I need someones help on understanding the concepts of stochastic dominance and mean preserving spread. I have an exercise which could be used for explanation.

Consider the following lotteries:

L1 ={70:0.5, 30:0.5}

L2 ={80:0.25, 60:0.25, 40:0.25, 20:0.25}

L3 ={90:0.2, 70:0.2, 40:0.4, 10:0.2}

L4 ={110:0.2, 50:0.4, 20:0.4}

Question 1: Compare L1 and L2 as well as L3 and L4 by the principle of stochastic dominance.

My solution: First of all we can calculate the expected values of this lotteries and after that check them for the first order stochastic dominance. Since, expected value equals to 50 for all 4 lotteries, we can't make any conclusion about the first order stochastic dominance. We can use graphical representation of the lotteries in order to check for the second order stochastic dominance.

Graph 1

According to the first graph we see that L1 second order stochastically dominates L2 (this is my educated guess, I still can't explain why). If this is true, the variance of L2 should be bigger then variance of L1 (it is true since Var(L1) = 400 and Var(L2) = 500).

Graph 2

On the second graph we see that the L3 second order stochastically dominates L4 (same story, I can't explain this). Again we check variance and Var(L3) = 760 < Var(L4) = 1080.

Question 2: Is any of the lotteries a mean preserving spread of another lottery? If yes, formulate that lottery as a mean preserving spread of the other lottery. If no, explain why not.

Sorry, but here I have no clue what to say. I can't really understand the criteria of mean preserving spread. I made a graph where all the lotteries are plotted and still don't understand how I should proof this. Graph of the all lotteries:

Graph 3


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