Game Theory Problem Research Articles

Artificial Intelligence, LLC: Corporate Personhood for AI

Our legal system has long tried to fit the square peg of artificial intelligence (AI) technologies into the round hole of the current tort regime, overlooking the inability of traditional liability schemes to address the nuances of how AI technology creates harms. The current tort regime deals out rough justice—using strict liability for some AI products and using the negligence rule for other AI services—both of which are insufficiently tailored to achieve public policy objectives. Under a strict liability regime where manufacturers are always held liable for the faults of their technology regardless of knowledge or precautionary measures, firms are incentivized to play it safe and stifle innovation. But even with this cautionary stance, the goals of strict liability cannot be met due to the unique nature of AI technology: its mistakes are merely “efficient errors”—they appropriately surpass the human baseline, they are game theory problems intended for a jury, they are necessary to train a robust system, or they are harmless but misclassified. Under a negligence liability regime where the onus falls entirely on consumers to prove the element of causation, victimized consumers must surmount the difficult hurdle of tracing the vectors of causation through the “black box” of algorithms. Unable to do so, many are left without sufficient recourse or compensation. This Article proposes a new framework to regulate AI technologies: bestowing corporate personhood to AI systems. First, the corporate personality trait of “limited liability” strikes an optimal balance in determining liability—it would both compensate victims (for instance, through obligations to carry insurance and a straightforward burden of causation) while holding manufacturers responsible only when the infraction is egregious (for instance, through veil-piercing). Second, corporate personhood is “divisible”—meaning not all corporate personality traits need to be granted—which circumvents many of the philosophical criticisms of giving AI the complete set of rights of full legal personhood. Third, innovation incentives weigh heavily in favor of shifting to a tort regime better tailored to the potential promises and unique harms of AI.

Multi-objective linguistic-neutrosophic matrix game and its applications to tourism management

Game theory plays an important role in numerous decision-oriented real-life problems. Nowadays, many such problems are basically characterized by various uncertainties. Uncertainties come to happen due to decision makers' collection of data, intuition, assumption, judgement, behaviour, evaluation and lastly, due to the problem itself. Fuzzy concept with membership degree made an initialization towards the treatment of uncertainty, but it was not enough. Intuitionistic fuzzy concept was evolved concerning with both membership and non-membership degrees but failed to express reality more accurately. Then, neutrosophy logic was developed with a new degree in uncertainty, say, indeterminacy degree besides membership and non-membership degrees. Multi-objective optimization is an area of multiple-criteria decision making related with mathematical optimization problems involving more than one objective function to be optimized at the same time. Game theory (matrix game) problems with imprecise, vague information, like neutrosophic, can be formed with multiple objective functions. We develop and analyse a matrix game with multiple objectives, and solve the problem under a single-valued neutrosophic environment in linguistic approach. The main achievement of our study is that we here introduce a problem-oriented example to justify our designed methodologies with a successful real-life implications using linguistic neutrosophic data rather than crisp data as used in previous researches.

Multi-AI competing and winning against humans in iterated Rock-Paper-Scissors game

Predicting and modeling human behavior and finding trends within human decision-making processes is a major problem of social science. Rock Paper Scissors (RPS) is the fundamental strategic question in many game theory problems and real-world competitions. Finding the right approach to beat a particular human opponent is challenging. Here we use an AI (artificial intelligence) algorithm based on Markov Models of one fixed memory length (abbreviated as “single AI”) to compete against humans in an iterated RPS game. We model and predict human competition behavior by combining many Markov Models with different fixed memory lengths (abbreviated as “multi-AI”), and develop an architecture of multi-AI with changeable parameters to adapt to different competition strategies. We introduce a parameter called “focus length” (a positive number such as 5 or 10) to control the speed and sensitivity for our multi-AI to adapt to the opponent’s strategy change. The focus length is the number of previous rounds that the multi-AI should look at when determining which Single-AI has the best performance and should choose to play for the next game. We experimented with 52 different people, each playing 300 rounds continuously against one specific multi-AI model, and demonstrated that our strategy could win against more than 95% of human opponents.

