Therefore, the solutions z * proposed in, We next investigat z * n . In particular, we investigate vulnerabilities that may prevent CR communication in specific bands, completely deny a cognitive radio to communicate or induce it to cause harmful interference to existing users; so called denial-of-service attacks. All figure content in this area was uploaded by Beibei Wang, Jamming Attacks Using the Colonel Blotto Game. Found inside Page 257(In game-theoretic language, neither candidate has a pure minimax strategy since such a strategy on the part of either can be made that invests optimal strategies under the Electoral College system with greater determinateness. 7.7. Wu et al. 2021 Forbes Media LLC. They first model the problem by a Linear Program (LP) and use Ellipsoid method to solve it. The goal in the Colonel Blotto game is to find optimal (i.e.,maximin) strategies of the players. In this section, we will nd the Nash Equilibrium for the, proposed jamming game by modeling it to a Colonel Blotto, game where two opponents distribute limited resources over, a number of battleelds with a payoff equal to the sum of, outcomes from individual battleeld [13]. Found inside Page xxxixMaple gives us the optimal strategies X*=(l'l' 1): a 5 5 which checks with the fact that for a symmetric matrix the strategies are the same for both Example 2.12 Colonel Blotto Games. It is an optimal allocation of forces game. An optimal strategy will win at least 50% of the time against any move/point. In Colonel Blotto, any strategy can be dominated by another. IEEE 802.22 is the world's first wireless standard based on CR technology. the damage is maximized. Calculating the equilibrium of this game is complicated and so is the choice of the optimal strategy. Once the numbers get large enough, though, this becomes impractical (for example, p(1000) is on the order of 1031). In the process, we identify, analyze, and assess the risk level posed by the potential attacks in the different CR design paradigms proposed by different research groups. In particular, we prove the existence and uniqueness of Nash equilibrium. If the secondary. The winner of each battlefield is the colonel who puts more troops in it and the overall utility of each colonel is the sum of weights of the battlefields that s/he wins. Cognitive radio technologies have become a promis-. Found insideIt is possible that there may exist an optimal strategy that uses a dominated row or column; if that is the case, Example: Colonel Blotto One of the sources for examples in game theory arises from military strategy. As secondary users may be selfish in nature and tend to be dishonest in pursuit of higher profits, we develop effective mechanisms to suppress their dishonest/collusive behaviors when secondary users distort their valuations about spectrum resources and interference relationships. Next, we manipulate parameters of the system to gain an, insight on how they affect the secondary user, payoff, that is, the average number of successful transmissions, per time slot. framework for wireless spectrum auctions. Simulation results are presented in section IV, followed by Section V that concludes this paper. There has been persistent e orts for nding the optimal strategies for the Colonel Blotto game. This leads to non-transitivity--Strategy A beats Strategy B beats Strategy C beats Strategy A (think rock, paper, scissors). As a reimburse- ment, the secondary users make payments to the primary network based on the service they receive. property is very important. In particular, we focus on investigating how incomplete information about the adversarys location can impact the rivals strategies. Now, p(100,5) is surely less than that by quite a bit, but it is still a rather large number. Only the algorithm knows. We analyze how users play the game and compare their behaviour with that reported . Brown, Hammer model threat assessment of. In fact, since we take into account the cost of transmission, we obtain even a generalization of the water-filling in the Found inside Page 92Blotto's optimal strategy : ( 1Is, 4/5, 0 ), Enemy's optimal strategy : ( 0, 3/5, 2l5 ), Value = 3/5. Such problems can occur in a variety of different guises, for example in the most efficient distribution of spare parts. The upshot is that analyzing strategies in Colonel Blotto isn't easy and is usually handled only via simulations. In this paper, focusing on defending against sweep jamming attacks and considering the inevitable transmission delay of an actual communication system, we propose a multi-step prediction Markov decision process (MPMDP) and set up a multi-step prediction Bellman iterative equation (MPBIE). The book presents choice theory, social choice theory, static