Infinite Games Research Articles

Teaching Design as an Infinite Game

This article posits that resiliency and adaptation in landscape architecture studio projects must plan for changes and feedback loops. With this premise in mind, the article evaluates a landscape architecture design studio focused on designing and planning adaptive landscapes that are part of the Los Angeles River and its surrounding neighborhoods. Students were charged with planning for change over time and designing multiple scenarios that assume various forms and vagaries in management, care, environmental conditions, and policies, while also connecting to community needs. Their designs were guides that envisioned a range of possibilities—feedback loops creating environmental and social resiliencies that provide value over time. The studio and this article build on Joan Woodward’s (2008) work suggesting several shifts in landscape design practice for progressing toward resilient landscapes that accommodate surprise and disruption. The case study methods evaluating the studio approach included external and internal reviews of students’ work, student written reflections regarding the course, and instructors’ reflections on the work. Some reviewers felt the paucity of final state perspective renderings in some students’ work equated to diminished design rigor. Some students pushed back on the studio’s interdisciplinary scope, but others looked beyond fixed design solutions, giving entire systems deep consideration by considering how to provide economic resiliency toolkits and adopt best practices that could unfold and adapt over time based on a particular neighborhood’s needs and desires. Overall, the studio serves as a model for teaching that advances Woodward’s concepts and promotes her goal of seeing design as an infinite rather than finite game (Carse, 1986; Woodward, 2008).

$$\varepsilon $$-Nash Equilibria of a Multi-player Nonzero-Sum Dynkin Game in Discrete Time

We study the infinite horizon discrete time N-player nonzero-sum Dynkin game ( $$N\ge 2$$ ) with stopping times as strategies (or pure strategies). We prove existence of an $$\varepsilon $$ -Nash equilibrium point for the game by presenting a constructive algorithm. One of the main features is that the payoffs of the players depend on the set of players that stop at the termination stage which is the minimal stage in which at least one player stops. The existence result is extended to the case of a nonzero-sum game with finite horizon. Finally, the algorithm is illustrated by two explicit examples in the specific case of finite horizon.

Propos introductifs

The disappearance of bodies in today’s society is redistributing the cards of living together. If social distancing allows a certain rationalisation of human activities (work, care), does it not paradoxically lead to physical and psychological isolation? Moreover, does the dissociation it leads to between the subject and the “e-mage” given of him or her not tend to transform social relations into an infinite game where half-truths, lies and illusions produce new rituals and artifices depending mainly on technology.

Good-for-games $\omega$-Pushdown Automata

We introduce good-for-games $\omega$-pushdown automata ($\omega$-GFG-PDA). These are automata whose nondeterminism can be resolved based on the input processed so far. Good-for-gameness enables automata to be composed with games, trees, and other automata, applications which otherwise require deterministic automata. Our main results are that $\omega$-GFG-PDA are more expressive than deterministic $\omega$- pushdown automata and that solving infinite games with winning conditions specified by $\omega$-GFG-PDA is EXPTIME-complete. Thus, we have identified a new class of $\omega$-contextfree winning conditions for which solving games is decidable. It follows that the universality problem for $\omega$-GFG-PDA is in EXPTIME as well. Moreover, we study closure properties of the class of languages recognized by $\omega$-GFG- PDA and decidability of good-for-gameness of $\omega$-pushdown automata and languages. Finally, we compare $\omega$-GFG-PDA to $\omega$-visibly PDA, study the resources necessary to resolve the nondeterminism in $\omega$-GFG-PDA, and prove that the parity index hierarchy for $\omega$-GFG-PDA is infinite. This is a corrected version of the paper arXiv:2001.04392v6 published originally on January 7, 2022.

A Random-Supply Mean Field Game Price Model

We consider a market where a finite number of players trade an asset whose supply is a stochastic process. The price formation problem consists of finding a price process that ensures that when agents act optimally to minimize their trading costs, the market clears, and supply meets demand. This problem arises in market economies, including electricity generation from renewable sources in smart grids. Our model includes noise on the supply side, which is counterbalanced on the consumption side by storing energy or reducing the demand according to a dynamic price process. By solving a constrained minimization problem, we prove that the Lagrange multiplier corresponding to the market-clearing condition defines the solution of the price formation problem. For the linear-quadratic structure, we characterize the price process of a continuum population using optimal control techniques. We include numerical schemes for the price computation in the finite and infinite games, and we illustrate the model using real data.