A nucleus for Bayesian Partially Observable Markov Games: Joint observer and mechanism design

An intelligent agent suggests an autonomous entity, which manages and learns actions to be taken towards achieving goals. The problem, reported as common knowledge in the literature in Artificial Intelligence (AI), is that it is a challenge to develop an approach able to compute efficient decisions that maximize the total reward of interacting agents upon an environment with unknown, incomplete, and uncertain information.To address these shortcomings, this paper provides a step forward: a nucleus for Bayesian Partially Observable Markov Games (BPOMGs) supported by an AI approach. Three fundamental topics conform the structure of the nucleus: game theory, learning and inference. First, we present a novel general Bayesian approach which is conceptualized for games that considered both, the incomplete information of the Bayesian model and the incomplete information over the states of the Markov system. In this new model, execution uncertainty is handled by using a Partially Observable Markov Game (POMG). Second, we extend the design theory, which now involves the mechanism design and the joint observer design (both unknown). The mechanism design results from the fact that agents act in their own individuals’ self-interest, and to induce agents to not reveal their private information and create a particular outcome. The joint observer design goal is related to represent the fact that agents may not be interested to provide accurate information of their states. In addition, agents follow a model that employs a Reinforcement Learning (RL) approach for estimating the transition matrices (also unknown) at each time step. Hence, as our final contribution, is an extended model of POMGs by introducing a new variable and proposing an analytical solution to compute both the observer design and the mechanism design (the two unknown). The proposed extension makes the game theory problem computationally tractable. We derive relations to recover analytically the variables of interest for each agent, i.e. observation kernels, joint observers, mechanism, strategies, and distribution vectors. The usefulness and effectiveness of the proposed nucleus is validated in simulation on a game-theoretic analysis of the patrolling problem designing the mechanism, computing the observers, and employing an RL approach.

An efficient characterization of submodular spanning tree games

Cooperative games form an important class of problems in game theory, where a key goal is to distribute a value among a set of players who are allowed to cooperate by forming coalitions. An outcome of the game is given by an allocation vector that assigns a value share to each player. A crucial aspect of such games is submodularity (or convexity). Indeed, convex instances of cooperative games exhibit several nice properties, e.g. regarding the existence and computation of allocations realizing some of the most important solution concepts proposed in the literature. For this reason, a relevant question is whether one can give a polynomial-time characterization of submodular instances, for prominent cooperative games that are in general non-convex. In this paper, we focus on a fundamental and widely studied cooperative game, namely the spanning tree game. An efficient recognition of submodular instances of this game was not known so far, and explicitly mentioned as an open question in the literature. We here settle this open problem by giving a polynomial-time characterization of submodular spanning tree games.

The emergence of cooperation in the thermodynamic limit

Predicting how cooperative behavior arises in the thermodynamic limit is one of the outstanding problems in evolutionary game theory. For two player games, cooperation is seldom the Nash equilibrium. However, in the thermodynamic limit cooperation is the natural recourse regardless of whether we are dealing with humans or animals. In this work, we use the analogy with the Ising model to predict how cooperation arises in the thermodynamic limit.

