Classify **simple** **games** into sixteen “types” in terms of the four conventional axioms: monotonicity, properness, strongness, and non- weakness. Further classify them into sixty-four classes in terms of finiteness (existence of a finite carrier) and computability. For each such class, we either show that it is empty or give an example of a game belonging to it. We observe that if a type contains an infinite game, then it contains both computable infinite **games** and noncom- putable ones. This strongly suggests that computability is logically, as well as conceptually, unrelated to the conventional axioms.

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In this work we have represent **simple** **games** (game theory) and molecules (Chemistry) from combinatorial structures (combinatorics). There is still much work to do with these topics (**simple** **games** and molecules v.s. combinatorial structures). For instance, from combinatorial structures we can study how many **simple** **games** or molecules can be generated with a specific structure and size (number of nodes). Some algorithms related with this problem are the so-called ranking, unranking, iteration or random generation of combinatorial structures [19, 20].

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Intuitively, a **simple** game describes in a crude manner the power distri- bution among observable (or describable) subsets of players. Since the cog- nitive ability of a human (or machine) is limited, it is not natural to assume that all subsets of players are observable when there are infinitely many play- ers. We therefore assume that only recursive subsets are observable. This is a natural assumption in the present context, where algorithmic properties of **simple** **games** are investigated. According to Church’s thesis [41, 35], the recursive coalitions are the sets of players for which there is an algorithm that can decide for the name of each player whether she is in the set. 8 Note that the class REC of recursive coalitions forms a Boolean algebra; that is, it includes N and is closed under union, intersection, and comple- mentation. (We assume that observable coalitions are recursive, not just r.e. (recursively enumerable). Mihara [33, Remarks 1 and 16] gives three reasons: nonrecursive r.e. sets are observable in a very limited sense; the r.e. sets do not form a Boolean algebra; no satisfactory notion of computability can be defined if a **simple** game is defined on the domain of all r.e. sets.)

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After defining the core for simple games with preferences, we extend Theorem 14 Nakamura’s theorem 1979 about the nonemptyness of the core: the core of a simple game with preferences is [r]

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Our aim in this thesis will be to investigate the three sides of the triangle of logic, social choice, and game theory, through these **simple** **games**. To this end, section 2 introduces the theory of **simple** **games** in a self-contained way, yet with clear focus on the links with objects more familiar to logicians. Section 3 highlights parallels between some results on **simple** **games** and the newer literature on judgement aggregation. Most of the results presented in this section are generalisations of work due to Monjardet [29] on **simple** **games** and social choice theory, which are then applied in the new context. There are good reasons that justify closer scrutiny of the judgement aggregation impossibility results, such as the analogue of Arrow’s theorem obtained by Dietrich and List [10], in this fashion. A basic motivation is that this approach helps to reveal the mathematical structure that leads up to impossibility results, and hence provides insight in how the judgement aggregation literature ties up with the older literature of social choice theory.

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In fact, if we restrict our attention to the infinite games games without a finite carrier, we can say this: Proposition 1 The axiom δ-computability is independent of monotonicity, prope[r]

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For example, a type 5 +−++ monotonic, nonproper, strong, nonweak computable game has Nakamura number equal to 2, whether it is finite or infinite.4 Note that the Nakamura number for a we[r]

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that do not provide unique predictions on the distribution of endogenous variables (see also Tamer, 2003, and Haile and Tamer, 2003). Aradillas-Lopez and Tamer’s paper shows that standard exclusion restrictions and large-support conditions are suﬃcient to identify struc- tural parameters despite the non-uniqueness of the model predictions. Though structural parameters can be point-identi ﬁ ed, the researcher still faces an identi ﬁ cation issue when he uses the estimated model to perform counterfactual experiments. Players’ behavior under the counterfactual scenario is not point-identiﬁed. This problem also appears in models with multiple equilibria. However, a nice feature of Aradillas-Lopez and Tamer’s approach is that, at least for the class of models that they consider, it is quite **simple** to obtain bounds of the model predictions under the counterfactual scenario.

