{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:16:14Z","timestamp":1760242574888,"version":"build-2065373602"},"reference-count":32,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2017,10,23]],"date-time":"2017-10-23T00:00:00Z","timestamp":1508716800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Games"],"abstract":"<jats:p>We survey the results on representations of committees and constitutions by game forms that possess some kind of equilibrium strategies for each profile of preferences of the players. The survey is restricted to discrete models, that is, we deal with finitely many players and alternatives. No prior knowledge of social choice is assumed: As far as definitions are concerned, the paper is self-contained. Section 2 supplies the necessary general tools for the rest of the paper. Each definition is followed by a simple (but nontrivial) example. In Section 3 we give a complete account of representations of committees (proper and monotonic simple games), by exactly and strongly consistent social choice functions. We start with Peleg\u2019s representations of weak games, and then provide a complete and detailed account of Holzman\u2019s solution of the representation problem for simple games without veto players. In Section 4 we deal with representations of constitutions by game forms. Following G\u00e4rdenfors we model a constitution by a monotonic and superadditive effectivity function. We fully characterize the representations for three kinds of equilibrium: Nash equilibrium; acceptable equilibrium (Pareto optimal Nash equilibrium); and strong Nash equilibrium. We conclude in Section 5 with a report on two recent works on representations of constitutions under incomplete information.<\/jats:p>","DOI":"10.3390\/g8040046","type":"journal-article","created":{"date-parts":[[2017,10,23]],"date-time":"2017-10-23T12:20:24Z","timestamp":1508761224000},"page":"46","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Representations of Political Power Structures by Strategically Stable Game Forms: A Survey"],"prefix":"10.3390","volume":"8","author":[{"given":"Bezalel","family":"Peleg","sequence":"first","affiliation":[{"name":"Institute of Mathematics and Center for the Study of Rationality, The Hebrew University of Jerusalem, 91904 Jerusalem, Israel"}]},{"given":"Ron","family":"Holzman","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Technion-Israel Institute of Technology, 32000 Haifa, Israel"}]}],"member":"1968","published-online":{"date-parts":[[2017,10,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Taylor, A.D., and Zwicker, W.S. (1999). Simple Games: Desirability Relations, Trading, Pseudoweightings, Princeton University Press.","DOI":"10.1515\/9780691223896"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1002\/bs.3830070104","article-title":"Simple games: An outline of the descriptive theory","volume":"7","author":"Shapley","year":"1962","journal-title":"Behav. Sci."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1007\/BF01753237","article-title":"Representation of simple games by social choice functions","volume":"7","author":"Peleg","year":"1978","journal-title":"Int. J. Game Theory"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/0304-4068(86)90022-4","article-title":"On strong representations of games by social choice functions","volume":"15","author":"Holzman","year":"1986","journal-title":"J. Math. Econ."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1016\/0165-4896(86)90033-8","article-title":"The capacity of a committee","volume":"12","author":"Holzman","year":"1986","journal-title":"Math. Soc. Sci."},{"key":"ref_6","unstructured":"Arrow, K.J. (1963). Social Choice and Individual Values, Wiley."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"341","DOI":"10.2307\/2215437","article-title":"Rights, games and social choice","volume":"15","year":"1981","journal-title":"No\u00fbs"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1016\/0304-4068(82)90009-X","article-title":"Cores of effectivity functions and implementation theory","volume":"10","author":"Moulin","year":"1982","journal-title":"J. Math. Econ."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"388","DOI":"10.1016\/0022-0531(74)90111-2","article-title":"A Pareto-consistent libertarian claim","volume":"7","author":"Gibbard","year":"1974","journal-title":"J. Econ. Theory"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1447","DOI":"10.2307\/1913838","article-title":"Construction of outcome functions guaranteeing existence and Pareto optimality of Nash equilibria","volume":"46","author":"Hurwicz","year":"1978","journal-title":"Econometrica"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Peleg, B., and Peters, H. (2010). Strategic Social Choice, Springer.","DOI":"10.1007\/978-3-642-13875-1"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Peleg, B. (1984). Game Theoretic Analysis of Voting in Committees, Cambridge University Press.","DOI":"10.1017\/CCOL0521259649"},{"key":"ref_13","unstructured":"Von Neumann, J., and Morgenstern, O. (1944). Theory of Games and Economic Behavior, Princeton University Press."