Shaping Emotions in Negotiation: a Nash Bargaining Solution

Modeling emotions in negotiations is an open challenge that attracted an increasing amount of attention from researchers. Bargainers look for achieving an agreement with the opposing parties and, at the same time, try to reach their own goals. This process consists of both bargaining and (game theory) problem solving. Game theory models seek to enlighten the rational negotiations between players, but these models lack the evidence of how emotional motives may influence individuals’ behavior. This paper suggests a model for shaping emotions in negotiation using Nash’s bargaining approach. We focus on the case where negotiation between players has motives of cooperating, considering eight emotions: anger, fear, joy, sadness, surprise, disgust, guilt, and disappointment. For representing the solution of the problem, we employ a homogeneous Markov game. The simplicity of the model relies on the fact that the emotions are represented by the states of the Markov chain. The relationship between the emotions is represented by a transition matrix that determines the probability of changing between the emotions (states) at any time. Because any emotion can be reached at any time with certain probability, the bargaining Markov game is ergodic. We represent naturally the emotional process of bargaining using a proximal method, which involves the bargaining Nash product for computing the equilibrium of the game. We show the convergence of the method to the emotional equilibrium point. The solution of the Nash bargaining game consists of cooperative emotional strategies, which are transformed in emotional probability distributions. Such emotional probability distributions are measured using an asymmetric distance function that determines the “emotional distance” between players in negotiations. Emotions are measured using an asymmetric distance function because they are different between players. We present a new approach for shaping emotions in negotiations employing Nash’s bargaining model. An application example shows the influence of expressing emotions in the relationship process, and those emotions are strategically selected to gain a benefit in negotiations. We show that the magnitude and direction of emotional distance matter and that feelings have an asymmetric effect on the negotiation process.

On the Approximability of the Stable Matching Problem with Ties of Size Two

The stable matching problem is one of the central problems of algorithmic game theory. If participants are allowed to have ties, the problem of finding a stable matching of maximum cardinality is an $$\mathcal{NP}$$ -hard problem, even when the ties are of size two. Moreover, in this setting it is UGC-hard to provide an approximation with a constant factor smaller than 4/3. In this paper, we give a tight analysis of an approximation algorithm given by Huang and Kavitha for the maximum cardinality stable matching problem with ties of size two, demonstrating an improved 4/3-approximation factor.

Shadow Investment Activity as a Factor of Macroeconomic Instability

This paper summarizes the arguments and counterarguments within the scientific discussion on the issue of countering shadow activities in the context of ensuring macroeconomic stability. The main purpose of the study is to determine the acceptable level of investment transactions with signs of fictitiousness, which corresponds to the balance between the level of shadowing of the national economy and its macroeconomic stability. The relevance of this scientific problem is that the shadow investment activities distort the market mechanism and make it impossible to attract financial resources for expanded reproduction in the country. The research of shadow investment activity in the paper is carried out in the following logical sequence: the dynamics of the level of shadowing of Ukraine’s economy, its macroeconomic stability and volume of investment activity with signs of fictitiousness are analyzed, the forms of functional dependence between the analyzed indicators are determined. national economy and maximizing its macroeconomic stability. Methodological tools of the research methods were linear and nonlinear regression analysis, iterative method of Brown-Robinson, which is the basis for solving the problem of game theory. The study period is 2010-2019. The paper proves the existence of nonlinear functional dependence of the level of investment operations with signs of fictitiousness on the level of shadowing of the national economy and its macroeconomic stability. The study empirically confirms and theoretically proves that the volume of investment activity with signs of fictitiousness at 14.76% of GDP is achieved by increasing macroeconomic stability to 0.840 shares and reducing the shadowing of the national economy to 36.30% of GDP. The results of the study can be useful for the executive branch in the formation of public investment policy aimed at improving the quality of the business environment, reducing administrative burdens, and increasing investment demand from foreign investors. Keywords: macroeconomic stability, shadow economy, shadow investment activity, optimization model, game theory.

Piece Problems: Component Valuation in Marketing and in Patent and Tort Law

The problems referred to in the title of this chapter concern evaluating a given variable when it is one of several that have combined to bring about a result. In some cases, there is an easy market solution. Imagine that you contract to buy a house and then the beautiful kitchen stove, one of many things that attracted you to the property, is destroyed before you close the transaction or occupy the property. How much should the price now be reduced? Here there is an upper limit based on the cost of a comparable replacement appliance. A more precise valuation would also be easy if identical houses, lacking this one feature, had recently been sold. The stove is just a piece of the larger transaction, and with these convenient facts, there is not much of a “component valuation problem.” Additionally, the stove is unlikely to have been of greater value because of its interaction with other items in the house; colors and sizes are fairly standardized. “Conjoint analysis” – a term that usually refers to survey evidence that tries to elicit the value of a component – is therefore unnecessary, or at least uncomplicated, because value does not depend on an interaction among variables in a way that is not directly observed. It is also interesting because it does not present a difficult Game Theory problem, or result that might be described in common parlance as something that depends on the relative bargaining skill of the parties.

