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.
We prove an n-EXPTIME lower bound for the problem of deciding the winner in a reachability game on Higher Order Pushdown Automata (HPDA) of level n. This bound matches the known upper bound for parity games on HPDA. As a consequence the mu-calculus model checking over graphs given by n-HPDA is n-EXPTIME complete.
The recent theory of sequential games and selection functions by Mar- tin Escardo and Paulo Oliva is extended to games in which players move simultaneously. The Nash existence theorem for mixed-strategy equilibria of finite games is generalised to games defined by selection functions. A normal form construction is given which generalises the game-theoretic normal form, and its soundness is proven. Minimax strategies also gener- alise to the new class of games and are computed by the Berardi-Bezem- Coquand functional, studied in proof theory as an interpretation of the axiom of countable choice…
The Robinson-Goforth topology of swaps in adjoining payoffs elegantly arranges 2x2 ordinal games in accordance with important properties including symmetry, number of dominant strategies and Nash Equilibria, and alignment of interests. Adding payoff families based on Nash Equilibria illustrates an additional aspect of this order and aids visualization of the topology. Making ties through half-swaps not only creates simpler games within the topology, but, in reverse, breaking ties shows the evolution of preferences, yielding a natural ordering for the topology of 2x2 games with ties. An ordinal…
We introduce the concept of budget games. Players choose a set of tasks and each task has a certain demand on every resource in the game. Each resource has a budget. If the budget is not enough to satisfy the sum of all demands, it has to be shared between the tasks. We study strategic budget games, where the budget is shared proportionally. We also consider a variant in which the order of the strategic decisions influences the distribution of the budgets. The complexity of the optimal solution as well as existence, complexity and quality of equilibria are analyzed. Finally, we show that the t…
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…
Continuous-time game dynamics are typically first order systems where payoffs determine the growth rate of the players' strategy shares. In this paper, we investigate what happens beyond first order by viewing payoffs as higher order forces of change, specifying e.g. the acceleration of the players' evolution instead of its velocity (a viewpoint which emerges naturally when it comes to aggregating empirical data of past instances of play). To that end, we derive a wide class of higher order game dynamics, generalizing first order imitative dynamics, and, in particular, the replicator dynamics.…
We study competition in a general framework introduced by Immorlica et al. and answer their main open question. Immorlica et al. considered classic optimization problems in terms of competition and introduced a general class of games called dueling games. They model this competition as a zero-sum game, where two players are competing for a user's satisfaction. In their main and most natural game, the ranking duel, a user requests a webpage by submitting a query and players output an ordering over all possible webpages based on the submitted query. The user tends to choose the ordering which di…
Games involving quantum strategies often yield higher payoff. Here, we study a practical realization of the three-player dilemma game using the superconductivity-based quantum processors provided by IBM Q Experience. We analyze the persistence of the quantum advantage under corruption of the input states and how this depends on parameters of the payoff table. Specifically, experimental fidelity and error are observed not to be properly anti correlated, i.e., there are instances where a class of experiments with higher fidelity yields a greater error in the payoff. Further, we find that the cla…
Beyond Outcome-Based Imperfect-Recall: Higher-Resolution Abstractions for Imperfect-Information Games
School of Artificial Intelligence（人工智能学院）；University of Chinese Academy of Sciences（中国科学院大学）；Institute of Automation（自动化研究所）；Chinese Academy of Sciences（中国科学院）
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.
Emergence of cooperation in nonlinear higher-order public goods games
Institute for Cross-disciplinary Physics and Complex Systems IFISC (CSIC-UIB)（交叉物理与复杂系统研究所）；Department of Network and Data Science, Central European University Vienna（中欧大学维也纳网络与数据科学系）；Human Evolutionary Ecology Group, Department of Evolutionary Anthropology, University of Zurich（苏黎世大学生物人类学系人类进化生态组）；Center for Networks and Complexity, Luiss University of Rome（罗马LUISS大学网络与复杂性中心）；Department of AI, Data and Decision Sciences, Luiss University of Rome（罗马LUISS大学人工智能、数据与决策科学系）
In a guessing game, players guess the value of a random real number selected using some probability density function. The winner may be determined in various ways; for example, a winner can be a player whose guess is closest in magnitude to the target or a winner can be a player coming closest without guessing higher than the target. We study optimal strategies for players in these games and determine some of them for two, three, and four players.
