{"history":[],"article":{"id":"42008d67-8b28-432d-9ce6-55f7fcd20c34","title":"Single peaked preferences","slug":"single-peaked-preferences","content":":::summary\n:::\n\n# \n**Single-peaked preferences** are a class of [preference relations](https://bliptext.com/articles/preference-relation).\nA group has single-peaked preferences over a set of outcomes if the outcomes can be ordered along a line such that:\nSingle-peaked preferences are typical of one-dimensional domains.\nA typical example is when several consumers have to decide on the amount of [public good](https://bliptext.com/articles/public-good-economics) to purchase.\nThe amount is a one-dimensional variable.\nUsually, each consumer decides on a certain quantity which is best for him or her, and if the actual quantity is more/less than that ideal quantity, the agent is then less satisfied.\nWith single-peaked preferences, there is a simple [truthful mechanism](https://bliptext.com/articles/truthful-mechanism) for selecting an outcome, which is to select the median quantity; this results in the [median voter theorem](https://bliptext.com/articles/median-voter-theorem).\nIt is truthful because the median function satisfies the [strong monotonicity](https://bliptext.com/articles/monotonicity-mechanism-design) property.\nThe notion was first presented by [Duncan Black](https://bliptext.com/articles/duncan-black) and later by [Kenneth Arrow](https://bliptext.com/articles/kenneth-arrow).\n\n# Definitions\nLet be the set of possible outcomes.\nLet be the set of agents.\nThe preference-relation of agent i is denoted by \\succ_i.\nThe maximum element of \\succ_i in X is denoted by x^*_i.\n\n## Definition using a common order\nThe group N is said to have **single-peaked preferences** over X, if there exists an ordering > of the outcomes such that, for every agent i in N:\nIn words, x^*_i is the ideal point for agent i.\nWhen the agent compares between two outcomes that are both to the right or to the left of his ideal point, he strictly prefers whichever option is closest to x^*_i.\nNote that the preference-relations \\succ_i are different, but the ordering > of the outcomes must be the same for all agents.\n\n## Necessary condition\nBallester and Haeringer proved the following necessary condition for single-peaked preferences.\nIf the group N has **single-peaked preferences** over X, then for every triplet of outcomes in X, there exists an outcome that is not ranked last by any agent in N.\n\n# Some examples\n\n## Single-peaked preferences\nThe following graph shows a set of three preferences that are single-peaked over outcomes {A,B,C,D,E}.\nOn the vertical axis, the number represents the preference ranking of the outcome, with 1 being most preferred.\nTwo outcomes that are equally preferred have the same ranking.\nThe ordering over the outcomes is A < B < C < D < E. The ideal outcome for the green agent is A, for the red it is B, for the blue it is C. For each agent, when we move away from his ideal outcome, the ranking decreases.\nIt can also be verified that, for each triplet of outcomes, one of them is never ranked last - the one in the middle.\nE.g., in {A,B,C}, B is never ranked last; in {C,D,E}, D is never ranked last; etc.\n# [300px | upload.wikimedia.org/wikipedia/commons/f/f0/Singlepeaked1.jpg]\n\n## Non single-peaked preferences\nIf each of the two preferences represented by the following two graphs is added to the three preferences above, then the resulting group of four preferences is not single-peaked:\nFor the blue preferences, it can be seen that the preference ranking spikes down for \"D\" and then spikes up for \"E\".\nThis proves that the blue preferences are not single-peaked with respect to the ordering A of the outcomes such that, for every agent i in N:\nIn words, x^*_i is the ideal point for agent i.\nWhen the agent compares between two outcomes that are both to the right or to the left of his ideal point, he strictly prefers whichever option is closest to x^*_i.\nNote that the preference-relations \\succ_i are different, but the ordering > of the outcomes must be the same for all agents.\n\n## Necessary condition\nBallester and Haeringer proved the following necessary condition for single-peaked preferences.\nIf the group N has **single-peaked preferences** over X, then for every triplet of outcomes in X, there exists an outcome that is not ranked last by any agent in N.\n\n# Some examples\n\n## Single-peaked preferences\nThe following graph shows a set of three preferences that are single-peaked over outcomes {A,B,C,D,E}.\nOn the vertical axis, the number represents the preference ranking of the outcome, with 1 being most preferred.\nTwo outcomes that are equally preferred have the same ranking.\nThe ordering over the outcomes is A < B < C < D < E. The ideal outcome for the green agent is A, for the red it is B, for the blue it is C. For each agent, when we move away from his ideal outcome, the ranking decreases.\nIt can also be verified that, for each triplet of outcomes, one of them is never ranked last - the one in the middle.\nE.g., in {A,B,C}, B is never ranked last; in {C,D,E}, D is never ranked last; etc.\n# [300px | upload.wikimedia.org/wikipedia/commons/f/f0/Singlepeaked1.jpg]\n\n## Non single-peaked preferences\nIf each of the two preferences represented by the following two graphs is added to the three preferences above, then the resulting group of four preferences is not single-peaked:\nFor the blue preferences, it can be seen that the preference ranking spikes down for \"D\" and then spikes up for \"E\".\nThis proves that the blue preferences are not single-peaked with respect to the ordering A

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