Support Set Invariancy for Interval Bimatrix Games

Traditionally, game theory problems were considered for exact data, and the decisions were based on known payoffs. However, this assumption is rarely true in practice. Uncertainty in measurements and imprecise information must be taken into account. The interval-based approach for handling such uncertainties assumes that one has lower and upper bounds on payoffs. In this paper, interval bimatrix games are studied. Especially, we focus on three kinds of support set invariancy. Support of a mixed strategy consists of that pure strategies having positive probabilities. Given an interval-valued bimatrix game and supports for both players, the question states as follows: Does every bimatrix game instance have an equilibrium with the prescribed support? The other two kinds of invariancies are slight modifications: Has every bimatrix game instance an equilibrium being a subset/superset of the prescribed support? It is computationally difficult to answer these questions: the first case costs solving a large number of linear programs or mixed integer programs. For the remaining two cases a sufficient condition and a necessary condition are proposed, respectively.

Why Habermas needs distributive equity principles: Heath's critique, game theory, and collective action problems

Why Habermas needs distributive equity principles: Heath's critique, game theory, and collective action problems

О ДВУХЭТАПНОМ ПРОКСИМАЛЬНОМ АЛГОРИТМЕ ДЛЯРЕШЕНИЯ ЗАДАЧИ О РАВНОВЕСИИ

In this paper, the weak convergence of an iterative twostage proximal method for the approximate solution of the equilibrium problem in a Hilbert space is investigated. This method was recently been developed by Vedel and Semenov and can be used to solve mathematical programming problems, variational inequalities and game theory problems. The analysis of the convergence of the method was carried out under the assumption of the existence of a solution of the equilibrium problem and under conditions weaker than the previously considered ones.

Случайное поведение участника как способ максимизации вероятности его выигрыша в парадоксе Монти Холла

In the issue we consider Monty Hall Dilemma. We use decision making approach and computer simulation to solve it. Monty Hall Dilemma is a three step problem involving a participant, a host and three doors with a valuable prize behind one of them and worthless prizes behind two others. The participant should guess where it is to win the valuable prize. After the participant makes an initial choice for one door, the host opens a non-chosen door with a worthless prize behind it. Then the participant is asked whether he wants to stay with his initial choice, or to switch to the remaining unopened door. The problem is quite old and still of much interest for mathematicians and psychologists because of counterintuitive solution and most humans erroneous situation behavior. Traditionally Monty Hall Dilemma is considered as a probability or a game theory problem. We choose an alternative approach and solve it as a decision making task. We determine possible participant’s and host’s actions and their probabilities and construct a problem decision tree. Its leaf nodes represent the situation outcomes. Then we evaluate the outcomes probabilities assuming that the participant sticks to his initial choice with a constant unknown probability. It allowed us to construct a total winning probability function. Its maximization allowed us to determine that an optimal participant behavior is to switch. We also performed computer simulation to verify our theoretical solution.

A Survey on the Combined Use of Optimization Methods and Game Theory

Game theory is a field of applied mathematics that studies strategic behavior of rational factors. In other words, game theory is a collection of analytical tools that can be used to make optimal choices in interactional and decision making problems. Optimization in mathematics and computer science is the choice of the best member of an existing collection for a specific purpose. Several optimization methods have been used in many problems to minimize costs or maximize profits. From a particular point of view, it can be said that the game theory is in fact a kind of optimization. In this paper, a combined use of game theory and optimization algorithms has been reviewed and a new categorization is presented for researches which have been conducted in this area. In some of these combinations, game theory has been used to improve the performance of optimization algorithms, and in some others, optimizations methods help to solve game theory problems. Game theory and optimization algorithms are also used together to solve some other problems.

