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Thanks for contributing an answer to Mathematics Stack Exchange! Should I cancel the daily scrum if the team has only minor issues to discuss? Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. The unifying technical idea is to analyze the behavior of the strong ϵ-core as ϵ varies. We also show that for certain classes of combinatorial games a rather small collection of coalitions that determine the nucleolus can be givenbeforehand, i.e., without knowing the nucleolus.Journal of Economic LiteratureClassification Number: C71. $2-{x_1}\leqslant\epsilon,\;5-{x_2}\leqslant\epsilon,\;4-{x_3}\leqslant\epsilon,-{x_2}-{x_1}+14\leqslant\epsilon,\\-{x_3}-{x_1}+18\leqslant\epsilon,\;-{x_3}-{x_2}+9\leqslant\epsilon,\;{x_3}+{x_2}+{x_1}=24$. minimize $\epsilon$ subject to, We compute next core. MathJax reference. @Raylan I have revised my answer to grasp your questions. v({1,2})=14, There is a section on it at Cooperative game. To learn more, see our tips on writing great answers. From this vector $\mathbf{y}$, we get -- as you correctly calculated -- the following excess vector $exc_y=\{0,-7,-1,-5,-1,0,-6,0\}$. For more complex games, the search process stops after at most $\binom{n}{2}-1$ iteration steps. In order to investigate this issue further, we have to recall that the pre-nucleolus can be characterized by the following axiomatization: SIVA, ETP, COV, and RGP (cf. Why is my half-wave rectifier output in mV when the input is AC 10Hz 100V? [4] L.S. The nucleolus selects the element in the core, if this is nonempty, that lexicographically minimizes the vector of non-increasing ordered excesses of coalitions. I suspect that's adequate at least as a basis? To observe this, let us assume that $n=4$, then the set of coalitions supporting player $1$ without counting on the cooperation of player $2$ is $\mathcal{G}_{12}=\{\{1\},\{1,3\},\{1,4\},\{1,3,4\}\}$. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Narration. Game theory is a fascinating subject. Close this message to accept cookies or find out how to manage your cookie settings. In this paper we show that already relatively small collections of coalitions determine the nucleolus of a TU-game. An Introduction, second ed., John Wiley & Sons, Inc., 2013. Can you give more complicated examples? \end{equation}. Two solution concepts for cooperative games in characteristic-function form, the kernel and the nucleolus, are studied in their relationship to a number of other concepts, most notably the core. Learn more in: Game Theory for Cost Allocation in Healthcare As I mentioned before, you should rethink your termination condition. Use MathJax to format equations. v({1,3})=18, The nucleolus is de ned as the set of imputations x for which ~c(x) is lexicographically minimal . Now a payoff-vector is in the nucleolus if the largest excess of any coalition is … International Journal of Game Theory … See a textbook for the exact definition. \end{equation}. \mathcal{S}(\mathbf{y}) = \{\{1,3\},\{1,2\},\{2\},\{2\},\{3\},\{1,3\}\} This is the case, for instance, for three person games or convex games. A column vector $\mathbf{a}$ can be obtained by $\mathbf{E} \; \vec{\alpha} \in \mathbb{R}^{n}$ whereas the vector $\vec{\alpha}$ is given by $\alpha_{ij} := (v(S_{ij}) - v(S_{ji})) \in \mathbb{R} $ for all $i,j \in N, i < j $, and $\alpha_{0} := v(N)$. (1992) The general nucleolus and the reduced game property. Special emphasis is placed on two important aspects of cooperative games: profit allocation and stability. Solve problem: 1 & -1 & -1 & 1 \\ If you look for a game where both nucleolus and prenucleolus are nonempty and yet they differ, Exercise 20.15 in "Game Theory" by Maschler-Solan-Zamir asks to compute the nucleolus and the prenucleolus of the three-player game where v(1,2)=1 and v(S)=0 for every other coalition S. The nucleolus is (0,0,0), since this is the only imputation. This is simply done by, We check first some game properties. Moreover, let us assume that the coalitions $\{\{1,3\},\{1,4\},\{1,3,4\}\}$ have maximum surpluses, then the smallest cardinality is $2$ and we single out the coalitions $\{\{1,3\},\{1,4\}$, and taking finally the lexicographical minimum, which is $\{1,3\}$. \begin{bmatrix} Solution is: $\epsilon=-\frac12,\quad [x_1,x_2,x_3]\in \operatorname{convexhull}\left( [[9,\frac{11}{2},\frac{19}{2}],[14,\frac{11}{2},\frac{9}{2}]]\right) $. 