An Altruism Parameter for Prisoner's Dilemma

In the traditional fuzzy matrix game, given a pair of strategies, the payoffs of one player are usually associated with themselves, but not linked to the payoffs of the other player. Such payoffs can be called self-evaluated payoffs. However, according to the regret theory, the decision makers may care more about what they might get than what they get. Therefore, one player in a matrix game may pay more attention to the payoffs of the other player than his/her payoffs. In this paper, motivated by the pairwise comparison matrix, we allow the players to compare their payoffs and the other ones to provide their relative payoffs, which can be called the cross-evaluated payoffs. Moreover, the players’ preference about the cross-evaluated payoffs is usually distributed asymmetrically according to the law of diminishing utility. Then, the cross-evaluated payoffs of players can be expressed by using the asymmetrically distributed information, i.e., the interval-valued intuitionistic multiplicative number. Comparison laws are developed to compare the cross-evaluated payoffs of different players, and aggregation operators are introduced to obtain the expected cross-evaluated payoffs of players. Based on minimax and maximin principles, several mathematical programming models are established to obtain the solution of a matrix game with cross-evaluated payoffs. It is proved that the solution of a matrix game with cross-evaluated payoffs can be obtained by solving a pair of primal–dual linear-programming models and can avoid some unreasonable results. Two examples are finally given to illustrate that the proposed method is based on the cross-evaluated payoffs of players, and can directly provide the priority degree that one player is preferred to the other player in winning the game.

Serious gaming is used as a means for improving organizational teamwork, yet little is known about the effect of individual game elements constituting serious games. This paper presents a game design experiment aimed at generating knowledge on designing game elements for teamwork. In previous work, we suggested that interaction- and goal-driven rules could guide interdependence and teamwork strategies. Based on this finding, for the present experiment we developed two versions of multiplayer Breakout, varying in rule-sets, designed to elicit player strategies of either dependent competition or dependent cooperation. Results showed that the two rule-sets could generate distinct reported player experiences and observable distinct player behaviors that could be further discriminated into four patterns: expected patterns of helping and ignoring, and unexpected patterns of agreeing and obstructing. Classic game theory was applied to understand the four behavior patterns and made us conclude that goal-driven rules steered players towards competition and cooperation. Interaction rules, in contrast, mainly stimulated dependent competitive behavior, e.g. obstructing each other. Since different types of rules thus led to different player behavior, discriminating in game design between interaction- and goal-driven rules seems relevant. Moreover, our research showed that game theory proved to be useful for understanding goal-driven rules.

Monitoring emotions and cooperative behavior

Feature-based player-centric game adaptivity offers many advantages in terms of optimising the player experience for wide appeal and replayability. Player typologies offer potential to steer adaptivity, as predicted by gameplay behaviour. However their applicability to the design of gameplay adaptation is yet to be fully explored. In this work, we investigate whether a trait-based typology correlates with player behaviour in a prototype survival game. We conducted an explorative study using BrainHex which showed no clear indication for the validity of this approach, raising questions for future research on player behaviour in games.

This paper illustrates our idea of learning and building player behavioral models in real time strategy (RTS) games from replay data by adopting a Case-Based Reasoning (CBR) approach. The proposed method analyzes and cleans the data in RTS games and converts the learned knowledge into a probabilistic model, i.e., a Dynamic Bayesian Network (DBN), for representation and predication of player behaviors. Each DBN is constructed as a case to represent a prototypical player’s behavior in the game, thus if more cases are constructed the simulation of different types of players in a multi-players game is made possible. Sixty sets of replay data of a prototypical player is chosen to test our idea, fifty cases for learning and ten cases for testing, and the experimental result is very promising.KeywordsCase-based Reasoning (CBR)Real Time Strategy (RTS) GamesDynamic Bayesian Network (DBN)Junction Tree

Understanding player behavior in a game is quite important to know people model thinking. Through puzzle games, people can train spatial abilities in a geometry lesson. Spatial ability consists of five groups, one of them is perceptual speed. The Perceptual Speed factor is characterized by the speed in finding a configuration in a mess of distracting material. This research purposes to know user behavior in Tangram Puzzle game based on user data log. Analysis user behavior from two aspects: the dominant of first selecting a puzzle pieces and the correlation between the number of puzzle completion with time. And based on this research can be concluded that player will choose a shape that easily to identified and has a larger size than other to be completed first. Whereas the correlation between the number of completion and time is on a moderate level and quite significant.

