arXiv subjects

Philipp Zahn

Publications and source records attributed to Philipp Zahn.

12 recordsLinked to original sources

arXiv2022

Robust Algorithmic Collusion

This paper develops a formal framework to assess policies of learning algorithms in economic games. We investigate whether reinforcement-learning agents with collusive pricing policies can successfully extrapolate collusive behavior from training to the market. We find that in testing environments collusion consistently breaks down. Instead, we observe static Nash play. We then show that restricting algorithms' strategy space can make algorithmic collusion robust, because it limits overfitting to rival strategies. Our findings suggest that policy-makers should focus on firm behavior aimed at coordinating algorithm design in order to make collusive policies robust.

econ.GN↗

arXiv2021

Composing games into complex institutions

Game theory is used by all behavioral sciences, but its development has long centered around tools for relatively simple games and toy systems, such as the economic interpretation of equilibrium outcomes. Our contribution, compositional game theory, permits another approach of equally general appeal: the high-level design of large games for expressing complex architectures and representing real-world institutions faithfully. Compositional game theory, grounded in the mathematics underlying programming languages, and introduced here as a general computational framework, increases the parsimony of game representations with abstraction and modularity, accelerates search and design, and helps theorists across disciplines express real-world institutional complexity in well-defined ways. Relative to existing approaches in game theory, compositional game theory is especially promising for solving game systems with long-range dependencies, for comparing large numbers of structurally related games, and for nesting games into the larger logical or strategic flows typical of real world policy or institutional systems.

cs.GT↗

arXiv2019

Bayesian open games

This paper generalises the treatment of compositional game theory as introduced by Ghani et al. in 2018, where games are modelled as morphisms of a symmetric monoidal category. From an economic modelling perspective, the notion of a game in the work by Ghani et al. is not expressive enough for many applications. This includes stochastic environments, stochastic choices by players, as well as incomplete information regarding the game being played. The current paper addresses these three issues all at once.

cs.GT↗

arXiv2018

Sequential games and nondeterministic selection functions

This paper analyses Escard\'o and Oliva's generalisation of selection functions over a strong monad from a game-theoretic perspective. We focus on the case of the nondeterminism (finite nonempty powerset) monad $\mathcal{P}$. We use these nondeterministic selection functions of type $\mathcal{J}^{\mathcal{P}}_R X = (X \rightarrow R) \rightarrow \mathcal{P} (X)$ to study sequential games, extending previous work linking (deterministic) selection functions to game theory. Similar to deterministic selection functions, which compute a subgame perfect Nash equilibrium play of a game, we characterise those non-deterministic selection functions which have a clear game-theoretic interpretation. We show, surprisingly, no non-deterministic selection function exists which computes the set of all subgame perfect Nash equilibrium plays. Instead we show that there are selection functions corresponding to sequential versions of the iterated removal of strictly dominated strategies.

cs.GT↗

arXiv2018

Sorting and filtering as effective rational choice procedures

Many online shops offer functionality that help their customers navigate the available alternatives. For instance, options to filter and to sort goods are wide-spread. In this paper we show that sorting and filtering can be used by rational consumers to find their most preferred choice -- quickly. We characterize the preferences which can be expressed through filtering and sorting and show that these preferences exhibit a simple and intuitive logical structure.

econ.TH↗

arXiv2018

On Rational Choice and the Representation of Decision Problems

In economic theory, an agent chooses from available alternatives -- modeled as a set. In decisions in the field or in the lab, however, agents do not have access to the set of alternatives at once. Instead, alternatives are represented by the outside world in a structured way. Online search results are lists of items, wine menus are often lists of lists (grouped by type or country), and online shopping often involves filtering items which can be viewed as navigating a tree. Representations constrain how an agent can choose. At the same time, an agent can also leverage representations when choosing, simplifying his/her choice process. For instance, in the case of a list he or she can use the order in which alternatives are represented to make his/her choice. In this paper, we model representations and decision procedures operating on them. We show that choice procedures are related to classical choice functions by a canonical mapping. Using this mapping, we can ask whether properties of choice functions can be lifted onto the choice procedures which induce them. We focus on the obvious benchmark: rational choice. We fully characterize choice procedures which can be rationalized by a strict preference relation for general representations including lists, list of lists, trees and others. Our framework can thereby be used as the basis for new tests of rational behavior. Classical choice theory operates on very limited information, typically budgets or menus and final choices. This is in stark contrast to the vast amount of data that specifically web companies collect about their users' choice process. Our framework offers a way to integrate such data into economic choice models.

