Fisher market information

Fisher market is an economic model attributed to Irving Fisher. It has the following ingredients:^[1]

A set of $m$ divisible products with pre-specified supplies (usually normalized such that the supply of each good is 1).
A set of $n$ buyers.
For each buyer $i=1,\dots ,n$ , there is a pre-specified monetary budget $B_{i}$ .

Each product $j$ has a price $p_{j}$ ; the prices are determined by methods described below. The price of a bundle of products is the sum of the prices of the products in the bundle. A bundle is represented by a vector $x=x_{1},\dots ,x_{m}$ , where $x_{j}$ is the quantity of product $j$ . So the price of a bundle $x$ is $p(x)=\sum _{j=1}^{m}p_{j}\cdot x_{j}$ .

A bundle is affordable for a buyer if the price of that bundle is at most the buyer's budget. I.e, a bundle $x$ is affordable for buyer $i$ if $p(x)\leq B_{i}$ .

Each buyer has a preference relation over bundles, which can be represented by a utility function. The utility function of buyer $i$ is denoted by $u_{i}$ . The demand set of a buyer is the set of affordable bundles that maximize the buyer's utility among all affordable bundles, i.e.:

${\text{Demand}}_{i}(p):=\arg \max _{p(x)\leq B_{i}}u_{i}(x)$ .

A competitive equilibrium (CE) is a price-vector $p_{1},\dots ,p_{m}$ in which it is possible to allocate, to each agent, a bundle from his demand-set, such that the total allocation exactly equals the supply of products. The corresponding prices are called market-clearing prices. The main challenge in analyzing Fisher markets is finding a CE.^[2]^{: 103–105}

^ Yishay Mansour (2011). "Lecture 10: Market Equilibrium" (PDF). Advanced Topics in Machine Learning and Algorithmic Game Theory. Retrieved 15 March 2016.
^ Vazirani, Vijay V.; Nisan, Noam; Roughgarden, Tim; Tardos, Éva (2007). "Chapter 5: Combinatorial Algorithms for Market Equilibria / Vijay V. Vazirani". Algorithmic Game Theory (PDF). Cambridge, UK: Cambridge University Press. ISBN 0-521-87282-0.

[m11-1] Yishay Mansour (2011). "Lecture 10: Market Equilibrium" (PDF). Advanced Topics in Machine Learning and Algorithmic Game Theory. Retrieved 15 March 2016.

[agt5-2] Vazirani, Vijay V.; Nisan, Noam; Roughgarden, Tim; Tardos, Éva (2007). "Chapter 5: Combinatorial Algorithms for Market Equilibria / Vijay V. Vazirani". Algorithmic Game Theory (PDF). Cambridge, UK: Cambridge University Press. ISBN 0-521-87282-0.

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