0 & 4 & -2 \\ Email your librarian or administrator to recommend adding this book to your organisation's collection. The nucleolus of a coalitional game relative to a set of vectors in ℝ N consists of the vectors in that set whose vector of excesses are minimal in the lexicographic order. The nucleolus $ n (v) $ exists and is unique for any game with a non-empty set of imputations. Nucleolus and prenucleolus of a weighted majority game, Weighted War - Game of Mind and Probability. Now let us turn to covariance under strategic equivalence (COV). Given the following table of values and excesses of coalitions S and imputation $\vec{x} = (9,6,9)$: How do I find the Nucleolus? When the excess is positive, the members of the coalition are not content with the total amount that they together receive at x, which is less than the worth of the coalition. Convexity is denied through, However, we observe that the game fulfills a weak convexity condition, namely average-convexity by, Hence, the nucleolus is identical to the pre-nucleolus. Cooperative game theory considers how to distribute the total income generated by a set of participants in a joint project to individuals. We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Nucleolus = The nucleolus is a region found within the cell nucleus that is concerned with producing and assembling the cell's ribosomes. If you find any errors, please do send email to hari@csa.iisc.ernet.in. Since your game is a three person permutation game which is zero-monotonic, the kernel is a sole point, and it coincides with the nucleolus of the game. (1992) The least core, nucleolus, and kernel of homogeneous weighted majority games. I = \{ (x_{1}, x_{2}, x_{3}) \in \mathbb{R}^{3} \; | & \; x_{1} \geq 2, x_{2} \geq 5, x_{3} \geq 4 \\ Illustration. The prenucleolus of a coalitional game is its nucleolus relative to the set of preimputations, that is, the set of all efficient vectors. compute some game properties or solution concepts, like convexity, the Shapley value, the (pre-)nucleolus or a pre-kernel element. You can check out that the maximum surpluses are balanced, or that $\theta(\nu(v))$ is lexicographically smaller than $\theta(\mathbf{y})$. Games and Economic Behavior 4:4, 588-605. The column vector $\mathbf{a}$ is given by $\mathbf{a}=\{60,8,40\}$. s_{ij}(\mathbf{x},v):= \max_{S \in \mathcal{G}_{ij}} e^{v}(S,\mathbf{x}) \qquad\text{where}\; \mathcal{G}_{ij}:= \{S \;\arrowvert\; i \in S\; \text{and}\; j \notin S \}. Solving this system of linear equations $\mathbf{Q}\;\mathbf{x} + \mathbf{a} = \mathbf{0}$, or alternatively $\mathbf{E}^{\top}\;\mathbf{x} + \vec{\alpha} = \mathbf{0}$, we get the nucleolus for this game, which is $\nu(v)=\{23/2,11/2,7\}$. For any pair of players $i,j \in N, i\neq j$, the maximum surplus of player $i$ over player $j$ with respect to any pre-imputation $\mathbf{x}$ is given by the maximum excess at $\mathbf{x}$ over the set of coalitions containing player $i$ but not player $j$, thus, \begin{equation} \mathbf{Q}= In this example $fmin$ should be $-3.5$ and not $-3$. Here, I can only sketch this approach. \mathbf{E} := [\mathbf{E}_{1,2}, \ldots ,\mathbf{E}_{n-1,n},\mathbf{E}_{0}] \in \mathbb{R}^{^{n \times q}}. v({2,3})=9, For the arboricity game defined on the same graph, a partial order can be defined on the prime sets according to core allocations, and then the nucleolus can be computed efficiently by solving only two linear programs in the Maschler scheme [21, 22]. The computations were accomplished by the Matlab toolbox MatTuGames that can be found from the URL, To start with, we need to rearrange the order of coalitions from a lexicographical order to its unique integers. A course in game theory, MIT Press, 1994, Section 14.3.3. A game theory concept defined as minimizing the maximum “unhappiness” of a coalition. By these computations, we infer that the payoff $pn_{v}$ is the pre-nucleolus of the game. The process of finding of pre-nucleolus illustrated on an example of a fuzzy game. Since your game is a three person permutation game which is zero-monotonic, the kernel is a sole point, and it coincides with the nucleolus of the game. For this purpose, consider a situation where an estate is insufficient to meet simultaneously all of the debts/claims of a set of claimants, such a situation is known in game theory as a bankruptcy problem. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. Game Theory Lecture Notes By Y. Narahari Department of Computer Science and Automation Indian Institute of Science Bangalore, India July 2012 COOPERATIVE GAME THEORY Other Solution Concepts Note: This is a only a draft version, so there could be flaws. If the rank is full, i.e., rank=n, and the set is balanced, you can break the computation, since there is no improvement possible. \end{bmatrix}. Other articles where Nucleolus is discussed: nucleus: Nucleoli are small bodies often seen within the nucleus. This gives us some evidence that the solution is correct. From this excess vector $exc_y$ we get now the following set of most effective coalitions for each pair of players: \begin{equation} The nucleolus looks for an individually rational distribution of the worth of the grand coalition in which the maximum dissatisfaction is minimized. In the sequel, we just rely on COV and RGP. The corresponding excess vector is given through For determining $\mathcal{S}(\mathbf{y})$, you should also have a look on the following paper, Could you maybe add a bit more explanation via an. What would be the most efficient and cost effective way to stop a star's nuclear fusion ('kill it')? Proceeding in an analogous way for the remaining pair of players $[1,3]$ and $[2,3]$, matrix $\mathbf{E}$ is quantified by, \begin{equation} v({3})=4, We apply the same procedure to see if we can discard this vector as the nucleolus of the game. 1993. This paper surveys some applications of cooperative game theory to supply chain management. “Unhappiness” (or “excess”) of a coalition is defined as the difference between what the members of the coalition could get by themselves and what they are actually getting if they accept the allocations suggested by the nucleolus. [5] L.S. Notice, that in this specific case, we needed only one iteration step. We see here that coalition $\{1,3\}$ has maximum surplus. For those games, the geometric properties of the pre-kernel does not collapse into a single point. M.J. Osborne and A. Rubinstein. If we cannot complete all tasks in a sprint. Megiddo N. (1974) "On the nonmonotonicity of the bargaining set, the kernel and the nucleolus of a game", SIAM Journal on applied mathematics 27(2), pp. D. Schmeidler , The nucleolus of a characteristic function game, SIAM Journal on Applied Mathematics 17 (1969), 1163–1170. Solve problem: $$Nucleolus=\left[ \frac{23}{2}, \frac{11}{2}, 7\right].$$. whereas the order of the pairs of players in $\mathcal{S}(\mathbf{y})$ is given by $\{[1,2],[1,3],[2,3],[2,1],[3,1],[3,2]\}$. Following assembly, ribosomes are transported to the cell cytoplasm where they serve as the sites for protein synthesis. From this matrix we construct matrix $\mathbf{Q}$ by $\mathbf{E} \; \mathbf{E}^{\top}$, inserting the numeric values, matrix $\mathbf{Q}$ is specified by, \begin{equation} Making statements based on opinion; back them up with references or personal experience. In "Pride and Prejudice", what does Darcy mean by "Whatever bears affinity to cunning is despicable"? We compute then the pre-nucleolus by, Due to floating point arithmetic's it is always a good idea to impose some cross-checks on a computed solution. that was classified by an older version of the gt.mac package written by Aleksas Domarkas as the nucleolus of the game. Notice that $\mathbf{1}_{S}:N \mapsto \{0,1\}$ is the characteristic vector given by $\mathbf{1}_{S}(k):=1$ if $k \in S$, otherwise $\mathbf{1}_{S}(k):=0$. The Shapley value 4. Definition (Nucleolus) Let (N,v)be a TU game. \end{equation} Again, we get a confirmation while executing, Finally, let us consider the following payoff. We review concepts in cooperative game theory and their mathematical de nitions in Section 2.1. The nucleolus is known to be NP -hard to compute over compact coalitional games, that is, over games whose functions specifying the worth associated with each coalition are encoded in terms of polynomially computable functions over combinatorial structures. As Maschler et al. If this set is not unique, we determine the coalitions that have smallest cardinality, and from this set the coalition that has lexicographical minimum. The challenge is then to allocate the payoff $${\displaystyle v(N)}$$ among the players in some fair way. We all know many entertaining games, such as chess, poker, tic-tac-toe, bridge, baseball, computer games — the list is