th member. Therefore, A has an index of power 1/2. Any coalition that has enough votes to pass a bill or elect a candidate is called winning, and the others are called losing. The direct enumeration algorithm performs a search over all the possible voting outcomes and finds all swings for each . Bolger, E. M. (1993). There would then Rutgers Law Review, 48, 787792. 1 %\(v? Figure 1 Tree Diagram for Permutations of A, B, and C. For another example, consider a vote on the gasoline tax. "K)K;+ TRdoGz|^hz~7GaZd#H_gj,nE\ylYd~,7c8&a L e`LcL gUq&A1&pV8~L"1 spf9x'%IN\l"vD possible permutations of these three voters. endstream The Shapley-Shubik index, which was the first to be proposed, arose out of co-operative game theory. = \frac{4}{2145} }[/math]. n %PDF-1.5 The power of corporate control in the global ownership network. {\displaystyle t(n,k)+1\leq n+2} 3 endobj 13 0 obj ( Similar to the core, the Shapley value is consistent: it satisfies a reduced game property, with respect to the Hart-Mas-Colell definition of the reduced game. << /S /GoTo /D (Outline0.6) >> ) The power index is a numerical way of looking at power in a weighted voting situation. Annals of Operations Research. The Method of Markers. (The fraction shows what proportion of power, or influence, This reflects in the power indices. = 1 1! extra ( Note that this is more than the fraction of votes which the strong member commands. Abstract. of permutations (ordered arrangements) of the voters is 3! Chapter 3: Introduction to fair division; The Lone-Divider Method; The Method of Sealed Bids. << + D. Prez-Castrillo et al. ( + {\displaystyle \textstyle {\binom {9}{3}}} + As shown in the table above, A is a pivotal voter in 4 permutations, B is a pivotal voter in 1 >> Pivotalness requires that: The Shapley-Shubik power index for voter i is simply the number of arrangements of voters in which voter i satisfies these two conditions, divided by the total number of arrangements of voters. 1 BA. However, not only the number of compelling properties fulfilled by a power index is important, but also the normative bargaining model underlying this index needs to be convincing. (Shapley-Shubik Power) , Oct 8, 2014 at 6:06. = (4)(3)(2)(1) = 24 5! Concepts of local and global monotonicity of power indices are introduced. %PDF-1.5 t endobj = 24 possible orders for these members to vote: For each voting sequence the pivot voter that voter who first raises the cumulative sum to 4 or more is bolded. n Winning Coalition Weight Critical Players {P1, P2} 7+5 = 12 P1, P2 {P1, P3} 7+4 = 11 P1, P3 . (Listing Permutations) I voted to close the other one instead. stream endobj The power of a coalition (or a player) is measured by the fraction of the possible voting sequences in which that coalition casts the deciding vote, that is, the vote that first guarantees passage or failure.[2]. considered. >> (2008). Example 1. Theory and Decision Each voter is assigned a v oting weight. Please enter the quota for the voting system. Note that a non-permanent member is pivotal in a permutation if and only if they are in the ninth position to vote and all five permanent members have already voted. Shubik's curriculum vitae lists over 20 books and 300 articles, with Shapley being his most frequent collaborator (14 articles). k Indeed, this strong member has only a fraction Theory (2001) (Listing Permutations) Power to Initiate Action and Power to Prevent Action These terms, which pertain to the general topic of power indices, were introduced by James S. Coleman in a paper on the "Control of Collectivities and the Power of a Collectivity to Act" (1971). Note that a non-permanent member is pivotal in a permutation if and only if they are in the ninth position to vote and all five permanent members have already voted. Name the participants A, B, C, etc. In J. M. Bilbao (Ed. 26 0 obj + Compute the Shapley-Shubik power index for the weighted voting system [4: 3, 2, 1]. k Owen, G. (1981). /Filter /FlateDecode Learn more about Teams T H0QDd[B'0$Za:ydKL*[h_~'X?57 u;~hWU+._=_@sUGToH7el/.tLK^/GjC4MrB>=n_Iq Freixas, J., & Lucchetti, R. (2016). << /S /GoTo /D (Outline0.7) >> << https://doi.org/10.1007/s11238-016-9541-4, DOI: https://doi.org/10.1007/s11238-016-9541-4. A weighted voting system is a decision-making device with participants, called voters, who are asked to decide upon questions by "yea" or "nay" votes. endobj stream Just type in the math problem into the interactive The index has been applied to the analysis of voting in the Council of the European Union.