Sharing Profit From Joint Offering of a Group of Wind Power Producers in Day Ahead Markets

Many current deregulated markets allow multiple wind power producers (WPPs) to jointly offer energy in the short-term electricity market via an external agent such as a wind power aggregator. This paper proposes a budget balanced, fair, and stable framework to share the profit due to the joint wind energy offering, which is modeled as a core selection problem in the cooperative game theory. In particular, this design problem can be formulated as a large scale linear program with an exponential number of implicit constraints whose parameters are the outcomes of the wind coalition's optimal energy offering strategies. We propose a novel constraint generation algorithm to optimally solve this large scale optimization problem with affordable computation efforts. Different from the traditional computation methods, our proposed algorithm can effectively address the complexity involved in generating a feasible cut by exploiting the linear structure of the wind power offering model and the weak duality theory of linear programing. Extensive numerical results are then presented to illustrate the efficiency of the proposed framework in dealing with large scale aggregation of WPPs in a complex market framework and flexible adoption of optimization objectives.

Chemical Game Theory

The purpose of this paper is to describe a new framework for representing and solving strategic game theory problems. This framework, called “Chemical Game Theory” (CGT), uses well-known, rigorous ...

Multiple stable states in microbial communities explained by the stable marriage problem.

Experimental studies of microbial communities routinely reveal that they have multiple stable states. While each of these states is generally resilient, certain perturbations such as antibiotics, probiotics, and diet shifts, result in transitions to other states. Can we reliably both predict such stable states as well as direct and control transitions between them? Here we present a new conceptual model-inspired by the stable marriage problem in game theory and economics-in which microbial communities naturally exhibit multiple stable states, each state with a different species' abundance profile. Our model's core ingredient is that microbes utilize nutrients one at a time while competing with each other. Using only two ranked tables, one with microbes' nutrient preferences and one with their competitive abilities, we can determine all possible stable states as well as predict inter-state transitions, triggered by the removal or addition of a specific nutrient or microbe. Further, using an example of seven Bacteroides species common to the human gut utilizing nine polysaccharides, we predict that mutual complementarity in nutrient preferences enables these species to coexist at high abundances.

On Lyapunov Game Theory Equilibrium: Static and Dynamic Approaches

This paper suggests a game theory problem in which any feasible solution is based on the Lyapunov theory. The problem is analyzed in the static and dynamic cases. Some properties of Nash equilibria such as existence and stability are derived naturally from the Lyapunov theory. Remarkable is that every asymptotically stable equilibrium point (Nash equilibrium point) admits a Lyapunov-like function and if a Lyapunov-like function exists it converges to a Nash/Lyapunov equilibrium point. We define a Lyapunov-like function as an Lp-norm from the multiplayer objective function to the utopia minimum as a cost function. We propose multiple metrics to find the Nash/Lyapunov equilibrium and the strong Nash/Lyapunov equilibrium. Finding a Nash/Lyapunov equilibrium is reduced to the minimization problem of the Lyapunov-like function. We prove that the equilibrium point properties of Nash and Lyapunov meet in game theory. In order to validate the contributions of the paper, we present a numerical example.

The Proximal Point Method for Locally Lipschitz Functions in Multiobjective Optimization with Application to the Compromise Problem

This paper studies the constrained multiobjective optimization problem of finding Pareto critical points of vector-valued functions. The proximal point method considered by Bonnel, Iusem, and Svaiter [SIAM J. Optim., 15 (2005), pp. 953--970] is extended to locally Lipschitz functions in the finite dimensional multiobjective setting. To this end, a new (scalarization-free) approach for convergence analysis of the method is proposed where the first-order optimality condition of the scalarized problem is replaced by a necessary condition for weak Pareto points of a multiobjective problem. As a consequence, this has allowed us to consider the method without any assumption of convexity over the constraint sets that determine the vectorial improvement steps. This is very important for applications; for example, to extend to a dynamic setting the famous compromise problem in management sciences and game theory.