Max Planck Institute of Molecular Cell Biology and Genetics（马克斯·普朗克分子细胞生物学与遗传学研究所）；Center for Systems Biology Dresden（德累斯顿系统生物学中心）；Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克科学数学研究所）
This work explores three-player game training dynamics, under what conditions three-player games converge and the equilibria the converge on. In contrast to prior work, we examine a three-player game architecture in which all players explicitly interact with each other. Prior work analyzes games in which two of three agents interact with only one other player, constituting dual two-player games. We explore three-player game training dynamics using an extended version of a simplified bilinear smooth game, called a simplified trilinear smooth game. We find that trilinear games do not converge on…
We introduce a class of extensive form games where players might not be able to foresee the possible consequences of their decisions and form a model of their opponents which they exploit to achieve a more profitable outcome. We improve upon existing models of games with limited foresight, endowing players with the ability of higher-order reasoning and proposing a novel solution concept to address intuitions coming from real game play. We analyse the resulting equilibria, devising an effective procedure to compute them.
This paper investigates first-order game logic and first-order modal mu-calculus, which extend their propositional modal logic counterparts with first-order modalities of interpreted effects such as variable assignments. Unlike in the propositional case, both logics are shown to have the same expressive power and their proof calculi to have the same deductive power. Both calculi are also mutually relatively complete. In the presence of differential equations, corollaries obtain usable and complete translations between differential game logic, a logic for the deductive verification of hybrid ga…
We study infinite two-player games where one of the players is unsure about the set of moves available to the other player. In particular, the set of moves of the other player is a strict superset of what she assumes it to be. We explore what happens to sets in various levels of the Borel hierarchy under such a situation. We show that the sets at every alternate level of the hierarchy jump to the next higher level.
This paper investigates mixed strategies in dynamic games with perfect information. We present an example to show that a player may obtain higher payoff by playing mixed strategy. By contrast, the main result of the paper shows that every two-player dynamic zero-sum game with nature has the no-mixing property, which implies that mixed strategy is useless in this most classical class of games. As for applications, we show the existence of pure-strategy subgame-perfect equilibria in two-player zero-sum games with nature. Based on the main result, we also prove the existence of a universal subgam…
In a two-player zero-sum graph game the players move a token throughout a graph to produce an infinite path, which determines the winner or payoff of the game. Traditionally, the players alternate turns in moving the token. In {\em bidding games}, however, the players have budgets, and in each turn, we hold an "auction" (bidding) to determine which player moves the token: both players simultaneously submit bids and the higher bidder moves the token. The bidding mechanisms differ in their payment schemes. Bidding games were largely studied with variants of {\em first-price} bidding in which onl…
This paper introduces a "green" routing game between multiple logistic operators (players), each owning a mixed fleet of internal combustion engine vehicle (ICEV) and electric vehicle (EV) trucks. Each player faces the cost of delayed delivery (due to charging requirements of EVs) and a pollution cost levied on the ICEVs. This cost structure models: 1) limited battery capacity of EVs and their charging requirement; 2) shared nature of charging facilities; 3) pollution cost levied by regulatory agency on the use of ICEVs. We characterize Nash equilibria of this game and derive a condition for i…
Two-player zero-sum repeated games are well understood. Computing the value of such a game is straightforward. Additionally, if the payoffs are dependent on a random state of the game known to one, both, or neither of the players, the resulting value of the game has been analyzed under the framework of Bayesian games. This investigation considers the optimal performance in a game when a helper is transmitting state information to one of the players. Encoding information for an adversarial setting (game) requires a different result than rate-distortion theory provides. Game theory has accentuat…
We study a distributed approach for seeking a Nash equilibrium in $n$-cluster games with strictly monotone mappings. Each player within each cluster has access to the current value of her own smooth local cost function estimated by a zero-order oracle at some query point. We assume the agents to be able to communicate with their neighbors in the same cluster over some undirected graph. The goal of the agents in the cluster is to minimize their collective cost. This cost depends, however, on actions of agents from other clusters. Thus, a game between the clusters is to be solved. We present a d…