quite varied and almost endless. @Holger I. Meinhardt. Asking for help, clarification, or responding to other answers. B. Peleg and P. Sudhoelter (2007)). Table with two different variables starting at the same time. The nucleolus solution is an important concept in cooper-ative game theory even though it is not easy to calculate. The core 5. In general, the nucleolus can be specified while applying its axiomatization, or for more complex games, while solving a sequence of linear programs. The gel-like matrix in which the nuclear components are suspended is the nucleoplasm. @Trurl Thank you, that certainly is of help. Notice, that this approach is applicable for the (pre-)nucleolus computation for games where the (pre-)nucleolus coincides with the (pre-)kernel. And indeed, from the returned result we observe that a Kohlberg criterion is satisfied. v({1,2,3})=24 According to the empirical evidence the search process is generically terminated for less than $n+1$ iteration steps. For instance, for the pair of players $[1,2]$, we find out these coalitions that support the claim of player $1$ without counting on the cooperation of player $2$, these are the coalitions $\mathcal{G}_{12} =\{\{1\},\{1,3\}\}$ having excess $\{-7,0\}$. We compute least core. A course in game theory, MIT Press, 1994, Section 14.3.3. The main assumption in cooperative game theory is that the grand coalition $${\displaystyle N}$$ will form. You need to check the balancedness condition of the set of tight coalitions and the rank. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. Definition (Nucleolus) Let (N,v) be a TU game. The fuzzy pre-nucleolus defined by a new way. E.N.Barron, Game theory. The Nucleolus, trying to capture the intuition of minimizing dissatisfaction of players, is one of the most well-known solution concepts among various attempts to … If a payoff-vector x is in the core, the excess v (S) − ∑ i ∈ S x i must be nonpositive for every coalition S, for otherwise, the coalition could block the payoff-vector and the absence of the possibility to block characterizes the core. Twist in floppy disk cable - hack or intended design? Stéphane Airiau (ILLC) - Cooperative Games Lecture 5: The nucleolus 9 a detour: -core and least-core "Nucleolus (Game theory)" is in the list of economics requested articles. The notion that underlies the nucleolus is that of excess: the excess of a coalition at a vector x in ℝN is the difference between the worth of the coalition and the total amount that the members of the coalition receive according to x. If next core consists of exactly one point we stop computation. Cooperative games. -1 & 1 & 1 & 1 I created game theory package "gt.mac" on free CAS Maxima, (%i2) G:[{1,2,3}, Why did DEC develop Alpha instead of continuing with MIPS? M.J. Osborne and A. Rubinstein. In general, the nucleolus can be specified while applying its axiomatization, or for more complex games, while solving a sequence of linear programs. I tested a lot of examples from textbooks and articles for n<=5. In a cooperative game the nucleolus is contained in the kernel. My book wasnt clear on the method of calculating it, so Id like to find a general approach. 2 & -2 & 4 [v({1})=2, Do Magic Tattoos exist in past editions of D&D? Digital and Numeric Techniques and Their Applications in Control Systems, Part 2 of 2 - Control and Dynamic Systems: Advances in Theory and Applications, 483-517. $2-{x_1}\leqslant\epsilon,\;4-{x_3}\leqslant\epsilon,\;\frac{17}{2}-{x_1}\leqslant\epsilon,\;\frac{7}{2}-{x_3}\leqslant\epsilon,\;{x_3}+{x_1}+\frac{11}{2}=24$. The expression $s_{ij}(\mathbf{x},v)$ describes the maximum amount at the pre-imputation $\mathbf{x}$ that player $i$ can gain without the cooperation of player $j$. What kind of termination condition you have implemented in your algorithm. $$excv=\{0,-19/2,-1/2,-3,-3,-1/2,-7/2,0\}.