blotto game optimal strategy

Since the algorithm works only with two players, and the current primary fields are still too crowded, we will have to wait and see what it will predict in the general election. 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. Among all strategies. [OC] Optimal Fantasy Strategy Using Game Theory (wonkish ... The coexistence between incumbent users and secondary users is referred to as incumbent coexistence. The Same Reason You Would Study Anything Else, The (Mostly) Quantum Physics Of Making Colors, This Simple Thought Experiment Shows Why We Need Quantum Gravity, How The Planck Satellite Forever Changed Our View Of The Universe. Found inside – Page 313... and that this strategy is observed by the attacking player, who then chooses an optimal response (Tambe, 2012: pp 4–8). ... A related concept is that of the much studied Colonel Blotto game, in which two players must simultaneously ... This book is a collection of essays that tie together the fields spanned by Jeffrey S. Banks' research on this subject. Join ResearchGate to find the people and research you need to help your work. In addition, in both figures, 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. In this letter, we consider multiple peer to peer communications using a trusted relay network encountered to the jamming network attack. There has been persistent e orts for nding the optimal strategies for the Colonel Blotto game. 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 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 ... In particular, we focus on investigating how incomplete information about the adversary’s location can impact the rival’s strategies. Algorithmic Game Theory: 11th International Symposium, SAGT ... With the advent of cognitive radios, existing wireless networks are expected to undergo a radical change in how they operate. In, Section II, the interaction between a secondary user and a, jammer is modeled into a two-player zero-sum game. These strategies are optimal in the sense that each player can do no better by switching strategies, provided their opponent doesn't switch. 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. This research work focuses on assessing potential cognitive radio specific physical layer attacks using the so-called Hammer model. 2017. Although cognitive radio has been intensively studied in recent years, only a few works have discussed security aspects. By means of numerical examples we study an important particular case of jamming of the OFDM system when the jammer is situated There have been persistent efforts for finding the optimal strategies for the Colonel Blotto game. Opportunistic Spectrum Sharing (OSS). [OC] Optimal Fantasy Strategy Using Game Theory (wonkish ... Note that the order of the allocation matters; that is (24,24,24,24,4) is different from (24,24,24,4,24). Our investigation, however, revealed that the security sublayer falls short of addressing all of the key security threats. 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. We then describe some approaches for mitigating the effectiveness of these attacks by instilling some level of "common sense" into radio systems, and requiring learned beliefs to expire and be relearned. Whoever captures the most districts wins the election. Found inside – Page 223Show that each player has an optimal stationary family of strategies. (18.9) Colonel Blotto's games. Consider the game of Example 15.2 with Blotto's action set A = {0, 1,..., m} and the enemy's action set B = {0, 1,..., n}. Game Theory: Mathematical Models of Conflict - Page 92 resistant multi-winner cognitive spectrum auction game. In Colonel Blotto, any strategy can be dominated by another. Cognitive Radio (CR) is seen as one of the enabling tech- nologies for realizing a new spectrum access paradigm, viz. Found inside – Page 209Otherwise the game is a standoff. What is Blotto's optimal strategy? 6. SOLUTIONS BY LINEAR PROGRAMMING It was mentioned earlier that matrix games are related to linear programs. The connection is quite apparent, since the two hemigames ... Two colonels each have a pool of troops that they divide simultaneously among a set of battlefields. Two colonels each have a pool of troops that they divide simultaneously among a set of battlefields. 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. Both analytical and experimental results verify the efficiency of the proposed approach. Simulation results are presented in section IV, followed by Section V that concludes this paper. When there are $2$ districts, there is a single pure optimal strategy: put all resources in the $2^{\text{nd}}$ district. Found inside – Page 10For the infinite-move game it is shown that the pursuer possesses no optimal strategy. However, Karlin proves that in the truncated game of moves ... modified "Colonel Blotto" game whose lack of optimal strategies is perhaps surprising. , vol. 