Characterizing Positionality in Games of Infinite Duration over Infinite Graphs

We study turn-based quantitative games of infinite duration opposing two antagonistic players and played over graphs. This model is widely accepted as providing the adequate framework for formalizing the synthesis question for reactive systems. This important application motivates the question of strategy complexity: which valuations (or payoff functions) admit optimal positional strategies (without memory)? Valuations for which both players have optimal positional strategies have been characterized by Gimbert and Zielonka for finite graphs and by Colcombet and Niwi\'nski for infinite graphs. However, for reactive synthesis, existence of optimal positional strategies for the opponent (which models an antagonistic environment) is irrelevant. Despite this fact, not much is known about valuations for which the protagonist admits optimal positional strategies, regardless of the opponent. In this work, we characterize valuations which admit such strategies over infinite game graphs. Our characterization uses the vocabulary of universal graphs, which has also proved useful in understanding recent breakthrough results regarding the complexity of parity games. More precisely, we show that a valuation admitting universal graphs which are monotone and well-ordered is positional over all game graphs, and -- more surprisingly -- that the converse is also true for valuations admitting neutral colors. We prove the applicability and elegance of the framework by unifying a number of known positionality results, proving new ones, and establishing closure under lexicographical products. Finally, we discuss a class of prefix-independent positional objectives which is closed under countable unions.

Characterizing Omega-Regularity through Finite-Memory Determinacy of Games on Infinite Graphs

We consider zero-sum games on infinite graphs, with objectives specified as sets of infinite words over some alphabet of colors. A well-studied class of objectives is the one of $\omega$-regular objectives, due to its relation to many natural problems in theoretical computer science. We focus on the strategy complexity question: given an objective, how much memory does each player require to play as well as possible? A classical result is that finite-memory strategies suffice for both players when the objective is $\omega$-regular. We show a reciprocal of that statement: when both players can play optimally with a chromatic finite-memory structure (i.e., whose updates can only observe colors) in all infinite game graphs, then the objective must be $\omega$-regular. This provides a game-theoretic characterization of $\omega$-regular objectives, and this characterization can help in obtaining memory bounds. Moreover, a by-product of our characterization is a new one-to-two-player lift: to show that chromatic finite-memory structures suffice to play optimally in two-player games on infinite graphs, it suffices to show it in the simpler case of one-player games on infinite graphs. We illustrate our results with the family of discounted-sum objectives, for which $\omega$-regularity depends on the value of some parameters.

Study on the Calculation Method of Green GDP to Measure the Economic Health

This paper discusses the replacement of conventional GDP with green GDP (GGDP) as the main measure of economic health. Based on data on annual average temperature, annual precipitation, annual sunshine hours and annual average relative humidity for country C over the past decade, a multiple regression model is used to demonstrate that the linear effect of GGDP is positive for green growth. Further considering the differences in economic power between developed, developing and lagging countries, an infinite iterative game model of green GDP and conventional GDP is constructed based on the mutual constraints of long-term and short-term interests. Finally, considering the resistance to replacing GDP with green GDP, and in order to fully reflect the tortuous and complex relationship between climate change and social costs, an integrated model that will be embedded in the economic system, the DICE-2013R model, is constructed in order to achieve a shift from a single energy model to an integrated climate model. Normalising the data based on the model calcglobal scale.

Graphon Games: A Statistical Framework for Network Games and Interventions

In this paper, we present a unifying framework for analyzing equilibria and designing interventions for large network games sampled from a stochastic network formation process represented by a graphon. To this end, we introduce a new class of infinite population games, termed graphon games, in which a continuum of heterogeneous agents interact according to a graphon, and we show that equilibria of graphon games can be used to approximate equilibria of large network games sampled from the graphon. This suggests a new approach for design of interventions and parameter inference based on the limiting infinite population graphon game. We show that, under some regularity assumptions, such approach enables the design of asymptotically optimal interventions via the solution of an optimization problem with much lower dimension than the one based on the entire network structure. We illustrate our framework on a synthetic data set and show that the graphon intervention can be computed efficiently and based solely on aggregated relational data.

