Blackwell’s theorem – relating majorization to the existence of a signal inducing a desired distribution of posterior means – has numerous applications in economics. We give a new, simple proof of this theorem via an explicit construction.  Our approach provides a concrete way to generate the signal: we show that if distribution can be induced by some signal, it can also be induced via a “downward uniform signal,” which simply provides a stochastic lower bound on the realized state. The results are applied to optimism in learning and product adoption in dynamic settings.

We characterize Nash equilibrium by postulating coherent behavior across varying games. Nash equilibrium is the only solution concept that satisfies the following axioms: (i) strictly dominant actions are played with positive probability, (ii) if a strategy profile is played in two games, it is also played in every convex combination of these games, and (iii) players can shift probability arbitrarily between two indistinguishable actions, and deleting one of these actions has no effect. Our theorem implies that every equilibrium refinement violates at least one of these axioms. Moreover, every solution concept that approximately satisfies these axioms returns approximate Nash equilibria, even in natural subclasses of games, such as two-player zero-sum games, potential games, and graphical games.

Whether equilibration to Nash equilibrium emerges, and how quickly, has been a productive research question for several decades. This talk will review some of that evidence, putting special focus on level-k models as a non-equilibrium alternative.

This paper uses a game-theoretic model to show that when a group of people has a greater need to coordinate their actions, the group uses coarser concepts, concepts that are more blunt and less specific. In other words, this paper understands concepts as resulting endogenously from the practical need of a group of people to coordinate, not just from individual cognitive processes. In the model, people can communicate using attributes that are correlated or not. Using several correlated attributes can maximize social welfare because it improves coordination. The argument is that a single attribute might not be seen or perceived. Hence when a person sees a single attribute, the person is not certain that another person sees it. But when a person sees several correlated attributes, the person knows that it is likely that other people see at least some of the attributes, and can thus infer the other attributes since they are all highly correlated. When people communicate using multiple correlated attributes, concepts are more coarse, and when people communicate using independent attributes, concepts are finer and more specific. Communicating using correlated attributes instead of independent attributes makes concepts coarser, but better generates the common knowledge necessary for coordinating actions.

AI agents are not like humans or groups of humans.  Their memories can be wiped, multiple copies of them can be spun up, they can be run in simulation, and their source code can be analyzed.  In principle, the framework of game theory is flexible enough to accommodate all these aspects; but historically, the game theory community has not focused on them, and some of the relevant work has taken place in philosophy.  I will discuss our work on these topics, with a focus on enabling AI agents to be cooperative in ways that humans cannot.

From playing complex games like Go at super-human level to training AI agents that interact with each other and with humans, many outstanding challenges in AI lie at its interface with Game Theory. Despite some impressive recent hits in this space, progress has remained slower. I will discuss this challenge and how it might be overcome using a combined game-theoretic, optimization, and learning-theoretic perspective.  I will conclude with an empirical investigation into how humans and Large Language Models can collaborate to develop game-theoretic models for studying literary stories, specifically the dilemmas facing characters, counterfactuals to how the story unfolded, and what makes a story interesting. I will illustrate the approach by looking at Romeo and Juliet.

Aumann, Maschler and Stearns (1968) introduced the notion of joint plan equilibrium (JPE) as a solution concept for two-person undiscounted infinitely repeated games with a single informed player. Such games may not have any nonrevealing Nash equilibrium. A JPE is a Nash equilibrium in which the informed player first sends a message to the uninformed one (message phase) and then, the players coordinate as in the standard “folk theorem” (payoff phase). Sorin (1983) and Simon, Spież, and Toruńczyk (1995) established that every game in the class above has a JPE. We investigate possible counterparts of JPE in static Bayesian games with contracts. JPE is a relevant solution concept in these games if

• no mediator is available, so that private information must be revealed through costless messages (cheap talk)

• contracts cannot be contingent on unverifiable information, so that contractible actions are based on messages only

• further private information can be used in case of disagreement.

Forecasts are “calibrated” if they are close to the long-run frequencies.  Calibrated and “continuously calibrated” forecasts can be obtained by “forecast-hedging” methods.  Best-replying to calibrated and continuously calibrated forecasts yield game dynamics that lead to correlated and Nash equilibria.

Research shows that generative AI improves productivity in specific, often experimenter-defined, tasks.  But understanding how these productivity enhancements will shift broader economic environments requires theoretical modeling and equilibrium analysis.  We study the impact of generative AI on content ecosystems, arguing that in equilibrium AI innovations may lead to disintermediation as consumers bypass intermediaries and use AI directly to create content. We then investigate the consequences of this disintermediation. While the intermediary is welfare-improving due to its ability to leverage economies of scale, the intermediary extracts all the gains to social welfare and its presence can raise or lower the equilibrium quality of consumed content. 

