We introduce a new variation of the m-player asymmetric Colonel Blotto game, where the n battles ... more We introduce a new variation of the m-player asymmetric Colonel Blotto game, where the n battles occur as sequential stages of the game, and the winner of each stage needs to spend resources for maintaining his win. The limited resources of the players are thus needed both for increasing the probability of winning and for the maintenance costs. We show that if the initial resources of the players are not too small, then the game has a unique Nash equilibrium, and the given equilibrium strategies guarantee the given expected payoff for each player.
Problem definition: Consider consumers who prefer to consume a good later rather than earlier. If... more Problem definition: Consider consumers who prefer to consume a good later rather than earlier. If the price is constant, then we would expect consumers to wait to buy the good. That does not hold if consumers are concerned that others will buy the good early, so that a shortage will later occur. When will consumers arrive when they fear a shortage? What is the profit-maximizing policy of a monopolist? Might the firm lose profits by offering advance sales? The timing of consumer arrivals is much studied. Little consideration, however, has addressed how anticipated shortages affect arrival times. The application is important: managers want to know when consumers will arrive, when they should make the product available, and what price to charge to maximize profits. Methodology/results: We use game theory. We analyze analytically outcomes when a single item is for sale: we give closed solutions for the equilibrium customer behavior and profit-maximizing firm strategy and conduct sensiti...
This work presents a variation of Naor's strategic observable model (1969), by adding a component... more This work presents a variation of Naor's strategic observable model (1969), by adding a component of customer heterogeneity induced by the location of customers in relation to the server. Accordingly, customers incur a "travel cost" which depends linearly on the distance of the customer from the server. The arrival of customers with distances less than x is assumed to be a Poisson process with rate λ(x) = x 0 h(y)dy < ∞, where h(y) is a nonnegative "intensity" function of the distance y. In a loss system M/G/1/1 we define the threshold Nash equilibrium
doi: 10.3389/fncom.2012.00016 Losing the battle but winning the war: game theoretic analysis of t... more doi: 10.3389/fncom.2012.00016 Losing the battle but winning the war: game theoretic analysis of the competition between motoneurons innervating a skeletal muscle
A consumer who wants to consume a good at a particular period may nevertheless attempt to buy it ... more A consumer who wants to consume a good at a particular period may nevertheless attempt to buy it earlier if he is concerned that the good will otherwise be sold. We analyze the behavior of consumers in equilibrium and the price a profit-maximizing firm would charge. We show that a firm profits by not selling early. If, however, the firm is obligated to also offer the good early, then the firm may maximize profits by setting a price which induces consumers to all arrive early, or all arrive late, depending on the good's value to the customer.
We apply a new Game Theoretical approach to analyze the competition between motoneurons (MNs) inn... more We apply a new Game Theoretical approach to analyze the competition between motoneurons (MNs) innervating a common muscle. At birth, each muscle-fiber is innervated by several MNs, of which the connection of only one becomes permanent. Each MN innervates initially many muscle-fibers and therefore engages in many competitions simultaneously, winning at some muscle-fibers and losing at others. The group of muscle-fibers that are eventually innervated by the same MN is called a “muscle unit”. In the adult system, MNs with successively higher activation-thresholds have successively larger muscle units. This is called “the size principle”, which is believed to be one of the most fundamental principles in the organization of motor-unit behavior Therefore it is important to understand how it evolves. In viewing the elimination period as a game in which MNs are competing to innervate a maximal number of muscle-fibers, the translation of the size principle is that less-active MNs (i.e., MNs ...
We introduce a new two-person multi-stage game in which players compete to win a maximal number o... more We introduce a new two-person multi-stage game in which players compete to win a maximal number of stages. At the start of the game each player is given resources which are needed for winning the competitions. Each such winning is penalized and the size of penalty may depend on the stage of winning. Thus the limited resources of players are needed both for increasing the probability of winning and for paying the penalties. We find the unique Nash equilibrium of this game. Surprisingly, we find that the equilibrium strategies are such that at each stage the decisions of the players do not depend on future penalties, but only on the penalty of the current stage, and on the remaining resources. We then generalize the game and the results to m players.
