Internet and Network Economics

Cournot and Bertrand oligopolies constitute the two most prevalent models of firm competition. The analysis of Nash equilibria in each model reveals a unique prediction about the stable state of the system. Quite alarmingly, despite the similarities of the two models, their projections expose a stark dichotomy. Under the Cournot model, where firms compete by strategically managing their output quantity, firms enjoy positive profits as the resulting market prices exceed that of the marginal costs. On the contrary, the Bertrand model, in which firms compete on price, predicts that a duopoly is enough to push prices down to the marginal cost level. This suggestion that duopoly will result in perfect competition, is commonly referred to in the economics literature as the "Bertrand paradox".In this paper, we move away from the safe haven of Nash equilibria as we analyze these models in disequilibrium under minimal behavioral hypotheses. Specifically, we assume that firms adapt their strategies over time, so that in hindsight their average payoffs are not exceeded by any single deviating strategy. Given this no-regret guarantee, we show that in the case of Cournot oligopolies, the unique Nash equilibrium fully captures the emergent behavior. Notably, we prove that under natural assumptions the daily market characteristics converge to the unique Nash. In contrast, in the case of Bertrand oligopolies, a wide range of positive average payoff profiles can be sustained. Hence, under the assumption that firms have no-regret the Bertrand paradox is resolved and both models arrive to the same conclusion that increased competition is necessary in order to achieve perfect pricing.

/pdf/no-regret-learning-in-oligopolies-cournot-vs-bertrand-3dpm5rij5w.pdf

No regret learning in oligopolies: cournot vs. bertrand

No Regret Learning in Oligopolies: Cournot vs. Bertrand

Algorithmic game theory attempts to mathematically capture behavior in strategic situations, in which an individual's success depends on the choices of others. Research in this area tends to focus on one of the following challenges: (1) Price of Anarchy: Characterize the inefficiency of equilibria vs the global optimum. (2) Computational Complexity of Equilibria (3) Algorithmic Mechanism Design : Design games with desirable properties that are efficiently implementable. 
Many strategic interactions are in their nature recurrent (e.g. financial markets) with the agents participating in them repeatedly. These agents learn over time to adapt to their environment as is defined by the game and the dynamic behavior of the other agents[96]. We show that incorporating the assumption of adaptive agents can lead to exciting new insights to long standing questions in all areas of algorithmic game theory.
A Learning Theoretic Refinement of the Price of Anarchy: It is well understood that competitive environments can result in losses to social welfare due to the lack of a central coordinator who could enforce the global optimum solution. Defined as the ratio of the cost of the worst possible (Nash) equilibrium over the cost of the optimum, price of anarchy (and its variants) are commonly used metrics for these inefficiencies. However, such worst case analysis can be rather misleading as worst case equilibria may never arise in practice. Can natural learning algorithms beat the price of anarchy?
We show that in the class of atomic congestion games natural learning algorithms can indeed learn to navigate away from such worst case equilibria. In some cases the implied performance bounds are exponentially better than the previously known worst case guarantees. Furthermore, such positive performance results are shown to be robust even when we impose restrictions on availability of accurate up-to-date information on which agents can base their decisions. 
Learning Inspired Equilibria and Computation: The plausibility of the Nash equilibrium, as a universal solution concept has been under attack, due to recent negative complexity results (e.g. for normal form games with constant number of players[27, 24]). To make matters worse, there exist games of constant overall size, for which Nash equilibria are unstable for most reasonable dynamics[88].
In this section we will look into learning inspired solution concepts such as strong CURB (Closed Under Rational Behavior) and CUBR (Closed Under weakly Better Replies) sets. A strong CURB (CUBR) set is a cartesian product of pure strategy sets such that each player's component contains all best (weakly better) responses to itself given any joint probability distributions of its opponents over the set. A strong CURB (CUBR) set is said to be minimal if it does not contain any proper subset that is also a strong CURB (CUBR) set. Such minimal sets exist in all finite games and are asymptotically stable for a great number of natural learning dynamics. Furthermore, we prove that we can also compute all minimal strong CURB (as well as CUBR) sets for any normal form game with a constant number of players in polynomial time. 
Mechanism Evolution: The goal of mechanism design is to design games that are uniquely solvable by reasonably self-interested players and which lead to socially optimal outcomes. Such guarantees are of critical importance in the case of economic interactions which are key revenue producers, such as ad-auctions. However, in many cases of social and economic interactions, centralized design is impractical and sometimes even undesirable (i.e. internet). Nevertheless, such mechanisms seem to be working fairly well in practise. How is that possible?
Most mechanisms that we participate in are not the product of "intelligent design". Instead mechanisms evolve. All games (mechanisms) that we participate in are in essence social contracts. The rules of these interactions change over time as the result of actions of the same individuals who participate in them. These changes are not random. On the contrary the involved agents are trying to bring on changes to the structure of the game, so as to improve their experience from participating in the game.
We formalize these intuitions and we analyze their implications in the context of oligopolistic markets. Specifically, we allow players one extra dimension in the pursuit of their strategic goals, the possibility of forming coalitions with other agents. We focus on the class of linear and symmetric Cournot games and we study the nature of stable coalition structures. We show than even the worst stable coalition can lead to a significant increase in market prices and profits for the firms.

