game and the equilibrium of the static game are equivalent in a two-person zero-sum game 1 . We extend this analysis to more general multi-players zero-sum game. We do not assume differentiability of payoff functions. However, we do not assume that the payoff functions are not differentiable. We do not use differentiability of payoff functions.
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For patch foraging, assuming that individuals are free to move between any of the available food patches, at effectively no cost, animals should follow the ideal free distribution (IFD) over the patches. The simplest single species model was developed by Fretwell and Lucas , Fretwell  (see also ), with a closely related model developed by Parker . More complex cases, including for multi-species [15,23], individuals with different abilities [19,40] and for more complex payoff functions, including the Allee effect [12,25], have also been considered.
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We consider a Stackelberg type symmetric dynamic three-players zero-sum game. One player is the leader and two players are followers. All players have the symmetric payoff functions. The game is a two-stages game. In the first stage the leader determines the value of its strategic variable. In the second stage the followers determine the values of their strategic variables given the value of the leader’s strategic variable. On the other hand, in the static game all players simultaneously determine the values of their strategic variable. We do not assume differentiability of players’ payoff functions. We show that the sub-game perfect equilibrium of the Stackelberg type symmetric dynamic zero-sum game with a leader and two followers is equivalent to the equilibrium of the static game if and only if the game is fully symmetric.
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The research we present in this work is directly motivated by  and considers a particular alteration of the payoff function in the respective model. In order to formulate our research questions, let us briefly introduce the central features of the model in . A repeated game between a team, consisting of a forecaster and an agent, and nature takes place. The sequence of the states of nature is assumed to be i.i.d. In each stage of the game, the team members are able to observe the past actions played and the past states of nature. In addition, unlike the agent, the forecaster is able to observe all future states of nature. The team receives a payoff in every stage, which depends on the current state of nature, as well as on the actions of the forecaster and the agent, which we call action triple in the following. The strategies of the team players induce an infinite sequence of random action triples and hence a limiting average distribution, Q, of an action triple. A distribution that is induced in such a way is also called an implementable distribution. The authors in  prove two important theorems that characterise the set of implementable distributions. This characterization involves an information theoretic inequality that applies the Shannon entropy function (see ), which is called the information constraint. This con- straint can be interpreted as the fact that the amount of information used by the agent cannot be greater than the information she actually receives from the forecaster. The first result in  states that every implementable distribution fulfills the information constraint. For the second result the authors show that for every distribution Q that fulfills the information constraint, there exist strategies of the forecaster and of the agent that implement Q.
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The simultaneous conversion implicitly assumes that coalitional payoffs orig- inate in two stages: a coalition formation stage, in which the coalition structure forms; a strategic form game, in which Nash strategies are played by each coalition. In fact, Nash strategies are a predictable outcome only if all elements of the game (the set of players, i.e., the elements of the newly formed coalition structure, their payoff functions and strategy sets) are commonly known. In other terms, deviations from a generally agreed joint strategy are carried out by first publicly abandoning the negotiation process (as, for instance, a group of countries leaving the international negotiation table) and then playing the Nash equilibrium strategies of the induced simultaneous game. Although appropriate in certain cooperative environments, the simultaneous conversion fails to capture the dynamic nature of coalition formation that we claimed is common to several economic problems. As we argued at the be- ginning, coalitions can often deviate by directly choosing an alternative strategy in the underlying game, as do firms defecting from an industrial cartel by directly and unexpectedly setting a lower price. In order to explore this idea, we construct a char- acteristic function formally expressing the assumption that forming coalitions can move first. We stress here that we do not attempt to endogenize the coalition struc- ture induced by a deviation, but we adopt the gamma assumption used in Chander and Tulkens (1997). 2 More precisely, we derive the coalitional value v φ (S) as the
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Another approach in continuous time is to model the trigger involving aggregate loss process. It is important to note that Vaugirard (2003) was the first to develop a simple arbitrage approach for evaluating catastrophe risk insurance-linked securi- ties, although they employed a non-traded underlying framework. In this paper, CAT bondholders have a short position on an option. Lin et al. (2008) applied a Markov- modulated Poisson process for catastrophe occurrences using a similar approach to that of Vaugirard (2003). Lee and Yu (2002, 2007) also introduced the default risk, moral hazard, and basis risk with stochastic interest rate. P´erez-Fructuoso (2008) developed a CAT bond with index triggers. Ma and Ma (2013) proposed a mixed approximation method to simplify the distribution of aggregate loss and to find the numerical solu- tions of CAT bonds with general pricing formulae. In addition, Nowak and Romaniuk (2013) expanded Vaugirard’s model and obtained CAT bond prices using Monte Carlo simulations with different payoff functions and spot interest rates.