We introduce a new variation of the m-player asymmetric Colonel Blotto game, where the n battles ... more We introduce a new variation of the m-player asymmetric Colonel Blotto game, where the n battles occur as sequential stages of the game, and the winner of each stage needs to spend resources for maintaining his win. The limited resources of the players are thus needed both for increasing the probability of winning and for the maintenance costs. We show that if the initial resources of the players are not too small, then the game has a unique Nash equilibrium, and the given equilibrium strategies guarantee the given expected payoff for each player.
Aim/Purpose: How does heterogeneous valuation of service affect optimal control of queues? Backgr... more Aim/Purpose: How does heterogeneous valuation of service affect optimal control of queues? Background We analyze this heterogeneity by adding a component of travel costs, which differ with distance from the service point. Methodology: Mathematical analysis of queuing theory. Analyzing the anarchy function. Contribution: Enabling consumers to make optimal choices based on knowledge about their status, and enabling better control of the organizer. Findings: In the arrival rate is bounded, there is no need of interference. If it is unbounded then in many cases the organizer should impose the socially optimal queue length. Recommendations for Practitioners: In the arrival rate is bounded, there is no need of interference. If it is unbounded then in many cases the organizer should impose the socially optimal queue length. Recommendations for Researchers: Explore the following points: What happens when there are more than one server, located at different point. How should consumers behave...
Traffic police faces the problem of enforcing speed limits under restricted budget. Implementing ... more Traffic police faces the problem of enforcing speed limits under restricted budget. Implementing high Enforcement Thresholds (ET) will ease the work load on the police but will also intensify the problem of speeding. We model this as a game between the police, which wishes that drivers obey the speed limits, and the drivers who wish to speed without getting caught. For the police we construct a strategy in which at each stage the ET is randomized between low and high values. This confuses the drivers who now need to consider the worst case of low ET. We have established analytically and by simulations that this strategy gradually reduces the ET until it converges to the desired speed limit without increasing the work load along the process. Importantly, this method works even if the strategy is known to the drivers. We study the effect of several factors on the convergence rate of the process. Interestingly, we find that increasing the frequency of randomization is more effective in expediting the process than raising the average amount of fines. 1 Introduction About 1.25 million people die every year as a result of traffic accidents worldwide and the injuries caused by road traffic accidents are the leading cause of death among young people, aged 15-29 years.
The purpose of this work is to offer for each player and any Nash equilibrium (NE), a measure for... more The purpose of this work is to offer for each player and any Nash equilibrium (NE), a measure for the potential risk in deviating from the NE strategy in any two person matrix game. We present two approaches regarding the nature of deviations: Strategic and Accidental. Accordingly, we define two models: S-model and T-model. The S-model defines a new game in which players deviate in the least dangerous direction. The risk defined in the T-model can serve as a refinement for the notion of “trembling hand perfect equilibrium” introduced by R. Selten. The risk measures enable testing and evaluating predictions on the behavior of players. For example: do players deviate more from a NE that is less risky? This may be relevant to the design of experiments. We present an Integer programming problem that computes the risk for any given player and NE. In the special case of zero-sum games with a unique strictly mixed NE, we prove that the risks of the players always coincide, even if the game...
This work presents a variation of Naor's strategic observable model (1969), by adding a component... more This work presents a variation of Naor's strategic observable model (1969), by adding a component of customer heterogeneity induced by the location of customers in relation to the server. Accordingly, customers incur a "travel cost" which depends linearly on the distance of the customer from the server. The arrival of customers with distances less than x is assumed to be a Poisson process with rate λ(x) = x 0 h(y)dy < ∞, where h(y) is a nonnegative "intensity" function of the distance y. In a loss system M/G/1/1 we define the threshold Nash equilibrium
We introduce a new sequential game, where each player has a limited resource that he needs to spe... more We introduce a new sequential game, where each player has a limited resource that he needs to spend on increasing the probability of winning each stage, but also on maintaining the assets that he has won in the previous stages. Thus, the players’ strategies must take into account that winning at any given stage negatively affects the chances of winning in later stages. Whenever the initial resources of the players are not too small, we present explicit strategies for the players, and show that they are a Nash equilibrium, which is unique in an appropriate sense.
Game theory is usually applied to biology through evolutionary games. However, many competitive p... more Game theory is usually applied to biology through evolutionary games. However, many competitive processes in biology may be better understood by analyzing them on a shorter time-scale than the time-course considered in evolutionary dynamics. Instead of the change in the "fitness" of a player, which is the traditional payoff in evolutionary games, we define the payoff function, tailored to the specific questions addressed. In this work we analyze the developmental competition that arises between motoneurons innervating the same muscle. The "size principle" - a fundamental principle in the organization of the motor system, stating that motoneurons with successively higher activation-threshold innervate successively larger portions of the muscle - emerges as a result of this competition. We define a game, in which motoneurons compete to innervate a maximal number of muscle-fibers. The strategies of the motoneurons are their activation-thresholds. By using a game the...