Equilibrium Resilience (ER) is a measure of stability that naturally explains n-player strategic phenomena not fully explained by Nash equilibrium (NE), Dominant Strategy equilibrium (DSE) and their refinements. For example, high ER explains the formation and persistence of competitive behavior in economic interactions and predicts confessions in new types of prisoners' dilemma games, even without dominant strategies. Conversely, low ER accounts for unstable unpredictable play in centrally controlled games, supply chains, other highly demanding participation games and in interactions that rely on mixed strategies. We focus on ER in normal-form games, where strategies and defections are well understood. In more complex interactions, such as extensive repeated or Bayesian games, ER inherently depends on their specified normal-form representation.

*Unlike theoretical illustrations, contemporary applications describe choices of real players in current games.

Norms – people’s common understanding of how to behave and interact with one another – shape economic and social life. Norms and identities are also intertwined, giving the acceptable behavior for different people in different contexts. This talk considers economists’ theories of norms, and how norms can affect outcomes in wide range of settings including labor markets, education, and firms. The concept of Nash equilibrium has been integral to this endeavor, and the talk highlights its contribution and insights. 

This talk investigates the relation between some strategic features of mixed Nash equilibria and their fixed point index in finite games. We present new results that deepen our understanding of how equilibrium structure relates to index theory:

1. A mixed Nash equilibrium x is isolated with index +1 if and only if it can be made the unique equilibrium of a larger game, constructed by adding strategies that are strictly inferior responses to x. This settles an open question posed explicitly by Hofbauer (2003) and implicitly by Myerson (1996).

2. A Nash component admits an equilibrium of index +1 in its neighbourhood under every perturbation of any strategically equivalent game if and only if the component itself has a positive index.

3. For any finite game, any selection of equilibria from each Nash component, and any assignment of indices ±1 to these equilibria such that their sum equals the index of the component, there exists a perturbation of a strategically equivalent game whose equilibria approximate the selected ones and preserve the assigned indices.

These results bridge equilibrium refinement, index theory, and robustness to strategic perturbations, offering new insights into the structure and stability of Nash equilibria.

We show that in a setting where voters’ preferences over candidates are weakly single-peaked (a relaxation of the requirement that voters rank candidates according to ideological position), a majority-rule-winning candidate (i.e., a Condorcet winner) exists, and there is a strategy-resistant voting system electing the Condorcet winner. Moreover, this system is (essentially) the unique strategy-resistant voting system among all voting methods satisfying anonymity (equal treatment of voters) and neutrality (equal treatment of candidates) for such preferences. These results are motivated by our analysis of voters’ rankings in Maine and Alaska (where ranked-choice voting is used) showing that weak single-peakedness has invariably held.

The first papers on game-theoretic learning—both on fictitious play—appeared in 1951, just one year after Nash's PhD thesis. Since then, 74 years and one booming body of literature later, only partial answers have been given to the driving questions in the field: When does learning lead to Nash equilibrium? When does it not? What other types of behavior emerge in the long run? 

In this talk, I will discuss some recent results concerning the long-run behavior of online learners that are involved in an unknown repeated game (that is, when any given player is not necessarily aware of the other players' actions or objectives). The talk will focus on a widely studied family of online learning methods known as "regularized learning", and we will pay special attention to the information available to the players—from full information, to payoff-based feedback. In this general context, I will describe a series of results characterizing the possible outcomes of the process, from convergence to a Nash equilibrium, to cyclic, (almost) periodic behaviors, and when the latter may be mitigated. 

Agents in populations often exhibit striking regularities of behaviour when encountering familiar situations. Such regularities are sometimes conventional in that incentives for compliance arise from an accurate expectation that others will comply. The formal study of conventions as emergent Nash equilibria in dynamic systems was famously initiated by Lewis (1969) and rigorously mathematically formalized for economics and the social sciences by Young (1993). This research program has since revealed deep connections between dynamics, equilibrium selection, and social stability. I survey key results, including recent advances and contributions of my own, and outline promising directions for future work on the formation, persistence, and control of conventions.

There has been a long history of failed efforts in economics and game theory to define natural game dynamics that converge to the game's Nash equilibria. Recent work has shown that this quest may be futile, and an alternative approach has been pursued, namely to embrace the asymptotic behavior of natural game dynamics as the meaning of the game.  I will discuss significant progress towards computing the attractors of the replicator dynamics of a game, as well as some intriguing remaining mathematical problems on this path.