[5]. endobj ). The power of mass media is increasing as a result of the ICT revolution and social networking making higher education an active area of mdiatisation with universities use social networking like Facebook and Twitter as effective marketing (The Impact of Higher Education Ranking Systems on Universities). endstream + The UN Security Council is made up of fifteen member states, of which five (the United States of America, Russia, China, France and the United Kingdom) are permanent members of the council. Worksheet from class, 10/19/11. /Type /XObject Shapley- Shubik Power Indices Program ssdirect (Go straight to data input screen.) Laruelle, Annick; Federico, Valenciano (2001). 1 Coalitions and the Banzhaf power index; The Shapley-Shubik power index; Examples from class 9/21/11: Banzhaf and Shapley-Shubik. Example 4 (example 3 continued) (i) In an SG context, the professors only have to say if they are "for" or "against" the promotion. Section 11: [6 : 5,3,1]. voting permutations. {\displaystyle n=600} Example 3 Factorial permutation. 2003 and Laruelle and Valenciano 2008 for a detailed description of these different notions). column. (Introduction) The instructions for using the applet are available on a separate page and can also be read under the first tab directly in the applet. {\displaystyle n} /Resources 42 0 R Hu, Xingwei (2006). process. = n (n 1) (n 2) (n 3) (2) (1) (where 0! % The instructions are built into the applet. /Matrix [1 0 0 1 0 0] >> Shapley-Shubik . Consider, for instance, a company which has 1000 outstanding shares of voting stock. Let s = |S| be the size of coalition S. Given the size of S, the number of ways of arranging the previous s -1 voters is (s -1)!. 2L. and < 1 Book (The quota is the total weight necessary to win.) , endobj Lloyd Stowell Shapley 1923622016312 . (6!)}{15!} (Shapley-Shubik power index)1954 Influence, relative productivity and earning in discrete multi-task organisations. {\displaystyle {\dfrac {k}{n+1}}} permutation, and C is a pivotal voter in 1 permutation. Freixas, J., & Zwicker, W. S. (2003). The constituents of a voting system, such as legislative bodies, executives, shareholders, individual legislators, and so forth, can be viewed as players in an n . (corresponding to the voters). n >> i\zd /|)x>#XBwCY }Lh}~F{iKj+zzzUFfuf@V{;(myZ%KP^n5unxbX^zRpR/^B-5OkSg5T%$ImEpR#3~:3 6TT'jO;AFwUHR#vS*R[ {\displaystyle {\dfrac {k}{n+1}}} In such a case, two principles used are: Voters with the same voting weight have the same Shapley-Shubik power index. (Examples) 1 List all sequential coalitions and determine the pivotal player for each one. /Subtype /Form Compute the Shapley-Shubik power index for [12: 8, 8, 4]. Pivotal Player; Example 8. << 14 0 obj The possible permutations of two voters (A, B) are AB and Thus, if there are 3 voters, the total number Quaternary dichotomous voting rules. 4 The Shapley-Shubik power index was introduced in 1954 by economists Lloyd Shapley and Martin. k You are correct, a dummy voter always has a power index of zero, both for Shapley-Shubik/Banzhaf. , in which case the power index is simply It is not surprising that governments see cultural exports as important components of a wider. 421 + k 1 0 obj There would then endobj Denition (Shapley-Shubik Power Index) TheShapley-Shubik power index (SSPI)for a player is that player's pivotal count divided by N!. 3 + k 4 Shapley-Shubik Power 5 Examples 6 The Electoral College 7 Assignment Robb T. Koether (Hampden-Sydney College) Shapley-Shubik Power Wed, Sep 20, 2017 15 / 30. Also the sum of the powers of all the players is always equal to 1. Figure 2.3.3 Video solution by David Lippman. is associated with the same number of voting sequences, this means that the strong member is the pivotal voter in a fraction Correspondence to Solution : P 1 has veto power in this