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In the study of farsighted coalitional behavior, a central role is played by the von Neumann–Morgenstern (1944) stable set and its modification that incorpo- rates farsightedness. Such a modification was first proposed by Harsanyi (1974) and was recently reformulated by Ray and Vohra (2015). The farsighted stable set is based on a notion of indirect dominance in which an outcome can be domi- nated by a chain of coalitional “moves” in which each coalition that is involved in the sequence eventually stands to gain. However, it does not require that each coalition make a maximal move, i.e., one that is not Pareto dominated (for the members of the coalition in question) by another. Consequently, when there are multiple continuation paths, the farsighted stable set can yield unreasonable pre- dictions. We restrict coalitions to hold common, history independent expecta- tions that incorporate maximality regarding the continuation path. This leads to two related solution concepts: the rational expectations farsighted stable set and the strong rational expectations farsighted stable set. We apply these concepts to **simple** **games** and to pillage **games** to illustrate the consequences of imposing rational expectations for farsighted stability.

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nipulations in the coalitional **games** and complexity of resource allocation on networks. The complexity of comparison of influence between players in **simple** **games** is characterized. The **simple** **games** considered are represented by win- ning coalitions, minimal winning coalitions, weighted voting **games** or multi- ple weighted voting **games**. A comprehensive classification of weighted voting **games** which can be solved in polynomial time is presented. An e ffi cient algo- rithm which uses generating functions and interpolation to compute an integer weight vector for target power indices is proposed. Voting theory, especially the Penrose Square Root Law, is used to investigate the fairness of a real life vot- ing model. Computational complexity of manipulation in social choice proto- cols can determine whether manipulation is computationally feasible or not. The computational complexity and bounds of manipulation are considered from var- ious angles including control, false-name manipulation and bribery. Moreover, the computational complexity of computing various cooperative game solutions of **simple** **games** in di ff erent representations is studied. Certain structural results regarding least core payo ff s extend to the general monotone cooperative game. The thesis also studies a coalitional game called the spanning connectivity game. It is proved that whereas computing the Banzhaf values and Shapley-Shubik in- dices of such **games** is #P-complete, there is a polynomial time combinatorial algorithm to compute the nucleolus. The results have interesting significance for optimal strategies for the wiretapping game which is a noncooperative game de- fined on a network.

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The leading meta-analysis of the field was published in 2018 (Lamb et al., 2018) where cognition, affect, and learning outcomes were all assessed when compared with serious **games** (SGs) and simulations. For example, in a review of 46 studies comparing SEG outcomes with traditional learning, the authors showed that effect sizes in learning for all three categories were no different from traditional learning but there were differences among them. In other words, consistent with earlier work on learning outcome studies, there is no demonstrable difference in course performance-related learning outcomes between SEGS and traditional instruction (Anetta et al., 2009; Fengfeng, 2008; Spires et al., 2011; Yang, 2012). Most of these studies were initially done with K-12 learners but the data seems to hold up well with adults. These data do not take into account the effects of simulation gaming in job performance rehearsal, which when assessed showed a strong effect size for cognition and affect. Affect, engagement, motivation, self-efficacy, cognition, skill development, and dimensionality were all assessed in Lamb’s study. Across 2151 articles entered into the analysis, there was a trend to show that greater overall understanding of subject matter ranked under the categories of cognition and skill development had a strong effect size. Affect, indicating emotional connection to learning, was also clearly increased during SEG conditions. We can conclude from these data that SEGS permit learners to put basic theory into practice more effectively than traditional methods. However, there was unequivocal evidence across numerous studies that SEGS showed a higher engagement level than traditional instruction and that, presumably, leads to more robust participation in learning activities.

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We gave a basic language and a minimal logic for epistemic game models and proved completeness. Furthermore we discussed several other modal formulas one might take as axioms for special subclasses of epistemic game models, such as almost bending back. By extending the language with inverse action relations and some iteration operators, we increased its expressiveness, e.g. it now became possible to define the property of being a single tree model by a modal formula. In comparing our static approach to the dynamic update systems of Baltag, Moss and Solecki [1] we found several means of extending their account to include the treatment of **games**.