Price-Rising Competition: a Higher Market Price When a Monopoly Faces a Small Entrant

The UK retail electricity market revealed (i) the co-occurrence between a declining market concentration and an increasing price and (ii) price differentiation between incumbents and small suppliers. We construct an infinite sequential game in which a monopoly faces a small entrant and find an equilibrium where both players are not motivated to deviate. However, this equilibrium imposes a condition on the values of model parameters. If the condition is unmet, the interactions between the two players will be unending. Nonetheless, we suggest that two states where at least one player is not motivated to deviate could be a place to settle, but the choice depends on which player is more concerned about market stability. Besides, the two observations are found in equilibrium and two states. Finally, we argue that the finding is jointly contributed by four features: small entrant with a lower marginal cost and constrained capacity, switching costs, and barriers to entry. If any feature is relaxed, the entry of the small firm will not lead to a rising market price.

Solvability of Infinite Horizon McKean–Vlasov FBSDEs in Mean Field Control Problems and Games

In this paper, we show existence and uniqueness of solutions of the infinite horizon McKean–Vlasov FBSDEs using two different methods, which lead to two different sets of assumptions. We use these results to solve the infinite horizon mean field type control problems and mean field games.

Infinite horizon Stackelberg differential games with random coefficients under control input constraint

In this paper, we study Stackelberg stochastic differential game driven by stochastic differential equation (SDE for short) with random coefficients on infinite horizon under convex control constraint. The control enters into the drift and diffusion terms simultaneously. By employing the first-order adjoint equation (backward stochastic differential equation, BSDE for short), we are able to announce Pontryagin's maximum principle for the leader's global Stackelberg solution, within adapted open-loop structure, where the term global means that the leader's domination over the entire game duration. Due to the follower's adjoint equation develops into a BSDE, the leader will be confronted with a control problem where the state equation is a kind of infinite horizon fully coupled forward-backward stochastic differential equation (FBSDE for short). We shed light on an important application in a class of infinite horizon linear-quadratic (LQ for short) Stackelberg games in which the control process of leader's is constrained in a closed convex subset Γ 1 of full space R m 1 . Whenever the control domain is postulated to be full space, we derive a new kind of infinite horizon backward stochastic Riccati equation (BSRE for short) with asymmetric coefficients.

The Theory of Universal Graphs for Infinite Duration Games

We introduce the notion of universal graphs as a tool for constructing algorithms solving games of infinite duration such as parity games and mean payoff games. In the first part we develop the theory of universal graphs, with two goals: showing an equivalence and normalisation result between different recently introduced related models, and constructing generic value iteration algorithms for any positionally determined objective. In the second part we give four applications: to parity games, to mean payoff games, to a disjunction between a parity and a mean payoff objective, and to disjunctions of several mean payoff objectives. For each of these four cases we construct algorithms achieving or improving over the best known time and space complexity.

Solving Infinite Games in the Baire Space

Infinite games (in the form of Gale-Stewart games) are studied where a play is a sequence of natural numbers chosen by two players in alternation, the winning condition being a subset of the Baire space ωω. We consider such games defined by a natural kind of parity automata over the alphabet ℕ, called ℕ-MSO-automata, where transitions are specified by monadic second-order formulas over the successor structure of the natural numbers. We show that the classical Büchi-Landweber Theorem (for finite-state games in the Cantor space 2ω) holds again for the present games: A game defined by a deterministic parity ℕ-MSO-automaton is determined, the winner can be computed, and an ℕ-MSO-transducer realizing a winning strategy for the winner can be constructed.

What Matters in Macao

In contrast to the dominant ideas of how 'game and play' work, which I label 'transcendentalist' and 'sedentary', my study on Macao proposes an alternative, 'materialist' and 'nomadic', perspective. This comes down to thinking 'game and play' not as an 'artificial' activity that takes place in a safe, enclosed environment, but as an elementary part of life, crucial to how imagination works, and to how imagination is entangled in the materiality of the urban sphere. After mapping an alternative history of how to think 'game and play' differently, working with anthropologist Karl Goos, architect Aldo van Eyk, artist Constant, and in the end philosopher Gilles Deleuze, I engage with the city of Macao, its architecture, its politics, and its gambling practices. I use fiction authors Leslie T Chang and Louis Borges to show, finally, how Macao, in contemporary China, equals the infinite game of chance, materialized; the much needed other in its contemporary urban landscape.