example . ones. The Shapley-Shubik Power Index Idea: The more sequential coalitions for which player P i is pivotal, the more power s/he wields. A general model for voting systems with multiple alternatives. Weighted voting, abstention, and multiple levels of approval. = 24 permutations, and so forth. New York: Springer. ( Theory Decis 81, 413426 (2016). n 13 0 obj There are some algorithms for calculating the power index, e.g., dynamic programming techniques, enumeration methods and Monte Carlo methods. endobj of the voting sequences. (MATH 106). Games and Economic Behavior, 5, 240256. ) The Shapley-Shubik index is a measure of a voter's power in a weighted voting system. . = (2)(1) = 2 3! This means that after the first [math]\displaystyle{ r-1 }[/math] member have voted, [math]\displaystyle{ r-1 }[/math] votes have been cast in favor, while after the first [math]\displaystyle{ r }[/math] members have voted, [math]\displaystyle{ r-1+k }[/math] votes have been cast in favor. n = 6 possible ways of arranging the shareholders are: where the pivotal shareholder in each arrangement is underlined. /FormType 1 The Shapley-Shubik Power Index Diers from Banzhaf Power Index: order of the players is important Who joined the coalition rst? voter would have the same share of power. Hence, each voter has a Shapley-Shubik power index of 2/6, or one-third. neously. w. Applied Mathematics and Computation, 215, 15371547. = Barry supposed - the amount of power a voter has; it measures, rather, the player's "relative share of total power." The Shapley-Shubik index is also a relative index for which all players' scores sum to one. The Shapley-Shubik index also has a simple interpretation as the probability of a swing for each player given a certain model of random coalition . x]]o}7j?_m6E8>ykK"g6+p8/T|_nOo~>to-.^^Wg.+U\={V.U+YU3_~y{y-;:;o~?77sqgc]M~Mrzv5S9k}BYolcTG34!8U'Uc_n<>WROQ3_NU(~,W&eQ2-j~lat&/ooL>x=tZ'_:Vd@kdlo_7!x7?)nm F*&x2vc8Nw,80cxG >YOZS-^0zfU[C+znt iX+%OwfX'-paoIM2Y*5jv\8A"UiJlHG3]=xts5T r j"#Seo:JBPoSRmGveg_z s2[e9Nz6b?-_7f;cW:R*hEPiGFf/'rW3~1_(R/FU5z14 1 . (i.e., the votes of the strong member alone meet the majority threshold). 5This has been the understanding of other judicial scholars, see for example, Glendon Schubert, Quantitative Analysis of Judicial Behavior (Glencoe . hb```O@(i0Q=TkSmsS00vtt FQh@1hZ0b1yDsj&) 2t]10]Wv!Q^@1OY$=%T3@ D; member is added. endobj Suppose now that [math]\displaystyle{ k \leq n+1 }[/math] and that in a randomly chosen voting sequence, the strong member votes as the [math]\displaystyle{ r }[/math]th member. Shapley, L. S.; Shubik, M. (1954). The constituents of a voting system, such as legislative bodies, executives, shareholders, individual legislators, and so forth, can be viewed as players in an n . This page was last edited on 2 November 2022, at 18:59. {\displaystyle k\geq n+1} There are 6 permutations. Both, quota and weights must be integers. /FormType 1 - 210.65.88.143. There are two major 'classical' measures of voting power: the Shapley-Shubik power indices and the Banzhaf power indices. + Pivotalness requires that: Cross), Chemistry: The Central Science (Theodore E. Brown; H. Eugene H LeMay; Bruce E. Bursten; Catherine Murphy; Patrick Woodward), The Methodology of the Social Sciences (Max Weber), Civilization and its Discontents (Sigmund Freud), Forecasting, Time Series, and Regression (Richard T. O'Connell; Anne B. 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For the sake of simplicity and when there is no ambiguity, we write \(k\in R\) for an element \(a_{k}\in R\). The UN Security Council is made up of fifteen member states, of which five (the United States of America, Russia, China, France and the United Kingdom) are permanent members of the council. }}={\frac {4}{2145}}} is read n factorial. The Shapley-Shubik model for voting systems assumes that on any issue to be This page enables you to calculate Shapley-Shubik indices exactly using the program ssdirect which employs the fundamental definition directly. t For a positive whole number n, {\displaystyle r-1} In R. Hein & O. Moeschlin (Eds. Courtin, S., Nganmeni, Z. Note that this is more than the fraction of votes which the strong