Ben Dhaou Bourheneddine, Ziad Abderrahmane

The paper proposes a solution for TU cooperative games in which solidary and non-solidary players coexist. The distinction between the two groups is that solidary players are able to support by consent at least one of their weaker partners, without disadvantaging non-solidary players. A value for games which takes into account the types of the players and satisfies some usual axioms is presented: Efficiency, Additivity, Symmetry among players who have the same type, Conditional null player, and a new axiom, the Unaffected Allocation of non-solidary players – (UA) – which is defined as follows: when players have the possibility of deciding freely in favor of solidarity, this should affect neither the allocations of non-solidary players nor the outcome of the game.

]]>Beidi Qiang

In this paper, the problem of multi-stage game playing is studied in a decision theory framework. The conditional strategies of payers are examined under the assumption of perfect information about previously played stages. The equilibrium is defined in analogy to the Nash equilibrium and the existence of such equilibrium is discussed mathematically through the idea of dynamic programming and backward induction. An example in a two-players zero-sum game setting is analyzed with details and the solution of the equilibrium strategy is presented.

]]>Olivier Lefebvre

One resumes the notion of “monopolistic competition” or product differentiation. The model used is Bertrand competition, the demands being deduced from the consumers’ utilities. A consumer is represented by a point u_{i} in the cube 0 ≤ u_{i} ≤ 1, u_{i}^{ }being the utility of the product i for him. The product differentiation is when the points representing the consumers are in the facets of the cube u_{i} = 1. One studies the mathematical properties of the equilibrium. Some of them correspond to the characteristics of product differentiation, which are: (1) all the consumers make a purchase. When the products are differentiated, the firms are innovative (since the preferred product of a consumer has a utility which is maximal). And innovation has been defined “struggle against no consumption” (Christensen) (2) each firm has its “garden”, consumers in the facet u_{i} = 1 for the firm E_{i}. A firm keeps the consumers in its “garden”, provided its price is not too high (3) given the existence of “gardens” the profits are sufficient. Product differentiation is symmetrical, and disruption is dissymmetrical: the disruptive firm has a “garden” with a utility higher than the utilities of the “gardens” of its competitors. One demonstrates that the profit of a disruptive firm increases. The goals of the paper are: - To set out a model which describes product differentiation and disruption; - To explore the possibilities of the model used (Bertrand competition, the demands being deduced from the consumers’ utilities). The author has already used this model to study a particular kind of merger: when the merger is profitable, the bought asset being closed down. It is a sign of saturated market. Even, it could be a criterium (for saturated markets) interesting for fintechs. Also, the model allows discriminating non differentiating innovation (when there is product differentiation and all the utilities increase) and differentiating innovation (disruption: only the utility of the product of the disruptive firm increases). One shows that differentiating innovation provides more profit, always, but non differentiating innovation could provide no more profit. Finally, the model is also useful to study the effects of cannibalization. In the paper, a tractable example is set out, allowing to answer this question: is it in the interest of a disruptive firm to buy and close down a competitor before disruption? If the competitor cannibalizes the product of the disruptive firm very much, it is better to buy and close down this competitor.

Issah Musah, Douglas Kwasi Boah, Baba Seidu

This paper presents a comprehensive review of solution methods and techniques usually employed in game theory to solve games with the view of demystifying and making them easy to understand. Specifically, the solution methods and techniques discussed are Nash Equilibrium Method; Pareto Optimality Technique; Shapley Values Technique; Maximin-Minimax Method; Dominance Method; Arithmetic Method; Matrix Method; Graphical Method and Linear Programming Method. The study has contributed significantly to knowledge by successfully filling or narrowing the knowledge or research gap of insufficient literature on reviews of solution methods and techniques usually employed to analyze or solve games in game theory.

