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A value oracle is an oracle that, given a set of items, returns the agent's value to this set. In this model: Dobzinski and Schapira [13] present a polytime / ()-approximation algorithm, and an (1-1/e)≈0.632-approximation algorithm for the special case in which the agents' utilities are set-coverage functions.
In mathematics, a subadditive set function is a set function whose value, informally, has the property that the value of function on the union of two sets is at most the sum of values of the function on each of the sets. This is thematically related to the subadditivity property of real-valued functions.
The value of A and C together is the less than the sum of their values. For instance two versions of the same CD may not be as valuable to a person as the sum of the values of the individual CDs on their own. I.e, A and C are substitute goods. The values of B and D together may be more than their individual values added.
Often, given a submodular set function that describes the values of various sets, we need to compute the values of fractional sets. For example: we know that the value of receiving house A and house B is V, and we want to know the value of receiving 40% of house A and 60% of house B.
Utility functions are also normally continuous functions. Such utility functions are also called von Neumann–Morgenstern (vNM). This is a central theme of the expected utility hypothesis in which an individual chooses not the highest expected value but rather the highest expected utility.
The maximum sum is 1, attained by giving one agent the item with value 1 and the other agent nothing. But the max-min allocation gives each agent value at least e, so the sum must be at most 3e. Therefore the POF is 1/(3e), which is unbounded. Alice has two items with values 1 and e, for some small e>0. George has two items with value e. The ...
This means that in both equilibria, all agents have exactly the same budget set (they can afford exactly the same bundles). In equilibrium, the utility of every agent is the maximum utility of a bundle in the budget set; if the budget set is the same, then so is the maximum utility in that set. b. The price vectors are not proportional.
The absolute value of the slope of the utility-possibility frontier showcases the utility gain of one individual at the expense of utility loss of another individual, through a marginal change in outputs. [2] Therefore, it can be said that the frontier is the utility maximisation by consumers given an economies' endowment and technology. [3]