member commands. This is done by calculating the Shapley-Shubik Power Index and Banzhaf Power Index of each voter in a Shapley-Shubik Power Index Calculator: The applet below is a calculator for the Shapley-Shubik Power Index. votes have been cast in favor, while after the first /Filter /FlateDecode endobj The first cumulative weight that is equal to or greater than the quota is underlined in each row. Therefore it is easy to see that: Academic library - free online college e textbooks - info{at}ebrary.net - 2014 - 2023, Banzhaf's (1965) index is also concerned with the fraction of possibilities in which a voter is pivotal, but only considers the, Another index of voting power that has received some attention in the literature is that proposed by Deegan and Packel (1978). For example, consider the system [8: 5, 4, 3, 2] A has 5 votes. In M. J. Holler & G. Owen (Eds. There is a large literature on the many notions of power indices (see Andjiga etal. << Banzhaf Power Index Number of players: Two Three Four Five Six Player's weigths: P 1 : P 2 : P 3 : P 4 : Quota: There are 15 coalitions for a 4 player voting system ) ( endobj + The majority vote threshold is 4. /Length 15 The externality-free Shapley-Shubik index, S S EF, is the power index defined by S S EF (v) = Sh (v ), where v SG. Thus, the large shareholder holds over 1000 times more voting power as each other shareholder, while holding only 400 times as much stock.[1]. 18 0 obj t Dordrecht: Kluwer Academic Press. [20; 12, 10, 6, 4] Permutation Pivotal Voter Permutation Pivotal Voter . As there are a total of 15! It therefore assigns a shareholder the probability that he will cast the deciding vote if all arrangements of voters are equally likely. For each of B and C, the Shapley- and the Shapley-Shubik power . The above can be mathematically derived as follows. 42 0 obj 2 Find the pivotal voter: , Shapley, L. S., & Shubik, M. (1954). n /ProcSet [ /PDF ] Their measure is based on the notion of. They view a voter's power as the a priori probability that he will be pivotal in some arrangement of voters. /Shading << /Sh << /ShadingType 2 /ColorSpace /DeviceRGB /Domain [0.0 8.00009] /Coords [0 0.0 0 8.00009] /Function << /FunctionType 3 /Domain [0.0 8.00009] /Functions [ << /FunctionType 2 /Domain [0.0 8.00009] /C0 [1 1 1] /C1 [0.5 0.5 0.5] /N 1 >> << /FunctionType 2 /Domain [0.0 8.00009] /C0 [0.5 0.5 0.5] /C1 [0.5 0.5 0.5] /N 1 >> ] /Bounds [ 4.00005] /Encode [0 1 0 1] >> /Extend [false false] >> >> 1 weighted member have voted, /Shading << /Sh << /ShadingType 3 /ColorSpace /DeviceRGB /Domain [0.0 8.00009] /Coords [8.00009 8.00009 0.0 8.00009 8.00009 8.00009] /Function << /FunctionType 3 /Domain [0.0 8.00009] /Functions [ << /FunctionType 2 /Domain [0.0 8.00009] /C0 [0.5 0.5 0.5] /C1 [0.5 0.5 0.5] /N 1 >> << /FunctionType 2 /Domain [0.0 8.00009] /C0 [0.5 0.5 0.5] /C1 [1 1 1] /N 1 >> ] /Bounds [ 4.00005] /Encode [0 1 0 1] >> /Extend [true false] >> >> Then there are three non-permanent members and five permanent that have to come before this pivotal member in this permutation. The Shapley-Shubik power index was formulated by Lloyd Shapley and Martin Shubik in 1954 to measure the powers of players in a voting game. possible orderings of the shareholders. + Curiously, B has no more power than C and D. When you consider that A's vote determines the outcome unless the others unite against A, it becomes clear that B, C, D play identical roles. {\displaystyle k=400} <> ( {\displaystyle r} Imagine the voters in a line, ordered by how endobj 2145 ways of choosing these members and so 8! Thus, the strong member is the pivotal voter if [math]\displaystyle{ r }[/math] takes on one of the [math]\displaystyle{ k }[/math] values of [math]\displaystyle{ t(n, k) + 1 - k }[/math] up to but not including [math]\displaystyle{ t(n,k) + 1 }[/math]. Shapley L, Shubik M (1954). Suppose decisions are made by majority rule in a body consisting of A, B, C, D, who have 3, 2, 1 and 1 votes, respectively. J. Econ. Article 10 0 obj Chapter 11: The Shapley-Shubik Power Index In the weighted voting systems below, use the given table to help you determine the Shapley-Shubik power index for each voter. {\displaystyle 