cs.GT↗

arXiv2016

Compositionality and String Diagrams for Game Theory

We introduce string diagrams as a formal mathematical, graphical language to represent, compose, program and reason about games. The language is well established in quantum physics, quantum computing and quantum linguistic with the semantics given by category theory. We apply this language to the game theoretical setting and show examples how to use it for some economic games where we highlight the compositional nature of our higher-order game theory.

cs.GT↗

arXiv2016

Compositional game theory

We introduce open games as a compositional foundation of economic game theory. A compositional approach potentially allows methods of game theory and theoretical computer science to be applied to large-scale economic models for which standard economic tools are not practical. An open game represents a game played relative to an arbitrary environment and to this end we introduce the concept of coutility, which is the utility generated by an open game and returned to its environment. Open games are the morphisms of a symmetric monoidal category and can therefore be composed by categorical composition into sequential move games and by monoidal products into simultaneous move games. Open games can be represented by string diagrams which provide an intuitive but formal visualisation of the information flows. We show that a variety of games can be faithfully represented as open games in the sense of having the same Nash equilibria and off-equilibrium best responses.

cs.GT↗

arXiv2015

Contextuality and the Weak Axiom in the Theory of Choice

Recent work on the logical structure of non-locality has constructed scenarios where observations of multi-partite systems cannot be adequately described by compositions of non-signaling subsystems. In this paper we apply these frameworks to economics. First we construct a empirical model of choice, where choices are understood as observable outcomes in a certain sense. An analysis of contextuality within this framework allows us to characterize which scenarios allow for the possible construction of an adequate global choice rule. In essence, we mathematically characterize when it makes sense to consider the choices of a group as composed of individual choices. We then map out the logical space of some relevant empirical principles, relating properties of these contextual choice scenarios to no-signalling theories and to the weak axiom of revealed preference.

cs.GT↗

arXiv2015

Higher-Order Game Theory

In applied game theory the motivation of players is a key element. It is encoded in the payoffs of the game form and often based on utility functions. But there are cases were formal descriptions in the form of a utility function do not exist. In this paper we introduce a representation of games where players' goals are modeled based on so-called higher-order functions. Our representation provides a general and powerful way to mathematically summarize players' intentions. In our framework utility functions as well as preference relations are special cases to describe players' goals. We show that in higher-order functions formal descriptions of players may still exist where utility functions do not using a classical example, a variant of Keynes' beauty contest. We also show that equilibrium conditions based on Nash can be easily adapted to our framework. Lastly, this framework serves as a stepping stone to powerful tools from computer science that can be usefully applied to economic game theory in the future such as computational and computability aspects.

cs.GT↗

arXiv2015

Higher-Order Decision Theory

Classical decision theory models behaviour in terms of utility maximisation where utilities represent rational preference relations over outcomes. However, empirical evidence and theoretical considerations suggest that we need to go beyond this framework. We propose to represent goals by higher-order functions or operators that take other functions as arguments where the max and argmax operators are special cases. Our higher-order functions take a context function as their argument where a context represents a process from actions to outcomes. By that we can define goals being dependent on the actions and the process in addition to outcomes only. This formulation generalises outcome based preferences to context-dependent goals. We show how to uniformly represent within our higher-order framework classical utility maximisation but also various other extensions that have been debated in economics.

cs.GT↗

arXiv2014

A Higher-order Framework for Decision Problems and Games

We introduce a new unified framework for modelling both decision problems and finite games based on quantifiers and selection functions. We show that the canonical utility maximisation is one special case of a quantifier and that our more abstract framework provides several additional degrees of freedom in modelling. In particular, incomplete preferences, non-maximising heuristics, and context-dependent motives can be taken into account when describing an agent's goal. We introduce a suitable generalisation of Nash equilibrium for games in terms of quantifiers and selection functions. Moreover, we introduce a refinement of Nash that captures context-dependency of goals. Modelling in our framework is compositional as the parts of the game are modular and can be easily exchanged. We provide an extended example where we illustrate concepts and highlight the benefits of our alternative modelling approach.

cs.LO↗