi.org/10.1073/pnas.1517760113.
Curello, Gregorio, and Ludvig Sinander. 2022. “The Comparative Statics of Persuasion.” arXiv. https://doi.org/10.48550/ARXIV.2204.07474.
Dobronyi, Christopher, Jiaying Gu, and Kyoo il Kim. 2021. “Identification of Dynamic Panel Logit Models with Fixed Effects.” arXiv. https://doi.org/10.48550/ARXIV.2104.04590.
Echenique, Federico, and Ivana Komunjer. 2013. “A Test for Monotone Comparative Statics.” Structural Econometric Models, December. https://doi.org/10.1108/s0731-9053(2013)0000032007.
Edlin, Aaron S., and Chris Shannon. 1998. “Strict Monotonicity in Comparative Statics.” Journal of Economic Theory 81 (July). https://doi.org/10.1006/jeth.1998.2405.
Gama, Adriana, and David Rietzke. 2019. “Monotone Comparative Statics in Games with Non-Monotonic Best-Replies: Contests and Cournot Oligopoly.” Journal of Economic Theory 183 (September). https://doi.org/10.1016/j.jet.2019.08.004.
“Game Theory.” 2010. https://doi.org/10.1057/9780230280847.
Henrich, Joseph, Robert Boyd, Samuel Bowles, Colin Camerer, Ernst Fehr, Herbert Gintis, and Richard McElreath. 2001. “In Search of Homo Economicus: Behavioral Experiments in 15 Small-Scale Societies.” American Economic Review 91 (May). https://doi.org/10.1257/aer.91.2.73.
Henrich, Joseph, Richard McElreath, Abigail Barr, Jean Ensminger, Clark Barrett, Alexander Bolyanatz, Juan Camilo Cardenas, et al. 2006. “Costly Punishment Across Human Societies.” Science 312 (June). https://doi.org/10.1126/science.1127333.
Jensen, Martin Kaae. 2017. “Distributional Comparative Statics.” The Review of Economic Studies 85 (May). https://doi.org/10.1093/restud/rdx021.
Kotani, Daisuke, Eiji Oki, Yoshiaki Nakamura, Hiroki Yukami, Saori Mishima, Hideaki Bando, Hiromichi Shirasu, et al. 2023. “Molecular Residual Disease and Efficacy of Adjuvant Chemotherapy in Patients with Colorectal Cancer.” Nature Medicine 29 (January). https://doi.org/10.1038/s41591-022-02115-4.
Kukushkin, Nikolai S. 2011. “Monotone Comparative Statics: Changes in Preferences Versus Changes in the Feasible Set.” Economic Theory 52 (November). https://doi.org/10.1007/s00199-011-0677-8.
Kumar, I. Elizabeth, Suresh Venkatasubramanian, Carlos Scheidegger, and Sorelle Friedler. 2020. “Problems with Shapley-Value-Based Explanations as Feature Importance Measures.” arXiv. https://doi.org/10.48550/ARXIV.2002.11097.
Lahiri, Somdeb. 2011. “Comparative Statics of Oligopoly Equilibrium in a Pure Exchange Economy.” SSRN Electronic Journal. https://doi.org/10.2139/ssrn.1744661.
Mitsilegas. 2012. “Immigration Control in an Era of Globalization: Deflecting Foreigners, Weakening Citizens, and Strengthening the State.” Indiana Journal of Global Legal Studies 19. https://doi.org/10.2979/indjglolegstu.19.1.3.
“Monotone Comparative Statics with Separable Objective Functions.” 2010. Economics Bulletin. https://doi.org/10.5167/UZH-38325.
Quah, John K.-H. 2007. “The Comparative Statics of Constrained Optimization Problems.” Econometrica 75 (March). https://doi.org/10.1111/j.1468-0262.2006.00752.x.
Roy, Sunanda, and Tarun Sabarwal. 2010. “Monotone Comparative Statics for Games with Strategic Substitutes.” SSRN Electronic Journal. https://doi.org/10.2139/ssrn.3323084.
Shirai, Koji. 2010. “Monotone Comparative Statics of Characteristic Demand.” SSRN Electronic Journal. https://doi.org/10.2139/ssrn.1553547.
“Structural Econometric Models.” 2013. Advances in Econometrics, December. https://doi.org/10.1108/s0731-9053(2013)31.
Strulovici, B. H., and T. A. Weber. 2007. “Monotone Comparative Statics: Geometric Approach.” Journal of Optimization Theory and Applications 137 (December). https://doi.org/10.1007/s10957-007-9339-1.
Tremblay, Carol Horton, and Victor J. Tremblay. 2010. “The Neglect of Monotone Comparative Statics Methods.” The Journal of Economic Education 41 (March). https://doi.org/10.1080/00220481003617293.
Vives, Xavier. 2005. “Complementarities and Games: New Developments.” Journal of Economic Literature 43 (May). https://doi.org/10.1257/0022051054661558.