We consider an infinitely repeated two-person zero-sum game with incomplete information on one si... more We consider an infinitely repeated two-person zero-sum game with incomplete information on one side, in which the maximizer is the (more) informed player. Such games have value v y ð pÞ for all 0 a p a 1. The informed player can guarantee that all along the game the average payo¤ per stage will be greater than or equal to v y ðpÞ (and will converge from above to v y ðpÞ if the minimizer plays optimally). Thus there is a conflict of interest between the two players as to the speed of convergence of the average payo¤s-to the value v y ð pÞ. In the context of such repeated games, we define a game for the speed of convergence, denoted SG y ðpÞ, and a value for this game. We prove that the value exists for games with the highest error term, i.e., games in which v n ðpÞ À v y ðpÞ is of the order of magnitude of 1 ffiffi n p. In that case the value of SG y ðpÞ is of the order of magnitude of 1 ffiffi n p. We then show a class of games for which the value does not exist. Given any infinite martingale X y ¼ fX k g y k¼1 , one defines for each n : V n ðX y Þ :¼ E P n k¼1 jX kþ1 À X k j. For our first result we prove that for a uniformly bounded, infinite martingale X y , V n ðX y Þ can be of the order of magnitude of n 1=2Àe , for arbitrarily small e > 0.
We apply a Game Theoretical approach to analyze the competition between motoneurons (MNs) innerva... more We apply a Game Theoretical approach to analyze the competition between motoneurons (MNs) innervating a common muscle. A typical skeletal muscle consists of many thousands of fibers. At birth each muscle-fiber is innervated by several MNs, but during the first couple of weeks after birth, a competitive mechanism-called "synapse elimination"-abolishes all inputs but one, which we term "the winner at the muscle-fiber". At birth, each MN innervates many muscle-fibers and therefore it
We introduce a new variation of the m-player asymmetric Colonel Blotto game, where the n battles ... more We introduce a new variation of the m-player asymmetric Colonel Blotto game, where the n battles occur as sequential stages of the game, and the winner of each stage needs to spend resources for maintaining his win. The limited resources of the players are thus needed both for increasing the probability of winning and for the maintenance costs. We show that if the initial resources of the players are not too small, then the game has a unique Nash equilibrium, and the given equilibrium strategies guarantee the given expected payoff for each player.
Problem definition: Consider consumers who prefer to consume a good later rather than earlier. If... more Problem definition: Consider consumers who prefer to consume a good later rather than earlier. If the price is constant, then we would expect consumers to wait to buy the good. That does not hold if consumers are concerned that others will buy the good early, so that a shortage will later occur. When will consumers arrive when they fear a shortage? What is the profit-maximizing policy of a monopolist? Might the firm lose profits by offering advance sales? The timing of consumer arrivals is much studied. Little consideration, however, has addressed how anticipated shortages affect arrival times. The application is important: managers want to know when consumers will arrive, when they should make the product available, and what price to charge to maximize profits. Methodology/results: We use game theory. We analyze analytically outcomes when a single item is for sale: we give closed solutions for the equilibrium customer behavior and profit-maximizing firm strategy and conduct sensiti...
This work presents a variation of Naor's strategic observable model (1969), by adding a component... more This work presents a variation of Naor's strategic observable model (1969), by adding a component of customer heterogeneity induced by the location of customers in relation to the server. Accordingly, customers incur a "travel cost" which depends linearly on the distance of the customer from the server. The arrival of customers with distances less than x is assumed to be a Poisson process with rate λ(x) = x 0 h(y)dy < ∞, where h(y) is a nonnegative "intensity" function of the distance y. In a loss system M/G/1/1 we define the threshold Nash equilibrium
doi: 10.3389/fncom.2012.00016 Losing the battle but winning the war: game theoretic analysis of t... more doi: 10.3389/fncom.2012.00016 Losing the battle but winning the war: game theoretic analysis of the competition between motoneurons innervating a skeletal muscle
A consumer who wants to consume a good at a particular period may nevertheless attempt to buy it ... more A consumer who wants to consume a good at a particular period may nevertheless attempt to buy it earlier if he is concerned that the good will otherwise be sold. We analyze the behavior of consumers in equilibrium and the price a profit-maximizing firm would charge. We show that a firm profits by not selling early. If, however, the firm is obligated to also offer the good early, then the firm may maximize profits by setting a price which induces consumers to all arrive early, or all arrive late, depending on the good's value to the customer.