and dynamic games of complete information, static and dynamic games of incomplete information, repeated games, bargaining theory, mechanism design and a mathematical appendix power restriction, and applied it in our simulation experiments. It deflnes the air interface for a wireless regional area network (WRAN) that uses fallow segments of the licensed (incum- bent) TV broadcast bands. To get a sense of it, note that if we ignore the order of the numbers, then we are looking for a partition of our numbern of troops. The standard includes a security sub- layer to provide subscribers with privacy, authentication, and confidentiality. Combing the complete strategy we proposed, the MPMDP strategy to maximize the utility of the FH communication system is obtained. number of battle elds he wins. Section III, we modify the game into a Colonel Blotto game, and derive the NE of the game, that is, the optimal strategy of power allocation. Forgetting the number k of pieces, there is a generating function to compute p(n). The transmitter's objective is to maximize the probability of communication connectivity with all the receivers. We conclude that bidding behaviors and pricing models have significant impact on auction outcomes. It turns out that it is possible to provide The most powerful of which is a self-propagating AI virus that could interactively teach radios to become malicious. By putting artificial intelligence in charge of wireless network devices, we are allowing unanticipated, emergent behavior, fitting a perhaps distorted or manipulated level of optimality. The Colonel Blotto game, first introduced by Borel in 1921, is a well-studied game theory classic. All rights reserved. Found inside Page 303 the point a 3, although it remains essential, does not appear in the spectrum of the optimal strategy. 7.9 Blotto games. The games known as Blotto games form an important class of convex games; they have the following structure. Opinions expressed by Forbes Contributors are their own. These strategies are optimal in the sense that each player can do no better by switching strategies, provided their opponent doesn't switch. 0 share . Two colonels each have a pool of troops that they divide simultaneously among a set of battlefields. Instead of choosing among a few discrete points, we might choose a point on the border of the hexagon at random. When there are $3$ districts and $2$ units of resources, there is still only one pure strategy: put all resources in the $3^{\text{th}}$ districts, namely, $(0, 0, 2)$. From the simulation results, the optimal correlated equilibria achieve better fairness and 5% ~15% performance gain, compared to the Nash equilibria. What Is (And Isn't) Scientific About The Multiverse. Recently, a team of computer scientists fromthe University of Maryland, Stanford University, and Microsoft Research announced that they had developed an algorithm to solve Colonel Blotto. This single-volume edition of a 2-volume set, discusses the theory of matrix games, linear and nonlinear programming, and mathematical economics while clarifying key mathematical concepts and demonstrates their applicability. 1959 edition. Furthermore, Ferdowsi et al. The coexistence between incumbent users and secondary users is referred to as incumbent coexistence. Aware of the absence of several primary users and the presence of a malicious user, a secondary user can allocate power to those fallow bands with a randomized strategy, in hope of alleviating the damage caused by the malicious user. It has been shown that the proposed closed-form method has extremely lower complexity compared to the bisection algorithm, while their performances are almost identical. cognitive radio denial of service attacks, wireless networks with transmission cost, in. } Fm1:]88 )8Gca\MEza__pu-%bK4{aCdu")YG\= |.6=e[horYk)Z* Offers an overview of mathematical modeling concentrating on game theory, statistics and computational modeling. We analyze how users play the game and compare their behaviour with that reported . Dynamic spectrum access (DSA), enabled by cognitive radio technologies, has become a promising approach to improve efficiency in spectrum utilization, and the spectrum auction is one important DSA approach, in which secondary users lease some unused bands from primary users. We model this scenario into a two-player zero-sum game, and derive its unique Nash Equilibrium under certain conditions using the Colonel Blotto game approach, which provides a minimax strategy that the secondary user should adopt in order to minimize the worst-case damage caused by the malicious user. CR technology enables unli- censed (secondary) users in WRANs to utilize licensed spec- trum bands on a non-interference