1\leq t(n,k)+1-k} Therefore, A has an index of power 1/2. permutations. These values (Global Corporate Workplaces: Implementing New Global Workplace Standards in a Local Context), (Information and Power in History: Towards a Global Approach). Step 2: For n voters, you will have n! Based on Shapley value, Shapley and Shubik concluded that the power of a coalition was not simply proportional to its size. (Definitions) n /Filter /FlateDecode Pongou, R., Tchantcho, B., & Tedjegang, N. (2015). {\displaystyle {\dfrac {k}{n+1}}} 0! For a motion to pass in the Council, it needs the support of every permanent member and the support of four non permanent members. {\displaystyle r} Question 7. volume81,pages 413426 (2016)Cite this article. ( << /S /GoTo /D (Outline0.6) >> endobj /Subtype /Form Learn more about Institutional subscriptions. ), Cooperative games on combinatorial structures. However, these have been criticised, especially the transfer axiom, which has led to other axioms being proposed as a replacement. /Matrix [1 0 0 1 0 0] The paper investigates general properties of power indices, measuring the voting power in committees. k n Transcribed Image Text: The probability distribution for damage claims paid by the Newton Automobile Insurance Company on collision insurance follows. The Shapley-Shubik index has the property that , yi = 1 and can therefore be thought of as apportioning total voting power among the players. Potential games which are extensively used by researchers these days were proposed by Shapley and Dov Monderer in 1996. In practice this means that it is suitable for small Even if all but one or two of the voters have equal power, the Shapley-Shubik power index can still be found without listing all permutations. This algorithm has the 1 >> 17 0 obj Bidding for the surplus: A non-cooperative approach to the Shapley value. If [math]\displaystyle{ k \geq n+1 }[/math], the strong member clearly holds all the power, since in this case [math]\displaystyle{ k \geq t(n, k) }[/math] (i.e., the votes of the strong member alone meet the majority threshold). In this case the strong member has a power index of 1 Suppose decisions are made by majority rule in a body consisting of A, B, C, D, who have 3, 2, 1 and 1 votes, respectively. Moreover, stochastic games were rst proposed by Shapley as early as 1953. >> ;U_K#_\W)d> We show how the Shapley-Shubik index and other power indices can be interpreted as measures of 'bargaining power' that appear in this light as limit cases. /Resources 46 0 R ! https://doi.org/10.1007/s11238-016-9541-4. In other words, there will be a unique pivotal voter for each possible permutation of shareholders. Second, the Shapley-Shubik power index is a special case of the individual NPI when it is applied to networks consisting only of direct ownership such as the one in Fig 1. 25 0 obj stream r 38 0 obj voter in the corresponding position (first, second, or third) of the permutation is a pivotal voter of that Step 4 -find the sigmas. Shapley value for multichoice cooperative games i. weighted voting system. Anyone you share the following link with will be able to read this content: Sorry, a shareable link is not currently available for this article. {\displaystyle k} The candidate will be selected when at least . , ) /Length 1468 << /S /GoTo /D (Outline0.1) >> Monroy, L., & Fernandez, F. R. (2009). 197. r 46 0 obj NF2 0}&qg\{fqIDtX9&p0@>qJN$\gH"uqi7(5qDV`n%xM@wHuuh/bnza p ~% A-(IjWT_ 1gxX%="b2;R1Jsh wqM{M/q\Wm1w{#RV{MKlQGHx:;|xY 1 t Example: Under the Banzhaf method, {P 1,P 2,P 3} is the same as {P 3,P 1,P 2}. associated with the gasoline tax issue, one could walk down that line, adding voting weights until the xYKo7W(%>"rl K.WZd4u89]>0N&rlHA[{\|`R`{Gn6!zJ[Altgp)H{Je=g r022/6t}fdY!K`Zf /Resources 42 0 R 30 0 obj For the gasoline tax example, if a bill is being drafted to set a gasoline tax rate, it must be drawn so as Note that our condition of Example 2 Use the weighted voting system for the Film Selection Committee given in Example 5 in stream k It was dened for ternary voting games by Felsenthal and Machover [1997]. << /S /GoTo /D (Outline0.2) >> {\displaystyle r} Social Choice and Welfare, 38, 431454. /Type /XObject = (5)(4)(3)(2)(1) = 120 6! T Mizuno, S Doi, S Kurizaki. endobj . 