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In this study, we have built an educational game about the introduction of fruits and vitamins named "Fruit Seller." This educational game has three different types of **games**, namely puzzles, catching fruit, and guessing pictures. In a puzzle game (figure 3), the player must be able to arrange the real problem to fit the desired image; if it is true, then the player will get the winning information and the bonus he gets. Otherwise, it will fail.

We have a large number of **Games** ( G), each game is located on different market level (i) with different number of players (j), strategies (n) and payoffs functions (JI) (See Expression 4.1.). A basic premise in the MD-**games** is that each game has its specific dimension into the mega-space coordinate system, at the same time, all players (j) are taking different speed of time ( ☼ ) to choose its strategy (See Expression 4.2.). Moreover, all players (j) have the freedom to choose any strategy (n) anytime and anywhere, but always exists the high possibility to have a coalition in some **games** (spaces or dimensions) simultaneously into the mega-space coordinate system.

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Thirdly, the **games** for which the main results of Kukushkin (2007) hold are naturally partitioned into two classes: “generalized congestion **games**” and “**games** with structured utilities.” In the former class, the players choose which facilities to use and do not choose anything else; in the latter, each player chooses how to use facilities from a fixed list. Here, both those classes are present too, but we also allow **games** combining both types of choice, i.e., “which” and “how.” It should be mentioned that, both here and in Kukushkin (2007), **games** with structured utilities form a representative subclass.

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Of course any upper bound on the value of a game with combinatorial search is also an upper bound for the traditional version of the game, with **simple** motion, so Lemma 1 applies to those **games** as well. For semi Eulerian networks like those covered by Lemma 1, the length of a minimal tour is given by + a; as arcs of total length a must be traversed twice. The upper bound on the value of V is better than that established by Gal (1979) for arbitrary networks of =2, because V can be written as =2+(a=2) (a= ) whereas Gal’s formula gives a higher upper bound of ( + a) =2; or =2 + (a=2) : This shows that networks satisfying the assumptions of Lemma 1 cannot be weakly Eulerian, where the =2 bound is tight. See Gal (2000).

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The Unique **Games** Conjecture has been shown to imply several inapproximability results, such as that the minimum edge-deletion bipartite subgraph problem is hard to approximate within any constant factor [11], that the Vertex Cover problem is hard to approximate within 2 − ε for every ε > 0 [14], that the Max Cut problem is hard to approximate within .878 · · · [12, 18], and that the Sparsest Cut problem [5] is hard to approximate within any constant. Extensions of the Unique **Games** Conjecture to subconstant γ have been used to prove that the Sparsest Cut problem is hard to approximate within Ω((log logn) 1/6 ) [15]. Several recent papers [23, 13, 2, 3, 20, 21, 17], too many to describe individually, have established inapproximability results for constraint satisfaction problems and other optimization problems by relying on the Unique **Games** Conjecture.

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A rail-shooter, or light-gun shooter game acts like a first- person shooter but the movement of the player is controlled by the game. Similar **games** that use this approach are **games** like Time Cop, Time Crisis, or even Duck Hunt, to some extent. By using a rail-shooter approach, the need to navigate through an environment is eliminated which simplifies the overall system but also allows the user to concentrate more on playing the game instead of trying to move. This approach also works well for the Kinect as it simplifies its usage and wouldn't require the user to move to navigate in the game. Due to the popularity of zombies and its simplicity, we decided that this game would be about zombies. Our intention was that this would allow visually impaired and sighted to both enjoy a zombie game together. The user would stand still in front of the Kinect and only need to move his arms. When the user moves his arms the guns also move. The game tries to mimic the user's movement accurately. It would be possible for the user to shoot the guns through the NIA's facial muscle movement recognition and extraction. By moving an eyebrow, jaw, or even winking, the guns are activated to fire providing superb control. Audio is provided by speakers placed in front of the user and is not intrusive as we continue to allow a hands- free system. Two **games** and a test demonstration were created using this approach.

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10 The Games will provide a lasting legacy of sports facilities 25 The Games will produce facilities which will not be used after the Games have finished 11 The Games have diverted money[r]

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