We apply a new Game Theoretical approach to analyze the competition between motoneurons (MNs) inn... more We apply a new Game Theoretical approach to analyze the competition between motoneurons (MNs) innervating a common muscle. At birth, each muscle-fiber is innervated by several MNs, of which the connection of only one becomes permanent. Each MN innervates initially many muscle-fibers and therefore engages in many competitions simultaneously, winning at some muscle-fibers and losing at others. The group of muscle-fibers that are eventually innervated by the same MN is called a “muscle unit”. In the adult system, MNs with successively higher activation-thresholds have successively larger muscle units. This is called “the size principle”, which is believed to be one of the most fundamental principles in the organization of motor-unit behavior Therefore it is important to understand how it evolves. In viewing the elimination period as a game in which MNs are competing to innervate a maximal number of muscle-fibers, the translation of the size principle is that less-active MNs (i.e., MNs ...
We introduce a new two-person multi-stage game in which players compete to win a maximal number o... more We introduce a new two-person multi-stage game in which players compete to win a maximal number of stages. At the start of the game each player is given resources which are needed for winning the competitions. Each such winning is penalized and the size of penalty may depend on the stage of winning. Thus the limited resources of players are needed both for increasing the probability of winning and for paying the penalties. We find the unique Nash equilibrium of this game. Surprisingly, we find that the equilibrium strategies are such that at each stage the decisions of the players do not depend on future penalties, but only on the penalty of the current stage, and on the remaining resources. We then generalize the game and the results to m players.
We introduce a new variation of the m-player asymmetric Colonel Blotto game, where the n battles ... more We introduce a new variation of the m-player asymmetric Colonel Blotto game, where the n battles occur as sequential stages of the game, and the winner of each stage needs to spend resources for maintaining his win. The limited resources of the players are thus needed both for increasing the probability of winning and for the maintenance costs. We show that if the initial resources of the players are not too small, then the game has a unique Nash equilibrium, and the given equilibrium strategies guarantee the given expected payoff for each player.
Aim/Purpose: How does heterogeneous valuation of service affect optimal control of queues? Backgr... more Aim/Purpose: How does heterogeneous valuation of service affect optimal control of queues? Background We analyze this heterogeneity by adding a component of travel costs, which differ with distance from the service point. Methodology: Mathematical analysis of queuing theory. Analyzing the anarchy function. Contribution: Enabling consumers to make optimal choices based on knowledge about their status, and enabling better control of the organizer. Findings: In the arrival rate is bounded, there is no need of interference. If it is unbounded then in many cases the organizer should impose the socially optimal queue length. Recommendations for Practitioners: In the arrival rate is bounded, there is no need of interference. If it is unbounded then in many cases the organizer should impose the socially optimal queue length. Recommendations for Researchers: Explore the following points: What happens when there are more than one server, located at different point. How should consumers behave...
Traffic police faces the problem of enforcing speed limits under restricted budget. Implementing ... more Traffic police faces the problem of enforcing speed limits under restricted budget. Implementing high Enforcement Thresholds (ET) will ease the work load on the police but will also intensify the problem of speeding. We model this as a game between the police, which wishes that drivers obey the speed limits, and the drivers who wish to speed without getting caught. For the police we construct a strategy in which at each stage the ET is randomized between low and high values. This confuses the drivers who now need to consider the worst case of low ET. We have established analytically and by simulations that this strategy gradually reduces the ET until it converges to the desired speed limit without increasing the work load along the process. Importantly, this method works even if the strategy is known to the drivers. We study the effect of several factors on the convergence rate of the process. Interestingly, we find that increasing the frequency of randomization is more effective in expediting the process than raising the average amount of fines. 1 Introduction About 1.25 million people die every year as a result of traffic accidents worldwide and the injuries caused by road traffic accidents are the leading cause of death among young people, aged 15-29 years.