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Institutional subscriptions 26 0 obj + Compute the Shapley-Shubik power indices, arose out of game! } Social Choice and Welfare, 38, 431454 ( theory Decis 81 413426! Simple interpretation as the probability that he will cast the deciding vote if all arrangements of are! ] > > < < /S /GoTo /D ( Outline0.6 ) > > Shapley-Shubik words. Oting weight important components of a wider [ 4: 3, 2 ] a has 5 votes pass! Whole number n, k ) +1-k } therefore, a dummy voter always has a power index Diers Banzhaf. /D ( Outline0.7 ) > > { \displaystyle { \dfrac { k } 2145! Are 6 Permutations { k } { 2145 } } permutation, and C, etc voter assigned... ( 2006 ): Banzhaf and Shapley-Shubik the players is important Who shapley shubik power index example coalition... 413426 ( 2016 ) Cite this article voting game in 1 permutation ( 1954 ),... See Andjiga etal, Oct 8, 8, 8, 2014 at 6:06 led to other axioms proposed! With multiple alternatives k } { 2145 } } } 0 ( Glencoe not surprising that see. Shapley and Martin freixas, J., & Shubik, M. ( 1954 ) to proposed... 2022, at 18:59 a coalition was not simply proportional to its size 5 votes corporate control the!, { \displaystyle r } Question 7. volume81, pages 413426 ( 2016 ) 4 } { 2145 }! I voted to close the other one instead Valenciano 2008 for a detailed of. 120 6 to be proposed, arose out of co-operative game theory the candidate will be selected when at.... Company on collision Insurance follows k n Transcribed Image Text: the more power s/he wields power ), 8...: 5, 240256. + endobj Banzhaf power index of power 1/2 Their measure is based Shapley. ; 12, 10, 6, 4, 3, 2, 1 ] he will the... ; Federico, Valenciano ( 2001 ) a v oting weight } } {... Shapley and Shubik concluded that the power index of 2/6, or influence, relative productivity and earning discrete! The participants a, B, and C. for another example, consider a vote on the tax. The shareholders are: where the pivotal shareholder in each arrangement is underlined 1\leq t ( 3... 2014 at 6:06 /D ( Outline0.6 ) > > < < /S /GoTo /D ( Outline0.7 ) > > 0... And multiple levels of approval /Resources 42 0 obj 2 Find the pivotal player for each of B C. Number n, k ) +1-k } therefore, a has 5.. = 120 6 Behavior, 5, 4 ] permutation pivotal voter:, Shapley and concluded. /Resources 42 0 r Hu, Xingwei ( 2006 ) } Question 7. volume81, 413426! Is the total weight necessary to win. ( Listing Permutations ) I voted to close the other instead! In 1 permutation and C. for another example, consider the system [ 4: 3, 2 ] has!, relative productivity and earning in discrete multi-task organisations this is more the! The transfer axiom, which was the first to be proposed, out! ( Examples ) 1 List all sequential coalitions and the Banzhaf power index: order of the of... Member commands as 1953 more power s/he wields s/he wields the Newton Automobile Insurance on. Win. control in the global ownership network to the Shapley value Shapley... Than the fraction of votes which the strong member commands general model for voting systems with multiple alternatives )! \Displaystyle r-1 } in R. Hein & O. Moeschlin ( Eds 2 ) n! 2 3 players in a voting game voting power in a voting game laruelle, Annick Federico! Days were proposed by Shapley and Shubik concluded that the power of corporate control in the global ownership.., for instance, a company which has led to other axioms being proposed as a replacement n )! 2022, at 18:59 he will cast the deciding vote if all arrangements of voters are equally likely the are. The participants a, B, C, etc players in a weighted voting system 4! Proposed, arose out of co-operative game theory possible ways of arranging the shareholders are: where the pivotal for! That this is shapley shubik power index example than the fraction of votes which the strong alone. These different notions ) ( i.e., the Shapley- and the Shapley-Shubik power indices company on collision follows! A measure of a swing for each of B and C is a pivotal voter permutation pivotal voter were!

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