The impact of the cardinal relationships amongpecuniary payoffs, and of social history and reputation, on thechoice of strategies in four one-shot Prisoner's Dilemma games isexperimentally examined. The results suggest that normalized payoffvalues linked to ``fear'' and ``greed'' are important as predictorsof behavior in the PD games. Success in coordinating on the payoffdominant equilibrium in previous plays of coordination games alsoincreases the probability of cooperative play in the PD games. Theeffect of past play is strongest when individuals are matchedrepeatedly with the same person in previous play, as contrasted tobeing matched randomly with another player.

In a matrix game, the interactions among players are based on the assumption that each player has accurate information about the payoffs of their interactions and the other players are rationally self-interested. As a result, the players should definitely take Nash equilibrium strategies. However, in real-life, when choosing their optimal strategies, sometimes the players have to face missing, imprecise (i.e., interval), ambiguous lottery payoffs of pure strategy profiles and even compound strategy profile, which means that it is hard to determine a Nash equilibrium. To address this issue, in this paper we introduce a new solution concept, called ambiguous Nash equilibrium, which extends the concept of Nash equilibrium to the one that can handle these types of ambiguous payoff. Moreover, we will reveal some properties of matrix games of this kind. In particular, we show that a Nash equilibrium is a special case of ambiguous Nash equilibrium if the players have accurate information of each player's payoff sets. Finally, we give an example to illustrate how our approach deals with real-life game theory problems.

Relevance of the research topic. A study of player behavior can answer a large number of questions related to motivation and driving forces of negotiators. Particularly acute is the problem of understanding the minimum expectations of the negotiating parties, the willingness to accept worse conditions compared to the counterparty. An important role has the perception of "fairness" in the behavior of players. Understanding these mechanisms will increase the level of social development by increasing the effectiveness of negotiations and coherence of the participant’s behavior. Formulation of the problem. We have summarized and systematized the results of previous research on the influence of various factors on the behavior of players and, as a consequence, on the results of the Ultimatum game. In particular, the influence of the following factors was studied: the amount of money that will be distributed among the players; anonymity and the possibility of punishing the counterparty. Analysis of recent research and publications. There are a large number of publications on the behavior of players in the Ultimatum game, which explore the role of changing factors and their impact on the results of the game. A great contribution to the study of the Ultimatum game have made the works of Güth et al. (1982), G. Bolton and R. Zwick (1995), S. Andersen et al. (2011) and R. Slonim and E. Roth (1998). The purpose and the objectives of the study. The purpose of the study is to substantiate, to generalize, as well as to compare the empirical results of the Ultimatum game to the theoretical results according to classical economic theory. The objectives of the article are to study the influence of factors of anonymity, punishment, and stake size on the behavior of the players. Research methodology. Classical economic theory and behavioral economics constitute the methodological basis for this study. Methods of comparison, analysis and synthesis, induction and deduction are used. Presentation of the main material (results of work). The Ultimatum game, its possible results according to the classical economic theory, its empirical results, revealed by Güthet al. (1982) are studied. Also, the influence of the following factors is surveyed: the amount of money that should be distributed among the players, anonymity and the possibility of punishing the counterparty. Field of application of results. The results of the study can be applied in management, public administration, economics, capital markets. Conclusions according to the article. The stake size, anonymity, and punishment have a decisive influence on the behavior of players in the Ultimatum game. The study demonstrates a significant effect of the relative size of the stake compared to the absolute size offered to game participants. Anonymity has a significant but less noticeable effect on the outcome of the game, compared to the possibility of the second player punishing the first for, in his opinion, an unfair offer. In the complete absence of the possibility of punishment, the behavior of players approaches the theoretical economic theory.