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Abstract. In this paper we apply the Complete Analysis of Differentiable Games (introduced by D. Carfì in , , , , and al- ready employed by himself and others in , , ) to the classic Cournot Duopoly (1838), classic oligopolistic market in which there are two enterprises producing the same commodity and selling it in the same market. In this classic model, in a competitive background, the two enterprises employ, as possible strategies, the quantities of the commodity produced. The main solutions proposed in literature for this kind of duopoly are the Nash equilibrium and the Collusive Optimum, without any subsequent critical exam about these two kinds of solutions. The absence of any critical quantitative analysis is due to the relevant lack of knowledge regarding the set of all possible outcomes of this strategic interaction. On the contrary, by con- sidering the Cournot Duopoly as a differentiable game (a game with differentiable payoff functions) and studying it by the new topological methodologies introduced by D. Carfì, we obtain an exhaustive and complete vision of the entire payoff space of the Cournot game (this also in asymmetric cases with the help of computers) and this total view allows us to analyze criti- cally the classic solutions and to find other ways of action to select Pareto strategies. In order to illustrate the application of this topological methodology to the considered infinite game, several compromise decisions are considered, and we show how the complete study gives a real extremely extended comprehension of the classic model.
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Hence, the underlying reasoning is analogous to the Bayesian one in statistics, according to which fixed parameters are viewed as random variables obeying a certain statistical distribution, and so on. Therefore, the global optimization process ”designs” a new game in an autonomous way so that the desired steady state becomes one of its Nash equilibria. Besides, by design, the optimization method used in the solution allows the strategist to limit the values of each ”incentive” present in the payoff functions in a particular numerical interval. In truth, even more constraints may be incorporated without too much effort, so as to make it easier to reach customized conditions for several types of problems - total amount available for use in a given strategic context, for instance.
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Closest to the present work, Wiedenbeck, Yang, and Well- man (2018) learn payoff functions of symmetric games. For each strategy, a regressor takes as input a pure-strategy opponent-profile and outputs a utility. They provide an ap- proximation method for deviation payoffs that allows for computation of role-symmetric -Nash equilibria. This ap- proach can learn complicated payoff functions and scales well in the number of players. However, the approxima- tion method, which is only valid for fully-symmetric games learned via RBF kernel Gaussian process regression, is pro- hibitively expensive, and thereby limited to small data sets from games with small numbers of pure strategies. We com- pare against this method in our experiments and demonstrate favorable results for deviation-payoff learning.
Although the problem of finding Nash equilibria of finite strategic games re- ceived a great deal of attention during the last decades and many solutions nowadays exist, the inverse problem of obtaining a complete set of payoff functions that realize one desired Nash equilibrium, with preestablished num- ber of agents and individual strategies is new, having already some proposals for finding solutions, obtained by means of evolutionary techniques . The present work aims to go further, also using stochastic global optimization to obtain games with multiple and predefined equilibria. To that end, some associated concepts must be precisely established.
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able elimination (VE) algorithm (Guestrin et al., 2002a), which is in essence identical to variable elimination in a Bayesian network (Zhang and Poole, 1996). The algorithm eliminates the agents one by one. Before an agent (node) is eliminated, the agent first collects all payoff functions related to its edges. Next, it computes a conditional payoff function which returns the maximal value it is able to contribute to the system for every action combination of its neighbors, and a best-response function (or conditional strategy) which returns the action corresponding to the maximizing value. The conditional payoff function is communicated to one of its neighbors and the agent is elimi- nated from the graph. Note that when the neighboring agent receives a function including an action of an agent on which it did not depend before, a new coordination dependency is added between these agents. The agents are iteratively eliminated until one agent remains. This agent selects the action that maximizes the final conditional payoff function. This individual action is part of the optimal joint action and the corresponding value equals the desired value max a u(a). A second pass
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In this paper, we have introduced a new solution concept called maximin equilibrium which extends von Neumann’s maximin strategy idea to n-player games by incorporating common knowledge of rationality of the players. The rationality assumption we use is stronger than the one of maximin strate- gist and weaker than the one of Nash strategist. Maximin equilibrium is a method for evaluating the uncertainty that players are facing by playing the game. In other words, maximin equilibrium is a strategy profile whose value (which is a vector) is maximized with respect to the methods of Pareto optimality and of Nash equilibrium. We showed that maximin equilibrium is invariant under strictly increasing transformations of the payoff functions of the players. Moreover, every finite game possesses a maximin equilibrium in pure strategies.
