# Uncertainty in an unfair gamble

If a risk averse person is given the option of a certain amount of 2000 or playing a lottery game giving him 10000 with 25% probability, and 500 with 75%, then what would he do ?.

The bet here is not a fair one, so wouldn't his choice depend on the degree of his risk aversion ?. Can somebody please help me out.

So let's decided that decision A is taking the certain sum and decision B is taking the risky sum, where $\bar{w}$ is current wealth and $\mathcal{U}$ is a utility function where $\mathcal{U}'>0$ and $\mathcal{U}''<0$, then you have two potential outcomes in utility. The first is $$\mathcal{U}(A)=\mathcal{U}(\bar{w}+2000)$$ and the second is $$\mathcal{U}(B)=\frac{\mathcal{U}(\bar{w}+10000)+3\mathcal{U}(\bar{w}+500)}{4}.$$

Given only this information, the decision depends upon $\bar{w}$, the shape of $\mathcal{U}$, the payoff outcomes, and the probabilities involved. If you wanted to determine how sensitive a decision would be to change, you could treat the currently fixed factors as variable factors one at a time. To provide an example, you could create a new variable $\pi$ so that option A would become $$\mathcal{U}(\bar{w}+\pi).$$

You could also do this to substitute for $\bar{w}$, the other payoffs, or the odds. If you added the assumption that the actors were indifferent, then you could also use the Envelope Theorem to look at local sensitivity.

Dave Harris gave a great mathematical explanation but I wanted to give a more intuitive version of what he's saying.

If you assume that the safe option is option A and the risky option is option B then you first need to determine the expected value for each option.

Option A's expected value is $2,000, meaning no matter what the benefit of option A is $2,000.

Option B's expected value is (10000 * .25) + (500 * .75) = 2875.

If the individual was risk neutral they would clearly choose option B because it has a \$875 higher expected value than option A.

This individual is risk adverse meaning they put some value on not facing risk or facing less risk. This value that they place on avoiding the risk is referred to as the risk-premium. The value of the risk premium depends on the factors mentioned by Dave Hariss but in essence it is the amount of additional money they most receive to take the risk.

In your example, for the individual to choose the safe option, their risk premium for this situation must be > 875. Otherwise they will choose the risky option despite being risk adverse.