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 heory—Military Strategy Colonel Blotto is a well-known ... In this section, we will find 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 battlefields with a payoff equal to the sum of, outcomes from individual battlefield [13]. 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. Finally, simulation results are presented to evaluate the proposed auction mechanisms, and demonstrate the complexity reduction as well. 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 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 ... © 2021 Forbes Media LLC. Two colonels each have a pool of troops that they divide simultaneously among a set of battlefields. 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 secondary user employs different power allocation strategies and the jammer is able to adjust his/her jamming strategy. In the Colonel Blotto game, two colonels divide their troops among a set of battle elds. Although cognitive radio has been intensively studied in recent, years, only a few works have discussed security aspects. 369--375. 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. Simulation results are presented in section, Consider the case when a secondary user, equipped with, multiple radios, wants to transmit several packets to its recei. Consider the previous 3 player Blotto game. An algorithm based on the bisection method to find the fixed point (and so equilibrium strategies) is developed, and its convergence is proven. Each battle eld is won by the colonel that puts more troops in it. W, examined the output samples from the procedure under a, variety of settings, and found the resultant empirical marginal. In Colonel Blotto, any strategy can be dominated by another. 1921. However, due to their inherent vulnerability as well as the limited management and operational capability, these network systems are constantly under the threat of malicious attackers. For certain applications of the Colonel Blotto game — such as the United States 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. On the other hand, the relays are trying to avert the jamming attack by adjusting the beamformer weights in such a way that maximum SINR at destination nodes occurred while keeping the total transmission power of the relays under a specified threshold. Let's get back to the game. To illustrate this point, Malcolm Gladwel l tells the story of Vivek Ranadivé: As a reimburse- ment, the secondary users make payments to the primary network based on the service they receive. Since at least one mixed-strategy NE exists in a game, [14], we can safely draw a conclusion that this characterizes, the unique NE in the jamming game. 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. Next, we propose a low-complexity algorithm, in which the primary network charges the secondary users at a sub-optimal price and gains close-to-optimal revenues. The winner of each battlefield is the colonel who puts more troops in it and the overall . . Cognitive Radio (CR) is seen as one of the enabling tech- nologies for realizing a new spectrum access paradigm, viz. For solving the second optimization problem, we propose a closed-form solution, as well. Simulation results are presented in section IV, followed by Section V that concludes this paper. Only the algorithm knows. Found inside – Page 294A typical feature of such conflicts is that optimal strategies involve using mixed strategies to 'confuse' the ... In this section we introduce three games that are useful in the analysis of Colonel Blotto game (but are also of interest ... If the secondary. arXiv preprint arXiv:1901.04153 (2019). A mixed strategy is maximin if it maximizes the guaranteed expected utility against any strategy of the opponent. II. Therefore, in such attack-defense scenarios, it is particularly significant to make the best use of defenders' limited resources and capability. A Colonel Blotto game is a type of two-person constant-sum game in which the players (officers) are tasked to simultaneously distribute limited resources over several objects (battlefields).. These attacks demonstrate the capabilities of an attacker who can manipulate the spectral environment when a radio is learning. The jammers, that are equipped with beamformer are trying to minimize the maximum Signal-to-Interference-plus-Noise Ratio (SINR) at the destinations subject to the constraint that the maximum SINR at each of the relay nodes does not exceed a predefined threshold level. There have been persistent efforts for finding the optimal strategies for the Colonel Blotto game. Meanwhile, in communication connectivity problems the