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In the Mining Day benchmark , a mining company mines gold and silver (objectives) from different mines (lo- cal payoff functions) spread throughout the region in which the company operates. The workers live in villages also spread throughout this region. The company has supplied one van to each village (agents) for transporting the workers of that village and must determine every morning to which mine each van should go (actions). A van can only travel to mines that are close to the village (graph connectivity). Workers are more efficient if there are many workers at a mine. Since the company aims to maximize profit, the best strategy depends on the fluctuating prices of gold and silver on the market, which are not known when the plan must be computed. 10
We consider concurrent games played by two-players on a finite-state graph, where in every round the players simultaneously choose a move, and the current state along with the joint moves determine the successor state. We study the most fundamental objective for concurrent games, namely, mean-payoff or limit-average objective, where a reward is associated to every transition, and the goal of player 1 is to maximize the long-run average of the rewards, and the objective of player 2 is strictly the opposite (i.e., the games are zero-sum). The path constraint for player 1 could be qualitative, i.e., the mean-payoff is the maximal reward, or arbitrarily close to it; or quantitative, i.e., a given threshold between the minimal and maximal reward. We consider the computation of the almost-sure (resp. positive) winning sets, where player 1 can ensure that the path constraint is satisfied with probability 1 (resp. positive probability). Almost-sure winning with qualitative constraint exactly corresponds to the question of whether there exists a strategy to ensure that the payoff is the maximal reward of the game. Our main results for qualitative path constraints are as follows: (1) we establish qualitative determinacy results that show for every state either player 1 has a strategy to ensure almost-sure (resp. positive) winning against all player- 2 strategies, or player 2 has a spoiling strategy to falsify almost-sure (resp. positive) winning against all player-1 strategies; (2) we present optimal strategy complexity results that precisely characterize the classes of strategies required for almost-sure and positive winning for both players; and (3) we present quadratic time algorithms to compute the almost-sure and the positive winning sets, matching the best known bound of the algorithms for much simpler problems (such as reachability objectives). For quantitative constraints we show that a polynomial time solution for the almost-sure or the positive winning set would imply a solution to a long-standing open problem (of solving the value problem of turn-based deterministic mean-payoff games) that is not known to be solvable in polynomial time.
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In this paper, a two person zero-sum fuzzy matrix game is introduced. The problem is considered by incorporating L R fuzzy numbers in payoff. Based on the fuzzy number comparison introduced by Rouben's method , the fuzzy payoff is converted into the corresponding deterministic payoff. Then, for each player, a linear programming problem is formulated. Also, a solution procedure for solving each problem is proposed.
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by using MATLAB, and it has been applied to several examples. The implementation of the algorithm gives the parametric expressions of the critical zone of any game in the family under consideration both in the bistrategy space and in the payoff space and the graphical representations of the disjoint union (with respect to the parameter set of the paramet- ric game) of the family of all payoff spaces. We have so the parametric representation of the union of all the critical zones. One of the main moti- vations of our paper is that, in the applications, many normal-form games appear naturally in a parametric fashion; moreover, some efficient models of coopetition are parametric games of the considered type. Specifically, we have realized an algorithm that provides the parametric and graphical representation of the payoff space and of the critical zone of a parametric game in normal-form, supported by a finite family of compact intervals of the real line. Our final goal is to provide a valuable tool to study simply (but completely) normal-form C 1 -parametric games in two dimensions.
