I'm working with this article by Bauermeister et al. that compares the risk attitude parameters found using two different risk elicitation models. The models each use a series of gambling options to determine at what point the subject is indifferent between the control game and the variable game, which increases the pay out at a fixed low probability (i.e. the control game is 40 at 30% and 20 at 70% and the variable game has increasing payout at fixed 10% and fixed payout 5 at 90%). Once this indifference point is identified, that should yield the risk attitude parameter $\alpha$ and the probability weighting factor $\lambda$. But I'm confused about how to solve for these variables.

Given in Bauermeister: $$ U({x}_{1},p,{x}_{2},1-p) = w(p)v({x}_{1})+(1-w(p))v({x}_{2}) $$ where ${x}_{1} \geq {x}_{2} \geq 0$, and with $p \in (0,1)$ $$ v(x) = {x}^{\alpha} \\ w(p) = exp[-{(-ln(p))}^{\gamma}] $$

Now if we find indifference between the games $$ p{x}_{1} \text{ or } (1-p){x}_{2} \\ p'{x}_{3} \text{ or } (1-p'){x}_{4} $$

how do we solve for $\lambda$ and $\alpha$?

Other supporting papers include: Abdelloui (2000) Tanaka (2010) Wakker and Deneffe (1996)



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