Dynamic Set Values for Nonzero-Sum Games with Multiple Equilibriums

Nonzero sum games typically have multiple Nash equilibriums (or no equilibrium), and unlike the zero-sum case, they may have different values at different equilibriums. Instead of focusing on the existence of individual equilibriums, we study the set of values over all equilibriums, which we call the set value of the game. The set value is unique by nature and always exists (with possible value [Formula: see text]). Similar to the standard value function in control literature, it enjoys many nice properties, such as regularity, stability, and more importantly, the dynamic programming principle. There are two main features in order to obtain the dynamic programming principle: (i) we must use closed-loop controls (instead of open-loop controls); and (ii) we must allow for path dependent controls, even if the problem is in a state-dependent (Markovian) setting. We shall consider both discrete and continuous time models with finite time horizon. For the latter, we will also provide a duality approach through certain standard PDE (or path-dependent PDE), which is quite efficient for numerically computing the set value of the game.

Studi Eksperimen Pada Unit Kegiatan Mahasiswa Petanque Universitas Islam Riau: Meningkatkan Kesegaran Jasmani Melalui Permainan Hadang dan Bentengan

The purpose of this study was to determine and test the increase in physical fitness through bentengan and obstacle course games. In this study the authors used an experimental method. From the results of paired samples test, it can be seen that the significance value is 0.038, which is smaller than α = 0.05. These results prove that using the game hadang has a significant effect on increasing the level of physical fitness of UIR Petanque UKM athletes. A different thing happened to the fortress game that it was not significantly proven to the level of physical fitness of the Petanque UIR UKM athletes with a value of 0.603. If seen in table 1 the mean value of the game hadang was 1.12, while the game of bentengan was -62, this could mean that the game hadang better than the clash of arms.

The Gaming Experience With AI

There are several uses of artificial intelligence in games that are useful for the better game design. With the help of AI, we can improve the games in different ways by simply playing them. In the game industry, when artificial intelligence of the game enhances to the profitable value of the game, this adds to better game reviews, which results to improve the experience of the player. By using AI, we can control both the player as well as non-player characters of the game. AI emphasizes on optimizing the performance of play, which means to measure the degree to which a player comes across the goals of the game, in case of player character. Whereas the role of AI in case of a non-player character emphasizes automatic game balancing mechanisms as well as allow dynamic difficulty adjustment. The use of AI for the empathetic player experience can improve and drive the design process of games. This chapter explores gaming with AI.

Minimax functions on Galton–Watson trees

We consider the behaviour of minimax recursions defined on random trees. Such recursions give the value of a general class of two-player combinatorial games. We examine in particular the case where the tree is given by a Galton–Watson branching process, truncated at some depth 2n, and the terminal values of the level 2n nodes are drawn independently from some common distribution. The case of a regular tree was previously considered by Pearl, who showed that as n → ∞ the value of the game converges to a constant, and by Ali Khan, Devroye and Neininger, who obtained a distributional limit under a suitable rescaling.For a general offspring distribution, there is a surprisingly rich variety of behaviour: the (unrescaled) value of the game may converge to a constant, or to a discrete limit with several atoms, or to a continuous distribution. We also give distributional limits under suitable rescalings in various cases.We also address questions of endogeny. Suppose the game is played on a tree with many levels, so that the terminal values are far from the root. To be confident of playing a good first move, do we need to see the whole tree and its terminal values, or can we play close to optimally by inspecting just the first few levels of the tree? The answers again depend in an interesting way on the offspring distribution.We also mention several open questions.

An Optimal Pursuit Differential Game Problem with One Evader and Many Pursuers

The objective of this paper is to study a pursuit differential game with finite or countably number of pursuers and one evader. The game is described by differential equations in l 2 -space, and integral constraints are imposed on the control function of the players. The duration of the game is fixed and the payoff functional is the greatest lower bound of distances between the pursuers and evader when the game is terminated. However, we discuss the condition for finding the value of the game and construct the optimal strategies of the players which ensure the completion of the game. An important fact to note is that we relaxed the usual conditions on the energy resources of the players. Finally, some examples are provided to illustrate our result.