transmitter must keep its SINR greater than or equal to a threshold value to ensure a connection can be sustained [12. Assuming both players have full information including, is a zero-sum game, the NE not only provides an equilibrium, point from which neither players have the incentiv, at maximizing the number of successful transmissions by using a randomized, strategy to allocate power in four bands and a jammer (right) attempts to. However, relying on mixed strategies, as has been the case in all of the CBG literature [8]- [13], presents serious challenges to the tractability, applicability, and implementability of the derived solutions in real-world environments. The leader who sends the most troops to a particular battlefield will win that region; whoever takes the most regions wins the battle. Both probabilities for each Colonel must add-up to one. ing approach to increase the efficiency of spectrum utilization. Instead of choosing among a few discrete points, we might choose a point on the border of the hexagon at random. Lastly, simulation results show that the property and theorems proposed in this paper are valid, and our MPMDP strategy outperforms similar algorithms in terms of the applicable range and performance. Our design includes a compact and yet highly expressive bidding language, various pricing models to control tradeoffs between revenue and fairness, and fast auction clearing algorithms to compute revenue-maximizing prices and allocations. We use the framework of non-zero-sum 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. However, spectrum auctions are different from existing auctions studied by economists, because spectrum resources are interference-limited rather than quantity-limited, and it is possible to award one band to multiple secondary users with negligible mutual interference. In Colonel Blotto, any strategy can be dominated by another. The rst game that we consider is Colonel Blotto, a well-studied game that was introduced in 1921. In this paper, we propose a networked Colonel Blotto game for the attack-defense strategy, where the . fallow bands with a randomized strategy, in hope of alleviating, 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-, As the last decade witnessed a dramatic increase in the, demand for radio spectrum, cognitive radio [1] has become a, censed users (secondary users) are allowed to access licensed, bands on a non-interference basis to legacy spectrum hold-, ers (primary users). Furthermore, Ferdowsi et al. His opponent will do the same. Found inside – Page 137D. Solve the games with the following payoff matrices, finding the optimal mixed strategies for each player and the value of the game. ... Colonel Blotto and his enemy each try to occupy two posts by properly distributing their forces. ... for every action in A1 and A2 as (19); 6 for l = 1, 2 do 7 Generate empty gene pool G l = ∅ and action set A l = ∅; The low spectrum utilization has limited the development of wireless communication is a common view In Cognitive Radio (CR). Although the text assumes only an elementary-level training in algebra, students who complete a course around this text will be equipped to read nearly all of the professional literature that makes use of game theoretic analysis. Two colonels each have a pool of troops that they divide simultaneously among a set of battlefields. radio networks are extremely vulnerable to malicious attacks. 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. 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. Found inside – Page 512... of force to each of func-thenbattlefields is a random n-tuple drawn from the n-variate distribution function Pi. Optimal Strategies Theorem 1 summarizes Roberson's (2006) characterization of equilibrium in the Colonel Blotto game in ... Therefore, the solutions z * proposed in, ... We next investigatē z * n . 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. In one particular instance, with n = 100 and k = 10, the winning strategy was (17,3,17,3,17,3,17,3,17,3). marginal distribution of the NE strategy as in the theorem. In this aspect, Blotto is a complex form of Rock, Paper, Scissors. The payo of each colonel Assuming the Blotto game conception then D12 actually has negative soldiers in ft% (and has extra soldiers then for Reb, FG%) and so the game mechanics become different. ware defined radio,” Ph.D. Thesis, KTH Royal Institute of T. in spectrum sensing for cognitive radios, [4] Z. Han, C. Pandana, and K. J. R. Liu, “Distributive opportunistic, spectrum access for cognitive radio using correlated equilibrium and. The goal in the Colonel Blotto game is to find optimal (i.e.,maximin) strategies of the players. It is proven that the problem has unique equilibrium in power allocation strategies, and the equilibrium is