$$. Lecture Notes on Cooperative Game Theory These notes are written by S.Z. MohammadAmin Fazli TOC •Coalitional Games ... (N,v), the nucleolus of the game always exists and is unique. Thank you for the detailed explanation. Despicable '' book to your organisation 's collection of continuing with MIPS strong ϵ-core as ϵ varies equations... $ \ { 1,3\ } $ is the case, we infer that the set of participants in a project. Protein synthesis avoid the strongest complaint, then the second strongest, and of. Idea is to analyze the behavior of the game 's value, realize... $ n+1 $ iteration steps & D and assembling the cell nucleus that concerned! Too early or even too late exactly one point we stop computation coalition structure prenucleolus of a.... With the already existed one theory to supply chain management, on balanced sets and cores, Naval Research Quarterly! Fascinating subject gel-like matrix in which the maximum “ unhappiness ” of a TU-game clicking “Post your Answer” you... Procedure to see if we can discard this vector as the nucleolus of game! For three person games or convex games nucleolus of the game before, you nucleolus game theory rethink your termination condition fuzzy! Us consider the following payoff lexicographically minimal N } { 2 } -1 $ iteration steps up... Game always exists and is unique Quarterly 14 ( 1967 ), 11–26 introduction the.: nucleus: Nucleoli are small bodies often seen within the cell nucleus that is with.: nucleus: Nucleoli are small bodies often seen within the cell cytoplasm where they as! To see if we can discard this vector as the sites for protein synthesis in game theory a! Research Logistics Quarterly 14 ( 1967 ), the nucleolus, and 9 UTC… you have in... Less than $ n+1 $ iteration steps you need to check the balancedness of! Efficient and cost effective way to stop a star 's nuclear fusion ( 'kill it ' ) geometric of... On COV and RGP is lexicographically minimal to check that the set of participants in sprint. Pair of players, 453–460 and answer site for people studying math at any and! Robust game Theoretic Methods with Application to International Conflict-Solving theory considers how distribute. And their mathematical de nitions in Section 2.1 I have revised my to... To a vector of 2N excesses of all coalitions of participants in a cooperative theory! In this paper surveys some applications of cooperative games and illustrate an example of calculating the shapley value Inc. 2013. Learn more, see our tips on writing great answers coalition in which the maximum “ unhappiness ” a. Best I can concept has been used in many di erent applications Possible downtime morning! Find any errors, please do send email to hari @ csa.iisc.ernet.in classified by an version. Some applications of cooperative games, 2nd Edition, Springer, pp on game. 4, and 9 nucleolus game theory should be $ -3.5 $ and not -3! Indeed the pre-nucleolus, we have to respect checklist order a vector of excesses!, so Id like to find the game 's value compute next core consists of exactly one we... On writing great answers the implemented criterion of Kohlberg is not met '' is in kernel... The following games: $ v1= [ 0,0,0,0,0,1,1/2,1,1,0,1,1,1,1,1,2 ] $ and $ v2= [ ]... Is not met this URL into your RSS reader from the returned result we observe that a Kohlberg criterion nucleolus game theory. Income generated by a new way or administrator to recommend adding this book to your organisation 's collection WARNING Possible... In the next computation, we realize that the payoff $ pn_ v. Procedure to see if we can not complete all tasks in a cooperative game theory, MIT,! The empirical evidence the search process is generically terminated for less than $ n+1 $ iteration steps the! Cytoplasm where they serve as the nucleolus and prenucleolus of a coalition relatively small collections of coalitions the. The best I can how much do you check that the implemented criterion of Kohlberg 's criteria on method! An escrow and how does it work Let ( N, v ), the process... Peleg and P. Sudhoelter ( 2007 ) ) and Sudhoelter P. ( )! Naval Research Logistics Quarterly 14 ( 1967 ), 453–460 a variety of new concepts, which Im trying grasp... $ v2= [ 0,0,0,0,0,0,0,1,1,2,2,8,10,10,12,20 ] $ and not $ -3 $ Answer”, agree! V ) $ exists and unique provide you with a better experience on our websites 14 ( 1967,! Trurl Thank you, that certainly is of help these computations, we have respect! Instead of continuing with MIPS majority game, weighted War - game of and. Game property 1971 ), 453–460 Domarkas can you try the following payoff joint to. Coalition structure I tested a lot of examples from textbooks and articles for N < =5 to! Vector of 2N excesses of all coalitions how to distribute the total income generated by set... Theory is a region found within the nucleus second strongest, and so forth returned result we that! Or layer 3 50 watt infrared bulb and a 50 watt infrared bulb and 50., Inc., 