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Players who use focal-point reasoning try to discriminate between pure-strategy Nash equilibria by using only informa- tion that is common knowledge. When describing focal-point reasoning, Schelling (1960, pp. 83, 96, 106, 163, 298) often uses the metaphor of a ‘meeting of minds’ between the players. The suggestion is that, in choosing their strategies, the players imagine themselves reasoning together about how to coordinate their behaviour. 12 A natural implication of this idea is that, in their reasoning, players use only information that is common knowledge between them. Items of such knowledge that can be used in this way will be called cues. Players will concentrate on cues that are salient – i.e., that can easily be recognised by both players – and discriminating – i.e., that can identify one of the Nash equilibria as the solution of the game. Given that player labels are symmetric, there are only two plausible types of cue that can pick out a speciﬁc pure-strategy equilibrium. There is a cue of label salience if one equilibrium stands out relative to the other by having a more salient label attached to the corresponding strategy. There is a cue of joint-payoff salience if one equilibrium stands out by having a pair of payoffs that is better for the players collectively, according to some salient criterion of ‘betterness’ that treats the players symmetrically. For our purposes, we can restrict the criterion to payoff dominance.
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Previous studies show that slight differences in information presentation may indeed have effects on subject’s behavior. Pruitt (1967), for instance, re- ports more cooperation in the prisoner’s dilemma game if the payoff structure of the game is presented to subjects in the decomposed form. In a public goods experiment, Saijo and Nakamura (1995) provide subjects either with a “rough” table containing basic payoff information or a “detailed” table that is comparable to the payoff table we provide. They find, if the marginal capita per return is high (and resp. low), average contributions to the public goods are higher with detailed tables (resp. lower) than the investments with rough tables. Huck et al. (1999) find that markets tend to become less competitive if more information about demand and cost conditions are present, while more information about competitors’ quantities and profits yields more competitive behavior. Bosch-Dom`enech and Vriend (2003) investigate imitation behavior in Cournot markets by varying the presentation of market information. They observe that the imitation frequency does not increase when the informa- tion retrieval gets more complex. In a gift-exchange experiment, Charness et al. (2004) find significant reductions in both wages and worker effort when subjects are provided with payoff tables compared to the baseline treatment without payoff tables. Requate and Waichman (2011) report no differences in behavior in Cournot duopoly experiments whether subjects are provided with payoff tables or use a payoff calculator. The studies of Charness et al. (2004)
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Our main results have related to exact payoff queries, though other query models are interesting too. A very natural type of query is a best-response query, where a strategy s is chosen, and the algorithm is told the players’ best responses to s. In general s may have to be a mixed strategy; it is not hard to check that pure-strategy best response queries are insufficient; even for a two-player two-action game, knowledge of the best responses to pure profiles is not sufficient to identify an -Nash equilibrium for < 1 2 . Fictitious Play (Fudenberg and Levine 1998, Chapter 2) can be regarded as a query protocol that uses best-response queries (to mixed strategies) to find a Nash equilibrium in zero-sum games, and essentially a 1/2-Nash equilibrium in general-sum games (Goldberg et al., 2013). We can always synthesize a pure best-response query with n(k − 1) payoff queries. Hence, for questions of polynomial query complexity, payoff queries are at least as powerful as best- response queries. Are there games where best-response queries are much more useful than payoff queries? If k is large then it is expensive to synthesize best-response queries with payoff queries. The DMP-algorithm (Daskalakis et al., 2009b) finds a 1 2 -Nash equilibrium via only two best-response queries, whereas Theorem 5 notes that O(k) payoff queries are needed.
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As different from others, the two-step communication protocol between the identity and the coin element allows, on the one hand, a coin element issuer to design digital coins that can be read only by a certain identity element, i.e. by a specific user/device. This means that even though the coin element is lost or stolen by an attacker, such an element will not work without the associated identity element hence providing a two-factor authentication for each transaction. It uses both symmetric and asymmetric cryptographic primitives‟ inorder to guarantee some security principles. The identity element can be used to be a protection against fraudsters. If an identity element is considered malicious and it is blacklisted, no matter which is the coin element used in the transaction, all payment requests will be rejected. The physical unclonable function was used only to authenticate core elements of the architecture, in this improved version multiple physical unclonable functions are also used to allow all the elements to interact in a secure way.