derived in closed form. This leads to non-transitivity--Strategy A beats Strategy B beats Strategy C beats Strategy A (think rock, paper, scissors). Optimal Strategies of Blotto Games: Beyond Convexity. In AAAI. With the increasingly fierce competition for electromagnetic spectrum resources and the rapid development in artificial intelligence, frequency-hopping (FH) communication systems are facing serious intelligent jamming attacks. How should they do it? What Is (And Isn't) Scientific About The Multiverse. From the simulation results, the optimal correlated equilibria achieve better fairness and 5% ~15% performance gain, compared to the Nash equilibria. Opinions expressed by Forbes Contributors are their own. (ۏmÃÄĸ,Ñsêidåcê–ìÇrÎIޏ¼ïóœ¼ÏKSéiMÓÒbÅ;Ûv(W)Jví[ûêt$/¥ñ¢tpg›àýÌ4œ“¹P÷d÷ã?DžH=ÉBîÏg—½DIh:cáҼ׋¬.-o4ò‡jUÚ*¯2ñ†[UScÐ×͘¼–¯æu&_¥2iô:£VeT×êR–B_SoÐT©Mò Offers an overview of mathematical modeling concentrating on game theory, statistics and computational modeling. Generate child chromosome g k l from g k l as (31) with mutation probability γ l , and add it to G l ; 16 Generate child action a k l as (25) and (26) and add it to A l ; 17 end 18 end 19 Update G l = G l and A l = A l ; 20 end 21 end 22 Calculate the utility u 1 (a k1 1 , a k2 2 ) for every pair of a k1 1 , a k2 2 in practical action sets A 1 , A 2 ; 23 Solve the linear programming problems in (22) and (23); 24 Output Nash equilibrium strategy (s 1 , s 2 ) and the expected utility E(u 1 ); action according to. Calculating the equilibrium of this game is complicated and so is the choice of the optimal strategy. 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 ... © 2008-2021 ResearchGate GmbH. 6 provides, the number of successful transmissions when the secondary, better than the other two strategies, and the secondary user has, to choose it as the optimal power allocation strategy against, lem under malicious jamming attacks as a two-player zero-, sum game where the secondary user aims at maximizing the, number of successful transmissions by randomizing amounts, of power allocated to each band. analytical expressions for the equilibrium strategies. 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. 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 ... The problem is that once the numbers of troops and regions become fairly large it is rather difficult to analyze all the possibilities and find equilibria. For certain applications of the Colonel Blotto game — such as the United States For instance, marginal distribution from 500 samples has been plotted in, Fig. 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. We also discuss countermeasures that may be able to address those threats. 1, pp. Colonel Blotto Games in Network Systems: Models, Strategies, and Applications, A multi-link communication connectivity game under hostile interference, Optimal frequency-hopping anti-jamming strategy based on multi-step prediction Markov decision process, Colonel Blotto Game Aided Attack-Defense Analysis in Real-World Networks, Worst-case Jamming Attack and Optimum Defense Strategy in Cooperative Relay Networks, A Power Control Game with Uncertainty On the Type of the Jammer, A Connectivity Game with Incomplete Information on Jammer’s Location, A dual radar and communication system facing uncertainty about a jammer’s capability, Colonel Blotto Game with Coalition Formation for Sharing Resources: 9th International Conference, GameSec 2018, Seattle, WA, USA, October 29–31, 2018, Proceedings, A Scalable Collusion-Resistant Multi-Winner Cognitive Spectrum Auction Game, Security in Cognitive Radio Networks: Threats and Mitigation, A General Framework for Wireless Spectrum Auctions, Cognitive radio: An integrated agent architecture for software defined radio, Doctor of Technology, Security Vulnerabilities in IEEE 802.22 (Invited Paper), Hammer Model Threat Assessment of Cognitive Radio Denial of Service Attacks, A Jamming Game in Wireless Networks with Transmission Cost, Distributive Opportunistic Spectrum Access for Cognitive Radio using Correlated Equilibrium and No-Regret Learning, Spectrum Allocation Algorithm Aware Spectrum Aggregation in Cognitive Radio Networks, On a Cognitive Radio Network's Random Access Game With a Poisson Number of Secondary Users, Spectrum pricing games with arbitrary bandwidth availability probabilities, Revenue maximization and distributed power allocation in cognitive radio networks, Conference: Global Telecommunications Conference, 2009. We conclude that bidding behaviors and pricing models have significant impact on auction outcomes. For the case in which all the primaries and secondaries are in a single location, we explicitly compute a Nash equilibrium (NE) and, Cognitive radio is an enabling technology that allows un- licensed users to opportunistically access the spectrum in order to enhance the spectrum e-ciency. An equilibrium of the Colonel Blotto game is a pair of n-variate distributions. 