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Decision under uncertainty is sometimes called a "game against chance", and can thus be modeled as a two-player normal form game: the decision-maker vs Nature/Chance. The possible states would form a set of pure strategies for Nature, and Nature commits to a publicly known mixed strategy that randomizes over those pure strategies (assume Nature is ...


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Pareto efficient allocations can be found by maximizing a weighted average of the utilities of the agents. Let $\lambda$ be the Pareto weight for agent 1 and $1 - \lambda$ the weight for agent 2 (where $\lambda \in [0,1]$). This then gives the following problem: $$ \max_{x_1, x_2} -\lambda \left[\pi_1(x_1 - b_{11})^2 + \pi_2(x_2 - b_{12})^2\right] - (1-\...


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Let $c$ be the cost per unit of insurance, so the premium is equal to $cn$. Then the agent maximises: $$ p(u(w - L - cn + n) + (1-p)u(w - cn). $$ The first order condition with respect to $n$ is given by: $$ (1 - c) p u'(w - L - cn + n) - c (1-p) u'(w - cn) = 0 $$ Rearranging gives: $$ \frac{p}{1 - p} = \frac{c}{1 - c} \frac{u'(w - cn)}{u'(w - L - cn + n)}. $...


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$p\circ x\oplus(1-p)\circ y$ is a lottery that gives you the prize $x$ with probability $p$ and the price $y$ with probability $(1-p)$. Unless $x,y$ can be identified with numbers, such as amounts of money, it makes no sense to take the expectation of this lottery. What is the expected value of a lottery that gives you a cow with a probability of $0.5$ and a ...


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The expected utility theorem (EUT), first and foremost, establishes a utility representation of the preference over lotteries. This is akin to establishing utility representation of preference over deterministic consumption bundles in consumer theory. The representation (in both cases) is valuable because it gives us tools like algebra and calculus to do ...


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I mispoke in the comments, this certainty equivalent should indeed not be negative. The certainty equivalent in your example is $w_0+c$, this certain payoff's utility is equivalent with the lottery's. The amount $c$ is not the certainty equivalent, but the amount the consumer is willing to forego in expected value in exchange for the certainty. Perhaps $-c$ ...


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We don't use the EUT for comparing lotteries with close probabilities, except if we are solving an exercise involving an Allais paradox type of question. We don't actually use it at all if we are not theorists trying to prove something. The EUT tells you that if you have preferences over lotteries satisfying the basic axioms, including independence, then ...


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In your last inequality, point $A$ is where $p_1=1$ and point $C$ is where $p_1=0$. So the condition $p_1<\frac34$ is captured by the line segment $\overline{CD}$.


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You have the right idea, make a 4 by 3 payoff matrix with fruit at the top and at the side weather/water yes/no, giving 12 different cells, then put the respective dollar payoffs. You could also make another table and find the expected payoff by multiplying each payoff by its probability, although since all the probabilities are the same, the table wouldn’t ...


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