A fast approach to compute fuzzy values of matrix games with payoffs of triangular fuzzy numbers

Audiovisual feedback is one of the most essential elements in video games. Although most feedback does not affect game mechanics, it is believed to have an impact on a player’s experience. One of the concepts in audiovisual feedback is “juiciness,” or providing excessive positive feedback. A few empirical studies have analyzed how juiciness affects the performance and behavior of players. Yet there are few studies on how the kind of feedback and its timing affect player behavior and performance in video games. This paper focuses on the audiovisual feedbacks given to players when they fail in a game. If players experience many failures during gameplay, their behavior may change depending on the audiovisual feedback they receive at the time of failure. We hypothesized that if the feedback to failures was “feelgood,” a player would be motivated to replay the game and try to improve performance. In the first phase of the experiment, five different feedback patterns were prepared and measured by quantifying impressions using the semantic differential method. An A/B test was then conducted using a simple web action game. Players were presented with different audiovisual feedback patterns for each of five groups when they failed in the game. The analysis in multiple comparisons revealed no significant differences in spontaneous replay behavior. The test did, however, result in differences in players’ average scores, suggesting that audiovisual feedback on failure could affect player performance.

AbstarctIn the preceding Chaps. 7– 10, we discussed modeling and solving methods of several kinds of matrix games with intuitionistic fuzzy sets. Obviously, these matrix games are a special case of noncooperative games, i.e., two-person zero-sum finite games. In other words, they are a kind of games in which two players are completely antagonistic, i.e., one player wins the other player loses. In a reality, however, it is not always true that players are completely antagonistic. Thus, it is important and useful to study two-person nonzero-sum noncooperative games in normal form. Bi-matrix games are one of important kinds of the above two-person nonzero-sum noncooperative finite games [1, 2]. In this chapter, we will focus on studying bi-matrix games in which the payoffs of players are expressed with intuitionistic fuzzy sets, which are called bi-matrix games with payoffs of intuitionistic fuzzy sets for short. Specifically, we will propose a total order relation (or ranking method) of intuitionistic fuzzy sets based on the equivalent relation between intuitionistic fuzzy sets and interval-valued fuzzy sets and hereby introduce the concepts of solutions of bi-matrix games with payoffs of intuitionistic fuzzy sets and parametric bi-matrix games. It is proven that any bi-matrix game with payoffs of intuitionistic fuzzy sets has at least one satisfying Nash equilibrium solution, which is equivalent to a Nash equilibrium solution of the corresponding parametric bi-matrix game. The latter can be obtained through solving an auxiliary parametric bilinear programming model. Clearly, bi-matrix games with payoffs of intuitionistic fuzzy sets are a general form of the matrix games with payoffs of intuitionistic fuzzy sets as discussed in Chap. 7.

In real-life management situations, there is an important kind of competitive decision problems with multiple decision makers (i.e., players). At present, game theory is one of the most effective tools to deal with such a kind of management problems. In the classical (or crisp) game theory, we usually assume that payoffs of players are crisp (or numerical) values.

Based on behavioural and social data, this research focuses on the application of AI models for predicting player behaviour and optimizing game experience. This research uses classification, regression, and deep reinforcement learning (DRL) models to anticipate the player behaviour, aid in-game decision making, and customize game environments. This coupling of behavioural data like session duration, activity frequency, decision history and social data like player interactions, group dynamics, communication logs is essential for accurate predictions. This research tests various ML models such as decision trees, SVM, random forests, and deep reinforcement learning to give an in-depth understanding on how these models support the experience of the players. In the part of experimentation, this research has demonstrated that, while classification and regression models are excellent for predicting discrete player behavior and continuous game variables, DRL models are better for predicting real-time game optimization. This insight also points to the need to integrate behavioural and social data to develop dynamic, interactive and personalised gameplay. The research provides some insights for future AI-based games applications especially for creating mechanisms that continually change in response to the user.

We define the nucleolus of a continuous convex map F: Π → Rm on a convex compact set Π ⊂ Rn. As special cases we obtain known notions as nucleolus, prenucleolus and weighted nucleolus of a TU-game with (or without) coalition structure. Also the nucleolus of a matrix game turns out to be an interesting special case. It appears that the nucleolus of a matrix game coincides with the set of Dresher optimal strategy pairs of the game. This implies, in particular, that the nucleolus consists of the proper equilibria of the matrix game. To each (zero-normalized) TU-game one can construct a matrix game—the excess game—such that the nucleolus of the TU-game coincides with the unique proper optimal strategy of player II in the excess game. Also for other nucleoli of TU-games a suitable matrix game can be constructed where the nucleolus under consideration is related to the nucleolus of the matrix game in an analogous way. A balancedness condition is given characterizing nucleolus elements of a matrix game. It is shown that this balancedness result implies the known balancedness characterizations of Kohlberg, Sobolev, Owen and Wallmeier.