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1090\/S0002-9904-1960-10418-1","article-title":"Von Neumann-Morgenstern solutions to cooperative games without side payments","volume":"66","author":"Aumann","year":"1960","journal-title":"Bull. Amer. Math. Soc."},{"key":"ref_15","unstructured":"Shubik, M. (1967). A survey of cooperative games without side payments. Essays in Mathematical Economics, Princeton University Press."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"153","DOI":"10.2307\/1913652","article-title":"Consistent voting systems","volume":"46","author":"Peleg","year":"1978","journal-title":"Econometrica"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1016\/0304-4068(81)90002-1","article-title":"The structure of exactly strongly consistent social choice functions","volume":"8","author":"Oren","year":"1981","journal-title":"J. Math. Econ."},{"key":"ref_18","unstructured":"Polishchuk, I. (1978). Monotonicity and Uniqueness of Consistent Voting Systems, Center for Research in Mathematical Economics and Game Theory, Hebrew University of Jerusalem."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1007\/s003550050092","article-title":"Effectivity functions, game forms, games, and rights","volume":"15","author":"Peleg","year":"1998","journal-title":"Soc. Choice Welf."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1016\/S0165-4896(01)00095-6","article-title":"Nash consistent representation of constitutions: A reaction to the Gibbard paradox","volume":"43","author":"Peleg","year":"2002","journal-title":"Math. Soc. Sci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"152","DOI":"10.1086\/259614","article-title":"The impossibility of a Paretian liberal","volume":"78","author":"Sen","year":"1970","journal-title":"J. Polit. Econ."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1016\/j.mathsocsci.2003.09.005","article-title":"Representation of effectivity functions by acceptable game forms: A complete characterization","volume":"47","author":"Peleg","year":"2004","journal-title":"Math. Soc. Sci."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Abdou, J., and Keiding, H. (1991). Effectivity Functions in Social Choice, Kluwer Academic Publishers.","DOI":"10.1007\/978-94-011-3448-4"},{"key":"ref_24","first-page":"167","article-title":"Binary effectivity rules","volume":"10","author":"Keiding","year":"2006","journal-title":"Rev. Econ. Des."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1007\/s00199-014-0816-0","article-title":"Representation of constitutions under incomplete information","volume":"57","author":"Peleg","year":"2014","journal-title":"Econ. Theory"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1007\/s00355-015-0891-6","article-title":"On existence of ex post Nash consistent representation for effectivity functions","volume":"45","author":"Peters","year":"2015","journal-title":"Soc. Choice Welf."},{"key":"ref_27","unstructured":"Peters, H. (2008). Game Theory: A Multi-Leveled Approach, Springer."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Maschler, M., Solan, E., and Zamir, S. (2013). Game Theory, Cambridge University Press.","DOI":"10.1017\/CBO9780511794216"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"587","DOI":"10.2307\/1914083","article-title":"Manipulation of voting schemes: A general result","volume":"41","author":"Gibbard","year":"1973","journal-title":"Econometrica"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1016\/0022-0531(75)90050-2","article-title":"Strategy-proofness and Arrow\u2019s conditions: Existence and correspondence theorems for voting procedures and social welfare functions","volume":"10","author":"Satterthwaite","year":"1975","journal-title":"J. Econ. Theory"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"11","DOI":"10.2307\/2296596","article-title":"Counter-threats and strategic manipulation under voting schemes","volume":"43","author":"Pattanaik","year":"1976","journal-title":"Rev. Econ. Stud."},{"key":"ref_32","unstructured":"Laslett, P., and Runciman, W.G. (1967). Values and collective decision-making. Philosophy, Politics and Society, Basil Blackwell."}],"container-title":["Games"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-4336\/8\/4\/46\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:48:14Z","timestamp":1760208494000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-4336\/8\/4\/46"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,10,23]]},"references-count":32,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2017,12]]}},"alternative-id":["g8040046"],"URL":"https:\/\/doi.org\/10.3390\/g8040046","relation":{},"ISSN":["2073-4336"],"issn-type":[{"type":"electronic","value":"2073-4336"}],"subject":[],"published":{"date-parts":[[2017,10,23]]}}}