The purpose of this work is to offer for each player and any Nash equilibrium (NE), a measure for... more The purpose of this work is to offer for each player and any Nash equilibrium (NE), a measure for the potential risk in deviating from the NE strategy in any two person matrix game. We present two approaches regarding the nature of deviations: Strategic and Accidental. Accordingly, we define two models: S-model and T-model. The S-model defines a new game in which players deviate in the least dangerous direction. The risk defined in the T-model can serve as a refinement for the notion of “trembling hand perfect equilibrium” introduced by R. Selten. The risk measures enable testing and evaluating predictions on the behavior of players. For example: do players deviate more from a NE that is less risky? This may be relevant to the design of experiments. We present an Integer programming problem that computes the risk for any given player and NE. In the special case of zero-sum games with a unique strictly mixed NE, we prove that the risks of the players always coincide, even if the game...
This work presents a variation of Naor's strategic observable model (1969), by adding a component... more This work presents a variation of Naor's strategic observable model (1969), by adding a component of customer heterogeneity induced by the location of customers in relation to the server. Accordingly, customers incur a "travel cost" which depends linearly on the distance of the customer from the server. The arrival of customers with distances less than x is assumed to be a Poisson process with rate λ(x) = x 0 h(y)dy < ∞, where h(y) is a nonnegative "intensity" function of the distance y. In a loss system M/G/1/1 we define the threshold Nash equilibrium
We introduce a new sequential game, where each player has a limited resource that he needs to spe... more We introduce a new sequential game, where each player has a limited resource that he needs to spend on increasing the probability of winning each stage, but also on maintaining the assets that he has won in the previous stages. Thus, the players’ strategies must take into account that winning at any given stage negatively affects the chances of winning in later stages. Whenever the initial resources of the players are not too small, we present explicit strategies for the players, and show that they are a Nash equilibrium, which is unique in an appropriate sense.
Game theory is usually applied to biology through evolutionary games. However, many competitive p... more Game theory is usually applied to biology through evolutionary games. However, many competitive processes in biology may be better understood by analyzing them on a shorter time-scale than the time-course considered in evolutionary dynamics. Instead of the change in the "fitness" of a player, which is the traditional payoff in evolutionary games, we define the payoff function, tailored to the specific questions addressed. In this work we analyze the developmental competition that arises between motoneurons innervating the same muscle. The "size principle" - a fundamental principle in the organization of the motor system, stating that motoneurons with successively higher activation-threshold innervate successively larger portions of the muscle - emerges as a result of this competition. We define a game, in which motoneurons compete to innervate a maximal number of muscle-fibers. The strategies of the motoneurons are their activation-thresholds. By using a game the...
We consider an infinitely repeated two-person zero-sum game with incomplete information on one si... more We consider an infinitely repeated two-person zero-sum game with incomplete information on one side, in which the maximizer is the (more) informed player. Such games have value v y ð pÞ for all 0 a p a 1. The informed player can guarantee that all along the game the average payo¤ per stage will be greater than or equal to v y ðpÞ (and will converge from above to v y ðpÞ if the minimizer plays optimally). Thus there is a conflict of interest between the two players as to the speed of convergence of the average payo¤s-to the value v y ð pÞ. In the context of such repeated games, we define a game for the speed of convergence, denoted SG y ðpÞ, and a value for this game. We prove that the value exists for games with the highest error term, i.e., games in which v n ðpÞ À v y ðpÞ is of the order of magnitude of 1 ffiffi n p. In that case the value of SG y ðpÞ is of the order of magnitude of 1 ffiffi n p. We then show a class of games for which the value does not exist. Given any infinite martingale X y ¼ fX k g y k¼1 , one defines for each n : V n ðX y Þ :¼ E P n k¼1 jX kþ1 À X k j. For our first result we prove that for a uniformly bounded, infinite martingale X y , V n ðX y Þ can be of the order of magnitude of n 1=2Àe , for arbitrarily small e > 0.
We apply a Game Theoretical approach to analyze the competition between motoneurons (MNs) innerva... more We apply a Game Theoretical approach to analyze the competition between motoneurons (MNs) innervating a common muscle. A typical skeletal muscle consists of many thousands of fibers. At birth each muscle-fiber is innervated by several MNs, but during the first couple of weeks after birth, a competitive mechanism-called "synapse elimination"-abolishes all inputs but one, which we term "the winner at the muscle-fiber". At birth, each MN innervates many muscle-fibers and therefore it
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