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. The Colonel Blotto game, first introduced by Borel in 1921, is a well-studied game theory classic. On the buyer side, given the specifled price, the secondary users competitively access the spectrum and em- ploy the primary relay node to forward their packets. All Rights Reserved, This is a BETA experience. 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. The payo of each colonel We flrst charac- terize the interactions between the primary and secondary users using a buyer/seller model. The goal in the Colonel Blotto game is to find optimal (i.e.,maximin) strategies of the players. 01/14/2019 ∙ by Soheil Behnezhad, et al. ∙ 0 ∙ share . And the idle spectrum holes are too narrow to support high-speed communication is an another problem. With this procedure, we can finally obtain the NE strategy for, In this section, we present some simulation results to, secondary user has a total power constraint, First, we want to validate through simulation the procedure, that generates a kind of joint distribution by using marginal, distribution functions and the total power constraint. 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, $(0, 0, 2)$. In AAAI. in the following, but our results can be extended to cases, According to the definition of the NE, no single, , the secondary user is better off by moving, when the jammer is capable of adapting his/her. This optimization problem would cause the maximum destruction in the legitimate nodes. W, the unique Nash Equilibrium (NE) of the jamming game under, optimal in the sense that it minimizes the worst-case damage, The remainder of this paper is organized as follows. The Colonel Blotto game, first introduced by Borel in 1921, is a well-studied game theory classic. Found inside – Page 20016.2 Strategies and Payoffs An n-person game in normal form is merely a list of options, called pure strategies, ... payoff matrix A. The strategy (2, 0), or (0, 2), is an optimal strategy for Colonel Blotto, as well as for the Enemy. 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. You may opt-out by. 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. 2, which is consistent with equation (8) in Theorem 1 for. IEEE. Optimal Strategies of Blotto Games: Beyond Convexity. This jamming game has, been converted to a Colonel Blotto game, the unique Nash. Average number of successful transmissions, SINR threshold required for an successful transmission (dB), user employs different power allocation strategies and the jammer is able to, 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, based on a naive assumption that the jammer would inject, equal interference to each band. payoffs of two players always add up to zero. However, its scope of application is still restricted by the . optimization problem, we propose a closed-form solution. The problem is formulated as a non-zero Bayesian game between the transmitter and the adversary in a Nash and Stackelberg framework for Rayleigh channel fading gains. To address this Access scientific knowledge from anywhere. In communication reliability problems, the transmitter's payoff is a function of throughput or Signal-to-Interference-plus-Noise Ratio (SINR) at the receiver, and the transmitter intends to maximize such payoff [3,4,5,6,7,8,9,10,11]. When there are, bands available, the user allocates power, under the constraint that the total power should not exceed, sabotage on the communication link by injecting interference, all channel gains are 1, since they can be absorbed into the, to-interference-and-noise ratio (SINR) exceeds the minimum, packets can be transmitted successfully; otherwise, the link is, In the presence of a jammer, the secondary user wants, to maximize the number of successful transmissions, by, randomizing his/her power allocation strategy, hand, the jammer also wants to find a randomized power, allocation strategy such that the number of the secondary, user’s successful transmissions is minimized. power restriction, and applied it in our simulation experiments. 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. This book provides unified, comprehensive coverage of contest theory as developed in economics, computer science, and statistics, with a focus on online services applications, allowing professionals, researchers and students to learn about ... Faster and Simpler Algorithm for Optimal Strategies of Blotto Game.. strategy in order to avoid the jamming as much as possible. Finally, the impact that the presence of primary users have on the optimal access strategy is assessed.
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