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Squaring the dominance matrix to obtain D2
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- The second order dominances are found by squaring the dominance matrix to obtain D2. If team A beat team E who then beats team C, then A has a second order dominance over team C. The numbers in the D2 matrix show how many second order dominances a given team has over another. The D 2 matrix is found by squaring the D matrix.
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Definition 2 (Second-Order Stochastic Domination). A cumulative distribution F second-order stochastically dominates another distribution G iff Zc −∞ F(x)dx ≤ Zc −∞ G(x)dx for all c with a strict inequality over some interval. If F second-order stochastically dominates G then EF(x) ≥ EG(x). They would have the same
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To find the mathematical concept of FOSD, begin by observing that for any distribution function F and its density function f, and for any utility function u. Z b. u(W ) f(W ) dW. a. Z b. = [ u(W ) F (W ) ]b − u0(W ) F (W ) dW. integrating by parts. Z b. = u(b) ∗ 1 − u(a) ∗ 0 − u0(W ) F (W ) dW.
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The full lesson and more can be found on our website at https://mathsathome.com/dominance-matricesIn this lesson we learn how to construct and read dominance...
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Second-order stochastic dominance: when a lottery F dominates G in the sense of second-order stochastic dominance, the decision maker prefers F to G as long as he is risk averse and u is weakly increasing.
Mar 26, 2018 · We say that F second-order stochastically dominates (SOSD) G if and only if ∫u(x)dF(x) ≥ ∫u(y)dG(y) for all increasing and concave u(⋅). In particular, let u(z) = z, which is increasing and concave. Then it follows that ∫xdF(x) ≥ ∫ydG(y) ⇔ E(X) ≥ E(Y).
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Oct 20, 2016 · Let $F$ and $G$ be two distributions with the same mean. $F$ is said to second order stochastically dominate (SOSD) $G$ if $$\int u(x)\mathrm dF(x)\ge \int u(x)\mathrm dG(x)\tag{1}$$ for all increasing and concave $u(\cdot)$.