basis to incumbent users. In fact, cognitive. Found inside Page 10For the infinite - move game it is shown that the pursuer possesses no optimal strategy . Blotto " game whose lack of optimal strategies is perhaps surprising . via a general result relating infinite games in extensive form 10 For example, spectrum, sensing techniques were investigated in [2], and [3] sho, that the sensing time could be reduced and spectrum agility, could be enhanced through user cooperation. A spectrum auction system must consider local demand and spectrum availability in order to maximize revenue and utilization. IEEE. While the wireless networks in which a cognitive radio device operates may implement device authentication, integrity checks and other higher-layer security mechanisms; the possibility of physical layer attacks, such as jamming attacks, still exists. In this, cognitive radio networks, where a malicious user wants to jam. It defines the air interface for a wireless regional area network (WRAN) that uses fallow segments of the licensed (incum- bent) TV broadcast bands. Found inside Page 382Determine the optimal strategy for each player and the expected value of the game. HINT [Use technology to do the pivoting in the associated linear programming problem] 0 Game T heoryMilitary Strategy Colonel Blotto is a well-known The MPMDP strategy and the optional intelligent sweep jamming attack strategy as a Nash equilibrium of the communication system and the jammer are proved. These two new collections, numbers 28 and 29 respectively in the Annals of Mathematics Studies, continue the high standard set by the earlier Annals Studies 20 and 24 by bringing together important contributions to the theories of games and When the equilibrium strategy is a mixed strategy, player must be indifferent among these values [14], namely, The similar argument can be applied to the jammer who, player can be better off by deviating unilaterally from the, From optimization problems (4) and (5), it is clear that, probability. The condition when the Stackelberg transmitter equilibrium strategy is non-sensitive to a priori information is derived. The average number of successful transmissions when the number of bands increases. Note that we only want partitions containing as many terms as the number k of battlefields. In Colonel Blotto, any strategy can be dominated by another. We prove theoretically, and numerically illustrate, that the Stackelberg equilibrium strategy for the transmitter can be non-sensitive to the a priori information, while the Nash equilibrium strategy for the transmitter is always sensitive to such information. Found inside Page 547( 1.24 ) In yet another pay - off game called Colonel Blotto game , the player P ( Colonel Blotto ) has an army of 4 Show that 4 1 4 T the value of this game is 14/9 and that 0 , 0 , is an optimal strategy for P , while 1 4 4 1 18 You may opt-out by. guarantees a bounded damage caused by the jammer. In this paper, we describe how adversaries can exploit or undermine such mechanisms to degrade the performance of 802.22 WRANs and increase the likelihood of those networks interfering with incumbent networks. The Colonel Blotto game, first introduced by Borel in 1921, is a well-studied game theory classic. 01/14/2019 by Soheil Behnezhad, et al. This seems to win 50% of the time against any move within the hex, but only wins ~33% of the time against any move barely outside the hexagon (found by . Found inside Page 415Determine the optimal strategy for each player and the expected value of the game. [HinT: Use technology to do the pivoting in the associated linear programming problem.] 45. Game Theory: Military Strategy Colonel Blotto is a In, Section II, the interaction between a secondary user and a, jammer is modeled into a two-player zero-sum game. An equilibrium of the Colonel Blotto game is a pair of n-variate distributions. Finally, relying on four real-world network systems, i.e., computer networks, Internet of vehicles, air transportation systems and social networks, simulation results show the effectiveness and feasibility of our proposed model, which is conducive to the design, management and maintenance of network systems. If Colonel Blotto is lazy in his thinking, he may just send 20 divisions to each field; denote this strategy by (20,20,20,20,20). Two colonels each have a pool of troops that they divide simultaneously among a set of battlefields. One is the, : both players allocate power to each band as. This text offers an exceptionally clear presentation of the mathematical theory of games of strategy and its applications to many fields including economics, military, business, and operations research. Found inside Page 131Show that the value of this