Kuifi kimün aukantun kimeltuwün meu. La enseñanza del juego mapuche desde las lógicas internas de su cultura (Kuifi kimün aukantun kimeltuwün meu. The teaching of the Mapuche game from the internal logic of its culture)

Este artículo trata la temática específica del juego ancestral mapuche y la enseñanza en el contexto actual de la revitalización cultural de este pueblo indígena en la sociedad chilena. Investigamos el impacto sociocultural de este aprendizaje en el contexto del kimeltuwün para el aporte de la transmisión del conocimiento mapuche en espacios y tiempos educativos de la propia comunidad. Estudio fenomenológico cualitativo, evidenció las narrativas co-construidas en dos comunidades mapuche y escuelas en la región de la Araucanía, Chile. Los resultados revelan el valor, uso del juego y orientaciones metodológicas.Summary. This article deals with the specific theme of the Mapuche ancestral game and its teaching in the current context of the cultural revitalization of these indigenous people in Chilean society. We investigate the sociocultural impact of this learning in the context of kimeltuwün as a contribution for the transmission of Mapuche knowledge in educational spaces and times within its community. The qualitative phenomenological study highlighted the narratives co-constructed in two Mapuche communities and schools from the Araucanía region, Chile. The results revealed the value of the game, its use, and the methodological orientations.

Solving Partially Observable Stochastic Games with Public Observations

In many real-world problems, there is a dynamic interaction between competitive agents. Partially observable stochastic games (POSGs) are among the most general formal models that capture such dynamic scenarios. The model captures stochastic events, partial information of players about the environment, and the scenario does not have a fixed horizon. Solving POSGs in the most general setting is intractable.Therefore, the research has been focused on subclasses of POSGs that have a value of the game and admit designing (approximate) optimal algorithms. We propose such a subclass for two-player zero-sum games with discounted-sum objective function—POSGs with public observations (POPOSGs)—where each player is able to reconstruct beliefs of the other player over the unobserved states. Our results include: (1) theoretical analysis of PO-POSGs and their value functions showing convexity (concavity) in beliefs of maximizing (minimizing) player, (2) a novel algorithm for approximating the value of the game, and (3) a practical demonstration of scalability of our algorithm. Experimental results show that our algorithm can closely approximate the value of non-trivial games with hundreds of states.

Ending the myth of the St Petersburg Paradox

Nicolas Bernoulli suggested the St Petersburg game, nearly 300 years ago, which is widely believed to produce a paradox in decision theory. This belief stems from a long standing mathematical error in the original calculation of the expected value of the game. This article argues that, in addition to the mathematical error, there are also methodological considerations which gave rise to the paradox. This article explains these considerations and why because of the modern computer, the same considerations, when correctly applied, also demonstrate that no paradox exists. Because of the longstanding belief that a paradox exists it is unlikely the mere mathematical correction will end the myth. The article explains why it is the methodological correction which will dispel the myth.

A Fictitious Play Algorithm for Matrix Games with Fuzzy Payoffs

Fuzzy matrix games, specifically two-person zero-sum games with fuzzy payoffs, are considered. In view of the parametric fuzzy max order relation, a fictitious play algorithm for finding the value of the game is presented. A numerical example to demonstrate the presented algorithm is also given.

ON "THE POWER OF VERIFICATION QUERIES" IN UNCONDITIONALLY SECURE MESSAGE AUTHENTICATION

In this paper, we consider authentication codes where the adversary has access to a verification oracle. We formally study two attack games: offline attack and online attack. In an offline impersonation attack with verification query of order i, the adversary launches its attack through two stages. In the first stage — the query stage — the adversary can adaptively choose i distinct messages to query the verification oracle. The verification oracle will answer whether these queried messages are valid or invalid under the secret encoding rule agreed by the transmitter and the receiver. In the later stage — the spoofing stage — the adversary creates a fraudulent message which is different from all its queried messages and sends this message to the receiver. The adversary wins if the receiver accepts the fraudulent message as a valid message. In an online impersonation attack with verification query of order i, the adversary has i + 1 chances to query the verification oracle and wins as soon as one of the queries is a valid message. We make use of strategy trees, which allow optimal strategies in both attack games to be identified, to establish a number of relationships between the value of the two games. This allows us to formally prove a relationship between the value of the game when the adversary has i queries, and the one in which he does not have any. The relationship, though widely believed to be true, was only recently proved for computationally secure systems. Our result complements this latter work for the information theoretic setting.