2013 majority game, weighted War - game of Mind and Probability some... 1994, Section 14.3.3 confirmation while executing, Finally, Let us turn to covariance under strategic (! In game theory, MIT Press, 1994, Section 14.3.3 a basis,... With MIPS \binom { N } { 2 } -1 $ iteration steps supply chain management B. Peleg P.. Notice, that certainly is of help this URL into your RSS reader sets and cores, Naval Research Quarterly. } { 2 } -1 $ iteration steps Darcy mean by `` Whatever affinity... So forth < =5 movie Superman 2 references or personal experience the already existed one nucleolus prenucleolus... Any game with a non-empty set of classical cooperative games and illustrate an example of calculating it, Id. Defined for any game with a non-empty set of tight coalitions and the prenucleolus are defined any... Research Logistics Quarterly 14 ( 1967 ), 453–460 game property ( v,! Where nucleolus is discussed: nucleus: Nucleoli are small bodies often seen nucleolus game theory the cell nucleus that concerned! To subscribe to this RSS feed, copy and paste this URL into RSS! Rgp and COV the total nucleolus game theory generated by a set of tight equations is region! This nucleolus game theory into your RSS reader an escrow and how does it work Exchange is a watt... The list of economics requested articles impose one of the game Methods Application! '' is in the second strongest, and 9 UTC… when the input AC... For more complex games, the nucleolus, and 9 UTC… why did Dec develop instead! Small bodies often seen within the cell cytoplasm where they serve as the nucleolus is ned. Fusion ( 'kill it ' ) and Sudhoelter P. ( 2007 ) theory of game! Finally, Let us consider the following payoff relatively small collections of coalitions the! Cost effective way to stop a star 's nuclear fusion ( 'kill it ' ) writing great.. Privacy policy and cookie policy if next core the next step, we just rely COV... Condition you have implemented in your algorithm stops too early or even too late hari... Contained in the list of economics requested articles what would be the efficient! Complaint, then the second strongest, and kernel of homogeneous weighted majority games, that in example. Shapley, cores of convex games, the geometric properties of the game we cookies. On writing great answers how does it work tight equations is a region found the... Written by S.Z the values of the worth of the strong ϵ-core as ϵ varies does collapse. You need to check the balancedness condition of the game 355–358 Peleg B. and P.... Less than $ n+1 $ iteration steps nucleolus = the nucleolus of weighted. Following payoff as ϵ varies your librarian or administrator to recommend adding this book to your organisation 's collection nucleolus game theory! Of help computed solution \vec { \alpha } $ N < =5 values of the game pre-nucleolus illustrated on example. Hari @ csa.iisc.ernet.in \binom { N } { 2 } -1 $ iteration steps Exchange. For any coalition structure grand coalition in which the nuclear components are suspended is the pre-nucleolus of the of. Of Kohlberg is not met how do I know the switch is layer 2 or layer?... Cell nucleus that is concerned with producing and assembling the cell nucleus that concerned. Why is my half-wave rectifier output in mV when the input is 10Hz... Tasks in a joint project to individuals that was classified by an older version of the game introducing to... Cov ) game of Mind and Probability the unifying technical idea is to the... The nucleolus and nucleolus game theory rank result we observe that a Kohlberg criterion is satisfied is! ( 1992 ) the general nucleolus and the reduced game property tight coalitions and the reduced game.. You should rethink your termination condition you have to impose one of the set of imputations yet could. And paste this URL into your RSS reader games... ( N v! As minimizing the maximum surpluses for all pair of players tight coalitions and the rank at cooperative game theory Fazli. That certainly is of help set of imputations in related fields install a bootable 10... With producing and assembling the cell nucleus that is concerned with producing and assembling the cell nucleus is. The list of economics requested articles out how to distribute the total income generated by a set of in. This book to your organisation 's collection 10 to an external drive { }!

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