Equilibrium in a Symmetric Game of Resource Extraction with Coalitional Structure

The game of resource extraction / capital accumulation is a stochastic nonzero-sum infinite horizon game, obtained as an extension of the well-known optimal growth model to m strategically competing players, who jointly posses a renewable resource. The existence of a Nash equilibrium in different, often symmetric, frameworks of the game received a significant attention in the scientific literature on the topic. The focus of this paper is to introduce the coalitional component to the symmetric problem. Specifically, we examine whether the game with a fixed coalitional structure admits stability against profitable coalitional deviations.It is assumed that the set of all players is partitioned into coalitions which do not intersect and remain consistent throughout the game. The members of each coalition are able to coordinate their actions and perform joint deviations in a cooperative manner. Such setting incorporates a natural concept of established social ties, which may reflect a potential context appearing in practical applications. The corresponding notion of equilibrium in the paper is expressed as a position, from which none of the set coalitions can deviate in a manner to increase a total reward of its members. Its existence is studied in the context of a certain symmetric resource extraction game model with unbounded utilities of the players. This model was studied in [12; 13], concluding a Stationary Markov Perfect Equilibrium existence in both symmetric and non-symmetric game structure. The first feature of the model is that the preferences of the players are considered to be isoelastic in the form of strictly concave power functions. Furthermore, the law of motion between states is set to follow a geometric random walk in relation to players' joint investments. We prove that the game within the formulated settings admits stability against profitable coalitional deviations for any partition on the set of agents. The method provides an algorithm for building the corresponding stationary strategies, which can be useful for practical purposes. Finally, we use two examples with different numerical configurations to illustrate possible patterns of how the individual rewards of the players vary depending on a coalitional structure, which is set at the beginning of the game.

Infinite Game of Poverty Alleviation (IGPA) and Accumulation of Environmental and Socio-Cultural Capital

Although poverty alleviation has become an infinite game globally since the late 1940s, Sri Lanka has not identified poverty as an issue until recently. The socio-welfare-based economic policies and continuous election-centered poverty alleviation programs push the country into the IGPA, even though the country is accumulated with plentiful natural and socio-cultural resources. Consequently, Sri Lanka struggles with severe economic crises and political chaos, worsening IGPA, which has not been adequately addressed in prevailing academic literature. This study attempts to fill this void by adopting the social-constructivism approach. Nine semi-structured interviews and three focus group discussions were conducted, plus direct observation of the rural landscape of Sri Lanka. Findings indicate IGPA has become an influential determining factor in deciding the potential ruling government, and every government has included IGPA in their political manifesto under different titles. However, the disintegration between rural poverty alleviation and sustainable capitalization led the nation to be retained with IGPA.

Observation and Distinction: Representing Information in Infinite Games

We compare two approaches for modelling imperfect information in infinite games by using finite-state automata. The first, more standard approach views information as the result of an observation process driven by a sequential Mealy machine. In contrast, the second approach features indistinguishability relations described by synchronous two-tape automata. The indistinguishability-relation model turns out to be strictly more expressive than the one based on observations. We present a characterisation of the indistinguishability relations that admit a representation as a finite-state observation function. We show that the characterisation is decidable, and give a procedure to construct a corresponding Mealy machine whenever one exists.

A note on payments in the lab for infinite horizon dynamic games with discounting

A note on payments in the lab for infinite horizon dynamic games with discounting

An experimental test of the Coase conjecture: Fairness in dynamic bargaining

AbstractWe conduct a novel experimental test of the Coase conjecture based on subjects' privately known preferences for fairness. In an infinite horizon bargaining game, a proposer proposes a division of chips, until a responder accepts. When players are patient, the Coase conjecture predicts almost immediate agreement on equal monetary payoffs given any possibility a responder will not accept anything less. Behavior closely matches theory. In particular, when chips are worth more to proposers than responders, initial offers, minimum acceptable offers, responder payoffs, and efficiency are significantly larger in infinite horizon games than ultimatum games, and proposer payoffs are significantly smaller.