We explore the role of memory for choice behavior in unfamiliar environments. Using a unique data set, we document that decision makers exhibit a “memory premium.” They tend to choose in-memory alternatives over out-of-memory ones, even when the latter are objectively better. Consistent with well-established regularities regarding the inner workings of human memory, the memory premium is associative, subject to interference and repetition effects, and decays over time. Even as decision makers gain familiarity with the environment, the memory premium remains economically large. Our results imply that the ease with which past experiences come to mind plays an important role in shaping choice behavior.

The framework of noncooperative game theory has yielded many elegant results for the expression and analysis of strategic interactions. However, capturing realistic interactions with games of sufficient size and richness cannot be done by hand, and requires the support of software tools. Gambit – a software package for computational game theory – allows users to specify game models, e.g. in strategic or extensive form, and then analyse them, e.g., by computing Nash, quantal response, or other types of equilibria. Building on Gambit, we are developing an automated analysis pipeline that starts with descriptions of strategic interactions in natural language. First, I will describe initial progress on using LLMs to generate extensive-form game representations from natural language via Gambit's pygambit API. Then I will describe Gambit's current suite of equilibrium computation methods, current efforts to benchmark game creation and the use of these methods to solve large games, and our coming efforts to support approaches that combine traditional equilibrium computation with AI methods such as Deep Reinforcement Learning.

Learning in games studies how agent strategies evolve over time in response to observed outcomes. A central question is whether such learning processes converge to classical solution concepts like Nash equilibrium (NE). This talk explores the interplay between “uncoupled learning”, where agents adjust strategies based only on their own observed payoffs, and “higher-order learning”, where agents utilize additional internal states, possibly representing trends, momentum, or forecasts, to inform their updates.

We revisit prior foundational impossibility results showing that standard-order uncoupled learning generally cannot converge to mixed-strategy NE. By contrast, we demonstrate that higher-order uncoupled dynamics can, in general, achieve convergence to mixed-strategy equilibria, albeit under dynamics that may appear non-standard or even irrational from a traditional perspective.

To analyze these dynamics, we adopt a feedback systems viewpoint. Specifically, we model learning rules as dynamical systems that take payoff signals as inputs and output strategy updates, with the game itself providing the feedback loop. This abstraction allows tools from feedback control to be applied to the stability, convergence, and robustness in learning dynamics.

We compare the two basic concepts of (minmax) value and equilibrium points in terms of:

• definitions, properties and interpretation,

• tools used in the proofs,

• main developments and recent advances.

We will in particular question the approach asserting:

• Equilibrium as extension of the value to non-zero sum games,

• Value as a special case of equilibrium for zero-sum games.

Over the last two decades we have developed good understanding how to quantify the impact of strategic user behavior on outcomes in many games (including traffic routing and online auctions) and showed that the resulting bounds extend to repeated games assuming players use a form of learning (no-regret learning) to adapt to the environment.

In this talk, we will review this line of work, and will focus on repeated interactions that have carry-over effects between rounds: when outcomes in one round effect the game in the future, as in repeated auctions with budgets, as well as queuing systems. We will consider this phenomenon in the context of a game modeling queuing systems: routers compete for servers, where packets that do not get served need to be resent, resulting in a system where the number of packets at each round depends on the success of the routers in the previous rounds. We study the resulting highly dependent random process and show bounds on the excess server capacity needed to guarantee that all packets get served in two different the queuing systems (with or without buffers) despite the selfish (myopic) behavior of the participants.   

If a player expects others to adhere to a certain Nash equilibrium, what is then the player’s incentive to use his or her equilibrium strategy rather than some other best reply if such exists? The question is particularly pertinent for equilibria in mixed strategies. We here analyze this question for finite two-player games and formulate an incentive criterion, “detection stability”, for players to stick to their equilibrium strategies. Strict equilibria, as well as equilibria in strictly competitive games are detection stable, and so are some equilibria in other games, but there are games with no detection stable equilibrium (DSE). We characterize DSE as subgame perfect equilibria of a "spying game" and show that the criterion is logically independent of other refinements.

We study whether a given Nash equilibrium can be improved within the set of correlated equilibria for arbitrary objectives. Our main contribution is a sharp characterization: in a generic game, a Nash equilibrium is an extreme point of the set of correlated equilibria if and only if at most two agents randomize. Consequently, any sufficiently mixed Nash equilibrium involving at least three randomizing agents can always be improved by correlating actions or switching to a less random equilibrium, regardless of the underlying objective. We show that even if one focuses on objectives that depend on payoffs, excess randomness in equilibrium implies improvability. We extend our analysis to symmetric games and coarse correlated equilibria, revealing a fundamental tension between the randomness in Nash equilibria and their optimality.