/pdf/a-learning-theoretic-approach-to-algorithmic-game-theory-2gxlhcvmpm.pdf

A learning theoretic approach to algorithmic game theory

The existence of a unique equilibrium is the classic tool for ensuring predictiveness of game theory Typical uniqueness results, however, are for Nash and Bayes-Nash equilibria and do not guarantee that natural game playing dynamic converges to this equilibrium In fact, there are well known examples in which the equilibrium is unique, yet natural learning behavior does not converge to it Motivated by this, we strive for stronger uniqueness results We do not only require that there is a unique equilibrium, but also that this equilibrium must be learnable We adopt correlated equilibrium as our solution concept, as simple and natural learning algorithms guarantee that the empirical distribution of play converges to the space of correlated equilibria Our main result is to show uniqueness of correlated equilibria in a large class of single-parameter mechanisms with matroid structure We also show that our uniqueness result extends to problems with polymatroid structure under some conditions Our model includes a number of special cases interesting on their own right, such as procurement auctions and Bertrand competitions An interesting feature of our model is that we do not need to assume that the players have quasi-linear utilities, and hence can incorporate models with risk averse players and certain forms of externalities

http://paulduetting.com/pubs/ec14-correlated.pdf

Mechanism with unique learnable equilibria

First-in-class versus best-in-class: an update for new market dynamics

This paper explores the relation between equilibrium coarsenings and equilibrium refinements via Bertrand competition example and similar situations, it shows that the typical equilibrium coarsening --- a unique correlated equilibrium --- is equivalent to the unique Nash equilibrium itself, is also equivalent to the equilibrium refinement, for the standard n-firms Bertrand competition model with linear demand and symmetric, linear costs in the most special and simplest case, and compares some wonderful and remarkable differences of the existence, uniqueness, stability, connectivity, and strategic property of Nash equilibrium and correlated equilibrium between Cournot and Bertrand model. We also propose some open questions.

Correlated Equilibrium of Bertrand Competition

Strategic correlativity and network games

Background: Mechanistic models are commonly employed to describe signaling and gene regulatory kinetics in single cells and cell populations. Recent advances in single-cell technologies have produced multidimensional datasets where snapshots of copy numbers (or abundances) of a large number of proteins and mRNA are measured across time in single cells. The availability of such datasets presents an attractive scenario where mechanistic models are validated against experiments, and estimated model parameters enable quantitative predictions of signaling or gene regulatory kinetics. To empower the systems biology community to easily estimate parameters accurately from multidimensional single-cell data, we have merged a widely used rule-based modeling software package BioNetGen, which provides a user-friendly way to code for mechanistic models describing biochemical reactions, and the recently introduced CyGMM, that uses cell-to-cell diﬀerences to improve parameter estimation for such networks, into a single software package: BioNetGMMFit. Results: BioNetGMMFit provides parameter estimates of the model, supplied by the user in the BioNetGen markup language (BNGL), which yield the best ﬁt for the observed single-cell, timestamped data of cellular components. Furthermore, for more precise estimates, our software generates conﬁdence intervals around each model parameter. BioNetGMMFit is capable of ﬁtting datasets of increasing cell population sizes for any mechanistic model speciﬁed in the BioNetGen markup language. Conclusion: By streamlining the process of developing mechanistic models for large single-cell datasets, BioNetGMMFit provides an easily-accessible modeling framework designed for scale and the broader biochemical signaling community.

BioNetGMMFit: a Parameter Estimation Tool for BioNetGen using Single-Cell Snapshot Data from Cell Populations Evolving over Time

Background Mechanistic models are commonly employed to describe signaling and gene regulatory kinetics in single cells and cell populations. Recent advances in single-cell technologies have produced multidimensional datasets where snapshots of copy numbers (or abundances) of a large number of proteins and mRNA are measured across time in single cells. The availability of such datasets presents an attractive scenario where mechanistic models are validated against experiments, and estimated model parameters enable quantitative predictions of signaling or gene regulatory kinetics. To empower the systems biology community to easily estimate parameters accurately from multidimensional single-cell data, we have merged a widely used rule-based modeling software package BioNetGen, which provides a user-friendly way to code for mechanistic models describing biochemical reactions, and the recently introduced CyGMM, that uses cell-to-cell differences to improve parameter estimation for such networks, into a single software package: BioNetGMMFit. Results BioNetGMMFit provides parameter estimates of the model, supplied by the user in the BioNetGen markup language (BNGL), which yield the best fit for the observed single-cell, timestamped data of cellular components. Furthermore, for more precise estimates, our software generates confidence intervals around each model parameter. BioNetG-MMFit is capable of fitting datasets of increasing cell population sizes for any mechanistic model specified in the BioNetGen markup language. Conclusion By streamlining the process of developing mechanistic models for large single-cell datasets, BioNetGMMFit provides an easily-accessible modeling framework designed for scale and the broader biochemical signaling community.

https://www.biorxiv.org/content/biorxiv/early/2022/12/12/2022.12.08.519526.full.pdf

BioNetGMMFit: Estimating Parameters of a BioNetGen Model from Time-Stamped Snapshots of Single Cells

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