]]>Subarna Roy

This paper attempts to address the issue of internal migration in India in the context of the COVID-19 outbreak with specific implications on health and productivity. We introduce a new parameter to quantitatively represent the COVID-19 scenario in India- the k factor which is the ratio of daily increase in the number of infections to the number of cumulative infected cases. Also, as a more intuitive alternative to k-factor, we suggest a parameter called doubling rate defined as the number of days it takes for the cases to double. The k-factor / doubling-rate show varying trend across states implying that different states in India are at different stages of the pandemic which would eventually delay the “flattening of the curve” of the country and thereby growth and productivity. One of the factors that led to such a situation we argue is the large outflow of migrants from destination to source during the initial phase of the outbreak. In an empirical study, we demonstrate that states with historically high out-migration show lower doubling rates in recent months as migrants start coming back. The doubling rate is almost 16 days lower on average on a weekly scale compared to states from which the reverse migration has taken place. This is concerning as those states initially had very few cases and the weekly doubling rate was also higher. Within an analytical game-theoretic framework that involves two hypothetical states, we show that when non-co-operative states anticipate that outflow of migrants to other states has a high positive impact on its K-factor, sending back the migrants to their respective states would be a probable Nash-equilibrium. However, under circumstances where expected loss from discontinuity in economic activities owing to the scarcity of labor far outweighs the health impact, retention would have been a strictly dominating strategy. Based on the analysis we come up with a multiple for K-factor below which it is prudent to retain migrants for a state with historically high in-migration. This is the case when the other state is indifferent between the retention and release of its migrants. Our simulation study shows that if due to reverse migration state domestic product drops from 5% to 4%, then the host state should retain migrants, this is even though their release would lead up to 17 days gain in doubling rate. We propose that the opportunity cost of foregoing that gain could have been corrected through investment in meticulous health-care planning and migration welfare. In turn, states to which reverse migration took place would have been spared of a rapid increase in infections. Therefore, retention as a strategy would have led India to a higher growth path during pandemic times while accelerating the “flattening of the curve”.

]]>Jason J. Wang

Between controlling the virus and boosting the economy, it has become a “President’s Dilemma”, in which determining the optimal time to open-up the economy while still being able to contain the virus is the president’s top priority. In this study, we utilize game theory examples to illustrate various scenarios of the President’s Dilemma and provide possible game theory tools to obtain the optimal solutions. For the normal simultaneous game, the Nash equilibrium is at (Open-up, Lock-down). For the GDP vs mortality game, Open-up is a more cost-effective option. In the Bayesian game, if the proportion of the governors siding with the president q>=1/3, the President would choose Open-up while if q<1/3, he would choose Lock-down. The Stackelberg game point out the possible study path to figure out the leader-follower roles and related factors.

]]>Shahd H. Alkaraz, Essam El-Seidy, Neveen S. Morcos

]]>Hassan El Kady, Essam El-Seidy

We introduce a class of impartial combinatorial game which is the multi-player Last Nim game, denoted by in which there are N piles of counters which are linearly ordered, the move will be, the n-player will remove any positive integer of counters from the last pile, we will introduce this with Shifted Standard alliance system by 1, denoted by in which each player will prefer winning for another player over himself. The Aim is to determine the game value of the positions of where is the number of piles and is the number of players and we will present the possible and determine the game value in this case.

]]>Wesam B. Zorik, Essam EL-Seidy, Entisarat M. Elshobaky

Zero Determinant (ZD) strategies is a new class of probabilistic and conditional strategies discovered by Press and Dyson which has been applied on two players. ZD strategies for multiple players will be explored where the application of ZD has been taken on Iterated three-player Prisoner’s Dilemma Game (IPDG) with two actions C or D. It is assumed that the results in the case of the three players correspond to what was obtained in the case of two players with different calculation methods. It is interesting, although the player who adopts ZD strategies can determine the opponent’s payoffs in a unilateral way. And with an extortion strategy, he can impose a linear relationship between the expected return for him and the expected payoff of other players, but he cannot determine his own score. Also, the yield matrix was determined in the case of three players.

]]>Essam El_Seidy, Hassan El Kady, Dieaa I. Nassr

The Last Nim game is an impartial combinatorial game studied only in the case of the standard alliance matrix. In this paper, we consider the Last Nim game with N linearly ordered nonempty piles containing counters, and n players for any alliance matrix. For this, we give an algorithm to get the winner, the tree of all possibilities of the game, and the best strategy for the winner. For the practical result, we implement this algorithm using C++ language and give some examples.

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