game is 0, and that the optimal mixed strategy for both sides is (1/3, 1/3, 1/3). Ans. Blotto and his enemy are each equally likely to choose any strategy, and the game value is 0.8. The multi-channel power distribution game, which is the most widely studied in antagonism game based on the combination of multi-domains, was typically modeled using water injection algorithm [36] or Blotto game [9. Each battle eld is won by the colonel that puts more troops in it. An optimal strategy will win at least 50% of the time against any move/point. maxmin strategy for these games. In addition, in both gures, jamming causes more damage when the jammer has a higher, In order to show that the NE strategy minimizes the worst-, case damage for the secondary user, we have run simulations, with two other possible strategies considered: one decides, the number of bands to access according to the NE strategy. After almost a century Ahmadinejad, Dehghani, Hajiaghayi, Lucier, Mahini, and Seddighin provided a poly-time algorithm for finding the optimal strategies. There have been persistent efforts for finding the optimal strategies for the Colonel Blotto game. This book covers topics such as two-person games in strategic form, zero-sum games, N-person non-cooperative games in strategic form, two-person games in extensive form, parlor and sport games, bargaining theory, best-choice games, co This is the classic work upon which modern-day game theory is based. The average number of successful transmissions with different strategies adopted by the secondary user, in the scenario where the number of fallow bands varies from time slot to time slot due to presence/absence of primary users. game is well known in Game Theory (See [1] for further discussion of the game and its origins), and has been used to model many political and economic situations. utilized, the primary network leases its unused bandwidth and the idle relay node to the secondary users. Based on the derived, parameters and (6) (7), it is straightforward to write down the. Cognitive radio technologies have been, intensively studied in recent years. ulation attacks in cognitive radio networks, [9] A. Sethi and T. X. arXiv preprint arXiv:1901.04153 (2019). After almost a century Ahmadinejad, Dehghani, Hajiaghayi, Lucier, Mahini, and Seddighin provided a poly-time algorithm for finding the optimal strategies. To maintain communication the transmitter must keep the SINR greater or equal to an SINR threshold, while the adversary aims to break the communication by making this SINR less than this threshold. Google Scholar; mile Borel. Within each battlefield, the player that allocates the higher level of force wins. In order to improve the spectrum utilization, a. Calculating the equilibrium of this game is complicated and so is the choice of the optimal strategy. the communications of secondary users by injecting interference. Let's get back to the game. In one particular instance, with n = 100 and k = 10, the winning strategy was (17,3,17,3,17,3,17,3,17,3). However, its scope of application is still restricted by the . In the Colonel Blotto game, two players simultaneously distribute forces across n battlefields. The average number of successful transmissions when the required SINR threshold increases. one jammer, and model their interaction into the Colonel, Blotto game which originates to solve the problem where, two players simultaneously distribute forces across multiple, jammer can cause to the secondary user, and what strategy, the secondary user should adopt. In AAAI. With the assumption that the jammer can learn the MPMDP strategy of the FH communication system, we furtherly study its optional sweep jamming attack strategy. The Colonel Blotto game, first introduced by Borel in 1921, is a well-studied game theory classic. In this paper, we consider a cognitive system wherein there exists a pri- mary relay network and a secondary network. In this paper we focus primarily on PHY-layer issues, describing several classes of attacks and giving specific examples for dynamic spectrum access and adaptive radio scenarios. Found inside Page 110This is an example of a two player zero-sum game, when Blotto wins one, Zappo loses one, and vice versa. However, the game does not have an optimal pure strategy for winning the game. There are many mixed strategy equilibria for the is the Lagrangian multiplier for the jammer. We consider jamming in wireless networks with transmission cost for both transmitter and jammer. There is no ideal way to allocate your soldiers because strategies are non-transitive; Strategy A may defeat Strategy B, and Strategy B may beat Strategy C, but then Strategy C may beat A.
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